Higher Spin Field Theory : Interactions [2] 9783110675443, 9783110675528

Table of contents :
Preface
Acknowledgments
Contents
1 Introduction
2 Notes on the history of higher spin interactions
3 Algebraic structures in higher spin theory
4 General theory of interactions
5 Covariant approaches in Minkowski space-time
6 Light-front interactions
7 Higher spin in AdS and the Vasiliev theory
8 Archaeology of the Vasiliev theory
Bibliography
Index

Citation preview

Anders Bengtsson Higher Spin Field Theory

Texts and Monographs in Theoretical Physics

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Edited by Michael Efroimsky and Leonard Gamberg

Anders Bengtsson

Higher Spin Field Theory �

Volume 2: Interactions

Physics and Astronomy Classification 2010 03.50.-z, 11.00, 02.40.-k Author Dr. Anders Bengtsson Sejernäsvägen 64 432 74 Träslövsläge Sweden [email protected]

ISBN 978-3-11-067544-3 e-ISBN (PDF) 978-3-11-067552-8 e-ISBN (EPUB) 978-3-11-067554-2 ISSN 2627-3934 Library of Congress Control Number: 2023938206 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Cover image: Raycat / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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To my granddaughter, Stella

Preface This is the second part of my two-volume project on higher spin field theory. The project was initially planned as a single book, but due to the size of the subject, and the time it takes to write, the project was split into two volumes. The first volume mainly concerned the free field theory and background material needed to start work on higher spin theory. It also included a long historical chapter to put the subject in context. The present volume will treat the theory of interacting higher spin gauge fields. Naturally, there were limitations set to the contents of Volume 1, and there will be limitations set to the present volume also. Most of these will be apparent from the list of contents. But let me point out one limitation, partly set by the historical development of the subject, and partly by my own interests. As the subject developed, there became a strong focus on the AdS theory, from its inception in the late 1980s by E. S. Fradkin and M. A. Vasiliev, throughout the 1990s when the theory was worked out by Vasiliev. Then the AdS/CFT conjectures turned out to be applicable to higher spin interactions in AdS space-time, making it possible to study the interactions in the AdS bulk through the CFT on the boundary. This, and other approaches, led to a gradual understanding that the no-go problems, so notorious for the Minkowski theory, actually had their counterparts in the AdS theory; not the least the problems with locality at the quartic order of interaction. This, and certainly other circumstances contributed as well, had the effect that there was a renewed interest in the Minkowski space-time approach to the theory of higher spin gauge fields, in particular, to the light-front formulation of the theory, initially discovered in 1983 by I. Bengtsson, L. Brink and the present author, then elaborated by I. Bengtsson, N. Linden and the present author in 1987, and then studied at the quartic order by R. Metsaev in 1990. Quite unexpectedly, there has been a swing toward studying the light-front formulation of higher spin gravity during the last years. This, coupled with my own interests and knowledge, is reflected in my choice of contents. I had initially planned a more extensive treatment of the Vasiliev AdS theory, but as the topic of higher spin interactions has developed over the last 10 years or so, I now realize that such an endeavor is quite unrealistic and perhaps not very judicious. Even if I had mastered the subject fully, there is simply too many complex concepts and formalisms involved to give the subject a fair treatment in a book that is meant to cover also other aspects of higher spin field theory. More importantly, since some of the issues at stake are still considered contentious, it is best to let another decade or two pass before passing a verdict. For all these reasons, and since I wanted to write a book—two books as it has turned out—that is helpful for the newcomer to the fascinating subject of higher spin gauge theory, I have decided to focus on the fundamentals. It has indeed been necessary, since many of the theoretical techniques employed in higher spin theory are book-size subjects in themselves. https://doi.org/10.1515/9783110675528-201

VIII � Preface Although during the first 20 years of the present millennium, there has grown a community around higher spin gauge theory; the subject is still characterized by a small number of dedicated researchers well versed in the technicalities of the subject. The, quite a few, reviews of the subject that exist are often advanced and specialized and not really suitable for the newcomer or the curious, at least that is so in my opinion. There does not yet exist any broad reviews of the basics of the subject. Furthermore, given the strong focus on the AdS theory in the extant reviews, it is certainly appropriate here to shift some of the focus to the Minkowski theory. Finally, the task of writing an introduction to the subject is made difficult by the fact that very many mathematical techniques, often quite sophisticated, are used or have been used in it. While mathematical rigor has its own merits, employing it a theoretical physics text, is not without problems. Doing it seriously requires space, not having infinite space, the result may be strings of definitions and nonproved theorems with very little intuition. That is not my style. I have tried to point out where my treatment is simplified, or perhaps even oversimplified, but the reader should be wary of the risk that there might very well be subtleties hidden under the text. Let it be said, the two books are elementary. They are the kinds of books you can read in the bathtub in the wintertime, or if you are a young student, bring with you on a train trip to a conference or on summer vacation. If you want to spend the vacation educating yourself in higher spin theory—in the Liberal Arts sense—I hope these are the first two books you need to bring. I do not like to speculate, but I think it is fair to the prospective reader to state my view on the role of higher spin gauge theory. It is my contention that although a huge amount of technical work has been done on higher spin theory during the last 40 years, conceptually, the subject is still far from understood. In particular, I do not subscribe to the standard “ultrahigh energy unification” scenarios, possibly with higher spin gauge theory as a high energy limit of string theory that are conventionally put forth in the literature, or the other way around, a higher spin theory breaking down to a string theory. Such scenarios are, in my opinion, too naive and too much colored by the post Standard Model 1970s projections of further unification and attempts at string theory unification. I do think that higher spin gauge fields will play a role in the description of fundamental physical reality, hopefully in a way that will surprise us. Let me also add that I have focused on non-supersymmetric higher spin theory in four space-time dimensions. There is a practical reason for this, simply to keep the development comparatively simple and focused. There is also a natural philosophy reason; that the world is four-dimensional and non-supersymmetric is a very well-established experimental and observational fact. I believe it will remain so. The overall main focus of the book is on the construction of interacting higher spin gauge theories as classical field theories. Quantum properties lie beyond our reach in the present volume.

Acknowledgments Over the years, I have now and then felt quite inadequate for this undertaking, given all the bright workers in the field. But then again, the higher spin community seems to me to be quite good humored and humble, with many people sharing their understanding and confusion without prestige. I would like to thank all researchers that I have communicated with during the work. For obvious reasons, such communications has been through email during the last few years. In particular, many thanks to Evgeny Skvortsov for his kind support of this project and sharing his knowledge of—and thinking on—the subject. His reading and commenting on the last two chapters was very valuable. Glenn Barnich, Xavier Bekaert, Nicolas Boulanger, Marc Henneaux, Karapet Mkrtchyan and Massimo Taronna have kindly answered questions on their work, in particular, concerning various aspects of the Minkowski Fronsdal program. For the light-front theory, I have had the opportunity to consult Ruslan Metsaev. Thanks for explaining various aspects of the theory. Mikhail Vasiliev read and commented on the long last chapter on the development of the AdS theory. His comments made for many improvements and clarifications in my text. Much of my higher spin thinking, after my come-back to the subject in the early 2000s, have been inspired by papers of Jim Stasheff. Although we have been to the same conference, I have never spoken to him in person, but we have had email exchanges over the years. I sent a copy of the second chapter—that contains a long section on two-particle relativistic mechanical models—to Narasimhaiengar Mukunda. Some of the work reviewed in that section was done by Mukunda while visiting the Institute for Theoretical Physics in Göteborg in 1979 and in the early 1980s. Unfortunately, our world-lines did not cross at that time. Thanks for reading and writing back with comments. Finally, many thanks to my friends Ingemar Bengtsson, for reading and encouragement, and Bo Sundborg, for discussions on my “nonconventional” views on the nature of higher spin. Thanks also to my late PhD supervisor Lars Brink—for always being supportive—but who sadly passed away last year. Having realized that the vision of my youth—to solve the higher spin problem— will not come through, I am more than happy at an older age if I succeed in putting together a coherent story of the endeavors of many researchers. The problem is so difficult that no single person is likely to solve it—finding a fully consistent mathematical theory and finding its role in nature—unless a deep thinker like Albert Einstein, or an innovator like P. A. M. Dirac, comes around with new ideas, or a real world physicist like Steven Weinberg.

https://doi.org/10.1515/9783110675528-202

Contents Preface � VII Acknowledgments � IX 1 1.1 1.2 1.3

Introduction � 1 Setting the stage: What is the problem? � 2 Indeed, what is higher spin? � 4 The programs � 5

2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9 2.2.10 2.2.11 2.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.5 2.5.1 2.5.2 2.5.3 2.6 2.6.1 2.6.2 2.6.3 2.7

Notes on the history of higher spin interactions � 8 The long 1960s weak and strong interaction physics � 8 A parallel strand of history: systems of particles � 9 Direct interaction particle mechanics and the no-interaction theorem � 11 Models derived from strings � 16 The Casalbuoni–Dominici–Gomis–Longhi series of papers � 18 Relativistic harmonic oscillator models � 27 Multiplicative potential models and the Kalb–Van Alstine preprints � 32 The Nagoya University series of papers � 34 Komar’s constrained dynamics approach � 36 Historical notes on action-at-a-distance theory � 40 Beyond the standard Dirac particle interaction program � 42 Bi-local theories, infinite component fields and dynamical groups � 46 Spinning and supersymmetric particles � 47 String theory, supergravity and higher spin theory � 47 Early coupling problems and classic no-go results � 48 Buchdahl’s gravity compatibility investigations � 49 The Velo–Zwanziger problem and Johnson–Sudarshan problem � 50 The Weinberg S-matrix argument � 51 The O’Raifeartaigh and Coleman–Mandula theorems � 54 The Weinberg–Witten theorem � 55 Gauge theories of gravitation � 56 R. Utiyama � 57 T. W. B. Kibble � 64 D. W. Sciama � 69 The Fronsdal program � 70 The early Fronsdal (generalized Gupta) program � 70 The BBvD general analysis � 72 The Fronsdal program in the 1980s after BBvD � 75 The nonlocality circle of problems � 76

XII � Contents 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.1.8 3.1.9 3.1.10 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.6 3.6.1

3.6.6 3.6.7

Algebraic structures in higher spin theory � 78 Abstract algebras and operations on algebras � 78 The concepts of Grassmann parity and Z2 grading � 79 Associative algebras � 81 Lie algebras � 81 Structure preserving maps and representations of algebras � 82 Automorphisms, conjugations and involutions � 83 Real and complex algebras � 84 Graded and filtered algebras � 85 Derivations and differentials � 86 Cohomology and homology � 87 Local cohomology in field theory � 90 Higher spin related algebras � 90 The Heisenberg algebra � 92 The Weyl algebra � 93 Star product � 95 The Weyl map � 96 The Moyal product � 98 The Weyl map and the Moyal product � 99 Integral form of the star product � 101 Universal enveloping algebras � 102 General transformations revisited � 105 Maurer–Cartan theory � 105 Some more differential geometry recollected � 106 Lie algebras of Lie groups � 109 The structure constants � 113 Frames, coframes and the structure of Lie algebras � 114 Higher-order differential operators � 116 Higher-order differential operators and tentative higher spin algebras � 116 Two (or more) approaches to higher spin gauge algebras � 118 Some general properties of higher-order differential operators � 121 The Weyl algebra and higher-order differential operators � 122 Hamiltonian cotangent operators and higher-order differential operators � 123 A few steps into the theory of higher spin diffeomorphisms � 124 Toward the Fradkin–Vasiliev higher spin algebras � 125

4 4.1 4.1.1 4.1.2

General theory of interactions � 130 The group manifold approach � 130 Maurer–Cartan theory � 131 Gauge vs. general coordinate transformations � 134

3.6.2 3.6.3 3.6.4 3.6.5

Contents �

4.1.3 4.2 4.2.1 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5

Horizontality and factorization � 136 The Fronsdal–BBvD general theory of deformation � 138 The BBvD general analysis analyzed � 138 BRST-BV antifield theory � 142 Gauge fields, antifields and transformations � 142 Solution of the master equation � 147 The Koszul–Tate and longitudinal differentials and homology � 148 Homological perturbation theory—Lagrangian version � 150 Solution of the master equation: summary � 152 Space-time locality of the BRST-BV formalism � 152 Deforming the master equation and locality of interactions � 153 The algebraic Poincaré lemma and the descent equations � 156 Some further notes on Noether coupling techniques � 159 The Stasheff synthesis of ideas � 159 Strongly homotopy Lie algebras � 161 Jet spaces, Lagrangian field theory and locality � 161 Syntactic–semantic approach to Lie∞ algebras in interacting field theory � 163 The Lie∞ mathematical elaboration of the BBvD theory � 169

5 5.1 5.1.1 5.1.2 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6

Covariant approaches in Minkowski space-time � 171 Spin-3 cubic covariant interactions—BBvD � 171 The spin-3 cubic self-interaction � 171 Gauge algebra problems � 173 The covariant BRST approach � 175 Vertex operator approach to interactions—a reminder � 175 The first mid-1980s BRST attempt at interactions � 176 The 1988 Yang–Mills producing higher spin vertex � 178 General analysis of the BRST approach � 183 Follow up papers on the BRST approach � 185 Mechanics BRST vs. field theory BV formalism � 185 Non-BRST approaches—post millennium investigations � 187 Obstructions to the Fronsdal–BBvD program � 195 Some general properties of interactions � 195 Gauge invariants � 196 Spin 3 revisited � 197 Strong obstruction of the BBvD vertex � 198 A few more references � 199 Chapter 5 epilogue � 200

6 6.1

Light-front interactions � 201 The cubic interactions: review � 201

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XIV � Contents 6.1.1 6.1.2 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6 6.3.7 6.3.8 6.3.9 6.3.10 6.3.11 6.3.12 6.3.13 6.4 6.4.1 6.4.2 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.6 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 7.2 7.2.1 7.3 7.3.1

All possible cubic interactions in four dimensions � 203 Vertex operator formalism for cubic vertices � 205 Nonlinear realization of the Poincaré algebra � 208 Transformations and Fock field commutators � 208 The 𝒟 − 𝒟 algebra � 209 Computing the cubic interactions � 210 The 𝒦 − 𝒟 algebra � 210 Computation of commutators to cubic order � 211 Immediate restrictions from the 𝒦 − 𝒟 algebra: cubic and arbitrary order � 212 Reminder on Fock space fields � 213 Preliminary ansatz for the vertex functions � 214 Further restrictions from the 𝒦 − 𝒟 algebra � 215 Restrictions on the dynamical Lorentz generators � 217 Definite ansatz for the cubic vertex functions � 219 Derivation of all cubic vertices in four dimensions � 220 The cubic differential equations � 222 Solution of the cubic differential equations � 223 Summary of cubic interactions and the exp Δ question � 225 Comparision to the formalism of Metsaev and Ponomarev–Skvortsov � 226 Some deeper properties of the Poincaré algebra � 228 “Goodness” split of the Poincaré algebra � 229 The little group and the Hamiltonians � 230 Short orientation on recent developments � 231 The Metseav quartic analysis � 231 The Ponomarev–Skvortsov elaboration � 233 The cubic chiral theory � 235 Short guide to higher dimensions and supersymmetry � 237 Chapters 5 and 6 epilogue and a research question � 237 Higher spin in AdS and the Vasiliev theory � 239 The geometry and algebra of anti-de Sitter space-time � 239 Construction of de Sitter and anti-de Sitter space-times � 240 Symmetries of anti-de Sitter space-time � 242 Jordan split of the AdS algebra � 243 Outline of representations � 245 Oscillator representations and singletons � 251 Gauging the AdS algebra � 251 The Vasiliev theory: introductory remarks � 253 Dropping the Vasiliev equations on the reader � 255 Vasiliev free field equations � 257 Higher spin fields in AdS � 257

Contents

7.3.2 7.3.3 7.3.4 7.3.5 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.4.6 7.4.7 8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.4 8.4.1 8.4.2 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.6 8.6.1 8.6.2 8.6.3 8.6.4 8.7 8.8 8.8.1 8.8.2

Star products (and symbol calculus) � 262 AdS background, free fields and linearized curvatures � 264 Higher spin algebras � 272 Free differential algebras � 274 The nonlinear theory � 277 The fields and the doubling of the oscillators � 277 The Vasiliev star product � 278 Klein operators, supersymmetry and the twisted adjoint representation � 280 The first three Vasiliev equations � 282 The second two “nonlinear” equations � 282 Perturbative expansion of the equations � 283 Epilogue: some comments and where to go from here? � 289 Archaeology of the Vasiliev theory � 291 Archaeology? � 291 The free fields sequence � 296 The paper FF1 � 297 Papers FF2b and FF2f � 304 The paper FF3 � 304 The algebra sequence of papers � 308 The A1 paper � 309 The A2 paper � 318 The A3 paper � 324 The A4 and A5 papers � 334 The cubic interaction papers � 335 The CI1 paper � 336 The CI2 paper � 338 The free differential algebra sequence of papers � 341 The FDA1 paper � 342 The FDA2 paper � 343 The FDA3 paper � 353 The FDA4 paper � 357 The Vasiliev equations sequence of papers � 358 The VE1 paper � 359 The VE2 paper � 361 The VE3 paper � 365 The VE4 paper � 366 The review and generalization era � 367 What was achieved during 1986–1992? � 368 Motivations for Vasiliev theory � 368 The Vasiliev theory and physics � 369

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XVI � Contents 8.8.3 8.8.4

Higher derivatives � 371 Epilogue and a personal reflection � 372

Bibliography � 373 Index � 387

1 Introduction Given the very restrictive no-go results that surround the theory of higher spin interactions—to be reviewed in Chapter 2—one may very well question the meaningfulness of pursuing the subject at all. The common view is to see the no-go results as something positive—almost as a prediction of general quantum field theory—that delimit the spin spectrum that one needs to consider regarding fundamental theory. This has been the mainstream view since the supergravity days in the late 1970s, although the experimentally verified spin spectrum of fundamental fields conspicuously does not contain spin 3/2. For those who anyway have chosen to pursue the subject, one may be curious as to their motivations. In an answer to that, one could then argue that it is an interesting problem of mathematical physics that need not have anything to do with reality. However, given that theoretical physics is ultimately about understanding fundamental physics as it presents itself to us through observation and experimentation, the hope would be that there is some role for higher spin gauge fields to play in nature. Let me begin by stating my own point of view regarding these questions. I do think that higher spin gauge field theory constitutes a very interesting and challenging area of mathematical physics. But I also sincerely hope that the theory has some role to play in the mechanics of observed nature. In our physical world, fundamental matter is described by spin 1/2 fermionic fields while the fundamental forces of electromagnetism and weak and strong nuclear interactions are described by spin-1 bosonic gauge fields. The universal force of gravity may be thought of as being mediated by a spin-2 gauge field. It is well understood macroscopically, but far from it microscopically. Fortunately, the weakness of gravity makes it possible to ignore it at the energies attainable in present day experimental high energy physics. Beginning in the 1950s, higher spin matter particles were discovered in the accelerator laboratories. Fundamental or composite, they prompted research into higher spin wave equations for massive fields. Massless fields were mentioned in some papers, but only for “completeness of treatment” it seems. They had no inherent interest. Wigner and Bargmann–Wigner of course treated them, as did Weinberg. Except for Weinberg’s early 1960s papers, it is not clear from the literature that I have seen, if there was a recognition that massless higher spin fields would be force fields rather than matter fields. It might have been too obvious to point out. In this dualistic picture of fundamental reality as being constituted of matter and interactions, one may be more tantalized by the interaction side of the picture. A new matter constituent may be interesting, but a new kind of interaction is really intriguing!

https://doi.org/10.1515/9783110675528-001

2 � 1 Introduction

1.1 Setting the stage: What is the problem? What then is the higher spin problem? Vague as the question is, the mathematical physics problem may stated as follows: Constructing interacting field theories for classical and quantum gauge fields of spin higher than 2, possibly together with lower spin fields.

This problem may be considered to have received a particular solution by the Vasiliev theory. The theory was developed almost single-handedly—if not in isolation—by Mikhail Vasiliev from 1987 to the end of the 1990s, when it started to attract interest from other theoreticians. The Vasiliev theory then received a lot of attention during the early 2000s, largely in the context of the AdS/CFT-dualities. This does not mean that the subject of higher spin interactions is anywhere near being closed. There are still conceptual and technical questions left in a state of confusion. In actual fact, it could be said, at least within the confines of local field theory, that the theory has failed its objectives in that it suffers from serious problems with locality. These problems were actually unearthed as a consequence of the interest the theory attracted after the millennium. This issue has become contentious. The question of interactions in Minkowski space-time has also received renewed attention in the last 10 years or so, in particular, in the light-front formulation. The lightfront cubic interactions found in 1983, when set in the context of quartic investigations of 1990 and 1991, has produced what is called the cubic chiral theory, until very recently only known in the light-front formulation. Inroads to a covariant theory have been made during the last couple of years using twistor theory. Since nonlocality is almost bound to appear in higher spin theories at the quartic level, the cubic chiral theory escapes this fate, at the price of nonunitarity. One may debate which is the worse, nonlocality or nonunitarity, but the conventional view is to deem both alternatives ample reasons to scrap a theory. Normally, nonunitary time evolution, leading to the nonpreservation of probabilities in quantum physics, is seen as something bad, indeed as unacceptable. A nonconventional view would be to instead try to make a virtue out of a vice, and ask what purpose a nonlocal or a nonunitary theory could serve. Let us pause this line of inquiry, and instead simply formulate a more phenomenological aspect of the higher spin problem. The theoretical physics problem is ultimately related to phenomenology, to what is observed or could possibly be observed in the future of physics. The problem may be stated as follows, allowing for a wider mindset than a narrow unification of forces narrative: What is the role played by higher spin gauge fields in the fundamental structure of the world? Where are they and what purpose do they serve?

From the point of view of experimental physics, it is clear that there are sub-microscopic phenomena that can be well described by the concept of spin and the concomitant quan-

1.1 Setting the stage: What is the problem? �

3

tum theory. As already alluded to, the fundamental particles and fields are all described by very low values of spin: 1, 1/2 and 0. This—in the form of the Standard Model—seems to cover all presently known phenomena in the microworld. The gravitational interaction stands apart. At the submicroscopic scales probed today, gravity plays no role. It is a theory of the macroscopic world. It sets the stage of modern cosmology. Therefore, gravity—in its still largely unknown quantum version—is also the force of the very early universe and of black holes and the very late universe. There is a dichotomy of description here. Gravity seems to lend itself naturally to a geometrical picture of the phenomena. This was the way the theory was developed by Einstein and the way most subsequent work on it has followed. But it can also be viewed as a classical field theory on flat space-time. It becomes highly nonlinear when expressed in terms of a spin-2 field. It may seem unnatural to view the theory in that way, but it can be done.1 On the other hand, the gauge theories of electromagnetic, weak and strong interactions can be quite easily formulated and manipulated with no particular reference to geometry. This is so, even though gauge theory can be rephrased in terms of the geometry of certain fiber bundles, and in this way the theory can be made to conform closely to generalized geometrical ideas. Doing this offers useful techniques for unraveling deeper properties of the theories, as well as investigating topological phenomena. From the existing approaches to the description of the fundamental forces of nature, we can extract at least three aspects of the theories of massless low spin fields, i. e., spin 2 and spin 1: (i) Gauging. Both theories can be seen as “nongeometrical” gauge theories where a global symmetry group is made local. For spin 1, the procedure is a standard textbook exercise. However, while technically possible for spin 2, it is conceptually not straightforward. (ii) Deforming. Both theories can be seen as nonlinear deformations of a local gauge symmetry. Again, for spin 1, the procedure is fairly straightforward, although seldom performed in textbooks. Performing it for spin 2 is not very easy, and authors are reluctant to write out the details. (iii) Differential geometric. Both theories can be seen as generalized differential geometric theories. These three aspects are not exclusive, although at low levels of mathematical sophistication, the first two aspects appear rather different in their philosophy and implementation.2 As the sophistication is increased, they look more and more as different ways

1 It is quite interesting to read the text related to this issue in the Weinberg textbook Gravitation and Cosmology; see page 147 in [1]. 2 Beyond spin 1, the gauging approach actually requires a step of deformation in order to find the full dynamics.

4 � 1 Introduction to follow the third aspect. On a high enough level of abstraction and using sufficiently powerful mathematics, the three viewpoints may be merged into one. This seems to be a general phenomena. Given that a certain theory of physics exists in the sense of being internally mathematically consistent and consistent with accepted basic tenets of theoretical physics (these can change of course), it would seem that the various formalisms (with objects, concepts, formulas, etc.) describing it must be equivalent or isomorphic. Still, different approaches may look very different indeed, varying not just in notation but also in underlying philosophy, concepts and methodology, including areas of mathematics employed. It should also be kept in mind that it is one thing to be in possession of the Lagrangian and its symmetries, and being able to mold it into different guises, and another thing entirely to find it in the first place. Theoretically, the Yang–Mills theory was constructed using the gauging method without knowing the answer beforehand. This is the only significant example of an entirely successful application of method (i). On the other hand, the deformation theoretic approach (ii) was applied to spin 1 only after the theory was already known. Spin-2 theory (referring to Einstein gravity) was discovered using method (iii). Methods (i) and (ii) have been applied to spin-2 theory only after the theory was already known. As we will see, the gauging method is far from straightforward for spin 2.3 There is no reason to expect a simple solution to the higher spin problem. Perhaps a theory will be so complicated, even if it exists, as to be without intrinsic interest. Anyway, the weight of the no-go results indicates that any viable higher spin theory— theoretically and observationally—will be quite different from the lower spin theories and have significantly differing properties. And in my opinion, any phenomenological application will differ from the lower-spin gauge fields.

1.2 Indeed, what is higher spin? One more aspect of the basic problem definition must be added. What indeed is higher spin? The question may be related to the strange divergence in the history of the subject from the first positive interaction results in the 1980s. The development of the lightfront theory stalled in the early 1990s as regards fundamental questions (i. e., not counting technicalities such as higher dimensions, supersymmetry and massive fields) while the AdS theory developed to the final set of Vasiliev equations and became almost synonymous to higher spin theory. In the light-front theory, spin in “higher spin” has al-

3 Another sobering circumstance is the fact that the spin-1 electromagnetic theory—classical and quantum—was constructed over the course of several centuries, even millennia, if one starts the clock in antiquity, based on observing natural phenomena and very extensive experimenting by many natural philosophers and researchers. The discovery and invention of the Yang–Mills theories of weak and strong nuclear forces are, of course, unthinkable without the data from the high-energy laboratories of the 20th century. The fiber bundle formulation of the theory was however known to mathematicians.

1.3 The programs

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ways referred to the traditional phenomenological concept of spin—or helicity—in four space-time dimension. A higher spin algebra—whatever that could be—should be some infinite-dimensional algebra connecting fields of all spin. While in the AdS theory, although it started out with the natural generalization of the concept of spin from the Poincare to the AdS group, in the later incarnations of the theory, the concept of spin has been broadened to almost any “quantum number” or “eigenvalue” of operators in an infinite-dimensional algebra. In order to “solve the problem”, more and more structure was added, finally ending up in a very general scheme, quite far removed form the initial intuition of the problem—nothing strange or wrong with that in itself, but is it higher spin? This is not an easy question to answer, and unless there will ultimately be an answer from phenomenology, one may end up in a quite comic exchange of Groucho Marx like arguments: “Whatever it is, it is higher spin!” countered by “That’s not higher spin!”.

1.3 The programs It has become popular to speak about “programs” within the theory of higher spin.4 To what extent this is consciously in reference to Imre Lakatos’ theory of research programs is not clear. Most likely, it is only meant in a more loose sense. Although it can be seen as just a play with words, it is actually convenient to bring some “top-down order” to such a historically diverse area of research as higher spin theory is. Of course, it is always possible to retrospectively group sets of papers together into programs, even when the authors themselves may not have thought of the research as belonging to a program at the time of writing. Some programs may therefore have been deliberately conceived and set up as such, perhaps with some delay, while others may be seen as coalescing out of history more or less naturally. Furthermore, we are here concerned with a theoretical subject where criteria for acceptance such as “truth” is—at the present time—not a comparison to observed reality but rather intertheoretical consistency.5 A research program aiming at solving a certain set of problems within an area of science, in the loose sense intended here, may be thought as consisting of the conceptual and mathematical constructs themselves—ideas and methods—as well as the community pursuing it. Be

4 It seems that the first author to write in terms of “programs” within higher spin research was actually C. Fronsdal in the conference paper [2]. 5 Lakatos’ theory was concerned with experimental or observational science, aiming partly to reconcile the views of K. Popper and T. Kuhn and to modify and extend these views. The Lakatos’ theory was developed in interaction with P. Feyerabend, whose critique of Popper and Kuhn, was in its turn criticized by Lakatos. A central theme of this discussion was the demarcation problem, trying to understand what sets science apart from other areas of human thought. This is not the place to discuss the problem of demarcation in modern fundamental theoretical physics, but it is certainly important. It has been debated with some vigor in connection to string theory. The interested reader may find an inroad in [3].

6 � 1 Introduction that as it may, the following programs within the broad area of higher spin research can be discerned with some imagination. First, we have the Fierz–Pauli program that largely concerned free fields or fields interacting with electromagnetism or gravity. It was designated as a program by C. Fronsdal in 1979. It emanated from the discovery by Fierz in the 1930s that minimal electromagnetic coupling of higher spin fields, in particular, spin 3/2, was inconsistent due to the noncommutativity of covariant derivatives. Researching the Fierz–Pauli remedy of introducing auxiliary fields and deriving all field equations from an action, became the program itself. It is often referred to in modern higher spin literature. It may now be considered a closed program. From our historical overview in Volume 1, one may name two more related programs. The wave-equation program, going back to the Dirac spin-1/2 equation itself and to a Majorana (1932) paper, was concerned with researching first-order relativistic wave equations aimed at describing the known elementary particles and their electromagnetic and nuclear interactions. This program was largely exhausted by the work of H. J. Bhabha, by Harish-Chandra and by H. Umezawa in the late 1940s. As such, it was part of main-stream physics and the development of early quantum mechanics into modern quantum field theory. The tensor field Lagrangian program saw a movement from firstorder wave equations to Lagrangians for second-order equations describing massive higher spin particles such as those seen in the accelerator laboratories of the 1950s and 1960s. The often referred to papers by the modern higher spin community of S-J. Chang and by L. P. S. Singh and C. R. Hagen, belong to this program. These papers can be seen as intermediaries between Fronsdal’s thesis paper from 1958 and the classic Fronsdal (1978) paper on massless higher spin gauge fields. This latter paper became the founding paper of the broad modern higher spin gauge field theory program. Let us then turn to the specific programs aimed at massless higher spin interactions. From our discussion in Volume 1, we can discern three major approaches to interactions among higher spin gauge fields. The first explicitly named program was the generalized Gupta program discussed by Fang and Fronsdal in 1979. It was actually proposed as a research program already in the Fronsdal (1978) paper, aimed at finding self-interactions for higher spin gauge fields. In the modern higher spin community, it goes under the name of the Fronsdal program. It concerns covariant self-interactions in Minkowski space-time. The first major advance beyond Fronsdal came with the Berends, Burgers and van Dam papers of the mid 1980s. It was later analyzed by mathematicians R. Fulp, T. Lada and J. Stasheff and has been studied by the modern higher spin community. To this program, one may also refer BRST-BV approaches such as by the present author and a few groups of authors running up to the present time. For a long time, the most conspicuous program was the Vasiliev program in antide Sitter space-time. Initiated by Fradkin and Vasiliev in the late 1980s and extensively developed by Vasiliev in the period 1986–1992, it is the most advanced program so far. The greater part of the research into higher spin theory, after the late 1990s renewed

1.3 The programs

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interest in the subject, has been within this program. This partly has had to do with the connection with the AdS/CFT duality conjectures and the computational tools provided by the dualities. I will now name the third program. The chronologically first positive self-interaction result came within the light-front approach in Minkowski space-time with two papers by Ingemar Bengtsson, Lars Brink and the present author in 1983. Apart from a 1987 paper by I. Bengtsson and myself, together with Noah Linden, the only other researcher working on it for a quite long time after that was R. R. Metsaev who studied the theory at quartic order. It was then taken up in the mid-2010s by D. Ponomarev and E. Skvortsov. I will designate this program the Dirac program rather than the light-front program. The rationale is the 1949 Dirac paper on “forms of relativistic dynamics” where Dirac explicitly wrote: [...] the problem of finding a new dynamical system reduces to the problem of finding a new realization of the Poincaré Lie algebra.

Dirac meant nonlinear realizations, i. e., dynamical systems with interactions. The lightfront higher spin theory is just such a system. Calling the program “Dirac” allows for further developments. It should also be mentioned that the term “Dirac program” has been used before, namely in the context of directly interacting particles in the paper [4]. In relation to this, one may see two branches to the Dirac program: a particle mechanics branch and a field theory branch. To summarize, we have three major programs for higher spin gauge interactions: the Fronsdal program, the Vasiliev program and the Dirac program. This volume is devoted to an attempt to understand them.

2 Notes on the history of higher spin interactions In this chapter, we will continue the historical overview of higher spin theory that was begun in Chapter 2 of our Volume 1. The focus will be on research pertaining to higher spin interactions. The story we will tell will include elements that are not so often included in short surveys in the higher spin literature. But as we saw in Volume 1, higher spin theory had its place in fundamental theoretical physics from the early 20th century up to the advent of the Standard Model in the 1970s. In these early days, the focus and connotation were different from those of today; these latter emerging only with the advent of supergravity theory, and in particular with Fronsdal’s shift of emphasis in 1978 from massive fields to massless higher spin gauge fields. Some historical notes on interaction research from the 1980s, namely the 1983 positive results on cubic interactions in the light-cone formulation, the Berends–Burgers– van Dam (BBvD) covariant cubic interaction for spin 3 and their general analysis, and the Fradkin–Vasiliev AdS theory, were included in Volume 1, and will not be repeated here.1 The actual theories will be treated in the appropriate chapters, and further historical comments will be made there. Since alternative approaches to gravity have been a source of much inspiration in higher spin research, we will recount the history of gauge approaches to gravitation. Hopefully, this will shed light and perspective to the technical sections in the following chapters. As in Volume 1, our historical notes refer to the published record.

2.1 The long 1960s weak and strong interaction physics In retrospect, the “long” 1960s (from late 1950s to early 1970s) stand out as the golden era of experimental elementary particle physics. Ever more powerful accelerators came online and more sophisticated detection apparatuses were constructed. This fascinating story is told in many places, for instance, in [5, 6, 7]. The parallel theoretical work in providing a rational theory describing and explaining the data eventually resulted in the unified U(1) × SU(2) × SU(3) Standard Model. However, if this was a “belle epoque” for particle phenomenology, it has also been described as a period “crisis” for theoretical particle physics. The success of renormalized quantum electrodynamics did not, at first, carry over to the weak and strong interactions. This lead to a distrust in quantum field theory, and the exploration of alternative theoretical schemes. Yang–Mills theory was discovered during this period and eventually became the model for the nuclear forces when it turned out that the theory was, in fact, renormalizable both in the unbroken case (strong forces) [8, 9] and Higgs bro-

1 See Section 2.12 of our Volume 1. https://doi.org/10.1515/9783110675528-002

2.2 A parallel strand of history: systems of particles

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ken case (weak forces) [10]. Asymptotic freedom also added strongly to its acceptance [11, 12].2 The Standard Model The shorthand notation U(1) × SU(2) × SU(3) hides the complexity of the model. One ought to write it as UY (1) × SUL (2) × SU(3)C . The SUC (3) quantum chromodynamics part of the theory only affects the color charged quarks and gluons building up the hadrons. The SUL (2) weak part of the theory affects all lefthanded particles, leptons as well as quarks, whereas the UY (1) part of the theory refers to the interactions between particles carrying hypercharge Y related to electric charge and isotopic spin. Spontaneous symmetry breaking—the Higgs field acquiring a vacuum expectation value—gives masses to all the fermions and to three of the resulting Uem (1) × SUweak (2) gauge fields. One of the electro-weak gauge fields remain massless, and that is what we see macroscopically as the electromagnetic field. The actual couplings between spin-1 bosonic gauge fields and spin-1/2 fermionic matter fields are therefore quite complicated, although in principle given by minimal coupling. A review of the model, with technical details, can be found in [16].

Part of the theoretical work leading to the Standard Model was to disentangle the effects of strong and weak interactions.3 Before the Yang–Mills description became convincing, there were other theoretical schemes developed, in particular, S-matrix theory and a focus on symmetries, for instance, in the form of current algebra since one did not have any model for the dynamics of the interactions. Regge theory, dual models and string theory were other approaches. To this mix also belongs the theory of dynamical groups, infinite component fields and directly interaction particle models. Most of these schemes were primarily concerned with the strong interaction, which was responsible to the spectrum of higher spin excitations found in the experimental data. Parts of this research may be of some relevance if one is interested in underlying dynamical principles for higher spin theory.

2.2 A parallel strand of history: systems of particles In the 1970s, the research into string theory inspired studies of simpler mechanical systems such as point particles and systems of point particles, for instance, two-particle systems of the harmonic oscillator type. As it became clear that the hadrons were composite particles, it was indeed natural to study particle mechanical models of composite systems. This strand of research also contains discretized string models and rigid string models. Although this strand of history evidently belongs to the history of strong interac-

2 See, for instance, [13] and [14] for history. The theory was initially put forth as a gauge theory of isotopic spin in the Yang–Mills paper [15]. We have some historical notes in Section 2.5. 3 M. Gell-Mann have elaborated on these difficulties in interviews that can be accessed through the Internet.

10 � 2 Notes on the history of higher spin interactions tion and hadron physics—and is seldom referred to in higher spin contexts—concepts and methods that were employed are interesting and may still offer insights into the higher spin interaction problem. In my opinion, the question of an underlying rationale for higher spin gauge theory must eventually be addressed, even though it may not be in the form of a simple underlying mechanical model. A historical study may nevertheless be inspiring. Foraging deeper into the garden of papers, it becomes clear that the story goes back further to the late 1940s. At this time, it had become clear that renormalized QED was a very successful theory, although there remained a skepticism over the fundamental validity of renormalization, regarded as subtracting infinity from infinity. It also seemed to be very difficult to explain the weak and strong interactions with a similar kind of quantum field theory. This is before Yang–Mills theory was discovered, although mediating massive vector fields had been considered in the case of weak interactions. Furthermore, bound states was a difficult problem in quantum field theory. So, as the story goes, field theory went out of fashion, and other approaches to the microscopic forces were investigated. Of these, S-matrix theory is perhaps the most prominent example, and what eventually became dual models and string theory. Another such strand of research, that will concern us here, was theoretical experiments with direct interaction theories for point particles, both classical and quantum. It was a return to what could be designated as “Poincaré invariant quantum mechanics”, to distinguish the research from “relativistic quantum mechanics”, which carries other connotations in its relation to quantum field theory. The practical implementation of such models was to what could be collectively designated as “constituent quark or parton models”. This literature is very extensive. One may ask what is the relevance of studying particle mechanical models for strong and nuclear interactions in relation to the higher spin problem? As long as one is only interested in low-spin gauge fields, i. e., of spin 1 and 2, then these can be considered one at a time if so desired. However, for spin higher than 2, an infinite tower of higher spin fields must be considered all together. It is then natural to think of them as one single entity, where it does not make sense to pick out any one spin in particular. The spectrum of spin could then be the consequence of some underlying mechanical system, and be governed by some “dynamical group” or “spectrum generating algebra”. It is not obvious where to start an exposition of this research. As we have described it above, it was phenomenology driven and quite disparate. There are considerable overlaps between some of the papers as well as parallel strands of papers not cross-referred. There is also a theory driven strand, starting in the late 1940s with papers on position operators in relativistic quantum mechanics [17] and mass center definitions for relativistic particle systems [18]. These papers, by Newton–Wigner and by Pryce, respectively, are contemporary with the Dirac 1949 paper on forms of relativistic dynamics. Further research into this area lead to the “no-interaction theorem” and the need to relate to that result.

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We will mention a few strands of research that one may happen upon when searching for underlying mechanical models for higher spin fields. Let us start with an overview of direct interaction theories, and then move on in the following sections to examples that may have some relevance for higher spin theory. After that, we will return to an attempt at a summary of the broader theoretical context of these theories in Section 2.2.8. As for connections between higher spin gauge field theory and string theory, we will have a few comments in Section 2.3.

2.2.1 Direct interaction particle mechanics and the no-interaction theorem To fix ideas, one may think of the mesons and baryons as made up of pairs or triples of partons or quarks, interacting directly without any mediating field. For non-relativistic Galilean invariant theories, there is nothing strange about such action-at-a-distance interactions, neither classically nor quantum mechanically.4 For Poincaré invariant relativistic theories, complications arise, in particular, in quantum theory, since the Lorentz transformations mix space and time. These complications are normally resolved, as is well known, by passing on to classical or quantum field theory. Nevertheless, one may wish to stay within the context of particle theory to do phenomenological calculations. A review article that recounts both the theory and the applications, as they stood in the 1980s, is [19]. To get a first handle on the problem of directly interacting particles in special relativity, it may be useful to recapitulate how free relativistic point particles may be described. Clearly, even in relativity, as the particle moves, it traces out a path in space, which in a certain “laboratory” frame of reference may be described by the 3-vector coordinates x i (t) where t is the time in the laboratory. However, in special relativity we prefer to use 4-vector coordinates x μ (τ) with τ a parameter along the world line of the particle. This leads to a redundancy of d. o. f.’s, which is manifested in a “gauge symmetry” of the action, or more exactly, a τ-reparametrization invariance. This, in its turn, is reflected in the presence of a first-class constraint in the Hamiltonian theory. The gauge can be fixed by choosing a particular time as a relation between x 0 and τ.5 μ Now, if there are two particles, free or interacting, there are two world lines x1 (τ1 ) μ and x2 (τ2 ) parametrized by two different parameters τ1 and τ2 . These must be related in some way. Therefore, one may suspect the presence of further constraints and gaugefixing conditions.

4 That is, “nothing strange” from a theoretical physics perspective. Philosophically, one may worry about action-at-a-distance—as the natural philosophers of old did—even in Newtonian physics: What is gravity? 5 Each massive particle has its own proper time, also often denoted by τ. This is the time measured in the rest frame of the particle, a frame of reference moving with the particle.

12 � 2 Notes on the history of higher spin interactions Relativistic point particles The action for a free relativistic point particle with mass m is taken as the length of the wordline S = m ∫ ds = m ∫ √−dxμ dx μ = m ∫ √−xμ̇ x ̇μ dτ = ∫ Ldτ

(2.1)

The momentum conjugate to x μ is pμ =

mxμ̇ 𝜕L =− 𝜕x ̇ μ √−x ̇2

(2.2)

from which follows the first class constraint p2 + m2 ≈ 0 (often referred to as the Einstein relation in this research). The canonical Hamiltonian H = p ⋅ x ̇ − L is zero, which signals that the action S is reparametrization invariant. It may be checked that S is indeed invariant under a transformation τ → τ ′ (τ). To accommodate the case of zero mass, one may write the action using an einbein e, S = ∫ dτ(−

x ̇2 m2 e + ) 2e 2

(2.3)

A way to describe two free particles now presents itself. A natural choice for an action is S = m1 ∫ √−dx1 ⋅ dx1 + m2 ∫ √−dx2 ⋅ dx2

(2.4)

This, however, raises the question of time parametrization of the two world-lines. Should one use two parameters τ1 and τ2 so that μ

dxi =

μ

dxi

dτi

dτi

for i = 1, 2

(2.5)

or just one common time τ with τ1 and τ2 related to it, in the simplest case as τ1 = τ2 = τ? In a one-parameter approach, a conventional Lagrangian treatment would take as an action for two free particles S = ∫(m1 √−x1̇ + m2 √−x2̇ )dτ

(2.6)

or a generalization of (2.3) with two einbeins e1 and e2 . As we will see, taking the action (2.4), with interparticle interaction introduced as addition to the masses mi , will (at least in the cases considered) lead to a transversality constraint P ⋅ r ≈ 0 that effectively reduces the model to one time. Here, Pμ and r μ are total momentum and relative coordinate formed out of the individual particle phase space variables. Intuitively, having two reparametrization invariances, one of them can be used to equalize parameters, corresponding to such a constraint. The remaining one-parameter invariance would then result in an Einstein relation primary constraint. This is indeed what happens in the detailed analysis of models of this type. A third secondary constraint generally occurs.

As we will see, there came to be developed basically two distinct (in their flavor), but related (in their end results) approaches to instantaneously directly interacting point particles. In the approach based on a Lagrangian, it is natural to have just one time parameter τ. The Lagrangians considered turn out to be singular, and in the course of their

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Dirac analysis, there appear constraints that correspond to relating any individual parameter τ1 and τ2 , had such been considered. In the constrained Hamiltonian approach, starting directly with judiciously chosen first-class constraints encoding interaction between the particles, it is a feature of the approach that one has individual time parameters for the particles. In order to have physical sensible models, one must chose gauge conditions, again such that the time parameters get related. It should be mentioned in this context that the Dirac 1949 paper [76] on “forms of relativistic dynamics” is central to this research.6 According to Dirac, any relativistically invariant system, whether mechanical or field theoretical, must furnish explicit expressions for the ten generators Pμ and Jμν of the Poincaré group in terms of the basic degrees of freedom of the theory. Dirac’s theory is set within Hamiltonian dynamics, and three basic choices of time evolution is given; the “instant form” (which is the most common one), the “front form” (the form employed in the light-front approach) and the not so common “point form”. The aim is to find new interesting models with interactions. In field theory, the light-front higher spin field theories are precisely such models. Here, we will focus on particle mechanics models with interactions. One such general scheme for mechanical models was found by H. Bakamjian and L. H. Thomas in 1953 [20]. The method of directly interacting particles was further explicated in a conceptual paper by L. L. Foldy [21]. However, in the early 1960s there appeared an obstacle to attempts at direct interaction theories: the so-called no-interaction theorem by D. G. Currie, T. F. Jordan and E. C. G. Sudarshan [22]. The problem was centered around the so-called world-line condition, the requirement that the world-lines for the individual particles should be “compatible” as viewed from different relativistic frames of reference. In essence, different observers should see the same world-lines. It had been observed already by Thomas in the paper [23], preceding the Bakamjian–Thomas paper, that the world-line condition may have to be given up in the case of particles interacting at a distance. Various proofs of the theorem, based on slightly different assumptions and methods, then appeared [24, 25, 26, 27] over the years. The root of the problem is succinctly formulated by H. Leutwyler in [24], from which we quote: [. . .] a Hamiltonian theory of two classical particles is unable to describe any interaction if the principle of relativity is satisfied, i. e., if a) there exists a set of ten generators of the inhomogeneous Lorentz group, and b) the observables of the theory (the coordinates of the particles) transform correctly under the inhomogeneous Lorentz group.

There is a clash between two principles. In the original paper by Currie et al., it is formulated as: 6 We have already referred to this paper in the Introduction as defining what we call the Dirac Program, in particular, as the foundation of the light-front approach to higher spin interactions.

14 � 2 Notes on the history of higher spin interactions

We distinguish two different kinds of assumptions that occur under the heading of “relativistic invariance” in theories constructed to give a relativistic invariant description of interactions between particles. The first of these reflects the principle of special relativity that the laws of physics should be invariant under changes of reference frame; it is formulated in terms of the symmetry of the theory under the group of frame transformations. The second is an assumption of “manifest invariance”; it requires that certain quantities transform under changes of reference frame in a particular manner that is intimately related to the Lorentz (or Galilean) transformations of space-time events.7

The first requirement is precisely what Dirac formulated in his 1949 paper. The second condition, called the world-line condition, becomes in practice in particle theories, the following three requirements in the instant form: [qi(n) , Pj ] = δij

[qi(n) , Jj ] = ϵijk qk(n)

[qi(n) , Kj ] = qj(n) [qi(n) , H]

(2.7)

where qi(n) are the spatial coordinates of the particle n, and Pi , Ji and Ki are the generators of translations, rotations and boosts, respectively. The first two requirements are unproblematic, but the third, involving the boosts, and therefore time evolution, clashes with the Poincaré algebra, forcing the velocities of the theory to be constants, i. e., no accelerations, and therefore, no interactions. The third requirement of (2.7) comes about [28] by demanding the coordinates to transform geometrically as qμ(n) → qμ′(n) = Λμ ν q(n)ν

(2.8)

at any one time, and quantum mechanically as 󵄨 qμ(n) (τ) → qμ(n)′ (τ ′ ) = U † (Λ)qμ(n) (σ)U(Λ)󵄨󵄨󵄨σ=τ ′

(2.9)

Disastrous as it may at first seem, there are several ways out of this dilemma, one of which is to deny the coordinates the status of observables, either in the mechanical theory itself as in the Bakamjian–Thomas approach, or by going to quantum field theory where the coordinates are no longer quantum operators. It gradually became clear that it might not always be possible to identify d. o. f.’s introduced at the outset of model building with their “intended meaning” as particle positions. Given the fact that the no-interaction theorem, or the world-line condition is seldom mentioned in phenomenological papers from the time, it can be gathered that it was not considered as any great hurdle. Indeed, given that direct interaction mechanical models were thought of as describing the internal dynamics of subnuclear particles, it made sense to assume that the positions of the constituents were not observable.8

7 Quotation marks in the original, emphases is ours. 8 Clearly, there was something a bit too pedantic about the no-interaction theorem, as direct interaction models were used for phenomenological calculations. I. T. Todorov in [29], compared the situation to Zeno’s proof of the impossibility of motion, where his opponent answered him by just walking.

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Another way of circumventing the dilemma of the no-interaction theorem is to consider constrained Hamiltonian systems. Indeed, if one wants to study the mechanics of relativistic point particles, free or interacting, one is naturally lead to reparametrization invariant systems, and consequently to constrained systems. It is mainly this type of theories that we will be concerned with in the following. We will, however, return to a brief review of the wider context in Section 2.2.8. The bosonic string There are numerous reviews of string theory. One such, that captures the flavor of the theory in the early days, is [30]. A classic textbook review is [31] from the 1980s era, and a more modern textbook is [32]. Here, we will just record the basic picture. The string is described by its coordinates in space-time x μ (σ, τ) where σ and τ are thought of as coordinates on the world-sheet that the string sweeps out as it moves. The action (the Nambu action) is the area of the world-sheet τ2

π

τ1

0

1 S=− ∫ dτ ∫ dσ √ (x ̇ ⋅ x ′ ) − x ′2 x ̇2 2πα′

(2.10)

where ̇ denotes τ derivative and ′ denotes σ derivative. The idiosyncratically denoted constant α′ has mass dimension −2 and is called the universal Regge slope parameter and can be thought of as the inverse of the string tension T , more precisely T = (2πα′ )−1 . The action is a reparametrization invariant, giving rise to an enumerable infinite set of constraints Ln satisfying the Virasoro algebra [Lm , Ln ] = (m − n)Lm+n

(2.11)

In the quantum theory, there is an anomalous term on the right-hand side that forces the theory to be consistent only in the unphysical number of 26 dimensions. Minimizing the action gives rise to equations of motion that are—under a suitable choice of gauge, for instance, the light-cone gauge—of the of vibrational type. This motivates the introduction of transverse harmonic oscillator modes ani , where negative n corresponds to creators and positive n to annihilators. In μ quantum theory, the L0 = 0 Virasoro constraint, that contains the “zero-mode” pμ ∼ a0 (the string center of mass momentum) then take the form pi pi = M 2 =

1 ∞ i i c ∑a a + α′ n=1 −n n α′

(2.12)

where c is a normal ordering constant. Since there are an infinite number of oscillators, the sum of zeropoint energies must renormalized [33]. The value of c can also be found by arguing that the first excited level i consists of states a−1 |0⟩ that describe the transverse modes of a massless particle of spin 1 [34]. This gives c = 1 in order that the mass is zero for these states. The ground state therefore has negative mass—the notorious tachyon. The formula (2.12) describes Regge trajectories. The “leading” trajectory being s = α′ M 2 +1. One speaks of the “zero-slope limit” where the string theory goes over to a low energy, low spin field theory. In the opposite limit, the infinite slope limit—or zero-tension limit—the oscillators decouple and the free string theory can be interpreted as a free higher spin field theory [35] (see our Volume 1, Section 5.4.4).

16 � 2 Notes on the history of higher spin interactions 2.2.2 Models derived from strings There are quite a few kinds of mechanical models derived from the bosonic string. String theory, as originally developed out of S-matrix theory and dual models, was thought of as a model for strong interactions between hadrons and mesons. Due to the complicated nature of the theory, simplifying schemes were devised. A common feature of these schemes was a drastic reduction in the number of degrees of freedom from the continuous infinity of string configurations x μ (σ, τ) and the corresponding momenta, to a finite number of discrete d. o. f.’s. For the discretized string, this is clearly in the nature of the very scheme. But also models where the string is constrained to move in a specified μ rigid way, the number of d. o. f.’s become finite. Then, given, for instance, coordinates x1 μ and x2 , related to the endpoints of the truncated string, and the corresponding momenta p1μ and p2μ , there is a small number of bilinears making up pieces of constraints. So, for this quite obvious reason, the sets of constraints that appear in the papers are similar to each other, but differ in their details and combinations. Let us however start with the zero-tension string where the number of discrete d. o. f.’s is countable infinite. Zero-tension limit string Starting from the bosonic string Virasoro algebra and performing the zero-tension limit, i. e., in terms of α′ , the limit α′ → ∞, one obtains [35] a simple first-class constraint † algebra spanned by an countable infinite set of oscillators (αnμ , αnμ ) G0 = −p2

G+n = αn ⋅ p

G−n = αn† ⋅ p

(2.13)

Since there is no longer any coupling between the oscillators, it is natural to truncate to just one oscillator. The resulting field theory, upon applying the Siegel mechanicsto-field-theory algorithm,9 is precisely Fronsdal’s theory of massless higher spin gauge fields in the triplet formulation. For the standard formulation with trace constraints on fields and parameters, one needs to add the second-class constraints, 1 T = α⋅α 2

1 T † = α† ⋅ α† 2

(2.14)

This model works very well for reproducing an infinite tower of integer spin gauge fields, but it has one physical drawback. In the limit process, the oscillators lose their nature as physical harmonic oscillator phase space variables, becoming rather formal Fock space generators for collecting all higher spin fields together into a master field |Φ⟩ = (ϕ + ϕμ α†μ + ϕμν α†μν + ⋅ ⋅ ⋅)|0⟩. For this reason, at least, it is interesting to look for other versions of underlying mechanical models for higher spin gauge fields. On the other hand, it is perhaps more likely that the singular nature of higher spin gauge fields,

9 See Sections 3.3.3 and 5.4 in our Volume 1.

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manifested in several ways, is indeed reflected in the somewhat singular nature of the underlying mechanical constraints. Discretized strings Let us study the discretized string of V. D. Gershun and A. I. Pashnev [36]. By a certain procedure, the authors discretize the bosonic string into n particles, and then specializes to two and three particles. In the two-particle case, the coordinates x1 and x2 naturally correspond to the endpoints of the string. Hamiltonian analysis of the resulting action leads to a set of four primary constraints Φ+ = p21 + p22 + 2ω2 r 2 ≈ 0 χ+ = (p1 + p2 ) ⋅ r ≈ 0

Φ− = p21 − p22 = P ⋅ Q ≈ 0

χ− = (p1 − p2 ) ⋅ r ≈ 0

(2.15)

where r = x2 − x1 , P = p2 + p1 and Q = p2 − p1 with ω related to inverse of the string α′ . Choosing the constraint Φ+ as Hamiltonian H, a further secondary constraint χ3 = 2 Q − 4ω2 r 2 results. Next, linearly combining the constraints into L±1 = Φ− ∓ 2iωχ+ and L±2 = Φ3 ∓ 2iωχ− , the algebra of constraints is worked out. This is such that if all the Lk are chosen as first class, then one must require also P2 ≈ 0 leading to a massless system.10 Thus, all the Lk are chosen as second class. The authors then proceed to impose the constraints between physical states, by requiring H|ψ⟩ = L1 |ψ⟩ = L2 |ψ⟩ = 0

(2.16)

Since the Hamiltonian describes harmonic oscillations of the relative coordinate r, it makes sense to introduce holonomic degrees of freedom that becomes creation and annihilation operators in the quantum theory. A further linear recombination then puts the constraints into the form

L−1

1 H = − (P2 + 8ωa† a + α0 ) 2 = 2i√2ωP ⋅ a† L1 = 2i√2ωP ⋅ a L−2 = −8a† ⋅ a†

(2.17)

L2 = −8a ⋅ a

with a normal ordering constant α0 in H. It is now clear that the field theory implementation of the conditions (2.16) leads to field equations for a tower of massive higher spin fields represented by transverse traceless symmetric tensors (see Section 2.5 in our Volume 1). The masses lie on a linear Regge trajectory. There is a follow-up paper by Pashnev [37]. The aim of this paper is to set up a free field theory action for the two-particle mechanical model. The presence of second-class 10 Or rather it is overconstrained, and unable to describe field theory degrees of freedom.

18 � 2 Notes on the history of higher spin interactions constraints make a BRST approach cumbersome, and the paper resorts to a method of dimensional reduction instead.11 The two-particle discretized string model discussed here is obviously a mechanical model for two particles interacting directly through a harmonic oscillator potential, and as such it could be set up and considered independently of any string backdrop. That was indeed done in a paper by A. Barducci and D. Dominici [38] referred to by Pashnev, which in its turn refer to a paper [39] by Y. S. Kim and M. Noz on harmonic oscillator models for hadronic physics. This strand of research will be treated in Sections 2.2.3 and 2.2.4 below. The transversality constraint and the issue of relative time The transversality constraint P⋅r ≈ 0 (where P is the total momentum and r the relative coordinate) or in terms of holonomic variables P ⋅ α† ≈ 0, tends to turn up in one form or another in most two-particle mechanics models. It is sometimes commented upon that this constraint is related to the question of time parameters for the two particles. One paper that does comment on it is [40] from which we quote. This constraint [P ⋅ α† ≈ 0] is welcome, since it is required at the classical level in order to eliminate the relative time variable, and at the quantum level in order to eliminate some unphysical states [removed reference]. The presence of this constraint is a feature of almost all constrained models proposed in the literature for the description of the interaction between two relativistic point particles.

Rigid strings or straight-line strings The rigid string strand of research grew out of a paper by A. Polyakov [41] that studied quantum properties of bosonic strings coupled to an extrinsic curvature (as embedded in a higher-dimensional manifold) on the worldsheet of the string. A few follow-up papers [42, 43] examined trajectories for the simplest cases of such rigid strings, in the open case, in the form of a rigid rod. See also [44] for the straight-line string. We will not say anything about this, but instead move on to a series of papers that are interesting for massless higher spin field theory. In these papers, concrete rigid string-type actions in terms of end-point coordinates are considered. 2.2.3 The Casalbuoni–Dominici–Gomis–Longhi series of papers There are quite a few papers written by a group of authors more or less centered in Firenze, Italy, in the mid to late 1970s. I will use the abbreviation CDGL12 to denote this 11 This method was to resurface in the late 1990s in connection with Pashnev’s work with M. M. Tsulaia on the rediscovered BRST approach to massless higher spin theory. See Section 2.11.6 of our Volume 1. 12 After the first letters in the names of the authors. There are papers by subsets of the authors, and may be occasional papers with additional collaborators. In our review of these papers, unless otherwise stated, we will conform to their notation and conventions, in particular, use a mostly negative metric.

2.2 A parallel strand of history: systems of particles

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research program. In this program, we find both more theoretical papers [38, 45, 46, 40] investigating the general relativistic two-body problem as well as more concrete models of the rigid string or the harmonic oscillator type [47, 48, 49]. These latter papers, I will denote by CL(I), CDL(II) and CD(III) respectively, (indicating the authors and the succession of papers). We will have a little more to say about the first set of papers, but start by focusing on the second set, where there is a shift from describing massive particles to massless fields. The first set of papers, denoted as BD(I), DGL(II), DGL(III) and DGL(IV) respectively—more concerned with constituents of massive hadrons—will then be discussed in Section 2.2.4 below, in particular their relation to the relativistic harmonic oscillator theory research subprogram as well as to the more theoretical action-at-adistance problem. According to the introduction to the central paper of the series CL(I) [47], the motivation came from, at that time, recent results on the zero-slope limit13 of bosonic strings. Let us quote a few defining sentences from the introduction: [. . .] in the limit of zero slope, the dual models give rise to the same results obtained in the theories usually used for the description of nonhadrons, such as Yang–Mills and gravitational field theories. This fact suggests that the nonhadronic interactions should be described in a form more similar to dual models than realized until now, with a different and much lower value of the intrinsic fundamental length. [. . .] we will choose an action, which corresponds to a particular motion of the relativistic string [removed references], namely that in which the string effects a rigid rotation. This state of motion can be described in terms of two-position four vectors, and the action will be taken proportional to the area swept out during the time evolution. [. . .] we get a number of primary and secondary constraints [. . .] The quantization will be performed covariantly following the method of Dirac [removed reference]. We will see that the model describes a tower of zero-mass states with all the integer spin values.

This introduction clearly reads very intriguing to anyone interested in higher spin gauge theory. Unfortunately, there are disappointments downstream, that we will come to. Let us, however, jump straight in and take a first look at the actual model. The authors consider two four-vectors x1 (τ) and x2 (τ) describing two world-lines depending on a common parameter τ. The infinitesimal area spanned by the difference x2 (τ) − x1 (τ) during dτ, under the condition that dx1 and dx2 lie on the same plane, is given by 1 2 dA = (√(ẋ1 ⋅ (x2 − x1 )) − ẋ12 (x2 − x1 )2 − (x2 ↔ x1 ))dτ 2

(2.18) τ

where the overdots denote τ derivatives. An action can be defined as S = − 2α1 ′ ∫τ f dA. i

13 The zero-slope limit refers to Regge trajectory diagrams with mass on the horizontal axis and spin on the vertical axis.

20 � 2 Notes on the history of higher spin interactions As commented on in the paper, this action can be obtained from the string action by considering a certain rigid motion of the string. Then x1 and x2 correspond to the endpoints of the string, and the difference x2 (τ) − x1 (τ) to the σ-derivative of the string coordinate x(σ, τ). Next, the authors introduce center14 and relative variables x and z, respectively, defined by x = 21 (x1 + x2 ) and z = 21 (x2 − x1 ). In terms of these, one can proceed to an analysis of the dynamics of the model. Physically, the zero-slope limit is also the infinite tension limit, where the string cannot perform any oscillations. The CDGL model: Dirac analysis The model offers a good example of fairly nontrivial Dirac analysis of reparametrization invariant particle dynamics. If one has not been through it before in detail, this is as good a place as any to dig in. We will stay close to the notation of the paper (with a mostly plus metric η instead of the papers mostly minus metric g). In particular, we will clarify some rather ad hoc choices made in the paper. The Lagrangian is L=−

2 1 √ ( ((x ̇ − z)̇ ⋅ z) − (x ̇ − z)̇ 2 z2 − (z → −z)) ≡ L+ + L− 2α′ μ

(2.19)

μ

in terms of x and z. Conjugate momenta can be defined by Px = 𝜕L/𝜕xμ̇ and Pz = 𝜕L/𝜕zμ̇ . Introducing the combinations Aμ = (x ̇ − z)̇ ⋅ zzμ − z2 (x ̇ − z)̇ μ and Bμ = (x ̇ + z)̇ ⋅ zzμ − z2 (x ̇ + z)̇ μ , the momenta can be computed to 1 Aμ Bμ 1 Aμ Bμ μ (2.20) Pxμ = ′ ( + ) and Pz = ′ (− + ) L+ L− 2α L+ L− 2α From these expressions, follow four primary constraints: Px ⋅ z ≈ 0

Px ⋅ Pz ≈ 0

Pz ⋅ z ≈ 0

Px2 + Pz2 +

1 2 z ≈0 α′2

(2.21)

̇ z ⋅ z−L ̇ is also weakly zero, as would be expected for a reparametrization The canonical Hamiltonian H = Px ⋅ x+P invariant theory. These calculations are greatly facilitated by noting that A2 = −z2 L2+ and B2 = −z2 L− . Poisson brackets are defined as {x μ , Pxν } = {zμ , Pzν } = ημν . Following the standard Dirac procedure (see, for instance, our Volume 1, Section 3.2), we can introduce an effective Hamiltonian that is a sum of the four primary constraints with indefinite multipliers. Using this Hamiltonian to compute the τ derivatives of the primary constraints and requiring them to be zero, one obtains two more secondary constraints Px2 ≈ 0

Pz2 ≈ 0

(2.22)

The secondary constraints that actually come out of the computation are Px2 ≈ 0 and Pz2 − 1′2 z2 . The ones α in (2.22) are obtained by linear recombination with the last one of (2.21). The paper obtains the secondary constraints by “closing the Poisson bracket algebra of the constraints”. Therefore, the theory now has six constraints in all. Clearly, except for linear recombinations, these are the maximum sets one can obtain from bilinears of the basic positions and momenta of the theory (excluding bilinears containing xμ ).

14 In special relativity, the concept corresponding to the nonrelativistic “center of mass” is not so clear cut. One could also speak of the “geometric center”. This was investigated by M. H. L. Pryce in [18]. Neither of the papers CL(I) and CDL(II) discuss it.

2.2 A parallel strand of history: systems of particles

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There is no explicit discussion in the paper as to the first- and second-class nature of these constraints, except the comment on using the method of “closing the Poisson bracket algebra of the constraints” to obtain the secondary constraints. But it is clear from the context that the six constraints form a first-class algebra. This is a problematic feature, because a mechanical phase space d. o. f. count would then offer (4 + 4) + (4 + 4) − 2 ⋅ 6 = 4, which is not enough to build any reasonable field theory on. A scalar field in four space-time dimensions needs a mechanical basis of 3 + 3 phase space d. o. f. Now, a mechanical d. o. f. count such as this can only give a heuristic hint about possible field theories built using the Siegel BRST algorithm (see our Volume 1, Section 3.3.3). A detailed analysis shows that a cubic “Yang–Mills type” spin-1 model can be constructed. We will come to this. Alternatively, one could argue that any interesting theory must be based on a subset of the six constraints. But then one loses the connection to the initial rigid string action. We will come to this also. The canonical Hamiltonian H is weakly zero, as we saw above, and the total Hamiltonian computed following Dirac is also zero; so, in this sense the time evolution is completely arbitrary. The paper presses on by choosing a Hamiltonian H=−

α′ 1 [λ P2 + λ2 (Pz2 + ′2 z2 )] 2 1 x α

(2.23)

No motivation is given for choosing this particular linear combination of the constraints, but the choice is reasonable. It lends itself to an interpretation of an oscillator ̈ with a uniformly moving center of motion, the equations of motion being x(τ) = 0 and ̈ z(τ) + λ22 z(τ) = 0, respectively. Next, it follows a discussion about the corresponding Lagrangian and a two-time quantization, with one-time parameter τ0 for x and one τR for z. We will forego the details of this, as it is also rather ad hoc (but see the box in Section 2.2.2), and just note that two Schrödinger equations for the one wave function ψ(x, z; τ0 , τR ) are set up with Hamilton operators H0 and HR corresponding to the two pieces of the classical Hamiltonian (2.23). A Feynman path integral kernel is computed, and not surprisingly, the result can be interpreted as the propagation of a zero-mass particle of particular integer spin. However, note that the Lagrangian corresponding to the Hamiltonian (2.23) implies no constraints. These must be supplied as extra conditions on the quantum states. Let us then move on to Section 4 of the paper where it turns to the physical states of the quantum version of the theory. First, the relative coordinates zμ and velocities żμ are reexpressed in terms of creation aμ† and annihilation aμ operators in a standard way. μ In terms of these, and the center of motion coordinate xμ and momentum Px , five of the constraints are Px2 ≈ 0

Px ⋅ a ≈ 0

while the sixth constraint

Px ⋅ a† ≈ 0

a⋅a ≈0

a† ⋅ a† ≈ 0

(2.24)

22 � 2 Notes on the history of higher spin interactions : HR : = −a† ⋅ a ≈ 0

(2.25)

acquires an ordering ambiguity in the quantum theory. The following conditions on the quantum states |ψ⟩ are chosen: Px2 |ψ⟩ = 0 (: HR : −α)|ψ⟩ = 0 Px ⋅ a|ψ⟩ = 0

a ⋅ a|ψ⟩ = 0

(2.26)

From what we know about the field theory interpretation of equations like these (see our Volume 1, Sections 3.3.3 and 5.4), they clearly indicate a theory of massless gauge fields of helicity α in the TT gauge (see our Volume 1, Section 5.1.1). The paper writes about a tower of fields: [. . .] in the quantum case the model describes a spin tower of zero-mass particles.

This, however, is too optimistic. Rather, the model may describe a zero-mass particle of arbitrary spin, not a full tower, but one particle. Indeed, the ensuing discussion in the paper, where the physical states are constructed in terms of polarization tensors—for low spin values—and internal space harmonic oscillator wave functions, supports this view. μ Normalized momentum q ground states |q; 0⟩ are introduced, satisfying Px |q; 0⟩ = aμ |q; 0⟩ = 0. The metric is indefinite, and the treatment of this is commented upon, but the paper continues to work with a Lorentz metric to “simplify the exposition”. Configuration space ground state wave functions are introduced according to ⟨x, z|q; 0⟩ = c0 exp[−iq ⋅ x] exp[

1 2 z] 2α′

where c0 =

1 (2π)2 πα′

(2.27)

These are harmonic oscillator ground state wave functions, adapted to a relativistic setting, and the corresponding eigenfunctions to : HR : with eigenvalues n are written as ⟨x, z|q; μ1 , . . . , μn ⟩ = cn exp[−iq ⋅ x] exp[

󵄨󵄨 1 2 𝜕(n) 1 󵄨 z ] exp[ ′ [J 2 + 2J ⋅ z]]󵄨󵄨󵄨 ′ μ μ 1 n 󵄨󵄨J=0 2α 𝜕J . . . 𝜕J α (2.28)

where the J μi derivatives produce Hermite polynomials15 when acting on the generating function exp[[J 2 + 2J ⋅ z]/α′ ]. The constants cn are given by (i√α′ /2)n c0 . The constraints of (2.26) must be imposed on the wave functions. The first two are satisfied by putting q2 = 0 and by choosing integer values n for α. The last two constraints generate transformations aμ† → aμ† + λqμ and aμ† → aμ† + λ′ aμ under which the states must be invariant. This can be achieved in a standard fashion by introducing polarization tensors. The paper treats the cases α = 0, 1, 2. 15 Again adapted to a D-dimensional relativistic setting.

2.2 A parallel strand of history: systems of particles

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The paper contains an interesting discussion on cubic vertices that we now turn to. Consider, as in Figure 2.1, two initial two-particle systems described by the “end-points” (x1i , x2i ) and (x3i , x4i ) at time τi , colliding to produce a third two-particle system described f f by (x1 , x3 ) at time τf : x3

x3i

x4i

f

x3

x2

x2i x1i

f

x1

x1

Figure 2.1: The Casalbouni–Longhi cubic vertex diagram.

The “boundary conditions” (i. e., coordinate continuity) at the interaction time τ are x1 (τ) = x1

x2 (τ) = x4 (τ) = x2

x3 (τ) = x3

(2.29)

The paper then defines a vertex operator16 V (τ) =

9π g ∫ dx1 dx2 dx3 |x1 , x3 ; τ⟩⟨x1 , x2 ; τ|⟨x2 , x3 ; τ| 16(α′ )2/3

(2.30)

In order to perform this integral over all interaction points, the states are represented by harmonic oscillator wave functions according to the generating function formula (2.28). For instance, the state ⟨x1 , x2 | is represented with the function f (x1 , x2 ; q, J) = exp[−iqx12 ] exp[

1 2 1 z12 ] exp[ ′ [J 2 + 2Jz12 ]] ′ 2α α

(2.31)

where x12 = 21 (x1 + x2 ) and z12 = 21 (x1 − x2 ). The other states entering into the vertex (2.30) are represented similarly. Based on (2.31), the paper defines the function V (qi , Ji ) = ∫ dx1 dx2 dx3 exp[−i(q1 x12 + q2 x23 − q3 x13 )] × exp[

1 1 2 [ (z + z213 + z223 ) + J12 + J22 + J32 + 2(J1 z12 + J2 z23 + J3 z13 )]] α′ 2 12

(2.32)

The idea is that derivating the expression (2.32) with respect to combinations of the sources Ji , one can get matrix elements for the vertex operator (2.30) for various spin states. But first the integrals must be performed. The authors chose x = x23 , z1 = z12 16 The powers of α′ are determined by dimensional analysis. See the box below.

24 � 2 Notes on the history of higher spin interactions and z2 = z23 , in terms of which all the other variables can be expressed, as integration variables. Performing the integrals, one then gets 4 V (qi , Ji ) = − (2π)8 gα′4 gδ(q1 + q2 − q3 ) 9 1 1 4 × exp[ ′ [− (J12 + J22 + J32 ) + (J1 J2 − J2 J3 − J1 J3 )] α 3 3 α′ 2 + i[J1 (q2 + q3 ) − J2 (q1 + q3 ) − J3 (q1 − q2 )] + (q12 + q22 + q32 )] 3 6

(2.33)

Grinding matrix elements out of the formalism and dimensional analysis Denoting a momentum state |q; μ1 , . . . , μn ⟩ with |q; n⟩, dimensionless matrix elements can be computed by bracketing the vertex operator V (τ) with three states |qi ; ni ⟩. As the formalism is devised, this bracketing corresponds to computing appropriate source derivatives, as in formula (2.28), of the vertex function V (qi , Ji ) and setting the sources to zero. The momentum ground states |q; 0⟩ are normalized according to ⟨q′ ; 0|q; 0⟩ = δ 4 (q′ − q). Working in mass dimensions, using this and formula (2.27), it follows that the momentum states have dimension −2 while the configuration states |x; z⟩ have dimension 4. This then explains the factor ′ 12/3 in the vertex V (τ) (α )

with the dimension of α′ equal to −2 (as in string theory) and taking g as dimensionless. When computing specific matrix elements for three particles with spins n1 , n2 and n3 , appropriate factors of α′ must be multiplied onto V (qi , Ji ). First, there is the factor ′ 13/2 ; then there is ( α1′ )3 from the constant (α )

c0 in formula (2.28). From this formula also comes factors of √α′ from cn , more precisely one factor of √α′ for each unit of spin, i. e., (√α′ )n1 +n2 +n3 . Finally, there is the factor α′4 from the integration. All in all one gets the coupling factor g(√α′ )n1 +n2 +n3 −1 . Further factors of α′ will result when computing the source derivatives.

The paper then use this vertex computing algorithm to compute vertices for three scalars, for scalar-photon-scalar and for scalar-graviton-scalar interactions. These vertices—noted to have the correct structure except for a nonlocal factor—are used to estimate values for α′ and g by matching to the fine structure constant and the gravitational coupling constant with the result that α′ ≈ 10−33 cm and g ≈ 0.6. The nonlocal ′ factor is exp[ α6 (q12 + q22 + q32 )]. Its origin from formula (2.33) for the integrated vertex function is clear, and it will occur as a factor in all vertices. It is commented in the paper that the factor is 1 on mass shell, and for the estimated value of α′ the factor deviates noticeable from 1 for off-shell momenta of the order of 1019 GeV. Since the paper is written within the context of strong interaction physics, there is some discussion of the model’s physical relevance, but let us end here and instead turn to the second paper, where a field theory is set up.17

17 The relation between the papers CL(I) and CDL(II) can be thought of as while the first paper is in a first quantized formalism, and the second paper is in a second quantized formalism. The practical results are the same though.

2.2 A parallel strand of history: systems of particles

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The second paper, CDL(II), treats the second quantization, i. e., field theory of the model. It quickly reviews the theory up to the constraints (2.26). Then, in a way that is now very familiar, it introduces higher spin field operators subject to all the constraints of the theory by introducing appropriate field creation and annihilation operators and the corresponding free field operators. The vertex computation is redone with the same end result. The example of three spin-1 fields interacting is done in this paper. Computing the derivative 󵄨󵄨 𝜕3 󵄨 V (qi , Ji )󵄨󵄨󵄨 󵄨󵄨Jμ 𝜕Jμ1 𝜕Jμ2 𝜕Jμ3

(2.34) 1

=Jμ2 =Jμ3 =0

and supplying the extra factors from the box above, produces the Yang–Mills cubic vertex, which we just quote from the paper f abc [−ημν (q1 − q2 )σ − ησν (q2 + q3 )μ + ησμ (q1 + q3 )ν ]

(2.35)

where the structure constants has been supplied. From a higher spin perspective, what we have reviewed here is quite interesting. One may hope to able to approach general higher spin cubic interaction along this road. Unfortunately, it does not work out. One may compare the vertex generating function (2.33) to the known structure of cubic vertices for higher spin, both in the light-front formulation, and in the covariant BRST formulation. Thinking of the currents J as playing a similar role to the oscillators, it is clear that the bilinear nature of the generating functions are nothing like the structure of the higher spin vertices that contain at least three oscillators. Since it is known that the higher spin BRST operator requires at least quadrilinear combinations of oscillators and momenta in the vertices, it is very unlikely that CDL-type vertices will work for higher spin.18 Furthermore, the next paper in the series, CL(III), strongly indicates that the theory is a pure spin-1 theory in four dimensions. This paper studies the mechanics BRSToperator constructed from the six constraints (2.21) and (2.22). The calculation is quite interesting, and we will study it in some detail. BRST quantization of the CDGL theory Let us use notation and conventions consistent with the BRST treatment of Fronsdal theory from our Volume 1, Section 5.4. The six first-class constraints will be denoted and written as G+ = α ⋅ p

G− = α† ⋅ p

G0 = −p2

18 Supported by unpublished computations of the present author.

(2.36a)

26 � 2 Notes on the history of higher spin interactions

T+ =

1 α⋅α 2

1 † † α ⋅α 2

T− =

T3 = α† ⋅ α +

D 2

(2.36b)

The first-class algebra is readily worked out, and recorded in formulas (5.82) and (5.83) in Volume 1. The G and T parts of the algebra form subalgebras, of which the G part is an invariant subalgebra. Ghosts and their anticommutators for the two subalgebras are taken as {c 0 , b0 } = 1 3

{e , d3 } = 1

{c + , b+ } = 1

{c − , b− } = 1

(2.37a)

{e , d+ } = 1

{e , d− } = 1

(2.37b)

+

−

The nilpotent higher spin BRST operator, formed according to the standard algorithm (see Volume 1, Section 3.3.3), is QHS = c 0 G0 + c + G+ + c − G− + c + c − b0

(2.38)

Hermiticity of QHS forces (c − )† = c + , (b− )† = b+ , (c 0 )† = c 0 and (b0 )† = b0 . For the T algebra by itself, one could introduce a BRST operator QT = e3 T3 + e+ T+ + e− T− − e+ e− d3 + 2(e+ d+ − e− d− )e3

(2.39)

Again, hermiticity forces (e− )† = e+ , (d− )† = d+ , (e3 )† = e3 and (d3 )† = d3 . Now, when checking the nilpotency of QT , there could be an ordering issue for the operator T3 therefore it should really be taken as T3 = α† ⋅ α − β with an ordering constant β to be determined. Requiring Q2T = 1 {Q , QT } = 0 yields β = − D2 , confirming the choice in (2.36b). 2 T For the full BRST operator Q6c including all six constraints, there are further structure constants of the algebra to incorporate. It is convenient, following the paper, to write Q6c in a string theory inspired way as Q6c = c 0 G0 + e3 T3̄ + MHS b0 + MT d3 + Ω

(2.40)

T3̄ = α† ⋅ α − β + 2(e+ d+ − e− d−) − b+ c + + b− c −

(2.41a)

with

MHS = c c

+ −

MT = −e e

(2.41b)

+ −

Ω = c G+ + c G− + e T+ + e T− − b− c e + b+ c e +

−

+

−

+ −

(2.41c)

− +

The computation of Q26c is facilitated by first ascertaining the, not all entirely trivial, commutators [T3̄ , MHS ] = [T3̄ , MT ] = [T3̄ , Ω] = [MHS , Ω] = [MT , Ω] = 0

(2.42)

D Q26c = G0 MHS + T3̄ MT + Ω2 = ( + β − 3)e+ e− 2

(2.43)

One gets

Requiring nilpotency then yields β = 3 − D2 , with the consequence that T3 = α† ⋅ α +

D 2

−3

The fact that the CDGL theory, when treated using BRST techniques, implies a definite value β = 3 − D2 , depending on space-time dimension D, for the ordering constant in

2.2 A parallel strand of history: systems of particles

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T3 = α† ⋅ α − β is quite interesting. The interpretation of the T3 constraint acting on the quantum states—to become fields in the second quantized theory—is to give the spin of the states/fields. Thus, in D = 4 the spin is fixed to 1. This same result can be found in the Meurice paper [50] (commented upon in our Volume 1, Section 2.11.4) from the same era that explicitly studies theories with an occupation number constraint. This paper also arrives at the condition β = 3 − D2 for the particular constraint structure of the CDGL paper. We mentioned above that a mechanical theory with six first-class constraints is problematic as a basis for a field theory, since in say four dimensions, the phase space d. o. f. count works out to 4, which is too few to support even a scalar field. The situation is quite puzzling. A little more light may be shed on this issue by looking at the field equations and gauge transformations of the model. But we leave it to the reader to dig deeper. Puzzling equations Quoting formulas from the paper, we have field equations for the vector field Aμ , p2 Aμ + pμ ψ̃ 1 = 0 1

(β − 1)Aμ + pμ ψ = 0

pμ Aμ + ψ̃ 1 = 0 ̃1

2 1

(β − 1)ψ − p ψ = 0

(2.44a) (2.44b)

where the fields ψ̃ 1 and ψ1 are scalar fields that appear in a component expansion over the ghosts c 0 and e3 in the usual manner of mechanics-BRST based field theory. The corresponding gauge transformations are δAμ = pμ Λ1

δ ψ̃ 1 = −p2 Λ1

ψ1 = (1 − β)Λ1

(2.45)

The interpretation of the formulas clearly depend on the value of β. Superficially, it looks like the vector field is pure gauge in the case β ≠ 1. The paper concludes that for β = 1, the equations describe a vector field in the Lorentz gauge and a constant scalar ψ1 . It is not entirely convincing.

2.2.4 Relativistic harmonic oscillator models We now turn to the first set of CDGL papers mentioned above. The first paper in the series, BD(I), refers back to the “nonhadron” papers already discussed, saying that the aim is now to enlarge the model in order to describe particles with mass and half-integer spin. Here, we will briefly review Section 2 of the paper on integer spin particles.19 The Introduction notes that the model studied, a relativistic harmonic oscillator model, has been studied by Kim and Noz in [39]. In the second paper, DGL(II), it is furthermore noted that generalizations of the model had already been studied by a group of theoreticians

19 The rest of the paper concerns a supersymmetric extension of the model.

28 � 2 Notes on the history of higher spin interactions at the Nagoya University in Japan. This work is simultaneous to the CDGL papers, with roots going back to work by Takabayasi in the 1960s (see paper in [91]) and to work by Yukawa [51, 52] in the 1950s.20 For reasons of presentation logic, we will however start with the theory from the point of view of the CDGL papers. Kim–Noz harmonic oscillator series of papers Y. S. Kim and Marilyn E. Noz wrote a series of papers in the mid 1970s on a certain approach to the quark model involving covariant harmonic oscillators. The series is theoretical but phenomenology oriented—based on wave equations—and does not discuss very much the problems with relativistic two-particle theories, but there are interesting comments. The first paper in the series [39] puts the subject in the context of hadron physics, referring back to Yukawa’s bilocal theory. The third paper [55] is perhaps the most interesting for a reader searching for ideas to be transferred to higher spin theory. There is also a reprint book [56] that might be interesting for the “paper-trotter”.

For the quantum relativistic oscillator, the paper BD(I) starts with the equation (p2 + q2 + k 2 z2 − m2 )|ψ⟩ = 0

(2.46)

where k is the frequency and (x, p) and (z, q) are center of mass and relative variables, respectively, with commutation relations21 [xμ , pν ] = −igμν

[zμ , qν ] = −igμν

(2.47)

The states also have to satisfy the subsidiary condition p⋅(

q + iz)|ψ⟩ = 0 k

(2.48)

Then the authors perform what could be described as a reverse engineering to find a classical Lagrangian that upon quantization yields this system. Reverse engineering of the quantum relativistic oscillator Corresponding to the equations (2.46) and (2.48), one should look for a Lagrangian producing the following constraints:

20 The number of papers on this type of mechanical models of hadrons is so huge that one can speak of a relativistic harmonic oscillator theory research program. It can be traced back to the end of the 1960s as is indicated by talks given at the Eight Nobel Symposium in 1968 [53]. In this context, the paper [54] by Feynman, Kislinger and Ravndal, is often mentioned. 21 We are using the minus-favoring metric of the original paper. This also explains some sign choices in the following.

2.2 A parallel strand of history: systems of particles

χ = p2 + q2 + k 2 z2 − m2 ≈ 0

χ1 = p ⋅ z ≈ 0

χ2 = p ⋅ q ≈ 0

�

29

(2.49)

where now all variables are classical functions of a time τ satisfying Poisson brackets corresponding to the quantum commutators (2.47). The constraints χ1 and χ2 are second class while χ is first class. Dirac brackets are computed and the first-class Hamiltonian H is given by H = λ(p2 + q2 + k 2 z2 − m2 )

(2.50)

with λ arbitrary. Hamiltonian equations of motion can then be computed in the standard fashion as Ȧ = dA/dτ = {A, H}Dirac , with the result xμ̇ = −2λpμ zμ̇ = −2λqμ

ṗμ = 0

q̇μ = 2λk 2 zμ

(2.51)

Using the equations of motion and the constraints, one may compute λ with the result 2λ = √ ± (x ̇2 + z2̇ )/(m2 − k 2 z2 ). From L = −px ̇ − qz ̇ − H, one then gets L = −√ (m2 − k 2 z2 )(x ̇2 + z2̇ )

(2.52)

choosing the negative solution for λ. This Lagrangian readily leads to the first constraint of the constraints in (2.49), but in no obvious way to the second and third. Instead, the authors, using the equations of motion (without noting it) write the second and third constraints as x ̇ ⋅ z ≈ 0 and x ̇ ⋅ z ̇ ≈ 0. The second of these equations follows from the first if one demands d(x ̇ ⋅ z)/dτ = 0, again using the equations of motion. Finally, the first equation x ̇ ⋅ z ≈ 0 can be satisfied identically through the transformation xμ̇ → xμ̇ − (x ̇ ⋅ z/z2 )zμ . Performing this transformation of the Lagrangian in (2.52) produces the Lagrangian L = −√ (k 2 −

m2 )[(x ̇ ⋅ z)2 − z2 (x ̇2 + z2̇ )], z2

(2.53)

which indeed leads back to the initial constraints (2.49).

The paper proceeds to quantization of the model. This is done by first introducing creation and annihilation operators (aμ , aμ† ) corresponding to the variables (zμ , qμ ). The

Hamiltonian constraint becomes [p2 + 2k(a† ⋅ a + α) − m2 ]|ψ⟩ = 0 with α an ordering constant. The two conjugated second-class constraints are imposed by the condition p ⋅ a† |ψ⟩ = 0, rather than basing the quantization on the Dirac brackets. By now, the reader has certainly noticed that the constraint p ⋅ z ≈ 0 tends to crop up in all models considered, as already noted in the box in Section 2.2.2 above, where we discussed its role. However, its appearance in a model describing two free relativistic particles can be doubted. In the wave equation (2.46), the interaction is represented by the term k 2 z2 , which then also occurs in the first constraint of equations (2.49). It is then clear that the transversality constraint p ⋅ z ≈ 0 remains in the noninteracting theory with k = 0. This is a question considered in the two papers DGL(II) and DGL(III).22 The

22 In the paper DGL(II), there is a first reference to the work [57] of the Nagoya University-based group and to a preprint by M. Kalb and P. van Alstine [58] that consider a class of models that has p ⋅ r ≈ 0

30 � 2 Notes on the history of higher spin interactions aim is to have a model where the constraint only holds where there is interaction. Then, for a finite-range potential, one could possibly describe both scattering (of initially and finally free particles) and bound states within the same model. It may be good idea first to collect some general comments and questions on twoparticle mechanical models in one place. The box below is in complement to the box above on the transversality constraint and the issue of the relative time variable. Two times or one time in two-particle mechanical models Consider two point particles, initially free and noninteracting. In a certain inertial frame, the two point partiμ μ cles will be described by two world-lines W1 and W2 given in terms of coordinates x1 (τ1 ) and x2 (τ2 ). The detailed description will vary with the chosen frame. With interaction, there appears the problem of correlating the points on the two world-lines and the corresponding time parameters τ1 and τ2 and the functional form of the interaction potential. Without mediating fields, the interaction is of the of direct action-at-a-distance type. Such interaction may be instantaneous or noninstantaneous along retarded/advanced light-cones.23 Here, we will consider the instantaneous case. Without in any way claiming encyclopedic knowledge of the way the question has been addressed, it seems to me that the Dirac theory of constrained Hamiltonian systems yields a systematic and consistent approach. After all, even a free relativistic point particle is described by a reparametrization invariant action where the “time” τ is just a gauge parameter corresponding to which there is a first-class constraint generating the reparametrization symmetry. It then makes sense if the interaction problem and the correlation of points on the world-lines can be treated as a problem in constrained Hamiltonian mechanics. As we see from the series of papers examined in the present section, this is also the approach employed. In the papers DGL(II) and DGL(III), the configuration space of the two particles is assumed to consist of the two-dimensional surface W1 × W2 in the product space M 4 × M 4 of two Minkowski spaces. A one-to-one correlation is chosen between W1 and W2 corresponding to selecting a line l in M 4 × M 4 among the infinitely many lying on the surface W1 × W2 connecting the initial (x1(0) , x2(0) ) and final (x1(1) , x2(1) ) pairs of events. In the words of the authors: of the infinitely many lines describing two free particles, only one describes the system when the particles are interacting. In these papers, the world-lines are parametrized by one and the same time τ.24

In DGL(II), in contrast to DGL(I), a different Lagrangian is postulated, that leads to the constraint p ⋅ z ≈ 0 only when the interaction is nonzero. After discussing this problem and alternative approaches by Droz-Vincent [60], by Currie [61] and by Hill [62],25 the paper states the basic action even for free particles. That model is the same as the model of BD(I) considered above, but there is no reference in DGL(II) to BD(I). 23 More comments on this in Section 2.2.8. 24 Without such a correspondence between the times, one has a world-surface rather than world-line as discussed by Kihlberg et al. [59]. 25 Ph. Droz-Vincent considered two times τ while D. G. Currie and R. N. Hill required simultaneity in all frames. The discussion in these papers in the wake of the no-interaction theorem, interesting as it undoubtedly is, is a bit out of line from our perspective.

2.2 A parallel strand of history: systems of particles

x1(1) ,x2(1)

2

S = −∑ i=1

∫

√Ui (r 2 )dxiμ dxiμ

�

31

(2.54)

x1(0) ,x2(0)

The integral is to be calculated along a line l in the product space M 4 × M 4 of two Minkowski spaces M 4 connecting the initial (x1(0) , x2(0) ) and final (x1(1) , x2(1) ) pairs of points. μ The coordinate differentials dxi are required to be time-like, future pointing, four vectors. The potential functions Ui are given by Ui (r 2 ) = mi2 − V (r 2 )

μ

μ

with relative coordinate r μ = x2 − x1

(2.55)

so as to take into account the masses mi of the particles. The paper then introduces a parameter τ to parameterize the line l and writes the τ action S = ∫τ 1 Ldτ with Lagrangian26 0

2

L = − ∑ √Ui (r 2 )ẋi2 i=1

(2.56)

This choice must be seen as a way of implementing the one-to-one correlation between the world-lines of the two particles. The paper DGL(II) continues with an analysis of the equations of motion of the model and shows that the constraint becomes (p ⋅ r)V ′ ≈ 0 where V ′ = dV /dr. The final, long Section 3 of the paper discusses scattering in a finite range potential, but let us turn to the canonical analysis of the model in the next paper DGL(III). From the Introduction and the list of references of the paper DGL(III), it is apparent that the authors had become aware of several, at that time, recent papers treating similar models, not the least by authors from the Nagoya University. Again, the common occurrence of the constraint p ⋅ r ≈ 0 (with p total momentum and r relative coordinate) in singular Lagrangian based two-particle models, is pointed out. This constraint, it is argued, should only hold in the interaction region. Thus, in the free region, the model is described by two first-class constraints, whereas in the interaction region, it is described by one first-class constraint and two second-class constraints as is indeed the case for the model considered in DGL(II) and DGL(III). The main part of the paper treats quantization based on an approach to the secondclass constraints via canonical transformations. Here, we will contend ourselves with a derivation of the constraint structure, showing that the Lagrangian has the desired properties (we are not following the paper in the box below).

26 In DGL(II), such a Lagrangian is called a “Lagrangian with a multiplicative potential”. The concept seems not to be otherwise defined, nor is the effect of changes of variables discussed. It is not clear whether or not a Lagrangian of the form of (2.53) should be included in the characterization. Presumably it should, because the Lagrangian of the CDGL model is also referred to as “multiplicative”.

32 � 2 Notes on the history of higher spin interactions Constraints for the multiplicative potential model From the Lagrangian (2.56) follows the momenta μ

pi = −

U (r 2 ) μ 𝜕L = √ i 2 xi̇ 𝜕xiμ̇ ẋ

(2.57)

from which follows the two, Einstein relation type, primary constraints χi = p2i − Ui ≈ 0. The canonical Hamiltonian H = −p1 x1̇ − p2 x2̇ − L is weakly zero. Following Dirac one then chooses an effective Hamiltonian H = λ1 (p21 − U1 ) + λ2 (p22 − U2 )

(2.58)

Next, demanding χ1̇ = {χ1 , H} ≈ 0, one computes the secondary constraint 2λ2 (p1 + p2 ) ⋅ rV (r) ≈ 0 and vice versa. Thus, the constraint p ⋅ r ≈ 0 is enforced for interacting particles but not for free particles. ′

The constraints of the multiplicative potential model with Lagrangian (2.56) are therefore χ1 = p21 − U1 ≈ 0

χ2 = p22 − U2 ≈ 0

(p1 + p2 ) ⋅ (x2 − x1 ) ≈ 0

(2.59)

where the third constraint is only effective for nonconstant potentials. Introducing total momentum p = p1 + p2 and relative momentum q = 21 (p2 − p1 ) and correspondingly center of motion coordinate x = 21 (x1 + x2 ) and relative coordinate r = x2 − x1 , the constraints can be linearly recombined into p2 + 4q2 − 2(m12 + m22 ) + 4V ≈ 0

1 p ⋅ q − (m22 − m12 ) ≈ 0 2

p⋅r ≈0

(2.60)

generalizing the harmonic oscillator constraints (2.49) to the case of unequal masses. The paper DGL(IV), which we will not review here, is an attempt to harmonize the singular Lagrangian/constrained Hamiltonian multiplicative potential models with the older predictive mechanics approaches of Bakamjian–Thomas, Foldy, Droz-Vincent, Kerner, Currie and Hill. It does so by working from equations of motion satisfying a set of postulates that are deemed reasonable. The interested reader is referred to the paper itself. We will instead summarize the situation regarding multiplicative potential models by reviewing the Kalb–Van Alstine papers, comment on the Nagoya University set of papers and then continue our investigation with the Komar approach and its subsequent analysis. 2.2.5 Multiplicative potential models and the Kalb–Van Alstine preprints The results of the preceding sections on multiplicative potential models certainly call for a general analysis of that kind of model. This can be found in two preprints by M. Kalb and P. Van Alstine [63, 58].27 27 I take the opportunity here to thank Professor M. Kalb for localizing and providing me with scanned copies of the papers.

2.2 A parallel strand of history: systems of particles

�

33

There does not seem to be any explicit general definition of the concept of a multiplicative potential, but from papers that treat the subject one can extract the common feature that the velocity term xi̇ ⋅ xi̇ for particle numbered i in the Lagrangian, is multiplied by a function Ui = mi2 − Vi . The mi are the masses of the particles and Vi are the actual interaction terms that may be functions of the coordinates and velocities. In the simplest cases, often the only ones considered in practice, we have Kepler or harmonic potentials that are functions of a relative coordinate. In these Lagrangian models, there is (in general) just one proper time parameter τ. To have a dimensionless action S = ∫ Ldτ, one thus needs a square root in the Lagrangian (there are other geometrical arguments for this also). The question then arises where to put the square root. Two natural alternatives are to write the Lagrangian either as Ls = √U1 ẋ12 + √U2 ẋ22 + ⋅ ⋅ ⋅ + √Un ẋn2

(2.61)

Ll = √U1 ẋ12 + U2 ẋ22 + ⋅ ⋅ ⋅ + Un ẋn2

(2.62)

or as

The first alternative, with “short” square roots, is the one employed in the papers DGL(II) and DGL(III), and which gives rise to potential dependent third constraint of (2.59) in the two-particle case. The second alternative, with the “long” square root, was employed in the paper DGL(I). Both forms were also thoroughly studied in the research program of the Nagoya University group. A most general version of the second alternative was investigated by M. Kalb and P. Van Alstine in [58]. There the action was written in the form μ

S = ∫ dτ(Mμνij q̇ i q̇ jν )

1/2

(2.63)

μ

where qi are generalized coordinates (for instance, center and relative coordinates). This action is the τ-reparametrization invariant and, therefore, the canonical Hamiltonian is zero. Computing the conjugate momenta piμ =

1 M q̇ ν L μνij j

(2.64)

μ it follows that H = piμ q̇ j − L = 0. Thus, there is at least one constraint present. By a systematic study of different forms of the matrix Mμνij , classes of models of this type— with various sets of constraints—can be analyzed. First, it is assumed that the inverse of Mμνij exists. Writing Mμνij = fij (r 2 )ημν +

gij (r 2 )rμ rν in terms of an unspecified relative coordinate r, its inverse is Mij

−1μν

fij−1 ημν −fik−1 gkl hlj−1 r μ r ν

2

=

where hij = fij +r gij . Thus, the inverse of M exists if and only if the

34 � 2 Notes on the history of higher spin interactions inverses of f and g exist. This is the basis of the further analysis. The case of an invertible matrix M is discarded since there is then just on the constraint, namely the Einstein −1μν relation piμ Mij pjν − 1 ≈ 0. This case corresponds to the Feynman–Kislinger–Ravndal model [54]. There are too few constraints to remove ghost states in the quantum theory. The authors then focus on cases where fij−1 and gij−1 does not exist, allowing for additional constraints and ghost removing mechanisms. A systematic treatment is based on a classification: 1. fij−1 exists 2. 3.

a. hij−1 exists

b. hij−1 is rank 1

c. hij = 0

a. hij−1 exists fij = 0 a. hij−1 exists

b. hij−1 is rank 1

c. hij = 0

b. hij−1 is rank 1

c. hij = 0

fij−1

is rank 1

Case 1b. turns out to be the most interesting, and it is this one that is analyzed at length in the paper. It covers the models already encountered above. Clearly, this general approach can be modified to cover Lagrangians with square roots of the “short” type and Lagrangians that may possibly be relevant to mechanical models for higher spin gauge fields.

2.2.6 The Nagoya University series of papers Much of the contents of the Nagoya series of papers (as we will refer to them28 ) run in parallel with the CDGL papers, but there are not many references to them, while the later CDGL papers refer to the Nagoya papers, probably when they became known to the authors. It may therefore seem that the Nagoya group worked partly in isolation.29 There are, however, references to the Kalb–Van Alstine preprints. It seems fair to conclude that the models considered in this series of papers are more or less equivalent to the CDGL and Kalb–Van Alstine models. However, two further circumstances set them a bit apart as a special branch of research. They are more phenomenology oriented in that the models are considered as describing mesons and baryons and that they refer back to the bilocal field theories of Yukawa, corresponding to the quantization of the models. There is indeed a stronger focus on quantization. There are also no discussions about the no-interaction theorem.30

28 Although there are papers from other universities as well in the series. 29 At least as can be discerned from the published record. 30 Why this is so, I do not know. It may have to do with a phenomenology driven interest, rather than a theoretical interest. And as we know, this type of model does not violate the no-interaction theorem anyway, as we have seen.

2.2 A parallel strand of history: systems of particles

�

35

There is a large number of papers, and we will only mention a few here. There is also a highly interesting review paper [64] from 1979 by T. Takabayasi—one of the central researchers in this subprogram—that contain further perspectives and historical material, in particular, on the Yukawa bilocal model and the author’s own group-theoretical work on constituent models from the 1960s. The first paper in this series is a paper [65] by T. Takabayasi from 1975. It does not start from a Lagrangian, but rather writes down equations of motion for two interacting particles. The description is in terms of two world-lines parametrized by two times as in the box above with the further condition that dx0(i) /dτi > 0 for the two particles. The particles are also assumed to move with velocities less than that of light so that |(dx (i) /dτi )2 | < 0. It is assumed that the theory is covariant under separate reparametrizations of the two times. Apart from this, the equations of motion are based on Poincaré invariance, conservation of momentum and angular momentum and supposed to have a Newtonian nonrelativistic limit. We will not review any details of the paper, except to note how the two points (for the particles), participating in the action-at-a-distance on the world-lines, are related. In the words of the paper:31 We define for that purpose the notion of “conjugate points” as follows: For any specified point xμ(1) (τ1 ) on W1 we consider a space-like plane passing xμ(1) (τ1 ) and write its intersection with W2 as

xμ(2) (τ2 ). We arrange this xμ(2) (τ2 ) such that the vector xμ(1) (τ1 ) − xμ(2) (τ2 ) is orthogonal to the average of the 4-velocities at τ1 and τ2 , namely (xμ(1) (τ1 ) − xμ(2) (τ2 ))(u(1)μ (τ1 ) + u(2)μ (τ2 )) = 0

(2.65)

and call those xμ(1) (τ1 ) and xμ(2) (τ2 ) the conjugate points. We then assume that an action-at-a-distance works just between conjugate space-time points.

Here, the u(i)μ (τi ) denote the 4-velocities of the particles. Although the definition at first appears rather ad hoc,32 it is clear that in a gauge where τ1 = τ2 = τ, the formula (2.65) takes on the form of a transversality constraint (see Section 4 of the paper). Compare it also to the analogous concept of a “one-to-one correlation” between the world-lines of the CDL papers. The second paper [66] in the series is a short letter announcing a Lagrangian for the model of the preceding paper. This Lagrangian is similar to the one considered in [38], referred to as BD(I) in the Firenze series. It is noted that the action is invariant under the transformation Ẋ μ → Ẋ μ + λξμ where Xμ is a center coordinate and ξμ a relative coordinate. This leads to a constraint Ẋ μ ξ μ = 0, which precisely translates into (2.65) for equal times.

31 In a somewhat modified notation and different equation numbering. 32 There is a further comment on this definition in the review paper [64]: “[the particles] interact with each other as if each of them knew its partner’s world-line. In this sense, the system has its unity character (emphasis of the original) even though it is a composite.”.

36 � 2 Notes on the history of higher spin interactions The third paper [57] with S. Kojima, expands considerably on the short preceding letter. What is interesting from the perspective of action-at-a-distance issues is the discussion of two times versus one-time approaches. This can be found in Section 3 of the paper. Also interesting from our perspective is the short discussion about basic cubic interactions between two-particle systems. Here, there is a reference to a paper by T. Goto and S. Naka [67] on a vertex function for a bilocal theory. The next paper in the series to be mentioned is [68] by M. Fujigaki and S. Kojima. Here, there is a switch from a “long square root” Lagrangian to “short square root” Lagrangian. The motivation is the same as in the DGL(II) and DGL(III) papers reviewed above, namely not to have the transversality constraint when the particles are free. The paper treats the case of different masses for the constituents. Two final papers to be mentioned here is [69] by Takabayasi and K. Takeuchi, that more or less caps the discussion on “long square root” models in the Nagoya branch of the research program, and a paper [70] by S. Kojima that does the same for “short square root” models.

2.2.7 Komar’s constrained dynamics approach Arthur Komar’s approach—developed in a series of papers [71, 72, 73]—to the problem of directly interacting particles is interesting for several reasons. It is a purely “constraint dynamics” formalism, working directly with an algebra of first-class constraints, not derived form an underlying Lagrangian. The method does not even single out any specific Hamiltonian. But then again, the actual constraints used for directly interacting particles, are similar, not surprisingly, to the ones found in the approaches reviewed here. There is, however, a point that could lead to some confusion in such a comparison. This point may be connected to a further reason for interest in Komar’s method. It leads to a series of papers containing criticism and clarifications, bearing on the whole area of constrained Hamiltonian dynamics of point particles. We will come to these papers in Section 2.2.9 below. The first paper in the Komar series is an outline of the general method. As the method does not start from a Lagrangian, there is no Dirac analysis.33 Although there does not seem to be anything that is new in the Komar approach, at least as seen from today’s perspective, there is a certain difference in flavor that makes it interesting to look at it a bit closer. 33 The theory of constrained Hamiltonian dynamics is almost everywhere attributed to Dirac. But there were precursors such as L. Rosenfeld (see [74]) and parallel development of the method in relation to the quantization of Einstein gravity by P. Bergmann and collaborators, of which A. Komar was one (see [75]). Perhaps it is anyway not unfair to say that the streamlined and general development of the method is due to Dirac. The analysis of constrained dynamical systems was in large part motivated by the desire to quantize general relativity, and that goal is also mentioned in the Komar papers reviewed here.

2.2 A parallel strand of history: systems of particles

�

37

The Komar constraint formalism Komar works within a phase space with coordinates qa and pb subject to the usual Poisson brackets {qa , qb } = {pa , pb } = 0 and {qa , pb } = −{pb , qa } = δab . The indices a, b, . . . (Komar uses capital letters) stand for spacetime indices and particle enumeration. The Poisson bracket between functions A(q, p) and B(q, p) defined on the phase space is defined in the usual way. This then defines the symplectic form on the space.34 Komar then defines a canonical mapping as a curvilinear coordinate transformation, which preserves the given symplectic form. Any function A on phase space may be regarded as a generator of an infinitesimal canonical transformation according to δA qm = {qm , A} and δA pm = {pm , A} from which follows δA B = {B, A} for functions A and B. Komar leaves out the infinitesimal parameter, but clearly any infinitesimal transformation formula should be written as δA B = ϵA {B, A}. On this basis, dynamics is defined by first giving a set of k phase space functions Ki (q, p) that are supposed to satisfy first-class type the relations k

{Ki , Kj } ≈ ∑ λm ij Km m=1

(2.66)

where the λm ij need not be constants. The Ki are next supposed to be constraints Kj ≈ 0 in the Dirac sense, in this way defining a lower-dimensional surface in phase space, the constraint hypersurface. This surface is invariant under infinitesimal canonical transformations generated by the constraints. Then, starting with an arbitrary point on the constraint hypersurface and mapping it iteratively using infinitesimal canonical transformations generated by the constraints, one obtains an equivalence class of points lying on the constraint surface. Such an equivalence class is defined to be a generalized dynamical trajectory. Komar then remarks that this definition makes no explicit reference to any time parameter and, therefore, is manifestly relativistic covariant. On the other hand, it is a kind of “frozen” geometrical dynamics. In the simple example of a single relativistic point particle with an 8-dimensional phase space, the constraint hypersurface given by K = p2 − m2 ≈ 0 is 7-dimensional, while the dynamical trajectories are 1-dimensional. Komar also defines the concept of an observable (clearly in a classical sense) as a function Q on phase space, which generates infinitesimal canonical transformations that leave the constraint hypersurface invariant. This is equivalent to demanding {Q, Ki } ≈ 0 for all the constraints. The constraints themselves are observables in this sense. The set of Q’s (other than the constraints) need not commute. While the K ’s generate trajectories, the Q’s (other than the constraints) permute them intact. Komar notes that the generators of the Poincaré group are observables in his sense, and that they leave the constraint hypersurface invariant. The coordinates q are, however, not observables.

In the second paper in the series, Komar turns to the actual problem of two interacting relativistic point particles. The intention with the paper was, as written in the conclusion “[. . .] to develop the simplest possible nontrivial relativistic action-at-a-distance theory in such manner that it could be easily quantized.”. The long discussion of the quantization of the model is interesting, but we will forego this, and only focus on the classical aspects.

34 Clearly, one may be more general in defining the concepts, and more specific on allowed function spaces, but this is the way Komar starts. It is good enough for his purpose.

38 � 2 Notes on the history of higher spin interactions μ

μ

Komar introduces a 16-dimensional phase space with coordinates q1 , q2 , p1μ , p2μ subject to standard Poisson brackets as in the general theory outlined in the box above. The Poincaré algebra is satisfied by the ten quantities μ

μ

Pμ = p1 + p2

and

μ

μ

μ

μ

Lμν = q1 pν1 − q1ν p1 + q2 pν2 − q2ν p2

(2.67)

At this stage in the development, Komar does not follow Dirac of 1949 in choosing a particular “form of dynamics”—something that would lead to Bakamjian–Thomas type of theories that are not manifestly covariant—but retains the manifest covariant form (2.67) of the Poincaré representation. Instead, covariant constraints are introduced on the phase space to represent the dynamics. In the case of two free particles, these constraints are taken to be KF1 = p21 − m12

and

KF2 = p22 − m22

(2.68)

These constraints commute, and together they define a 14-dimensional constraint hypersurface in phase space. The trajectories are 2-dimensional,35 and factoring them out, yield the ordinary 12-dimensional phase space of two free particles. Interactions can now be introduced by adding a potential term to the mass terms in (2.68) as we have seen in several cases above. Komar choses to first reexpress his theory in terms of center (Pμ , X μ ) and (pμ , χ μ ) relative variables. Then, by linear recombination, the constraints are brought to the form KFα = P2 + p2 − 2m12 − 2m22

and KFβ = P ⋅ p − m12 + m22

(2.69)

For the interacting theory, Komar requires the first-class nature of the constraints to be maintained, and that they remain manifestly covariant. An interaction potential V , depending (for simplicity) only on a relative separation between the particles, can then be added to the first constraint KFα = P2 + p2 − 2m12 − 2m22 − 8μV (r)

(2.70)

the second constraint being unaltered in this formulation. The constant μ is the nonrelativistic reduced mass, the choice motivated by the desire that the function V (r) correspond to the usual classical potential in the nonrelativistic limit. Other variants of interaction are discussed in the paper. At this stage, the reader may worry, because a relative separation r does not in general commute with the relative momentum p and center momentum P. Komar solves this by choosing the relative separation as r = 2((P ⋅ χ)2 /P2 − χ 2 ) 35 This is a significant point to which we will return below.

1/2

(2.71)

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which is such that the crucial bracket {V (r), P⋅p} is zero. This is necessary in order to have a model with only two constraints, otherwise there would appear further constraints of the form P ⋅ r ≈ 0 and p ⋅ r ≈ 0 as we have seen. This point is related to the critique, or elaboration one should perhaps say, of the Komar approach that we come to in the papers reviewed in Section 2.2.9 below. We now turn to the third paper in the series that specifically treats the question how the theory evades the no-interaction theorem. The third paper concerns the question of the transition form phase space trajectories to space-time orbits, or world-lines. It is in this transition where one may expect complications leading to no-interaction results. The issue is thoroughly discussed in the set of papers to be discussed in Section 2.2.9. Here, we will see how Komar treats the problem. As we saw above, since the dynamics in the Komar models is governed by two commuting constraints, rather than one Hamiltonian, the trajectories are 2-dimensional. Komar poses the question of how “[. . .] can we recover the covariant space-time particle orbits, which the “no-interaction theorem” claims cannot exist?”. Komar’s resolution is centered on a conceptual distinction between syntactic and semantic observables. By a syntactic observable, Komar means a classical observable in the sense of the box above. A semantic observable, on the other hand, refers to the intended meaning of the variable. In the words of the author “[. . .] distances, times and velocities as measured by rulers and clocks in the frame of reference of some observer.”. We will not review any details of this, but just exemplify the meaning of semantic observables in the case of two free particles. For two free particles, Komar notes that the generators of the Poincaré group (2.67) are syntactical observables, and are constant on any trajectory. By assigning fixed values to them, a specific trajectory is then selected. One such choice is psi = ais and L0s i = qi0 psi − qis p0i = bsi (with constants ais and bsi and s a space index). Next, using also the μ constraints (2.68), one may solve for the coordinates qis . Then, writing x1 = (x1s , t) and μ x2 = (x2s , t) for the space-time coordinates of the particles for an observer, the intended semantic meaning of the phase space variables is then taken to be qis = xis and ps1 = mi vsi (1 − v22 )−1/2 for the 3-dimensional coordinates and momenta, and q10 = q20 = t the time. This is not Lorentz invariant. In the words of the paper, [. . .] the pairs of orbits in space-time, which are associated with a given phase-space trajectory depend critically on the frame of reference in which the semantic observables are related to the phase-space variables, and through them to the syntactic variables. We see in fact that the phasespace trajectories form a two-dimensional domain precisely due to the extra freedom required to permit different pairings of linear space-time orbits to be associated with a given trajectory. The differing pairings result from the exploitation of the frame-dependent nature of the relationship between the semantic observables and the syntactic observables.

The paper goes on to a quite lengthy implementation of this idea to the case of two interacting particles. Although the paper does not work explicitly in terms of gauge choices, there are vestiges of such in the treatment, and perhaps not surprisingly, the transversality condition appears toward the end. But we leave the rest to the reader.

40 � 2 Notes on the history of higher spin interactions 2.2.8 Historical notes on action-at-a-distance theory It was observed early on that concepts such as positions for particles [17] and mass centers for systems of particles [18] that are straightforward in Newtonian–Galilean mechanics, become problematic in relativistic mechanics, even more so in quantum theory. The root of the problems are the velocity boost transformations that in passing to special relativistic mechanics become Lorentz transformations that mix space and time. This can be considered as one branch of the research program of directly interacting relativistic particles. Another more phenomenological branch can be traced back to papers by H. Yukawa [51, 52] from the early 1950s on nonlocal field theory, while a third, theoretical branch, starts with Dirac’s 1949 paper [76] on “forms of relativistic dynamics.”. Theoretically, much of the research came to be centered on the question of actionat-a-distance interactions and the no-interaction theorem. So, while we can discern the three historical roots mentioned above: (i) questions of positions and mass centra in relativistic mechanics, (ii) nonlocal field theory and phenomenological models for hadrons and (iii) nonlinear realizations of the Poincaré group in Hamiltonian mechanics, theoretically the research branched out in other ways. Phenomenologically, it seems that one did not care much about the no-interaction theorem at all, but rather focused on harmonic oscillator potential forces between particles as these were most workable and semirealistic.36 It should also be noted that research into methods that we here designate as “theoretical” were used in phenomenological calculations.37 Theoretically, the subject split into research on non-instantaneous and instantaneous interactions, respectively.38 The classic papers in the non-instantaneous branch are the Feynman–Wheeler papers [81, 82] on electrodynamics where interactions are mediated by retarded and advanced potentials, and the van Dam–Wigner papers [83, 84] where one particle may be affected by influences from the second particle from the whole world-line segment between the retarded and advanced points.39

36 We mentioned such models in Section 2.2.4 above. 37 See the review article [77]. It is quite old, but our main interest is not nuclear phenomenology, and it does give a view of the subject as it stood at the time of research into direct interaction theory that we are here interested in. 38 We are here following the structure of the subject as outlined in the Introduction to the reprint collection [78] and the Introduction to the proceedings [79] and the contribution [80] to the latter. This characterization of the subject is a bit old, but it may still serve to point out in what directions we have followed it here. 39 The Feynman–Wheeler papers are interesting—and a good read—for other reasons as well. They were apparently conceived as a part of a greater project to rethink classical electrodynamics in an actionat-a-distance way as a preparation to a quantum theory. The story is told in J. Mehra’s biography on R. P. Feynman [85]. The project was never completed, the quantum part being made redundant by the well-known work on QED of R. P. Feynman, J. Schwinger and S. Tomonaga, although it probably was an

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The instantaneous branch of papers is the most extensive one, and this is also where the approaches reviewed in some detail in Sections 2.2.3–2.2.7 belong. Conceptually, the instantaneous branch can be thought of as originating in the 1949 Dirac paper on “forms of relativistic dynamics.”. Dirac’s theory demands single time dynamics (as opposed to the noninstantaneous approach, which is inherently a many-time approach) where the Poincaré generators should be realized in terms of the basic dynamical variables of the model in such a way that the Poincaré algebra is satisfied. This ensures relativistic invariance, but not (manifest) relativistic covariance, which is a further demand on any model. As we have already noted above in Section 2.2.1, Bakamjian and Thomas proposed a concrete implementation in 1953 [20], later further clarified by Foldy in [21]. Foldy also introduced the problem of cluster decomposition: that particles or clusters of particles, far from each other should behave as separate units, i. e., effectively be noninteracting. Somewhere at this stage, the no-interaction theorem appeared, and threatened to tear down the whole building. At least, it forced a need to circumvent it—if not simply ignore it—since the direct interaction models were so useful for phenomenological calculations, and also for gaining theoretical insight into relativistic dynamics. The nointeraction theorem is based on assumptions, that when taken together, implies that the particles must be free. Following [80], one may discern four key ideas involved in the theorem: 1. Canonical (unitary in the quantum case) realization of the Poincaré group. 2. World-line invariance. 3. Equivalence of physical coordinates and canonical coordinates. 4. Equations of motion hold for all time. These ideas are incompatible with interacting particles. One of the ideas must be given up to escape the conclusion of the theorem. The ideas therefore also point to ways of circumventing the theorem [28]. One could of course give up on particle mechanics and work with interacting fields. Then the coordinates X μ are no longer physical variables, or operators, but merely a set of parameters. The fields are the new dynamical variables, and in quantum theory the number of particles is no longer constant. Not choosing this option, which runs counter to the research program as such, one could give up the first condition or the second condition. This corresponds to an historical split of the subject [80], one branch giving up the world-line condition, the other branch giving up on the Hamiltonian scheme, instead working with particle accelerations given as functions of positions and velocities in a “Newtonian style”. To this latter

inspiration for both the diagrammatic and path integral approaches of Feynman. The historical notes in the Feynman–Wheeler papers go back to attempts by C. F. Gauss on non-instantaneous a-a-a-d and also refers to early work by K. Schwarschild, by H. Tetrode and by A. D. Fokker among others.

42 � 2 Notes on the history of higher spin interactions branch, which was designated as “predictive relativistic mechanics” belongs the work of D. G. Currie, R. N. Hill and E. H. Kerner. The approach had electrodynamics as its benchmark.40 The other option, giving up the world-line condition, is more interesting for our purposes. Interaction is then possible, but one cannot identify physical coordinates with canonical coordinates. The Bakamjian–Thomas and Foldy approach belongs to this option. In its further development, it leads to the constrained Hamiltonian approach to singular Lagrangian models, some of which we have reviewed above.

2.2.9 Beyond the standard Dirac particle interaction program It seems that the fog surrounding the area of directly interacting relativistic particles lifted in the early 1980s, at least within the approach of constrained Hamiltonian dynamics. As we have seen, one may work directly from a set of consistent constraints on phase space—as in the Komar approach—or one may start from a Lagrangian and derive the constraints via Dirac analysis—as in the Lagrangian multiplicative potential approach. One set of papers that should definitely be read by anyone interested in this branch of the Dirac program is the four consecutive papers [59, 86, 4, 87] in the Physical Review. I will refer to them as the “Mukunda series”, as N. Mukunda was a coauthor on all four of them.41 From the Introductions of the papers, it is clear that they were motivated (at least partly) by the appearance of papers—by Komar, I. T. Todorov, F. Rohrlich and Kalb–Van Alstine,42 claiming the success in introducing direct interaction for relativistic particles, thereby bypassing the no-interaction theorem. All of which work is based on constrained Hamiltonian concepts and formalism, with or without an underlying Lagrangian. The aim of the series is to clarify how this is possible. There are many interesting ideas and clarifying remarks to be found in these papers. These papers more or less cap the discussion of the constrained Hamiltonian approach of Komar—working from constraints not derived from a Lagrangian—to directly interacting particles. It may seem that the no-interaction theorem is circumvented in the constrained Hamiltonian approach of the Dirac program by giving up on the world-line condition. Many of the papers, some reviewed above, indicate this circumstance more or less explicitly. According to the authors of the four consecutive papers, the situation

40 References can be found in the review article [80]. A Hamiltonian formulation can, later on in the development of the theory, be recovered. 41 The other authors being A. Kihlberg, R. Marnelius, E. C. G. Sudarshan and J. N. Goldberg in various configurations. 42 Some of which we have mentioned here. There are no references to the Firenze series papers. For the work of Todorov, see [29].

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is, however, more deep than this. The WLC can be satisfied in Komar-type models by judiciously choosing “gauge conditions” conjugate to the first-class constraints. In this process, the constraint structure of the multiplicative potential models reappears for certain gauge choices. The Mukunda series follow Komar up to the point of choosing the two constraints (2.69) and the phase space analysis in terms of the 14-dimensional constraint hypersurface and its foliation by 2-dimensional trajectories. It departs from Komar when it comes to the transition to orbits in space-time.43 In contrast to Komar, the Mukunda papers introduce gauge choices χi conjugate to the Ki constraints, making the combined set of four conditions on the system, second class. The approach goes “beyond Dirac” in the sense that one of the gauge choices, the one involving a choice of time slice through space-time, does not need to be any of the types corresponding to the Dirac forms of dynamics.44 Curiously enough, the Hamiltonian is not one of the ten Poincaré generators in this approach, but instead an eleventh generator not part of the Poincaré algebra. The authors make a point of this. The world-line condition and the assumptions behind the no-interaction theorem A world-line for a relativistic point particle, moving freely or under the influence of forces, is the path through space-time represented by the four-vector x μ (τ) as a function of some evolution parameter τ. Relativistic covariance means that observers in different frames of reference agree as to the shape of the world-lines, even though they are described by different numerical functions in each frame. These are called the WorldLine Condition(s) (WLC), also referred to as the “objectivity of world-lines”. A world-line, in a certain frame, may also be given in terms of 3-dimensional coordinates x i and a clock x 0 as x i (x 0 ) through the gauge choice τ = x0. The assumptions behind the no-interaction theorem—within a canonical formalism—may be given as: (i) the set of 3-dimensional position coordinates of all the particles form one-half of a system of phase space variables of the system, (ii) the canonical and geometrical transformations of the position variables coincide under the Euclidean subgroup (in the instant form of dynamics) of the Poincaré group, (iii) the world-line condition.

The first paper in the series [59] starts by reviewing the free relativistic particle in order to explain the general method. By examples, it shows that “[. . .] the world-line condition depends on the choice of gauge constraint. This is expected, since the Dirac canonical

43 The Komar papers are the stepping stone, so to speak, but the analysis clearly applies to all approaches of the Lagrangian/Hamiltonian constrained dynamics type. 44 It is quite interesting historically; Dirac invented his “forms of relativistic dynamics”, often also called the “Dirac canonical generator formalism” and he coinvented the analysis of constrained Hamiltonian dynamics. It was left to others to put these two methods together into one scheme, indeed going, a bit at least, beyond Dirac.

44 � 2 Notes on the history of higher spin interactions transformation R⋆ (Λ, a), which represents a Poincaré group element in the final form of the theory, is built to preserve the value of τ, and τ has different space-time meanings in different gauges. All this is notwithstanding the fact that in building up a world-line we assign spatial position x j (τ) at physical (or laboratory) time x 0 (τ) whatever the gauge.”. The paper then continues with a lengthy analysis of two interacting particles of the Komar type. A point is made of the fact that the Komar theory results in world sheets for the two particles, rather than world-lines. The 2-dimensional trajectories S of Komar may be coordinatized by x10 and x20 , each point of S then determines two-world points in space-time. “However, except in the noninteracting situation, we expect that each of j j j x1 and x2 will depend on both of x10 and x20 as we wander over S. Hence, it seems as if x1 j

and x2 in general will trace out worldsheets instead of world-lines.”. This leads to the need to fix gauge conditions in order to obtain unique world-lines for the particles. The paper proceeds to study such gauge conditions χ(x1 , x2 , p1 , p2 , τ, σ) ≈ 0 with explicit dependence on two parameters τ and σ. This analysis shows that the gauge conditions must only depend on one common parameter τ ′ in order to have a satisfactory theory. The conclusion is that, effectively, only one of the gauge constraints has an explicit τ dependence. The authors state “The other constraint [not depending on τ] determines then a curve C on the sheet S and the choice of such a condition can no longer be viewed as a gauge choice giving different descriptions of “the same theory” since in general it leads to different pairs of world-lines. Rather such a choice is a part of a complete theory.”.45 A further analysis then shows that “[. . .] the gauge condition which has no explicit τ dependence must be manifestly Poincaré invariant.”. This is in order to accommodate both the world-line condition and allowing for interactions. Two examples are then given, one noncovariant and one covariant. The covariant gauge choice is χ1 = P ⋅ r and χ2 = P ⋅ X − Mτ with M = (P2 )1/2 . In this gauge, the world-line conditions can be satisfied for any potential V . The authors write “In conclusion, the world-line conditions are not always violated in the presence of interaction, but the situation depends on the gauge chosen and this gauge choice is then part of the specific theory [. . .].”. In a last section, the paper studies the relativistic harmonic oscillator model. In the conclusions, the authors return to the particular, and ubiquitous, covariant gauge constraint P ⋅ r ≈ 0, and notes that it leads to a theory with a simple Lagrangian L(τ) =

2 1 2 ẋα + mα2 Vα ) − m(V1 + V2 )V (r 2 ) ∑( 2 α=1 Vα

(2.72)

with V1 and V2 two einbein variables and m the reduced mass of the two particles. When the equations of motion for the einbeins are substituted into the Lagrangian, it

45 This result is indeed very interesting, as such τ-independent constraints generally appear in Lagrangian based models.

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takes the form of a Kalb–Van Alstine model [58] (see Section 2.2.5). It is thus clear that there are close relationships between Komar-type models, their further analysis as in the Mukunda series and the singular Lagrangians of the CDGL and Kalb–Van Alstine type. One paper that explores this further is [88].46 As for the three following papers in the series, it would take us too far to review them in any detail. We will contend ourselves with an outline of their basic approach and general results and refer the interested reader to the papers themselves. The Introduction to the second paper offers a brief but very informative history of the Hamiltonian approach to direct interaction, so does the Introduction to the third paper, with a thorough background discussion of the approach of the series of papers. In order to understand the drift of the argument in the papers, one needs to keep a set of thoughts in mind simultaneously. First, there are the two distinct aspects of special relativistic invariance: one is the invariance of physical laws, the other is the manifest covariance of certain dynamical variables. The Dirac generator formalism asks for canonical representations of the ten Poincaré group generators. This is an expression of relativistic invariance of the dynamical laws. The Poincaré generators contain the “Hamiltonians”—in the interesting cases with interaction terms—and therefore the generators also provide the dynamical evolution of the system within an inertial frame, as well as the transformations between the frames. On the other hand, manifest relativistic covariance is an additional requirement that the canonically transformations and the independent geometrical transformations, for a selected set of variables, are compatible. Second, the implementation of the constrained Hamiltonian approach is in terms of eleven generators. [. . .] a definite parameter of evolution must be specified, and eleven distinguished functions on phase space must be given. The first of these is the generator of the canonical transformations describing dynamical evolution in any inertial frame; the remaining ten generate a canonical realization of the Poincaré group and so describes changes of frame.

Third, there is the no-interaction theorem, the assumptions of which—within a canonical formalism—are given as in the box [The World Line condition...] above. The WLC is of central importance here, and expresses the requirement of relativistic covariance. The Introduction to the third paper writes: In retrospect, one can see that it is the fact that the ten generators have to do double duty, namely obey the Lie relations of the Poincaré group on the one hand, and obey the WLC on the other, that is the fundamental origin of the no-interaction theorem.

46 Apart from the “short square-root” (CDGL) and “long square-root” (Kalb–Van Alstine) Lagrangians, there is one more type of Lagrangian for this kind of model. The three possible Lagrangians correspond to three possible primary Hamiltonians made out of two or three of the constraints (2.60) (taken with equal masses, and always including the first constraint).

46 � 2 Notes on the history of higher spin interactions Not prepared to give up the “objective reality of world-lines”, the authors outline the third thought. One must go beyond the boundaries of Dirac’s program for relativistic dynamics, and envisage choices of evolution parameter that are dynamically, not kinematically, determined. In such a framework, all the eleven generators for a relativistic Hamiltonian theory enter with independent status.

Thus, the strictures of the no-interaction theorem is weakened by releasing the Poincaré generators from their double duty of both generating dynamical evolution and transformations of frame in accordance with the WLC. In the implementation of these ideas, the Mukunda series of papers arrives—in the final analysis—in the use two covariant gauge fixing conditions χ1 (x, p) ≈ 0

and

χ2 (x, p, τ) ≈ 0

(2.73)

as in the first paper of the series. The third paper gives the example χ1 = P ⋅ r

and

1 χ2 = P ⋅ (x1 + x2 ) − τ 2

(2.74)

for which the WLC is obeyed whatever the interaction potential. One last item can be mentioned regarding the second Mukunda paper. In it, the authors study an approach to the N-particle problem that uses redundant N + 1 sets of phase space variables. In addition to center variables (Q, P), there is N relative variables (ξa , ηa ), one pair for each particle. The redundancy must then be removed by further constraints. The study of this kind of model was inspired by such an approach advocated by F. Rohrlich in [89].

2.2.10 Bi-local theories, infinite component fields and dynamical groups When it comes to the topics of the title of this subsection, there is an enormous literature from the early 1960s to the late 1970s. These were also the days of theoretical exploration of strong interaction physics, which was the motivation behind the various models and investigations. We have touched on some of the papers in the sections above—those pertaining to Lagrangian mechanical models—the ones that seem to be closest to some relevance for the higher spin problem, but there may very well be more knowledge to be extracted from this research. It was naturally geared toward the physics of hadrons and their constituents, and not to massless gauge fields, except for the CDGL nonhadron program reviewed above. The related concepts of bilocal fields and infinite component fields are clearly also relevant for massless higher spin theory as we indeed know from modern research into

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the subject. See the review [90] that gives one perspective on this. Indeed, many of the papers on mechanical models also discuss bilocal interpretations of the mechanical models, in particular, when second quantized. That said, limitations of space and time forces us to leave this area of the history for future study. For the interested reader, here are some further inroads to the literature from the old times when this research was active [91, 92]. 2.2.11 Spinning and supersymmetric particles There was also the subresearch program of spinning and supersymmetric particles. Such models are based on extension of the coordinates, or phase space variables, with Grassmann or fermionic variables. They are able to describe particles with spin, but apparently not the kind of towers of spin that we are interested in within higher spin theory. A reference that offers an overview, historically and technically, of this kind of mechanical particle is the PhD thesis [93].

2.3 String theory, supergravity and higher spin theory That string theory and higher spin theory are related is, in some sense, obvious. Mathematical techniques used in string theory have been borrowed in higher spin theory. The propagating free string contain excitations that can be interpreted as an infinite tower of massive fields above the massless spin-1 and spin-2 fields of open and closed (super)strings, respectively. But the differences must be noted. Strings have a critical dimension, while—as far as we know—that is not so for the present higher spin gauge theories. Although higher spin gauge theories also have infinite towers of fields, the spectra of string theories are “infinitely” larger. In the context of the Vasiliev theory, it has been conjectured that higher spin gauge theories may be ultrahigh energy phases of string theory, and that string theories may correspond to broken phases of higher spin gauge theories.47 If, in a very rough history, the higher spin resonances discovered in the laboratories during the 1950s and 1960s had encouraged the continuing interest in the subject during these decades, then the discovery of supersymmetry and supergravity and the increasing interest in string theory was a main motivation behind the renewed interest in higher spin in the late 1970s and early 1980s. The focus then also changed from higher spin matter to higher spin massless gauge fields. Parts of this story was told in Section 2.10 of our Volume 1. For content and references relevant to the status of these theories at the time, see the reviews [94, 95] for supersymmetry and supergravity and the review [34] for superstring theory.

47 I am personally not convinced by such conjectures.

48 � 2 Notes on the history of higher spin interactions It must be born in mind that for theoreticians at this time, higher spin theory started already with spin 3/2. Rarita and Schwinger had published on this field in 1941 [96].48 In 1969, G. Velo and D. Zwanziger showed that minimal coupling to an electromagnetic field was inconsistent in that it lead to superluminal propagation. This added to the already perceived problems with higher spin fields discovered by Fierz and Pauli. Supergravity, solved this problem of spin 3/2 in that the Rarita–Schwinger field fitted naturally into the N = 1 supergravity multiplet (and higher N multiplets). The question naturally arose if higher than spin-3/2 fermions could be coupled supersymmetrically, in particular, spin 5/2. Such a theory was called hypergravity and was considered a natural extension of supergravity theory. Furthermore, the connection between supergravity multiplets and the number of supersymmetries implied that the largest multiplet not going beyond spin 2 was N = 8. The concomitant O(8) group was too small to contain the Standard Model gauge group SU(3) × SU(2) × U(1). To find room for a phenomenologically viable model, one would then need to go beyond spin 2. This lead to more detailed studies of spin-5/2 fields [97, 98] and a renewed interest in higher spin fields in general. However, hypergravity theories turned out to suffer from gravity interaction incompatibility problems, as analyzed in several papers [99, 100, 101]. Weyl tensors appear in the computation of commutators of covariant derivatives that cannot be compensated by other terms. This is so in Minkowski space, but Fradkin and Vasiliev discovered, in an unpublished paper, that this obstruction could be circumvented in AdS space-time. This was one inspiration to what became the Vasiliev AdS higher spin theory, to be extensively reviewed in Chapter 8 Gauge theory approaches to supergravity were soon tried after the inception of the supergravity. An early reference is that of MacDowell and Mansouri [102], which in the elaboration of Stelle and West [103], came to be an inspiration and role model for the first attempts at finding an action within the Vasiliev approach to higher spin. We will have an occasion to review a related approach to supergravity—the group manifold approach—that is of relevance for the higher spin problem in Section 4.1.

2.4 Early coupling problems and classic no-go results As discovered by Fierz and Pauli, coupling higher spin fields—massive or massless— to electromagnetism and gravity, is not straightforward; rather, it is potentially inconsistent. In the particular cases investigated by Fierz and Pauli, coupling to electromagnetism, the difficulties could be overcome by deriving field equations and subsidiary conditions from a well-chosen action.49 But this is not the full story, as became apparent

48 The following letter in the journal, by S. Kusaka, discusses a spin-3/2 theory of the neutrino. 49 See Section 2.1.5 in our Volume 1.

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through the investigations of G. Velo and D. Zwanziger in 1969. However, let us remain in the late 1950s and early 1960s first. In the early and mid-1960s, when the strong and weak interactions were researched experimentally and theoretically, there appeared a number of “theorems” that were collectively denoted as no-go theorems. At this time, Yang–Mills theory had not yet been successfully applied to these interactions. Rather, field theory was generally distrusted as a fundamental tool for understanding subnuclear interactions, and instead there was a toolbox of other approaches. One of them was the quite successful SU(3) eight-fold way of M. Gellman [104] and Y. Ne’eman [105] that united particles of the same spin. Naturally, there were attempts to extend this global symmetry of particles to bigger groups uniting particles of different spin, one of them being SU(6) [106, 107, 108] and tentative relativistic generalizations.50 The group was meant to be big enough to potentially contain the Poincaré group as a nontrivial subgroup. However, all such attempts failed in various ways, and there soon appeared results [111, 112, 113] showing the impossibility of any such “unification” of internal and space-time symmetries. These results were, however, not that general but rather relied on particular assumptions and applied to particular groups, and there soon appeared more general and encompassing no-go theorems. These theorems are often cited in connection with the question of higher spin gauge interactions. Authors with a inclination to rule out higher spin gauge fields take the no-go theorems as arguments for the nonexistence and nonrelevance of higher spin fields for fundamental physics. While authors within the higher spin positive community have to think in terms of circumventing the theorems or finding loop-holes in the proofs. A more modern view is to “accomodate” the no-go results, i. e., be consistent with them while still having an interesting theory. However, before deciding on how to react to the theorems, one must try to understand what and if they have anything to say about higher spin gauge interactions at all. In this endeavor, we will state the theorems and their assumptions, but we will not attempt to review their proofs, which are often quite technical. Our focus will be on how—and to what extent—the theorems might apply to higher spin gauge fields.

2.4.1 Buchdahl’s gravity compatibility investigations H. A. Buchdahl’s early investigation in 1958 and again in 1962 of coupling spin 3/2 to gravity is interesting, because the difficulties he encountered are related to supergravity, in which theory the problems were overcome. In the first paper [114], Buchdahl considers the spin-3/2 massive field equations in two-component spinor formulation of Dirac. Replacing the derivatives with covariant derivatives, he derives further conditions, not present in the free theory to the effect that the Ricci tensor must satisfy Rμν = λgμν .

50 References can be found in Chapter 24 of [109] referring to the reprint volume [110].

50 � 2 Notes on the history of higher spin interactions Therefore, the solution is an Einstein space, or Ricci-flat in the case of zero λ. In this paper, neither any nonminimal coupling terms, nor any back-reaction of the spin-3/2 field on the gravitational field is considered. Thus, the gravitational field is considered as a passive background. Buchdahl interpreted this result as meaning that spin 3/2 could not interact with gravity, since in that case the Ricci tensor cannot be zero. The reason is that the righthand side of Einstein’s equations is the spin-3/2 energy-momentum tensor, and that contributes to the Ricci tensor. In the second paper [115], Buchdahl considers coupling fields of spin higher than 3/2 to a gravitational background, and finds that it is only possible for space-times of constant Riemann curvature. In both papers, the problem is the generic one: commuting general covariant derivatives brings up curvatures, in the spin-3/2 case the Ricci tensor, and for higher spin, the Riemann curvature. Both Buchdahl papers are written on the level of field equations. One may ask how supergravity escapes Buchdahl’s conclusion. The answer is that in supergravity the spin 3/2 is a source to the gravitational field, and that there is an extra coupling through torsion.51 The field equations so obtained by varying the vierbeins and spin connections independently, together conspire to make the contributions to the Ricci tensor vanish.

2.4.2 The Velo–Zwanziger problem and Johnson–Sudarshan problem The so-called Velo–Zwanziger problem concerns the propagation of charged massive spin 3/2 in an electromagnetic background. In [118], G. Velo and D. Zwanziger showed that the classical Rarita–Schwinger field, when coupled minimally to an external electromagnetic potential, have solutions that propagate faster than light. A related problem, in the quantum theory, had been noted earlier in 1961 by K. Johnson and E.C.G Sudarshan [119]. These authors showed that properties of the equal time commutators of the theory are not compatible with relativistic covariance. The problem of superluminal propagation is not specific to the Rarita–Schwinger field, but affects all massive higher spin fields. Another related type of problem is changes in the number of propagating degrees of freedom. This was noted early on for massive spin 2 in a short note [120]. Velo and Zwanziger studied this problem in a second paper [121]. As they write in the abstract “A constraint is converted into an equation of motion, so that there are six degrees of freedom instead of the desired five”; see also [122].

51 This is explicit in the first-order formulation of S. Deser and B. Zumino [116]. For a recent history and explanation of this, see [117].

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The superluminal problem for the Rarita–Schwinger field can, however, be remedied as shown in [123] by the appropriate choice of nonminimal couplings. This paper contains an interesting discussion of the problem and sets it in a larger context of supergravity. Further modern comments and work can be found in [124] and in the review article [125].

2.4.3 The Weinberg S-matrix argument In the early 1960s, S. Weinberg began a series of papers on S-matrix theory [126, 127]. This is before it was understood how the Yang–Mills theory could be applied to the weak and strong interactions, but the papers are not concerned particularly with the S-matrix approach to strong interactions.52 Rather, in a rational, pragmatic no-nonsense style, the papers set out to develop “[. . .] explicit Feynman rules for perturbation calculations, in a formalism that varies as little as possible form one spin to another.”. These are the Feynman rules for any spin-papers that we briefly looked at in Section 2.6 in our Volume 1. Part of this work then became the basis for the bottom-up approach to quantum field theory that is employed in the first volume [129] of the Weinberg quantum theory of fields textbooks, already outlined in a conference report [130]. Then came a set of papers about photons and gravitons in perturbation theory [131, 132, 133]. It is in these papers that we find the no-go result. But from the text of the papers, one cannot conclude that the no-go theorem was the major objective. Rather, the focus was on deriving the spin-1 Maxwell and spin-2 Einstein theories from S-matrix considerations without recourse to Lagrangian field theory and gauge invariance. In particular, the goal was to understand charge conservation and the equality of gravitational and inertial mass. This is what we read in the Introduction to [132]: The purpose of this article is to bring into question the need for field theory in understanding electromagnetism and gravitation. We shall show that there are no general properties of photons and gravitons which have [emphasis in the original] been explained by field theory, which cannot also be understood as consequences of the Lorentz invariance and pole structure of the S matrix for massless particles of spin 1 or 2. We will also show why there can be no macroscopic fields whose quanta carry spin 3 or higher.

The no-go result is a byproduct of a general approach to the interactions of massless particles. It can be found in Section IV of the paper (with supporting theory in the preceding sections), or in Section 5.3 of the lecture notes [130]. Let us try to review the main steps in some more detail than is normally done in the higher spin review literature. We will stay close to the original, dividing up the argument in four logical steps (i)–(iv). Much of the underlying S-matrix theory will be taken for granted.

52 A third paper [128] in the series came in 1969.

52 � 2 Notes on the history of higher spin interactions Weinberg’s argument in some detail (i) Weinberg first discusses (Section II of the paper) tensor amplitudes for emission of massless particles of integer spin. Let S±j (q, p) denote the S-matrix element for a process where a massless particle with momentum q and helicity ±j is emitted. The momenta and helicities of all other particles are represented collectively by p. Weinberg shows that S±j (q, p) can be written as μ ∗

μ∗

S±j (q, p) = (2|q|)−1/2 ϵ±1 (q) ⋅ ⋅ ⋅ ⋅ ⋅ ϵ±j (q)M±μ1 ...μj (q, p)

(2.75) μ∗

in terms of a, the so-called, M function. The function M±μ1 ...μj is a symmetric Lorentz tensor, and the ϵ± are

μ∗ μ ∗ polarizations. The polarization ϵ±1 ⋅⋅ ⋅ ⋅⋅ϵ±j for the massless particle does not, however, transform as a Lorentz μ tensor since the Lorentz transformation for the polarization ϵ± (q) contains a term qμ (see Section 3.5.5 in our

Volume 1). Therefore, in order for S±j (q, p) to be Lorentz invariant, S±j (q, p) must vanish when any one of the polarizations is replaced by a momentum factor, i. e., qμ1 M±μ1 ...μj (q, p) = ⋅ ⋅ ⋅ = qμj M±μ1 ...μj (q, p) = 0

(2.76)

This may be interpreted as “mass-shell gauge invariance” as Weinberg writes.53 (ii) The rest of the argument is based on a general property assumed to hold for the S-matrix, namely that the S-matrix has poles corresponding to Feynman diagrams in which a virtual particle, with fourmomentum nearly on the mass-shell, is exchanged between two sets 1 and 2 of incoming or outgoing particles. The residue at the pole factors into the product of two amplitudes Γ1 and Γ2 .54 The vertex amplitudes depend only on the “quantum numbers” of the particles in the respective sets 1 and 2. As a consequence, it is possible to give a “[. . .] purely S-matrix-theoretic definition of the vertex amplitude Γ for any set of physical particles, as a function of their momenta and helicities; the coupling constant or constants define the magnitude of Γ.”.55 (iii) Based on this, Weinberg defines charge and gravitational mass as the coupling constants determining the magnitudes of soft photon and graviton vertex functions. Then go back to the formula (2.75) applied to the emission of a low energy massless particle by a spin J = 0, mass m particle with momentum pμ = (p, E) and energy E = √p2 − m2 . Weinberg argues that the only tensor that can be used to form the M-function is pμ1 . . . pμj . More precisely, he takes the vertex amplitude μ ∗

μ∗

pμ1 . . . pμj ϵ±1 (q) ⋅ ⋅ ⋅ ⋅ ⋅ ϵ±j (q) 2E(p)(|q|)1/2

(2.77)

53 In the lecture notes [130], this is phrased as “M is conserved”. 54 Weinberg writes “vertex amplitudes” with citation marks, likely thinking of them in this context as just “vertex amplitudes” for the interaction of the participating particles. Both in the paper [132] and in the lecture notes [130], there are cautionary remarks as to the rigor of the arguments. However, “Our purpose is to explore the implications of the generally accepted ideas about the pole structure.”. Remember from complex analysis that the residue at an isolated pole z0 for an analytic function f (z) is the coefficient for the 1/(z − z0 ) term in the Laurent expansion of the function. 55 It is very interesting to read these passages with modern hindsight, in particular, with regard to the very close connections between the light-front vertices for massless higher spin particles and the spinorhelicity approach to scattering amplitudes [134].

2.4 Early coupling problems and classic no-go results

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There is then an argument that this form holds also for an emitting particle with spin J ≥ 1. Specializing to photons and gravitons, we have μ∗

epμ ϵ± (q)

2E(p)(|q|)1/2 μ∗

fG1/2 (pμ ϵ± (q))2 2E(p)(|q|)1/2

(2.78a) (2.78b)

where we have suppressed numerical constants not essential for the argument. Newton’s constant is introduced in order to make the gravitational coupling f dimensionless. (iv) The actual argument, leading to charge conservation and universality of the gravitational coupling as well as the no-go result for higher spin, starts by considering an S-matrix element Sαβ for a reaction α → β where α (incoming) and β (outgoing) stands for the states of the participating particles. These particles may be charged or uncharged, massive or massless. The same reaction may then occur with the emission of an extra very soft photon or graviton of momentum q and helicity ±1 or ±2—or for the higher spin case—a very soft ±j higher spin massless particle of helicity ±j. Such matrix elements are denoted by Sαβ (q) with j = 1, 2, 3, . . . . The extra soft massless particle may be emitted from an internal or external line. However, an internal particle line will not be on mass-shell, therefore not producing pole.

q p±q

p

Sβα

Figure 2.2: Left figure: Vertex for the emission of a soft massless particle of momentum q from an external particle line of momentum p. The + is for p outgoing and the − for p incoming. Right figure: The initial S-matrix element. The emission can occur from any of the external lines.

It is therefore assumed that the extra particle is emitted from one of the incoming or outgoing particles ±j in the states α and β (see Figure 2.2). This means that the new matrix element Sαβ (q) may be considered to consist of two amplitudes, one corresponding to the original matrix element Sαβ and one corresponding to the coupling of a soft massless particle to an external particle, connected by a virtual particle line. The result is the product of the amplitudes and a propagator for the virtual particle line. For the full modified amplitude, one must sum over the external lines. The propagator for the connecting line becomes 1 1 →± 2p ⋅ q (p ± q)2 + m2

in the limit q → 0

(2.79)

since the external particles are on-shell. For a spin 1 massless particle, the amplitude is, up to inessential factors

∑ n

ηn en pn ⋅ ϵ±∗ (q) Sαβ pn ⋅ q

(2.80) μ∗

where ηn = 1 for outgoing particles and ηn = −1 for incoming particles. Then, replacing the polarization ϵ± with qμ , we get charge conservation ∑n ηn en = 0.

54 � 2 Notes on the history of higher spin interactions

For a spin-2 massless particle, the amplitude is, up to inessential factors, G1/2 ∑ n

ηn fn (pn ⋅ ϵ±∗ (q))2 Sαβ pn ⋅ q

(2.81)

μ∗

Now, replacing a polarization ϵ± with qμ we get μ∗

G1/2 (∑ ηn fn pnμ )ϵ± (q) = 0 n

(2.82)

Compared to momentum conservation ∑n ηn pn = 0, we conclude that all gravitational couplings fn must be equal. Finally, performing the corresponding steps for emission of a massless spin j particle leads to the demand μ∗

j−1

∑ ηn gn(j) (pnμ ϵ± (q)) n

=0

(2.83)

with higher spin coupling constants gn . But this equation cannot be satisfied for generic momenta. There(j) fore, all the couplings gn must vanish, at least in the low energy limit. A slightly more streamlined argument based on Feynman diagrams can be found in the paper [135] and in Section 13.1 in the textbook [129]. (j)

The Weinberg no-go result is interpreted in two related ways. Higher spin massless particles may exist, but they cannot couple to other particles at low energy-momentum (soft limit). This is in sharp contrast to spin 1 and 2 where we can interpret the result as a derivation of charge conservation and universality of the gravitational coupling through soft coupling. This also means that higher spin particles cannot mediate long range, inverse-square forces. Note, however, that the cubic vertices themselves, involving j momentum factors, are not prohibited; rather, they are demanded by Lorentz invariance. This is indeed consistent with the light-front analysis.

2.4.4 The O’Raifeartaigh and Coleman–Mandula theorems The theorems of O’Raifeartaigh and of Coleman–Mandula were reactions to attempts to find larger symmetry groups for the strongly interacting particles that unified with the Poincaré symmetry in a nontrivial way (so that particles of different spin could be united in the same multiplet), perhaps also explaining the mass splittings among multiplets. The theorems proved this to be impossible. The apparent contradiction with supersymmetry, discovered less than 10 years later, was explained by O’Raifeartaigh and Coleman–Mandula not allowing for algebras with anticommuting elements. There is also a supersymmetric extension of the Coleman–Mandula theorem, the Haag– Lopuszanski–Sohnius theorem [136]. For a good review of the historical context of

2.4 Early coupling problems and classic no-go results

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these developments, see the introductory chapter in Weinberg’s third textbook [109]. That chapter also contains a streamlined proof of the Coleman–Mandula theorem. While the conclusions of the two theorems pointed in the same direction, that no nontrivial unification of space-time symmetries and internal symmetries was possible, their methods were different. The O’Raifeartaigh theorem is group theoretical whereas the Coleman–Mandula theorem is within S-matrix theory. The Coleman–Mandula theorem is quite easy to state. Why not let the authors speak for themselves? Theorem: Let G be a connected [removed footnote] symmetry group of the S-matrix, and let the following five conditions hold: 1. (Lorentz invariance.) G contains a subgroup locally isomorphic to P [Poincaré group]. 2. (Particle finiteness.) All particle types correspond to positive-energy representations of P. For any finite M [mass], there are only a finite number of particles with mass less than M. 3. (Weak elastic analyticity.) Elastic-scattering amplitudes are analytic functions of center-ofmass energy, s, and invariant momentum transfer, t, in some neighborhood of the physical region, except at normal thresholds. 4. (Occurrence of scattering.) Let |p⟩ and |p′ ⟩ be any two one-particle momentum eigenstates, and let |p, p′ ⟩ be the two-particle state made from these. Then T|p, p′ ⟩ ≠ 0 except perhaps for certain values of s. Phrased briefly, at almost all energies, two plane waves scatter. [T is the nontrivial piece of S with momentum conservation factored out.] 5. (An ugly technical assumption.) The generators of G have distributions for their kernels. [Elaborated in the paper.] Then G is locally isomorphic to a direct product of an internal symmetry group and the Poincaré group.

While the authors worried about the fifth, unphysical assumption, it is the relevance of the second assumption that may be doubted for higher spin gauge theory. Indeed, there are infinitely many zero-mass particles below any finite mass, and it can be doubted whether an argument performed for massive particles has any relevance for massless particles. However, the theorem holds also for massless particles, according to Weinberg in the textbook [109] mentioned above, albeit for a finite number of them. Nonetheless, as argued in [137] (Section 2.3), the drift of the argument is to rule out global higher spin symmetry generators, with nontrivial commutators with the Poincaré generators, that could have come from large distance scattering. 2.4.5 The Weinberg–Witten theorem The Weinberg–Witten theorem [138] was conceived as an answer to proposals to describe the graviton as a composite of massless particles. That is not possible according to the theorem. The theorem is often mentioned also as a higher spin no-go theorem. We will not try to explain or prove this theorem since there is an excellent exposition of it in [139] explicating both the proof and explaining its relevance for lower and higher spin. Somewhat differing views on the theorem have been expressed within the higher spin community; see [137] and [140].

56 � 2 Notes on the history of higher spin interactions

2.5 Gauge theories of gravitation Gauge theories of gravitation—quite an extensive subject in itself—has been an important source of ideas and techniques to be generalized to the higher spin interaction problem. Furthermore, the subject has its own conceptual difficulties that may very well be of fundamental importance for higher spin gauge theory. For these reasons, we will here attempt a review of a few central papers in this tradition. The focus will be on aspects that seem to be of most interest from a higher spin perspective: transformations and actions. Notation will be close the original papers, but some homogenization will be done in order to facilitate easier comparison. Such simplifications will be indicated in the text. The contents of this section can be seen as historical background to our discussion in Section 4.6 in Volume 1. Notes on notation Notation for local vs. total variations—or transformations—may vary between authors. Kibble uses δ for total variations and δ0 for local. Some authors are not so explicit and may also use only one of them, which one often has to be surmised from the context. One may also come across the notation δ ⋆ —designated substantial variation—for local variations. I will use Δ for total variations and δ for local (for further clarification, see our Volume 1, Section 3.14), and consequently change the notation when citing formulas from the original papers. Both Utiyama and Kibble use commas to indicate differentiation, a fairly common usage in those days, particularly in relativity. Utiyama also uses commas to separate different kinds of indices on the same object. These we have removed as they are unnecessary and may cause confusion.

Before starting, let us establish some common ground. In order to accommodate halfinteger fermionic matter into gravitational theory, one must reformulate general relativity in terms of vierbeins or tetrads, as basic gravitational fields, the metric becoming secondary. The coframe tetrads eμa can then be thought of as “moving special relativistic

coordinate systems”.56 Local Lorentz transformations, i. e., varying from point to point, can be performed in these coordinate systems. In modern language, we have a tangent bundle with the structure group the Lorentz group. The connection, relating fibers at different space-time points, is the spin connection ωμab , which in standard GR has an explicit expression in terms of the vierbeins. Lorentz covariant derivatives are also introduced. This is all standard and well understood. The historical reference is H. Weyl’s (1929) paper [141].57 Now, interestingly enough, as regards local Lorentz transformations, the resulting theory looks precisely as a Yang–Mills gauge theory with the gauge group, the homogeneous Lorentz group, albeit in a local inertial system. Thus, from this point of view, the

56 This corresponds to Cartan’s moving frames. 57 Translated and reprinted in [142] and [14].

2.5 Gauge theories of gravitation

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spin degree of freedom for the fields behaves just as an internal group degree of freedom, despite the fact that spin ultimately is a space-time based property via the Poincaré group of space-time symmetry. This is also a reflection of the fact that angular momentum comes in two kinds: orbital angular momentum and spin angular momentum. The spin, once its existence and properties are developed from the representation theory of the Poincaré group, very much behaves as an internal degree of freedom. This leads us to our first historical paper to examine, the Utiyama paper [143] of 1956. The reader may want to first read Section 2.9.2 in our Volume 1 as an introduction. There are two main issues: the relation between the diffeomorphism group and the group of local translations, and the question of where and with what motivation to introduce the vierbeins into the theory.

2.5.1 R. Utiyama Utiyama is, in some places, criticized for not really having found a gauge theory for gravity in that he introduces the vierbein field ad hoc. While true, the situation is more complex than that, and the allegation is a bit unfair, since he does not actually state his goal in the paper as finding a full gauge theory of gravity. Gravitation is mentioned in the context of “gauge theory” but nowhere is he claiming to have derived full Einstein gravity from any gauge principle.58 The following quote is more true to the contents and explicit aims of the paper: In the usual textbooks of general relativity, the covariant derivative of any tensor is introduced by using the concept of parallel displacement. On the other hand, we shall see in Section 4 that the covariant derivative of any tensor or spinor can be derived from the postulate of invariance under “generalized Lorentz transformations” derived by replacing the six parameters of the usual Lorentz group with a set of six arbitrary functions of x. In deriving such covariant derivatives, it is unnecessary to use explicitly the notion of parallel displacement.

The quote comes at the end of the Introduction, where the electromagnetic field and the newly found—by Yang and Mills in 1954—isospin field Bμ are listed as examples of “interactions of the first kind”, interactions derived by “postulating invariance under a certain group of transformations”. Gravitation is included in this class of interactions, but again, what is actually written explicitly is the following: We shall find an analogy [our emphasis] between the transformation characters of the electromagnetic field Aμ , the Yang–Mills field Bμ , and Christoffel’s affinity Γμνλ of the general relativity. Furthermore, we shall understand the reason why in the Yang–Mills field strength the quadratic term,

58 Remember also, that we are in 1956, and the concept of non-Abelian gauge theory had not had time to settle as it has today. What Utiyama may have thought about gauge theory of gravitation, we have to leave for deeper historical analysis; see [14].

58 � 2 Notes on the history of higher spin interactions

Bμ × Bν , appears which is quite similar [our emphasis] to that occurring in the Riemann–Christoffel tensor Rλμνρ , namely the term ΓΓ − ΓΓ i R.

It is therefore quite clear that it is the appearance of the Lorentz gauge symmetry of gravitation that Utiyama wants to understand in a way similar to the electromagnetic and Yang–Mills type of theories. More serious, however, for Utiyama’s claim to have found a gauge theoretical—“invariant theoretical” in his terminology—way of deriving the need for connections, is the fact that not only does he introduce vierbeins, he actually also introduces the affine connection by hand at one stage of his development. It is this, rather than the introduction of vierbeins, that goes against his own stated goal. We will return to the question about the degree to which Utiyama captures the gauge properties of gravitation after having looked at the contents of the paper. Before going more deeply into Utiyama’s paper, let us record five questions that he explicitly sets out to answer. Utiyama’s questions Let us here note (almost verbatim) the five questions that Utiyama poses in the Introduction. The new group of local transformations with parameters ϵ a (x) is called G′ . Q may be thought of as a matter field. 1. What kind of field A(x) is introduced on account of the invariance under G′ ? 2. How is this new field A transformed under G′ ? 3. What does the interaction between the field A and the original field Q take? 4. How can we determine the new Lagrangian L′ (Q, A) from the original one L(Q)? 5. What type of field equations for A are allowable?

Section 1 of the paper has the title “General theory”. It can be read as just that: a general theory of Yang–Mills types of interactions. But rather than reiterating the nowadays well-known details, we will highlight what is actually “general” and what is not present in the development. For that, consider Utiyama’s (matter) fields QA transforming as δQA = T(a)AB ϵa QB

(2.84)

with ϵa constant infinitesimal parameters with the index a running from 1 to n. We have removed commas separating different kinds of indices,59 but retained the parenthesis (a) on the T matrices. In the case of Lorentz transformations, there will be an index group (kl) in this place. Utiyama names the T(a)AB as “constant coefficients”. Nowadays, we would straight away think of them as representation matrices of a semisimple Lie algebra. Utiyama only notes that the transformation δQA “is assumed to be a Lie group G depending on the n parameters ϵa ”. There is no mentioning of the semi-simplicity requirement. From there on, the treatment is “standard” (meaning that we would do the same today) up to the introduction of arbitrary infinitesimal parameters ϵa (x). 59 As they are unnecessary and may be confused with commas used to denote differentiation.

2.5 Gauge theories of gravitation

� 59

Now, in order to preserve the invariance of the matter Lagrangian, Utiyama argues that it is necessary to introduce a new field A′J (x), where the index J runs from 1 to M with the range M unspecified so far.60 We note that there is no space-time vector index on the new field. Wanting to be general, Utiyama writes the ansatz for the transformation of the new field as J

δA′J = U(a) K A′K ϵa (x) + C Jμa

𝜕ϵa xμ

(2.85)

where U and C are “unknown constants” to be determined. From a modern vantage point, we know that the gauge field A will carry a spacetime vector index μ and an adjoint representation index a so that in some way we want to perform a transition A′J → Aaμ . Indeed, we would say that the matter fields Q reside in the fundamental representation of the gauge group, and the gauge field in the adjoint representation, with indices chosen accordingly. But Utiyama does not use that terminology. Rather, it is more true to the context and time of writing to view his objective as analyzing how that very index structure is forced upon us from an “invariant theoretical” analysis of the situation. What Utiyama did (I): general theory Consider now a new Lagrangian L′ (QA , QA ,μ , A′J ) and the transformation formula (2.84) (with x dependent ϵ) and (2.85). Requiring the Lagrangian to be invariant, one gets δL′ =

𝜕L′ A 𝜕L′ 𝜕L′ δQ + A δQA ,μ + ′J δA′J = 0 𝜕A 𝜕QA 𝜕Q ,μ

(2.86)

Inserting the explicit transformation formula into this requirement, one gets two identities, one for ϵ a and one for 𝜕ϵ a /𝜕x μ . Let us record the second one 𝜕L′ 𝜕L′ T(a) AB QB + ′J C Jμa = 0 A 𝜕A 𝜕Q ,μ

(2.87)

These are 4n equations (in four dimensions). Therefore, Utiyama argues, in order to uniquely determine the Jμ A′J dependence of L′ , the number of components of A′J must be M = 4n. Furthermore, the matrix C a must be nonsingular, and its inverse is defined by the equations C Jμa C −1aμK = δ JK

and

C −1aμJ C Jνb = δ ab δ ν μ

(2.88)

This is clearly a judicious choice of requirements with the J index type to be traded for the group of indices aμ.

60 Utiyama writes the field A′J (x), with the prime to disambiguate it from other fields A defined later on in the development, and from the matter fields QA , whose indices A, B, . . . are of a different nature.

60 � 2 Notes on the history of higher spin interactions

The inverse matrices can now be used to perform the map A′J → Aaμ : Aaμ = C −1aμJ A′J . Then equation (2.87) can be written 𝜕L′ 𝜕L′ + T A QB = 0 𝜕Aaμ 𝜕QA ,μ (a) B

(2.89)

This implies that the gauge field should be contained in L′ only through the combination ∇μ QA =

𝜕QA − T(a) AB QB Aa μ 𝜕x μ

(2.90)

The transformation law in terms of the Aaμ fields become δAaμ = S(c) aμ νb Abν ϵ c (x) +

𝜕ϵ a 𝜕x μ

where S(c) aμ νb = C −1aμJ U(c) J K C Kν b

(2.91)

Utiyama then examines the consequences of (2.86) regarding the terms containing ϵ a . The key idea is that the new Lagrangian must now depend on QA and ∇μ QA only, and that it can be obtained from the original Lagrangian by substituting ∇μ QA for 𝜕μ QA . This leads to a determination of the coefficients S, S(c) aμ νb = δ ν μ fa c b

(2.92)

With this, we have arrived at well-known Yang–Mills formulas. It also follows that the covariant derivative (2.90) indeed transforms covariantly. Next, Utiyama sets out to “[. . .] investigate the possible type of Lagrangian for the free A-field.”. By “free” he means independent of the matter field Q. He denotes the Lagrangian by L0 (Aaμ , Aaμ,ν ). The invariance under the gauge transformations now lead to three equations (correcting a misprint in the second one) corresponding to the parameters ϵ a , ϵ a,μ and ϵ a,μν . 𝜕L0 a c 𝜕L0 a c f A + f A =0 𝜕Aaμ b c μ 𝜕Aaμ,ν b c μ,ν

(2.93a)

𝜕L0 𝜕L0 c b + f A =0 𝜕Aaμ 𝜕Ac μ,ν a b ν

(2.93b)

𝜕L0 𝜕L0 + =0 𝜕Aaμ,ν 𝜕Aaν,μ

(2.93c)

The last equation demands that the derivative of A only appear in the antisymmetric combination Aaμ,ν − Abν,μ . This, in its turn, used in the second equation shows that A appears in L0 only through the well-known combination 1 F aμν = Aaμ,ν − Aaν,μ − fb ac (Abμ Ac ν − Abν Ac μ ) 2

(2.94)

Finally, from the first equation, Utiyama surmises that L0 is a function of F aμν only and must satisfy the identity 1 𝜕L0 a b f F =0 2 𝜕F aμν c b μν

(2.95)

following from (2.93a) and using the Jacobi identities for the structure constants. Furthermore, F aμν transforms covariantly under gauge transformations.

2.5 Gauge theories of gravitation

� 61

So, from a minimal number of assumptions, Utiyama derives the general theory of Yang– Mills fields, and answers his own five questions from the Introduction to the paper. True, needed assumptions regarding the possible gauge groups are overlooked, but Utiyama’s paper contains a full treatment of the “algebraic approach” to Yang–Mills theory including discussion of conserved currents. This approach has now become standard, although many of the details are often left out.61 It is curious though that he does not write down the standard FF action of Yang–Mills theory, although it is of course allowed by the identity (2.95) above. The general theory is exemplified by electromagnetism and the Yang and Mills isospin theory before turning to the gauge theory of the Lorentz group in Section 4 of the paper. Let us stress again that there is no attempt to gauge the translation subgroup of the Poincaré group in the Utiyama paper. It is only the homogeneous Lorentz group that is intended. It is then no wonder that local Lorentz frames has to be introduced from the start. More than that, it turns out that not only vierbein fields have to be introduced, but also Christoffel connections. So, one may ask what Utiyama actually tries to achieve? The answer that may come to mind upon a first reading of his Section 4 is that it is what we now call the spin connections that he wants to understand from a gauge point of view. A deeper reading shows, however, that what he actually does is to set up a gauge theory of the Lorentz group on a curved manifold. We know that we can have curved manifolds without gravity—it is just differential geometry. We should therefore not dismiss Utiyama’s efforts too lightly. But we are running far ahead. Let us see what is actually in the paper and if there is even a deeper layer to it. What Utiyama did (II): gauging the Lorentz group The starting point is an action I = ∫ L(QA , QA ,k )d 4 x invariant under Lorentz transformations. A curvilinear

system of coordinates uμ is introduced besides the rectilinear system x k that Utiyama calls a “local Lorentz frame”. The metric in the x-system is g∗11 = g∗22 = g∗33 = −g∗44 = 1 with all other components zero. The metric in the curvilinear system is given by gμν (u) =

𝜕x i 𝜕x k ∗ g 𝜕uμ 𝜕uν ik

(2.96)

Furthermore, what we now call frames and coframes are introduced μ

hk (u) =

𝜕uμ 𝜕x k

and

hkμ (u) =

𝜕x k 𝜕uμ

(2.97)

Then Utiyama lists all the usual relations between these objects. Let us only record one for future reference g∗kl hkμ hl ν = gμν (u)

61 We reviewed it in Section 4.2 of our Volume 1, also leaving out details.

(2.98)

62 � 2 Notes on the history of higher spin interactions

So, it is clear what we have here is curvilinear coordinates on a flat manifold. The local frames, at every point, are transformed in the same way under global Lorentz transformations x k → x k + ϵ kl x l μ

μ

hk → hk + δhk

with ϵ kl = −ϵ lk

μ

μ

(2.99a) l

with δhk = −ϵ k hl

μ

(2.99b)

Considering tensor fields, they can now be transferred between the “world coordinates” and “local coordinates” using the frames and coframes. Utiyama writes Qk (u) = hkμ (u)Qμ (u)

and

μ

Qμ (u) = hk (u)Qk (u)

(2.100)

“where the abbreviation Qk (u) = Qk {x(u)} has been used.”. What to do with nontensor indices A on the (matter) fields comes after introducing an action integral I = ∫ Ld 4 u with μ

L = L(QA (u), QA , μ (u), hkμ (u)) = L(QA (u), hk (u)QA , μ (u))h

(2.101)

with h = det(hkμ (u)) and QA , μ = 𝜕QA (u)/𝜕uμ . The index A, in this context thought of as a spinor index, has no world representation, and can only be defined with respect to a Lorentz frame. We will now follow Utiyama’s arguments in some detail. The action I is invariant under the following two kinds of transformations: (1) (Global) Lorentz transformations with uμ unchanged δhkμ = ϵ k l hl μ δQA =

(2.102)

1 T A QB ϵ kl 2 (kl) B

where the T(kl)A B are representation matrices for the appropriate finite-dimensional representation of the Lorentz group to which the fields QA belongs. (2) General point transformations (this is Utiyama’s terminology) uμ → uμ + λμ (u) = u′ δhkμ = −

with λμ (u) an arbitrary function of uμ

ν

𝜕λ k h 𝜕uμ ν

(2.103)

ΔQA (u) = Q′A (u′ ) − QA (u) δQA, μ = −

𝜕λν A Q 𝜕uμ ,ν

Then comes a crucial passage in Utiyama’s paper that has to be read carefully in order to understand what is subsequently achieved and not achieved. Let us quote in extenso. Now our Lagrangian [number (2.101) here] has the suitable form for the application of the general method stated in [the Introduction to the paper], if the given functions, hkμ , are regarded as a set of field quantities satisfying the condition: 𝜕hkμ 𝜕uν

=

𝜕hkν 𝜕uμ

(4.5)

(2.104)

and having the transformation character [(2.102)] under the Lorentz group. Though we will omit the condition (4.5) [(2.104)], the invariance of I [the action] under the transformations [(2.102)] and [(2.103)] still holds. The only role of (4.5) is to guarantee the possibility of finding the simplest and most convenient

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system of coordinates (x 1 , . . . , x 4 ). In fact, if we replace the parameters, ϵ ik , with a set of arbitrary functions, ϵ ik (u), after the Lorentz transformation depending on such ϵ(u)’s, the relation (4.5) is destroyed. The condition (4.5) is inconsistent with the application of the general procedure of Section 1 to the present problem. Accordingly, we shall consider the h’s as a set of 16 independent given functions [emphasis of the original]. The condition (2.104) is what we now refer to as holonomy of the frames (see our Volume 1, Section 4.5.1). Utiyama then continues with “generalized Lorentz transformations” of the form (2.102) but with arbitrary function ϵ ik (u). He notes that in order to retain invariance of the action I, it is necessary to introduce a new field Akl μ = −Alk μ transforming as δAkl μ = ϵ k m Amlμ + ϵ l m Akmμ +

𝜕ϵ kl 𝜕uμ

(2.105)

Next, a covariant derivative ∇μ QA is introduced and a new Lagrangian where the ordinary derivatives in (2.101) are replaced covariant ones. Before leaving the details of Utiyama’s approach, let us quote in full once more. Because of the “general Lorentz transformation”, under which each local frame at each world point is transformed differently, the relation (4.5) was abandoned. Since this relation is satisfied only when the basic world is flat, we are forced to take as our basic space-time some Riemannian space with the metric gμν (u) = hkμ hkν and the affine connection Γμν λ =

1 λσ 𝜕gσμ 𝜕gσν 𝜕gμν g ( ν + − ) 2 𝜕u 𝜕uμ 𝜕uσ

Accordingly, we would expect that there exists some relationship between Akl μ and hkμ . [Our emphasis] Here, we will leave Utiyama, because now he is set on a path that will lead to Einstein gravity in vierbein formulation and the now well-known formulae. The connection he has introduced in the quote above is not really what we would nowadays introduce as a (general) affine connection, but rather the Christoffel symbol. In the subsequent development of Utiyama, he introduces another connection Γ′μν λ related to Γμν λ by the addition of what we now call torsion terms. In a footnote (no. 7), these terms are set to zero.

At one stage in the above reasoning (after equation (2.101)), one might pause and ask what would happen if one dispensed with the curvilinear coordinates, and proceeded to gauge the Lorentz transformations in just the same way as for Yang–Mills semi-simple groups? The Lorentz group is admittedly noncompact, but it is simple, so the procedure should work. However, in this case, there would be no Lorentz transformations of the coordinates (2.99a) and no frames or coframes (barring trivial constant ones simply amounting to identifying the μ and k indices). The spinor-tensor indices on the fields Q would Lorentz transform according to the appropriate finite representation of the Lorentz group. One would have local spin, but no local angular momentum. There would be no trace of gravity. So, curvilinear coordinates cannot be disposed of. The question is with what motivation to introduce them.

64 � 2 Notes on the history of higher spin interactions The place where Utiyama introduces the vierbeins, and the motivation for doing so, is in the first quote above. This is also the source of the criticism that there is no gauge motivation for their introduction. We also see, in the second quote, that not only vierbeins, but also Christoffel connections are introduced, ad hoc, as it were. In conclusion, the failure of Utiyama, if indeed it is a failure, shows the very slippery nature of the gauging procedure. In the modern fiber bundle approach to Yang– Mills theory, which is geometrical in its conception—rather than the algebraic nature of gauging—it is indeed quite hard to maintain a distinction between “gauging” and “parallel transport between fibers”. What is clear is that gravity is not a gauge theory of the homogeneous Lorentz group.

2.5.2 T. W. B. Kibble The Kibble paper [144] is central in the literature on gauge theory of gravitation and we will study it in detail. Indeed, it can be said to mark a watershed, together with the Sciama papers.62 Still it came quite early in the history of the subject.63 It is a complex paper, conceptually and technically, as we will see. To begin, the abstract of the Kibble paper is so well written and informative that it is helpful to quote it at full length. An argument leading from the Lorentz invariance of the Lagrangian to the introduction of the gravitational field is presented. Utiyama’s discussion is extended by considering the 10-parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables. It is then unnecessary to introduce a priori curvilinear coordinates or a Riemannian metric, and the new field variables introduced as a consequence of the argument include the vierμ bein components hk as well as the “local affine connection” Ai jμ . The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system. The free Lagrangian for the new fields is shown to be a function of two covariant quantities analogous to the Fμν for the electromagμ

netic field, and the simplest form is just the usual scalar density expressed in terms of hk and Ai jμ . This Lagrangian is of first order in the derivatives, and is the analog for the vierbein formalism of Palatini’s Lagrangian. In the absence of matter, it yields the familiar equations Rμν = 0 for empty space, but when matter is present there is a difference from the usual theory (first pointed out by Weyl), which arises from the fact that Ai jμ appears in the matter Lagrangian, so that the equaμ

tion of motion relating Ai jμ to hk is changed. In particular, this means that, although the covariant

derivative of the metric vanishes, the affine connection Γλ μν is nonsymmetric. The theory may be reexpressed in terms of the Christoffel connection, and in that case additional terms quadratic in the “spin density” S kij appear in the Lagrangian. These terms are almost certainly too small to make any experimentally detectable difference to the predictions of the usual theory.

62 As remarked by F. W. Hehl in a lecture series on the subject (National Center for Theoretical Sciences, Hsin-chu, Taiwan). 63 See Chapter 3 of [142] for contemporary work—with Kibble—on the subject.

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Kibble starts in Section 2 by considering Yang–Mills type “linear transformations”. This is essentially a summary (as Kibble notes) of Utiyama’s work “[. . .] although the emphasis is rather different, particularly with regard to the covariant and non-covariant conservation laws.”. Let us jump in at Section 3, which treats global Lorentz transformations. What Kibble did in his Section 3 Kibble works mainly with total variations denoted by δ, as opposed to local variations denoted by δ0 . I will instead use the notation Δ and δ, respectively (in accordance with our Section 3.14 of Volume 1), thus, to translate back to Kibble’s notation, do Δ → δ and δ → δ0 .64 In our notation, Kibble starts with x μ → x ′μ = x μ + Δx μ

(2.106a)

χ(x) → χ (x ) = χ(x) + Δχ(x)

(2.106b)

′

′

where the matter fields are denoted by χ. The relation between total and local transformations are given as δχ = χ ′ (x) − χ(x) = Δχ − Δx μ χ,μ ν

Δχ,μ = (Δχ),μ − (Δx ),μ χ,ν

(2.107a) (2.107b)

For the variation of the Lagrangian, Kibble writes ΔL =

𝜕L 𝜕L μ 𝜕L Δχ + Δχ + Δx 𝜕χ 𝜕χ,μ ,μ 𝜕x μ

(2.108)

thus including, for the time being, an explicit dependence on the coordinates. The action integral is invariant if ΔL + L(Δx μ ),μ = δL + (LΔx μ ),μ = 0,

(2.109)

which is a transformation law for an invariant density.65 Then Kibble specializes to constant Lorentz transformations Δx μ = ϵ μν x ν + ϵ μ

and

Δχ =

1 μν ϵ Sμν χ 2

(2.110)

No explicit Lorentz indices are used on the matter fields, instead the action of the Lorentz group on the “spin” indices is implicitly given by the multiplication by the representation matrices Sμν . It also follows, using (2.107b), that the derivative of the field transforms as

64 It is often convenient to work with local variations since they commute with derivatives, whereas the total variations obey formula (2.107b) involving the transport term. However, there is a downside to this choice since then δx = 0. This follows by consistency by applying formula (2.107a) to the coordinates themselves, and is also conceptually clear, as a coordinate cannot vary locally. It seems that one often makes an exception for x and writes δx = Δx, a habit that I also slipped into in my Volume 1 on page 215. 65 See formula 3.426 in our Volume 1.

66 � 2 Notes on the history of higher spin interactions

Δχ,μ =

1 μν ϵ Sμν χ,μ − ϵ ρμ χ,ρ 2

(2.111)

Now, since (Δx μ ),μ = 0 for constant Lorentz transformations, the equations (2.108) and (2.109) for the variation of the Lagrangian implies 10 identities, 4 corresponding to translational invariance with parameter μ ϵ μ and 6 to Lorentz invariance with parameter ϵ ν . Since Δχ does not involve ϵ μ , the translational identities imply that L has no explicit dependence on x. The Lorentz invariance identities are 𝜕L 𝜕L S χ+ (S χ + ημρ χ,σ − ημσ χ,ρ ) = 0 𝜕χ ρσ 𝜕χ,μ ρσ ,μ

(2.112)

These identities are assumed to be satisfied throughout. Kibble notes that they are the analogs of the corresponding identities for “linear transformations”, i. e., Yang–Mills of his Section 2. There is a difference though, that he does not comment on, in that the second term involves a contribution from orbital angular momentum. However, there is a related interesting remark in this context. After writing down the standard formulas for the conservation laws of energy-momentum and angular momentum, Kibble rewrites the transformation of χ in terms of local variations 1 ρσ δχ = −ϵ ρ 𝜕ρ χ + ϵ (Sρσ + xρ 𝜕σ − xσ 𝜕ρ ) 2

(2.113)

The role played by the matrices Ta (in the Yang–Mills case) is now played by the differential operators −𝜕ρ and (Sρσ + xρ 𝜕σ − xσ 𝜕ρ ). There is one more point to note. The conserved currents are stressed throughout Kibble’s paper, but the treatment is not entirely clear. The best thing for the reader is probably to consult Kibble’s paper directly. Here, we will just comment on a minor item concerning the identity (2.112). In Section 4 on generalized transformations, the coordinates are transformed Δx μ = ξ μ with no explicit Lorentz part. If that is done also for the rigid transformations, the orbital terms in (2.112) drop out. Still it is the identity (2.113) that is assumed to hold in the generalized case, since the appropriate covariant derivative is assumed to transform analogously to (2.111).

Thus, Kibble’s Section 3 ends with the observation that, as is well known, the generator of translations, are derivatives. Here, it appears in the guise of the transport term when comparing total and local transformations of the fields. Kibble’s Section 4 treats “generalized Lorentz transformations”, i. e., Poincaré transformations with varying parameters. This is where the “real action” starts, so to speak. The criticism of Utiyama’s approach was that he introduced curvilinear coordinates and the vierbein fields a priori, without any “gauge theory” motivation. Such a motivation is what Kibble now has to provide. Kibble does that in what can be divided into two steps, first a step where the basic set up of the theory in terms of transformations and coordinates is defined. Then a second step, itself divided into two stages, where the appropriate covariant derivatives are introduced. We start with the first step. It seems that any gauge theory of gravity must in some way set up an initial distinction between two sets of coordinates, or indices, without actually at the same time introducing too much, or all, of general relativity, as that would render the whole endeavor pointless. Let us see how Kibble approaches that problem.

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What Kibble did in his Section 4 (I) The (Poincaré) transformations (2.110) are generalized by letting the parameters become arbitrary functions of position. However, Kibble argues that rather than taking ϵ μν and ϵ μ as parameters, one should regard ϵ μν and ξ μ = ϵ μν x ν + ϵ μ

(2.114)

as independent functions. This avoids explicit appearance of x. One could then also consider transformations with ξ μ = 0 but nonzero ϵ μν . In this way, Kibble writes, coordinate and field transformations can be completely separated. This observation is then taken as motivation for introducing Latin indices for ϵ ij and for the matrices Sij and Greek indices for ξ μ and x μ .66 The transformations become Δx μ = ξ μ

Δχ =

1 ij ϵ S χ 2 ij

(2.115)

or 1 δχ = −ξ μ χ,μ + ϵ ij Sij χ 2

(2.116)

Kibble notes, as we have already remarked on above, that this notation emphasizes the similarity of the ϵ ij transformations to the “linear transformations” (Yang–Mills transformations), the only ones considered by Utiyama. Kibble then writes, “Evidently, the four functions ξ μ specify a general coordinate transformation.”. The last item in Kibble’s notational and conceptual setup concerns derivatives. The differential operator 𝜕μ must have a Greek index according to the just chosen convention of writing x μ . It would however be “inconvenient” Kibble writes to have two kinds of indices in the Lagrangian. Therefore, L is regarded as a given function of χ and χk satisfying the invariance identities from following from rigid translation and Lorentz μ invariance (see box above). The original Lagrangian is obtained by setting χk = δk χ,μ . Two comments are made by Kibble. (i) All other tensor components of χ must have Latin indices, as well as any Dirac gamma matrices occurring for spinors. (ii) There is no comma in χk . This is presumably in view of the fact that χk will μ not survive the subsequent development. The δk signifies that both sets of indices still denote flat coordinate systems.

Any reader has to judge for herself to what extent Kibble here succeeds in setting up a separation between two sets of coordinates, a. k. a. indices, without assuming too much of what is to be derived subsequently. Observe that nothing of this sort has to be done for Yang–Mills theory. The internal indices are cleanly separated form the space-time indices, i. e., the internal group space has nothing to do with space-time. This is so, an eventual reformulation in terms of fiber bundles notwithstanding. The fiber bundle language may help out here to underline the problem: the fibers in gravitation theory are just tangent spaces.67 Let us continue and analyze how Kibble reasoned concerning derivatives. The argument is quite delicate and focuses on the different nature of the two terms in the invariance requirement ΔL+L(Δx μ ),μ = 0 for the Lagrangian (see equation (2.109)). 66 Kibble writes “[. . .] it is convenient to use Latin indices [. . .].”. 67 This point is made in [145].

68 � 2 Notes on the history of higher spin interactions What Kibble did in his Section 4 (II) With position dependent parameters, one gets for the transformation of the derivatives of the field (from (2.115) using (2.107b)) Δχ,μ =

1 ij 1 ϵ S χ + ϵ ij S χ − ξ ν ,μ χ,ν 2 ij ,μ 2 ,μ ij

(2.117)

The original Lagrangian then transforms as 1 μ ΔL = −ξ ρ,μ J μ ρ − ϵ ij ,μ S ij 2 where S T

μ

ρ

=

μ

ij μ −J ρ

with J μ ρ = −

is the spin angular momentum current and J μ

μ

𝜕L χ 𝜕χ,μ ,ρ

and S

μ

ij

=−

𝜕L S χ 𝜕χ,μ ij

(2.118)

is related to the energy-momentum current via

ρ

− δ ρ L. The invariance requirement is however that

1 μ ΔL + L(Δx μ ),μ = −ξ ρ,μ T μ ρ − ϵ ij ,μ S ij = 0 2

(2.119)

Kibble remarks that the second term on the left-hand side, also occurring on the right-hand side via the relation between J and T , “[. . .] is of a different kind than those previously encountered, in that it involves L and not 𝜕L/𝜕χk . Thus, it is clear that we cannot remove it by replacing the derivative by a suitable covariant derivative.”. Kibble then proceeds in two stages: in the first obtaining ΔL = 0 and in the second obtaining ΔL + ij L(Δx μ ),μ = 0. The first stage itself is divided in two steps: first, treating the inhomogeneous ϵ ,μ term, and ρ μ μ then treating the ξ ,μ term (remember Δx = ξ ). For the first stage, a new Lagrangian L′ is introduced, obtained from the original L by replacing χk by a covariant derivative χ;k proclaimed to transform as Δχ;k =

1 μν ϵ Sμν χ;k − ϵ i k χ;i 2

(2.120)

The invariance of the Lagrangian ΔL′ then follows immediately from the identities (2.112) without calculation. This is so because the covariant derivative χ;k transforms just as the ordinary derivative χ,μ in (2.111) μ (remember the trivial correspondence χk = δk χ,μ ). In order to enforce the transformation law (2.120), Kibble introduces an intermediate covariant derivative denoted by χ|μ defined by 1 χ|μ = χ,μ + Aij μ Sij χ 2

(2.121)

ij

with 24 new field variables A μ . The covariant derivative χ|μ is devised so as to precisely compensate for the inhomogeneous term ϵ

ij

,μ

in the transformation of χ,μ . For this, Kibble imposes the condition Δχ|μ = ij

The transformation properties of A

μ

1 μν ϵ Sμν χ|μ − ξ ν ,μ χ|ν 2

(2.122)

can then be determined to be j

ΔAij μ = −ϵ ij ,μ + ϵ i k Akj μ + ϵ k Aik μ − ξ ν ,μ Aij ν

(2.123)

It now remains to handle the last term in (2.117), or equivalently in (2.122). Now, as Kibble notes, this term is already homogeneous as it contains a derivative of the field χ rather than just the field χ. Therefore, one should add to χ|μ , not a term in χ, but a term in χ|μ itself. In practice, this then amounts to a multiplication by

2.5 Gauge theories of gravitation

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a new field according to μ

χ;k = hk χ|μ

(2.124)

μ

The transformation of the field hk can now be determined from equations (2.120) and (2.122). It becomes μ

Δhk = ξ μ,ν hk ν − ϵ i k hi

μ

(2.125) μ

ij

So far, before field equations are considered, the 16 field components hk and the 24 field components A μ are unrelated. Now we have achieved ΔL′ = 0 for the new Lagrangian introduced above, but what we really need is μ ′ ΔL + L′ ξ ,μ = 0. Since this is the transformation law of an invariant density, Kibble introduces such a density μ L′ through the definition L′ ≡ HL′ where H itself is an invariant density so that ΔH + ξ ,μ H = 0. The only function of the new fields that obeys this transformation law, and does not involve derivatives, Kibble writes, μ μ is H = (det(hk ))−1 . It then follows that indeed ΔL′ + ξ ,μ L′ = 0. The final form for the gauged Lagrangian μ

ij

is then written by Kibble as L(χ, χμ , hk , A μ ) ≡ HL(χ, χ;k ).68

With this, the formal, kinematical step of gauging, or localizing, a matter field Lagrangian with respect to Poincaré transformations is complete. It is formal in the sense that the issue of interpretation has not been addressed. It is kinematical in that no dynamics for the new fields have been discussed. Let us take a preliminary look at the transformation laws so obtained. The first three ij terms in the transformation (2.123) of the field A μ show that it transforms as one would expect for a gauge field for the spin part of a localized Lorentz symmetry. The fourth term can be interpreted as a general coordinate transformation of a vector field if we think of the Greek indices as symbolizing a curved space-time. We are not yet quite μ there. The transformation law (2.125) for the field hk can be likewise interpreted. The first term again as a general coordinate transformation of a vector field, while the second is a Lorentz transformation.69 The rest of the Kibble paper treats the conserved currents and their appearance in the field equations, the action from which the field equations follow, and the geometrical interpretation of the objects in the theory—fields, their transformations and covariant derivatives. 2.5.3 D. W. Sciama The Sciama 1962 paper “The analogy between charge an spin in general relativity” [146], is written before but published after the Kibble paper. The two papers are complemen68 There is a discussion in the paper about the uniqueness of the gauge matter Lagrangian (it is not unique), but we refer the reader to the original paper (Section 4 and the Appendix) for this. 69 The transformation laws obtained here are the same, up to signs and index conventions, as those derived in our Volume 1, Section 4.6.1. For the reader’s convenience, the covariant derivatives correspond as follows: |μ ↔ Dμ and ; k ↔ Dk .

70 � 2 Notes on the history of higher spin interactions tary in that they arrive at the same results, but from different points of view. In contrast to Kibble, Sciama takes general relativity in the vierbein formulation as a given starting point, and “on top of that”, gauges the Lorentz group in a certain way. This leads to an asymmetric connection containing a torsion piece that couples to the spin current inside massive matter. However, we will stop here and refer the reader to the excellent reprint and commentary volume [142].70 This book contains a fair amount of follow-up research and literature on various gauge theory approaches of gravitation.

2.6 The Fronsdal program In his 1978 paper [148], Fronsdal explicitly proposed the generalized Gupta program of constructing interactions for higher spin massless fields. It seems that no one took up the challenge at the time—not until the Berends, Burgers and van Dam series of papers in the early mid-1980s—and the name did not stick to the program. When it was finally taken up by the modern higher spin community, it became known as the Fronsdal program.71

2.6.1 The early Fronsdal (generalized Gupta) program The period from 1978 to 1984 is interesting in higher spin theory. It actually began a few years before the Fronsdal and Fang–Fronsdal 1978 papers on free massless higher spin fields with the discovery of supergravity in 1976 and it ends with ascension of superstring theory in 1984 to the paradigm of theoretical-conjectural high energy physics. The positive interaction results in light-front and covariant Minkowski space-time of the early and mid-1980s did not receive attention, and neither did the late 1980s positive results in AdS space-time. The AdS Vasiliev theory was then developed almost single handedly by M. Vasiliev during the 1990s while the Minkowski space-time theory laid more or less dormant. This historical circumstance may not turn out have such a large effect on the subject in the long run, but it has produced a certain lopsidedness to the perception of it, as we have already noted. Be that as it may, let us return to the years 1978 to 1984. Workers in the supersymmetry community were interested in exploring the theory for massless fields beyond spin 2. This interest partly stemmed from the early realization that N = 8 supergravity with its 70 There is also another Sciama paper [147] from 1964 that is also interesting in the context. 71 Arguably, the light-front approach to massless higher spin, initiated in 1983, may be referred to the Fronsdal program, but I chose to refer that to a separate program, the Dirac program; see the Preface. For work on higher spin particles in the Soviet Union during the 1940s and 1950s, see an article by V. L. Ginzburg in I. A. Batalin, C. J. Isham and G. A. Vilkovisky (eds.), Quantum Field Theory and Quantum Statistics: Essays in Honour of the Sixtieth Birthday of E. S. Fradkin, Volume 2: Models of Field Theory.

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limitations to fields of spin ≤ 2 could not accommodate the standard model multiplets of fields (see Section 2.10.2 of Volume 1) in a simple way. Before the Fronsdal 1978 paper, there had been very few explicit forays into the massless terrain beyond spin 2. True, the mass term in the Dirac–Fierz–Pauli twocomponent spinor wave equations could be set to zero, but the consequences were not discussed. Wigner comments on the drop in the number of physical components when the mass term in set to zero, and Bargmann and Wigner discuss wave equations for the massless representations of the Poincaré group, but gauge invariance is not mentioned.72 As far as I am aware, the only explicit mention of higher spin gauge invariance is in H. Umezawa’s 1956 textbook [149] where the equations are given in the TT-gauge (see Section 2.4.5 in our Volume 1). Christian Fronsdal had been interested in both wave equations and representations of the Poincaré and AdS space-time isometry groups for a long time. As was natural for that period, the interest was grounded in a wish to understand the phenomenology uncovered in the high energy laboratories of the time. Then there was a shift of theoretical interest from massive fields to massless fields with the advent of supergravity. Furthermore, J. Schwinger had studied massless spin-3 fields in connection with his “source theory”.73 It must have been clear at the time, for workers interested in the subject that massless integer higher spin fields would be some kind of force fields rather than matter fields. I am however not aware of any explicit comments to the effect, apart from S. Weinberg’s S-matrix work from 1963. Not very much was written about higher spin interactions in the early papers of the mid to late 1970s, concerned as they were with developing the free field theory. But there was in some papers a last section on electromagnetic or gravitational interactions, or interactions with scalar fields or point particles. In Section 2.8 in our Volume 1, we reviewed parts of the Fang–Fronsdal deformation theory paper [150]. Apart from completing the Gupta program for spin 2, this paper also contains historical notes and references on the program. As for the higher spin Fronsdal program, very little is published by Fronsdal himself. Conference papers are sometimes a place to look for background thoughts behind research programs, and this is where we find the following quite interesting attempt. The Fronsdal 1979 conference paper In the conference proceedings [2], Fronsdal briefly outlines an attempt to set up a non-Abelian higher spin gauge algebra. In Section 4 of the paper, he writes “For spins exceeding 2, it would seem to be very difficult to find a non-Abelian gauge algebra without including all spins or at least all integer spins. Thus, the question calls for a single, unifying gauge algebra for all integer spins.”

72 See our Volume 1, Section 2.3. 73 J. Schwinger, Particles, Sources and Fields, three volumes.

72 � 2 Notes on the history of higher spin interactions

He considers a cotangent phase space over space-time with the usual symplectic structure and coordinates (x μ , πν ). Then he considers a set of traceless symmetric gauge parameters {ξ μ1 ⋅⋅⋅μs−1 }∞ s=0 and the corresponding set of gauge fields {ϕμ1 ⋅⋅⋅μs }∞ s=1 . These are collected into formal power series 1−s/2

∞

Ξ(π, x) = ∑ (π 2 ) s=0

πμ1 ⋅ ⋅ ⋅ πμs−1 ξ μ1 ⋅⋅⋅μs−1 1−s/2

∞

2 Φ(π, x) = π 2 + ∑ (π ) s=0

πμ1 ⋅ ⋅ ⋅ πμs ϕμ1 ⋅⋅⋅μs

(2.126) (2.127)

Transformations can now be calculated δΞ Φ = {Ξ, Φ} using the Poisson bracket {f , g} =

𝜕f 𝜕g 𝜕x μ 𝜕πμ

get the free theory transformations

−

(2.128)

𝜕f 𝜕g . 𝜕πμ 𝜕x μ

The presence of the π 2 term in H is needed in order to

∞

1−s/2

{Ξ, π 2 } = −2 ∑ (π 2 ) s=0

μ μ ⋅⋅⋅μ πμ1 ⋅ ⋅ ⋅ πμs 𝜕 1 ξ 2 s

(2.129)

Again, the Jacobi identity allows us to extract structure equations [δΛ , δΞ ]Φ = {Λ, {Ξ, Φ}} − {Ξ, {Λ, Φ}} = δ{Λ,Ξ} Φ

(2.130)

We get a field independent gauge Lie algebra. It is interesting to quote what Fronsdal writes about this construction. This, however, is a fake success. The trouble is that, since the massless fields are only double-traceless but not traceless, the “superfield” Φ does not determine its “components” and δΦ does not determine the variations δϕ uniquely. What is needed is a proper definition of a “superfield” that determines its double-traceless components. Nevertheless, this and similar attempts have tantalizing features and it is too early to give up on this line of search for the interacting theory.

We will return to the Fronsdal construction in Section 3.6.

2.6.2 The BBvD general analysis The first general analysis of the higher spin self-interaction problem was given in a paper [151] by F. A. Berends, G. J. H. Burgers and H. van Dam in 1985 (submitted in late December 1984). We will refer to this paper as BBvD-G henceforth. It came some time after their first paper [152] (BBvD-3) where they constructed the covariant spin-3 cubic interaction term. The thesis [153] by G. H. J. Burgers is also a good reference on the BBvD approach (denoted by BBvD-T here).

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Ingemar Bengtsson and I had both worked on spin-3 theory after our light-front work with Lars Brink. When the BBvD-3 paper arrived, we both realized that we had been too naive in our approaches. At least my ansatz was too restrictive, and neither of us had taken field redefinitions into account. The BBvD cubic interaction was consistent with our arbitrary spin light-front cubic interactions. Thus, if one had summarized the situation in 1984, one could have written: Cubic self-interactions exist for any single massless spin in a light-front formulation, and most likely also covariantly. However, the existence of a gauge algebra, and what form it would take, is unknown.74 After having secured the cubic interactions, this is indeed the next crucial question to ask: What is the gauge algebra? This was clear already from the Fang–Fronsdal paper on deformation theory [150], indeed even from early papers within the Gupta program for spin 2 (as reviewed in the Fang–Fronsdal paper). In BBvD-G, the authors, after having reviewed the free field theory of massless higher spin of Fronsdal, Curtright and de Witt–Freedman, set up an iterative approach to finding interactions and nonlinear gauge transformations. They first discuss the problem in a scattering theory context: Let us assume that asymptotic regions exist, where the particles participating in a collision are free. There are reasons to doubt the validity of this assumption when massless particles are involved, but let us not worry about that. Then in the asymptotic region we may use the free field theories. Such theories of the type [Fronsdal’s theory] described are gauge invariant because otherwise there is a conflict with Lorentz invariance. It is reasonable to assume that this asymptotic invariance induces gauge invariance of the full Lagrangian in the interaction region. In this way, one may search for the gauge-invariant full Lagrangian and for the gauge transformation of this Lagrangian.

The authors comment that learning from spin 1 and spin 2, we would expect the gauge transformations to be different from those of the free theory, and no longer commute. The searched for Lagrangian and gauge transformations are written as power series in a coupling constant g, L = L0 + gL1 + g 2 L2 + ⋅ ⋅ ⋅

(2.131) 2

ϕ → ϕ = ϕ + T(ϕ, ξ) = ϕ + 𝜕ξ + gT1 (ϕ, ξ) + g T2 (ϕ, ϕ, ξ) + ⋅ ⋅ ⋅ ′

(2.132)

Here, L0 denotes the quadratic free theory Lagrangian, L1 the cubic interaction, L2 the quartic interaction and so on. All terms in the transformation are linear in the parameter ξ. The inhomogeneous free theory gauge transformation is denoted by 𝜕ξ and T1 (ϕ, ξ) is linear in the field ϕ and its derivatives, T2 (ϕ, ϕ, ξ) is bilinear in the field ϕ and its derivatives, and so on. Rather than working directly with the gauge transformed Lagrangian, the authors consider the source constraint (see Section 3.14.5 in our Volume 1) implied by gauge invariance 74 The situation had improved, though, since Fronsdal wrote in the 1979 conference paper “[. . .] for s > 2 nothing is known.”.

74 � 2 Notes on the history of higher spin interactions 0 = B(ϕ, L,ϕ ) = 𝜕L,ϕ + gB1 (ϕ, L,ϕ ) + g 2 B2 (ϕ, ϕ, L,ϕ ) + ⋅ ⋅ ⋅

(2.133)

where L,ϕ stands for the field equations derived from L. Since the interaction terms in higher spin theory contain more than two derivatives, the Euler–Lagrange equations will contain more than the standard two terms.75 The Euler–Lagrange equations obtained from (2.131) are substituted into the source constraint (2.133) and terms are collected order by order. In the zeroth order of g, one then gets the source constraint of the free field theory 𝜕 ⋅ L0,ϕ = 0. To first order in g, one gets76 𝜕 ⋅ L1,ϕ + B1 (ϕ, L0,ϕ ) = 0

(2.134)

Rather than trying to solve this equation in one go, with ansätze for L1 and B1 , one may instead first solve 𝜕 ⋅ L1,ϕ = 0 for fields that satisfy the free field equations L0,ϕ = 0, and then determine the corresponding B1 term. Next, the corresponding first-order gauge transformation T1 can be determined form B1 .77 In performing these steps, there are two complications that must be accounted for. (i) In the first step, in determining L1 , one must exclude fake interactions that result from nonlinear field redefinitions ϕ → ϕ + gΦ(ϕ, ϕ) of the free Lagrangian. (ii) In the second step, there is an ambiguity in the determination of B1 and T1 corresponding to parameter redefinitions ξ → ξ + gΞ(ξ, ϕ). At this point, one may want to press on to the second order in the coupling g. However, as pointed out already by W. Wyss [154] in 1965, and stressed by Fang–Fronsdal in 1978, it may be possible to discern the gauge algebra already from commutator of the first-order gauge transformations. It is certainly so for spin-1 Yang–Mills theory and turned out to be so also for spin-2 gravity. This is also the strategy of BBvD-G. As this gets quite technical, we will defer the discussion to Section 4.2.1. One may ask about the level of generality of the BBvD analysis. As it stands, the formalism is not very well suited for a general approach to interactions among an infinite tower of higher spin fields. One can of course think of ϕ as denoting a multiplet of fields, possibly with different spins, as the authors indeed do when they derive all cubic couplings of fields with spin s1 -s2 -s3 with si ≤ 3, but then one has to descend to explicit ansätze in terms of tensors and spinor-tensors.

75 In this abstract formalism, derivatives are not explicitly indicated. Clearly, in any concrete implementation of the program this must be done in an ansatz. 76 I am clarifying the notation somewhat by writing 𝜕 ⋅ to indicate a divergence, distinguished from 𝜕 without the dot, denoting a symmetrized derivative. This is in accordance with modern condensed notation (see Section 5.1 in Volume 1). 77 How all this is done in practice will be outlined in Section 5.1.

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A few personal spin 3 (and higher) recollections After 37 years, it is difficult to patch together a clear recollection of what Ingemar Bengtsson and I did, together and separately, after the 1983 light-front papers. Judging from folders with calculations still kept, I spent a considerable amount of energy and time on trying to derive a covariant cubic spin-3 interaction. This was during the fall of 83 before writing up my thesis, and before the BBvD spin-3 paper arrived. Needless to say, I failed. I then did something which was a bit easier. I studied what I considered to be the most likely, but fairly general, gauge algebra for spin 3 inspired by the Fang–Fronsdal deformation theory paper. I managed to show that no such algebra was likely to exist involving only spin-3 fields [155]. From correspondence with Ingemar, I know that he also worked on covariant theory of higher spin around this time. But then the BBvD-G paper arrived, showing that we both had been too naive in our attempts. After my thesis in the spring of 1984, I did not do much research until I arrived at Queen Mary College in London in the fall of 1985. Ingemar and I then took up higher spin again and wrote two papers. One paper was on 1- and 2-dimensional toy models of higher spin [156]. The other paper was a renewed study of tentative higher spin gauge algebras [157]. This work was partly inspired by the Fang–Fronsdal deformation theoretic paper and by the Fronsdal conference report mentioned above. The paper also contains a proof that a pure quartic spin-3 interaction does not exist. After this, I went on to do the BRST approach to free and interacting higher spin theory in Minkowski space-time, as well as returning to the light-front theory with I. Bengtsson and with N. Linden.

2.6.3 The Fronsdal program in the 1980s after BBvD As it happened, the next concrete development in the Fronsdal–BBvD program came a few years later within the string theory inspired BRST approach. Remember, in the mid-1980s we are in the high tide of superstring theory, one vigorous branch of which was string field theory, in particular the quest for a covariant string field theory. As we described in Section 2.11 of our Volume 1, this was done using the “mechanics to field theory” algorithm. The first attempt at a covariant BRST approach to higher spin gauge interactions was a paper [158] by I. G. Koh and S. Ouvry soon after the BRST approach to the free field theory had been discovered. The string theory inspired cubic vertex of this paper did not reproduce the spin-1 Yang–Mills interaction for reasons explained in Section 5.2.2 where the paper will be reviewed. The first BRST approach that actually produced the correct Yang–Mills cubic interaction was found by the present author [159] in 1987. The paper was based on an insight gained from the corresponding light-front approach [160] with I. Bengtsson and N. Linden. There it was found that the crucial factors in the vertex operators involved products of three oscillators and one momenta, in distinction to the case of string theory, where the vertex operators involved separate factors with products of two oscillators and one oscillator and one momenta, respectively. This paper and the approach itself will be reviewed in Section 5.2.3. Also in the late 1980s and early 1990s, general aspects of the BRST approach to higher spin gauge interactions, were investigated in a few papers. Gravitational interactions

76 � 2 Notes on the history of higher spin interactions were investigated by Koh and Ouvry in [161] reproducing the known problems with higher spin gauge fields in gravitational backgrounds. In [162], Witten-type cubic BRSTinvariant vertices—of the Koh and Ouvry type—were studied. Then in [163], more general polynomial BRST-invariant theories were considered. These two latter references did not, however, result in more concrete vertex solutions. We will have a few more comments on these papers in Section 5.2.4. For the subsequent history of the Fronsdal–BBvD program, see the following Sections 4.2, 4.4, 5.1, 5.3 and 5.4.

2.7 The nonlocality circle of problems Locality seems not to have been discussed much in the early days of interacting higher spin gauge field theory. True, the high number of derivatives, expected and found, in the cubic self-interactions, was noted. But for any finite spin, the number of derivatives was finite. Furthermore, the kinetic term in the action was quadratic and the higherorder time derivatives in the cubic interactions were not thought to produce problems with unitarity. This was indeed the case in the light-front approach where the cubic interactions did not contain any light-front time derivatives at all. The issue came to the fore with work by X. Bekaert and N. Boulanger in various collaborations with S. Leclercq, S. Cnockaert and P. Sundell in the period 2005–2010 (to be reviewed in Section 5.4), building on earlier general work by G. Barnich, F. Brandt and M. Henneaux, using the BRST antifield method emphasizing the importance of locality. Without locality, the master action can be solved to arbitrary order, not necessarily so with locality imposed, as explained by G. Barnich and M. Henneaux (to be reviewed in Section 4.3.7). With locality imposed, obstructions tend to occur already at the quartic level, the level probed by Jacobi identities for the would be higher spin gauge algebra. This work was a sign of a new generation of workers entering the field of higher spin gauge theory. While the Fronsdal and Dirac programs had stalled in the early 1990s, the Vasiliev program was pursued vigorously by M. Vasiliev. There existed no higher spin research community at the time. As far as I know, no one worked on the Fronsdal program, and the only active worker in the Dirac program was R. Metsaev extending the fourdimensional light-front cubic interactions to more general contexts such as higher dimensions, massive fields and fermions. This work did not really push the program forward. However, Metsaev had also, in the very early 1990s, analyzed the light-front quartic order (to be reviewed in Section 6.5.1). That work hinted at serious nonlocalities at the quartic order, but also at a result that the theory could be truncated to cubic order, thus circumventing the non-locality. It took 25 years for these, rather implicit insights, to resurface within the higher spin research community. In the very late 1990s, the Vasiliev theory also attracted the attention of the new generation of researchers, to begin with by E. Sezgin and P. Sundell. This lead to clarifications and simplifications of the theory, in particular to a focus on the simplest four-

2.7 The nonlocality circle of problems

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dimensional purely bosonic theory, where it was feasible to expand the equations to some orders and study concrete interactions. Furthermore, then, at that time recent, AdS/CFT duality (Maldacena conjecture), added to the interest. AdS space-time has a boundary at space-like infinity, so one may talk about a “bulk” and a “boundary” and dualities between quantum field theories in the bulk and conformal field theories on the boundary. One initial strong force behind the construction of the Vasiliev theory was the hope to circumvent the no-go theorems in Minkowski space-time, in particular, the Aragone–Deser problem. The way the Vasiliev theory evades the Aragone–Deser problem is briefly described in the review [164]. In this review, from the early 2000s, it was also written that the AdS theory evades the Coleman–Mandula theorem since there is no S-matrix in AdS space-time. However, it soon became clear that the very AdS/CFT duality—that added impetus to AdS higher spin theory—a bit ironically, also transferred the Weinberg and Coleman–Mandula no-go from Minkowski space-time to AdS space-time in an interesting way. One may indeed ask why there were no no-go theorems for AdS space-time? The answer is that no-one was interested, with the exception of Fronsdal and some co-workers. The early no-go theorems of O’Raifeartaigh and Coleman–Mandula were concerned with the problems in the particle physics of their times. The Weinberg theorem was indirectly connected to such concerns. The Weinberg–Witten theorem was a response to proposals for composite gravity. The Aragone–Deser analysis was within the supergravity unification program. However, in step with people becoming interested in the Vasiliev theory, it was soon discovered that AdS higher spin theories suffered from analogous problems. It turned out that the Weinberg and Coleman–Mandula no-go results had their AdS counterpart. There is no S-matrix in AdS space-time, but the very AdS/CFT duality provides an analogous result. Higher spin gauge theory in the AdS bulk, of the type of the Vasiliev theory, is dual to a free (or critical) conformal field theory on the boundary. This may be seen as the AdS analogue to S = 1 in Minkowski space-time. It also became clear that there was no control of locality in the way the Vasiliev theory was constructed. The theory works by adding infinite chains of supplementary fields that are derivatives of arbitrarily high order of the physical fields. These derivatives of fields are treated as independent degrees of freedom, to be reduced by the equations of the theory. The procedure is called “unfolding”. In this way, the problem of constructing interactions seems to become an algebraic problem. But there is no control of locality. Subsequently, actual nonlocalities were discovered in the theory, even at the cubic order. A few inroad references, supporting the picture painted here, are [165, 166, 167, 168, 169, 170]. The work is highly technical, and is beyond the reach of the present book. Hopefully, the following chapters will be helpful as an introduction to further studies.

3 Algebraic structures in higher spin theory Many authors in modern theoretical physics—not just higher spin theorists—with the inclination to approach questions of the structure of fundamental theory stress the geometrical nature of the subject. Still, when it gets to actually manipulate the objects and processes of the theory or model, it comes very much down to algebra. Of course, these two aspects are intrinsically related in that, for instance, the theory of continuous transformation groups is the theory of Lie groups and, therefore, the theory of local transformations is captured by the theory of Lie algebras.1 Many algebraic concepts crop up time and again in higher spin theory such as involutions, conjugations, gradings and filtrations, derivations and differentials, cohomology and relations between associative algebras and Lie algebras. In this chapter, we will outline a few such aspects of algebra that are relevant for the interactions. Furthermore, particular algebras such as the Heisenberg and Weyl algebras and the closely related concepts of star products and universal enveloping algebras, are very prominent in higher spin theory. These concepts are, in my opinion, often introduced in a little bit too mathematically slanted way in the extant higher spin literature. What the practicing higher spin theorist often needs is intuition and concrete calculation methods. That will be the point of view in the present chapter. It can be read as a companion chapter to Chapter 3 in our Volume 1 upon which it builds and elaborates. We will start with something quite concrete, simple operations performed over algebras.2

3.1 Abstract algebras and operations on algebras First, remember that an algebra is a vector space, complex or real, upon which there is defined a binary multiplication operation that distributes over the addition of the vector space in the natural way. The axioms may be written in a few alternative ways—one way is given in Section 3.7.7 in our Volume 1—here we will use slightly more concrete defining equations. The elements of a general algebra A will be denoted by a1 , a2 , . . . while the real or complex numbers over which the underlying vector space is defined will be denoted by λ1 , λ2 , . . . . The product a1 ⋅ a2 of two elements in the algebra is assumed to satisfy (λ1 a1 + λ2 a2 ) ⋅ a3 = λ1 a1 ⋅ a3 + λ2 a2 ⋅ a3

a3 ⋅ (λ1 a1 + λ2 a2 ) = λ1 a3 ⋅ a1 + λ2 a3 ⋅ a2

(3.1)

1 R. Penrose’s Chapter 13 in [171] gives a nice intuitive introduction to this as well as refers to the basic theorems underlying it. In our Volume 1, Section 3.11, we noted that a Lie group is also a manifold. 2 This first section is partly inspired by Section 2 of the Vasiliev paper [172]. We will therefore, in particular, have occasion to use this when reviewing the Vasiliev theory in Chapter 8. https://doi.org/10.1515/9783110675528-003

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These equations essentially amounts to distributive laws, but note that the equations also encode how the multiplication by the scalars interact with the algebra multiplication. Of course, all the basic equations for the underlying vector space are assumed to hold. Nothing is as yet assumed regarding associativity or commutativity of the algebra multiplication. Consider now an arbitrary but given set {ej } of basis elements for the vector space. For a concrete given algebra, we expect to have relations ei ⋅ ej = uij k ek

(3.2)

where uij k are the structure coefficients of the algebra. Basic equations such as these are the ones that allow us actually compute in the algebra, for instance, making “simplifications”. Associativity, when it is present, amounts to certain conditions on the structure coefficients. If there are no relations of the type (3.2), the algebra is said to be free. In some free algebras, the multiplication may be thought of as just concatenation of the elements of the algebra (see Section 3.7.12 in our Volume 1).3 Since arbitrarily long words can be made, a free algebra is necessarily countable infinite-dimensional. An algebra is said to be unital if it possesses a unit element under multiplication. Before continuing, let me comment on a notational issue. Up until now, we have used a dot ⋅ to denote the multiplication in the algebra. I will continue to write a⋅b or just ab for the product of two elements in a generic algebra, trying to state explicitly whether the product is to be associative or not. For abstract Lie algebras, not deriving from an associative algebra, the product will be denoted by a bracket [⋅ , ⋅]. This comment holds also in cases of graded algebras and when the elements carry parities, a circumstance that we now come to.

3.1.1 The concepts of Grassmann parity and Z2 grading Let us start with a potentially confusing issue. A potentially confusing issue Graded algebras, in particular Z2 graded, are common in theoretical fundamental physics. In an associative algebra, such a grading can be brought to the fore by introducing parities π(x) for the elements x taking values

3 Think of stringing together the letters of an alphabet into words. Note, however, that such free algebras are necessarily associative since word concatenation associates by its very nature. To get a nonassociative free algebra, parentheses can be introduced into the words. Such structures are called free magmas. They can be represented by binary trees and are relevant for programming language syntax. Terminology varies in this area of abstract algebra.

80 � 3 Algebraic structures in higher spin theory

0 and 1 for even and odd elements, respectively. The grading is then captured by writing x ⋅y = (−1)π(x)π(y) y ⋅x. All is well so far. However, if we now want to construct a Lie algebra by introducing commutators in the standard way for the even elements (and anticommutators for the odd elements) we find that the Lie algebra is necessarily Abelian.

The problem is to reconcile a noncommutative product law (3.2) with a definition of grading of the elements. First, for a Z2 graded algebra, the elements a come in two kinds—even and odd— conveniently labelled by 0 and 1. This will be called the Grassmann parity π(a) of the elements, often shorted to parity where no confusion with other “parities” can occur. The parity is indeed a Z2 number set. Referring to the product law (3.2), we now define the parities for the basis elements by the following condition on the structure coefficients: uij k (1 − (−1)π(ei )+π(ej )+π(ek ) ) = 0

(3.3)

The definition may be usefully explicated as in Table 3.1 where we see that the definition works as expected. Table 3.1: Grassmann parity assignments and associative algebra structure coefficients. The standard bracket notation for super-Lie algebras is also recorded in the cases when it may be nonzero. π(ei )

π(ej )

π(ek )

structure coefficient k

− uji − uji − uji + uji

0 1 0 1

0 0 1 1

0 1 1 0

uij uij k uij k uij k

0 1 0 1

0 0 1 1

1 0 0 1

zero zero zero zero

k k k k

nonzero nonzero nonzero nonzero

standard super-Lie bracket [⋅, [⋅, [⋅, {⋅,

⋅] ⋅] ⋅] ⋅}

Nothing is implicitly assumed about the symmetry or antisymmetry of the structure coefficients, and thus non-Abelian Lie algebra anticommutators and commutators may be introduced in the standard fashion.4 It remains to note that definite parities can be assigned to general elements in the algebra that are linear combinations of basis elements of definite parities.

4 Of course, commutative or anticommutative associative algebras are also covered by this definition of Grassmann parity.

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One may also note that when elements of an algebra carry parities of various sorts, concepts and formulas that we are going to outline below, in general need to be adopted to such circumstances. Rather than trying to write formulas covering “all” possibilities, it is in my opinion better to sort such things out when the need arises.

3.1.2 Associative algebras Associativity is a very natural, commonly occurring and important property for algebras. In simple terms, it means that a(bc) = (ab)c. Without associativity, one would have to be careful with parentheses throughout all computations. It is probably safe to say that almost all algebraic structures that we will encounter in higher spin theory are associative; with one important exception, which is differential operators, where we have to be careful with the “scope” of the derivatives.

3.1.3 Lie algebras Lie algebras may be thought of, and often occur, independently of any underlying associative algebra. Then the commutators, and anticommutators in the case of super-Lie algebras, satisfy basic axioms by themselves, and need not be related back to any product law in an associative algebra. Indeed, the Lie bracket is the product in the Lie algebra. It then satisfies “skew symmetry” [x, y] = −[y, x] and the Jacobi identity (which in a sense replaces the associative law of the associative algebras). The corresponding statements hold in the case of super-Lie algebras. Let us for the sake of exactness state the equations. An algebra is a super-Lie algebra if it is Z2 graded and if for any elements a, b and c having definite Grassmann parities, the following equations hold for the product written as [ ⋅ , ⋅ }: [a, b} = −(−1)π(a)π(b) [b, a} (−1)π(a)π(c) [[a, b}, c} + (−1)π(b)π(a) [[b, c}, a} + (−1)π(c)π(b) [[c, a}, b} = 0

(3.4a) (3.4b)

The bracket notation, as it is written in the concrete cases, is recorded also in Table 3.1. When the super-Lie algebra is constructed based on an underlying associative algebra, then the brackets are defined in the well-known way as commutators for even–even and even–odd combinations of elements and anticommutators for odd–odd combinations. This could be referred to as a commutator Lie algebra.5 The inverse problem, of

5 Following [173], we will always reserve the word “commutator” for an explicit definition of a Lie algebra bracket defined from an associative algebra product.

82 � 3 Algebraic structures in higher spin theory constructing an associative algebra from a Lie algebra, is much more difficult and interesting, leading to the concept of universal enveloping algebras. It will be treated in Section 3.4. As for notation, we will try to be consistent in using notation like a (lowercase fractured font) for Lie algebras and 𝒜 (uppercase calligraphic font) for associative algebras. For a Lie algebra that explicitly derives from an associative algebra, we will write [𝒜]. Perhaps a note on the special case of commutative and supercommutative algebras may be clarifying. Then the only fact we know about the product of two elements a and b is that ba = (−1)π(a)π(b) ab. This is the algebra of the basic variables of classical mechanics and classical field theory, enhanced with odd variables. In field theory, we multiply fields, perhaps without thinking that we are doing algebra when we do simple rearrangements like fabc ϕaμ (x)ϕbν (x) = fabc ϕbν (x)ϕaμ (x). Of course, nonlinear transformations involving terms of the form fabc ϕaμ (x)ξ b (x) are noncommutative in general. Sens moral: The underlying variables we work with live in (super)commutative algebras, the transformations they suffer live in non-commutative algebras in general.

3.1.4 Structure preserving maps and representations of algebras In theoretical physics, we most often work with representations of the algebras rather than the abstract algebras themselves. In generic terms, we think of the elements of the algebra to act on some space, thus performing transformations on the elements of the space. Since algebras are also vector spaces, the structure preserving maps that serve as morphisms (also called homomorphisms) between algebras are therefore naturally linear maps. For structure preserving maps serving as representations, one may use more general kinds of maps such as affine maps. Let us capture these words in some definitions. Let 𝒜 and ℬ be two generic (associative) algebras. An algebra homomorphism from 𝒜 to ℬ is a linear map L:𝒜→ℬ

such that

L(x ⋅ y) = L(x) ⋅ L(y) for all x, y ∈ 𝒜

(3.5)

where, for simplicity, the same multiplication sign is used in both algebras. For representations, let us add a generic vector space V to the brew. Thinking of an algebra 𝒜 as acting on the vector space V , moving vectors around, it is natural to consider endomorphisms (morphisms to the same set) End(V ) in the vector space. The definition of a linear representation RV then reads RV : 𝒜 → End(V )

(3.6)

the arrow → denoting a structure preserving map. The vector space V is called the representation space or often an 𝒜-module.

3.1 Abstract algebras and operations on algebras � 83

Now all this simply serves to fix ideas, and is quite natural. What is interesting is the concrete actual representations that we can construct for algebras of interest. This involves choosing vector spaces and concrete implementations of the action of algebra elements a on vectors v. Abstractedly, one can write av, but that does not get us very far. We need to set it up so that we can actually compute the action of algebra elements on vectors, but such examples we have already seen. A nontrivial example is the action of the Poincaré algebra on particle states that we reviewed at length in our Volume 1, Section 3.5. An interesting set of algebra representations are the oscillator representation of algebras on Fock spaces. We will see examples of this, for instance, in Section 7.1.5.

3.1.5 Automorphisms, conjugations and involutions An automorphism of an algebra is a structure preserving map to the algebra itself. Denoting the automorphism by τ, its structure preserving properties can be captured by the equations6 τ(λ1 a1 + λ2 a2 ) = λ1 τ(a1 ) + λ2 τ(a2 ) τ(a1 ⋅ a2 ) = τ(a1 ) ⋅ τ(a2 )

(3.7)

It is implicit in using the language of morphisms7 that τ is invertible. This means that it is an isomorphism. If iterating τ twice gives the identity mapping, i. e., if τ 2 = I, then τ is said to be involutive. Concretely, τ(τ(a)) = a for elements a in the algebra. Automorphisms may generate subalgebras. Namely, if for a certain automorphism τ there is a set of elements for which τ(a) = a, then these elements form a subalgebra. Complex and Hermitean conjugation are very important operations in quantum theory. We can take the opportunity to capture their properties slightly more generally. Note that they are not automorphisms for complex algebras. A conjugation σ of an algebra is a mapping of the algebra onto itself defined by the following three properties: σ(λ1 a1 + λ2 a2 ) = λ̄1 σ(a1 ) + λ̄2 σ(a2 ) σ(a1 ⋅ a2 ) = σ(a1 ) ⋅ σ(a2 ) σ(σ(a)) = a

(3.8a) (3.8b) (3.8c)

The distinguishing property here is the second one, the preservation of the order of the operators. This we recognize from standard complex conjugation.

6 Here, the algebraic operations are actually the same on both sides of the equations. 7 See Section 3.7.1 of our Volume 1 for general morphisms.

84 � 3 Algebraic structures in higher spin theory A involution μ of an algebra is a mapping of the algebra onto itself defined by the following three properties: μ(λ1 a1 + λ2 a2 ) = λ̄1 μ(a1 ) + λ̄2 μ(a2 ) μ(a1 ⋅ a2 ) = μ(a2 ) ⋅ μ(a1 ) μ(μ(a)) = a

(3.9a) (3.9b) (3.9c)

The distinguishing property is again the second one, in this case the inversion of the order of the operators. This is typically what characterizes the standard Hermitian conjugation of quantum mechanics. Automorphisms and Z2 gradings are related in the following way. If there is a Z2 grading π, then one can define a simple automorphism τ with the property τ(a) = (−1)π(a) a, compatible with the product law of the algebra. This is clear from Table 3.1. On the other hand, given an involutive automorphism τ, consider the eigenvector equation τ(b) = λb b. Since the automorphism is assumed to be involutive it follows that λ2b = 1 so that λb = ±1. Then one may choose parities according to (−1)π(a) = λa . 3.1.6 Real and complex algebras The relation between the complex and the real forms of an algebra is a potentially confusing subject. To fix ideas, let us focus on matrix Lie algebras. One source of confusion is the fact that the basis generators of a such Lie algebra in general are given as matrices with complex, or at least imaginary, entries. Still one may speak about a real Lie algebra, or a Lie algebra over R, when the coefficients used for linear combinations of the generators are real. Likewise, we have a complex Lie algebra, or a Lie algebra over C, when the coefficients used for linear combinations of the generators are complex. Now, the basis generators of a given Lie algebra may be chosen in different ways and it is still the same Lie algebra. Once the complex numbers are brought in as coefficients of linear combinations, there is only one complex Lie algebra. There are, however, many real forms of the same complex Lie algebra.8 A good example to help keeping the thinking straight here, is the angular momentum Lie algebra [Li , Lj ] = i ∑3k=1 ϵijk Lk with Hermitian generators. First, there is the fact that there are representations of the generators corresponding to different spin, which translates into the rank of the generating matrices. Second, the generators may be linearly combined into a raising and lowering operator of the form L± = L1 ± iL2 and L0 = 2L3 with L± Hermitian conjugates of each other. Third, working over the complex numbers we have the algebra sl(2, C) no matter what form the generators are given in.

8 See Section 8.4 of [174].

3.1 Abstract algebras and operations on algebras � 85

However, working over the real numbers and using the raising and lowering form of the generators, we have the real form sl(2, R) (with real structure constants). This is not the same real Lie algebra as the real form su(2) using iLi as generators (and imaginary structure constants).9

3.1.7 Graded and filtered algebras If the concept of graded algebras can be likened to sliced bread, the concept of filtered algebras can perhaps be likened to the layers of an onion. Remember that in a graded algebra 𝒜, the elements can be sorted into disjoint subsets 𝒜i according to some index i, running over some index set I, often Z or N and one writes 𝒜 = ⨁ 𝒜i i∈I

with the unit element 1 ∈ 𝒜0

(3.10)

This is actually a definition that belongs to the underlying vector space.10 What is algebraically interesting is how the grading “interacts” with the product ⋅ in the algebra. For this, we have 𝒜i ⋅ 𝒜j ⊆ 𝒜i+j

(3.11)

The role model is monomials in one variable. For a filtered algebra, the role model is polynomials in one variable. Thus, the subsets 𝒜i are not disjoint, but rather satisfies 𝒜i ⊆ 𝒜i+1

(3.12)

with (3.11) still holding for the product. Gradation is a stronger requirement on an algebra than filtration, but can be obtained by a process of dividing out the lower than leading terms (we restrict our attention here to N for simplicity). Let 𝒜 be a filtered algebra and denote by gr(𝒜) the associated graded algebra and likewise its disjoint subsets by gri (𝒜). Then we would have gr(𝒜) = ⨁ gri (𝒜) where gri (𝒜) = 𝒜i /𝒜i−1 and i ≠ 0 i∈N

Conversely, any graded algebra 𝒜 = ⨁i∈N 𝒜i has a filtration ℱn given by 9 See also Section 3.11.2 in our Volume 1. 10 See Section 3.7.8 in our Volume 1.

(3.13)

86 � 3 Algebraic structures in higher spin theory n

ℱ n = ⨁ 𝒜i i=0

(3.14)

Filtered algebras naturally appear in the context of higher-order differential operators and Weyl algebras that we will have occasion to discuss below. Note also that when working with morphisms between filtered or graded algebras, in particular with automorphisms, one must demand that the filtration/gradation is also preserved—since it is part of the structure of the algebra.

3.1.8 Derivations and differentials A derivation D on an algebra is any linear operator that satisfies the Leibniz rule on products. When the derivation itself, and the elements that it acts on carries a Grassmann parity, this means the graded Leibniz rule D(fg) = (Df )g + (−1)π(D)π(f ) fDg

(3.15)

Here, the derivation acts from the left toward the right (as in calculus).11 An example of a derivation in a Lie algebra is the adjoint action. If x and y are elements in the Lie algebra, then the adjoint action of x is defined and computed as adx y = [x, y]

(3.16)

The Leibniz rule, as well as other required properties of a derivation, is satisfied through the properties of the Lie bracket. This is an example of an inner derivation. The Poisson bracket of mechanics also defines an inner derivation in a similar fashion. See Section 3.2.1 below. It is important to realize that the concept of a derivation is broader than the usual derivative of calculus. It may be put into the context of linear transformations on a vector space. These are endomorphisms (see Section 3.1.4) and forms a unital associative algebra.12 One may study the commutator of two such transformations which satisfies the Jacobi identity and, therefore, forms a Lie algebra. If the linear transformation in addition obeys the Leibniz rule, it is a derivation. If the vector space has a well-defined grading, the derivation may “interact” with the grading, for instance, by changing the grade ±1.

11 Some authors, in particular, the important textbook [175], adopts a convention of “right action”. The formula may then look a bit “unfamiliar”. See Section 8.2.3 of that book. 12 There is a “unit” transformation, and they can be composed associatively.

3.1 Abstract algebras and operations on algebras

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Example 1 (A very simple example). Consider the countable infinite-dimensional vector space of all polynomials in one variable x. This space is N-filtered by the degree of the polynomials. The ordinary derivative d/dx is a derivation of this algebra. 󳶣 Another example is the exterior derivative d of the exterior algebra of p-forms on a manifold. This derivation increases the form degree by 1. A differential Δ is an odd (π(Δ) = 1) derivation that is nilpotent in the sense 1 Δ2 = {Δ, Δ} = 0 2

(3.17)

This leads immediately over to the concept of cohomology algebra.

3.1.9 Cohomology and homology The concept of homology has its origin in topology in the classification of manifolds according to their topological structure.13 From there derives the language of chain complexes, cycles and boundaries. These concepts have a quite clear geometric interpretation as submanifolds. Cohomology, although more general than so, is in a certain sense dual to homology, working with functions defined on the submanifolds. There are many different homology and cohomology theories, and the reader should be aware of the fact that this is a huge advanced area of mathematics that has found applications in areas far removed from the initial geometric/topological origins.14 The differential cohomology theory most familiar to the general physicist is probably the de Rahm cohomology of differential forms, which can be seen as a distillation of classical mathematical physics of fields [177]. In that case, the differential is the exterior derivative d.15 An example of a cohomology theory familiar to theoretical physicists is the BRST theory where the differential is the BRST charge Q. The crucial question is what kinds of spaces nilpotent operators like these act in. The basic theory may be developed in very general terms.16 We will opt for a context that is general enough for our purposes in higher spin theory. For that, we will work in associative supercommutative algebras. These are freely generated algebras based on vector spaces containing even and odd elements. We will think of finite-dimensional examples—as in mechanics—such as having even elements

13 In three space dimensions, one may think of the distinctions between spheres, toruses and Klein bottles. 14 For an advanced history of algebraic and differential topology that clearly shows the wide range of the subject, see [176]. 15 See Section 3.8.3 of our Volume 1. For a thorough exposition, see [178]. 16 One reason why there are so many cohomology theories.

88 � 3 Algebraic structures in higher spin theory xi with i = 1, . . . m and odd elements ck with k = 1, . . . n satisfying the relations xi xj = xj xi , xi ck = ck xi and ck cl = −cl ck . General elements of such algebras are arbitrary, real or complex, polynomials in the variables xi and ck . Occasionally, notation such as R[xi , ck ] and C[xi , ck ] may be used for such algebras.17 Closer to higher spin field theory, we will also think of infinite-dimensional examples with freely generated supercommutative algebras of C ∞ functions defined on manifolds M. Here, we most often combine features of the finite-dimensional example with functions carrying indices valued in various vector spaces and algebras (in the latter case we actually leave the realm of supercommutative algebras). Most importantly, in higher spin theory we have countably infinite towers of fields with increasing spin.18 Finally, it is assumed that the spaces 𝒜 that we are working in are graded, either N-graded or Z-graded. The differential Δ is assumed to act either as Δ𝒜i ⊆ 𝒜i+1 or Δ𝒜i ⊆ 𝒜i−1 . In the first case, the differential is assigned a degree +1 and called a coboundary operator (cohomology) and in the second case, a degree −1 and called a boundary operator (homology). In the cohomology case, this results in a cochain complex or differential complex Δ

Δ

Δ

Δ

⋅ ⋅ ⋅ → 𝒜i−1 → 𝒜i → 𝒜i+1 → ⋅ ⋅ ⋅

(3.18)

and correspondingly with a “decreasing” chain complex in the homology case. A graded algebra with a differential is called a graded differential algebra. When the grading is of the N type, the sequence “stops” or “starts” at one end. For the de Rahm cohomology of differential forms, the complex is finite. It is the nilpotency of the differential that makes this interesting. As the notation suggests, the differential Δ is a map (an endomorphism) in the graded space 𝒜. One may therefore consider the image and kernel of Δ. The image is defined as x ∈ Im(Δ) ⇔ ∃y ∈ 𝒜 : x = Δy

(3.19)

Simply put, the image of Δ is the subspace of those elements that can be written as the differential of some element in 𝒜. The kernel is defined as x ∈ Ker(Δ) ⇔ Δx = 0

(3.20)

Simply put, the kernel of Δ is the subspace of those elements that are mapped to zero by the differential. Such elements are said to be closed. 17 In mechanics, the xi would include positions and momenta, while the ck would be the associated ghost coordinates and momenta. 18 In mathematics, one must of course be pedantic with definitions here, but for our purposes we prefer to be a little vague and supply the proper definitions in the particular circumstances that arise in higher spin theory.

3.1 Abstract algebras and operations on algebras � 89

The intuition is now that it is the elements in the kernel that are interesting for some reason. The equation Δx = 0 may, for instance, represent highly abstracted field equations. Since the differential is nilpotent, the elements in the image satisfies the equation Δx = 0 by default, and may therefore be considered as of secondary importance. Such elements are called exact. It then makes more sense to work with equivalence classes of solutions to Δx = 0, and so we arrive at a study of the cohomology of H ⋆ (Δ) defined as the quotient algebra H ⋆ (Δ) ≡ Ker(Δ)/ Im(Δ)

(3.21)

Now suppose a certain x in Ker(Δ) belongs to 𝒜k . Then it is mapped to zero in 𝒜k+1 . The corresponding equivalence class, denoted by H k (Δ), results from dividing out the elements x that are Δ-images of elements is 𝒜k−1 . We see that the cohomology space H ⋆ (Δ) inherits the grading and one can write H ⋆ (Δ) = ⨁ H k (Δ) k

(3.22)

The elements in H ⋆ (Δ) are called cocycles. They are the closed elements modulo the exact ones. Although elements in the image solve the equation Δx = 0 trivially, it does not mean that the image itself is trivial. Indeed, in the BRST approach to higher spin gauge fields, elements that can be written as Δy are gauge transformations, not at all uninteresting. One phenomena that often occurs is that the interesting cohomology is concentrated in the lower part of the cochain, perhaps even in 𝒜0 . This means that the cohomology is given by H 0 (Δ), i. e., by closed elements in 𝒜0 modulo exact elements. This is the case for de Rahm cohomology, and analogous cases occur in higher spin contexts. The method of contracting homotopies Consider a differential Δ in a graded differential complex. So, we have vector spaces 𝒜i making up a complex with a grading i. Suppose also that we have a linear operator L that can be diagonalized on the full space of all 𝒜i in the sense that Lxλ = λxλ where xλ is an eigenvector in 𝒜′λ . The subspaces 𝒜′λ are not necessarily the same as the 𝒜i , but suppose further that the subspace 𝒜′0 associated with the zero eigenvalue is contained in the subspace of degree zero, i. e., 𝒜′0 ⊂ 𝒜0 . Now, a linear operator σ whose anticommutator with Δ yields L, i. e., σΔ + Δσ = L

(3.23)

is called a contracting homotopy for L with respect to Δ. The intuition behind this definition is the following. We want to compute the cohomology Hk (Δ) of Δ. If we can contract the cohomology down to the zero eigenspace 𝒜′0 , it will follow that the only nonzero Hk (Δ) is H 0 (Δ). The argument goes as follows. From (3.23), we get [L, Δ] = 0. Consider a Δ-closed element x, and split up the vector x along the eigenvectors of L. The zero commutator [L, Δ] = 0 implies that the components of x are separately Δ-closed. Therefore, the cohomology of Δ can be analyzed for fixed eigenvalues of L. Then

90 � 3 Algebraic structures in higher spin theory

choose an element such that Lx = λx with λ ≠ 0 and Δx = 0. The crucial computation is (motivating both the definition of σ and the method itself) x=

1 1 1 Lx = (σΔ + Dσ)x = Δ( σx) λ λ λ

(3.24)

We see that x is Δ-exact for all λ ≠ 0. Thus, any nontrivial cocycles must lie in 𝒜′0 , i. e., in 𝒜0 .

3.1.10 Local cohomology in field theory Local cohomology is an important concept because of its relation to the question of locality for higher spin interactions at the quartic order of interaction. We will develop it fully in Section 4.3.6 in the BRST-BV antifield formulation of gauge field theory, and in Section 5.4 where it will be applied to the BBvD spin-3 theory. But the core of the concept can be outlined in general terms. The differential in the BRST-BV antifield formulation of gauge field theory is denoted by s. It is implemented in field theory as an antibracket sΦ ≡ (Φ, S) where S denotes the action—a function of the fields in the theory—enhanced with ghost fields (for gauge symmetry) and antifields, but the details of this need not concern us here (see Section 4.3.6). In field theory, we work with functions of the fields defined on space-time, and with functionals on space-time, i. e., functions integrated over space-time regions, of which the action itself is a prominent example. The exterior differential algebra naturally appears here since what are integrated are 4-forms, and surface terms, that typically appear when varying the action, can be written as ∫ dj with d the exterior derivative and j some 3-form current. The cohomology of the differential s may be denoted by H ⋆ (s) as above, and the space within it works is the space of functionals, i. e., the cochains of the theory are functionals over the fields, ghost fields and antifields of the theory and derivatives of all these. The extra requirement of locality in space-time means that the number of derivatives of the fields are finite.

3.2 Higher spin related algebras In this section, we will examine a few concrete algebras that are prominent in interacting higher spin theory. The reader should perhaps be aware of the fact that the concepts introduced below sometimes have more abstract and general meanings in mathematics. Often this is related to choosing coefficients from abstract rings, rather than from R or C and having modules as underlying objects rather than vector spaces.19 That said, we here work with R, C and vector spaces. 19 See our Volume 1, Section 3.7. As indicated in Section 3.1.4 above, we use “module” to mean “vector space”.

3.2 Higher spin related algebras

� 91

The motivating problem and its limitations It might be good to here remind ourselves of the basic motivation behind this study, especially since it also underlines its inherent limits. Remember in Chapter 4 of our Volume 1 we studied some aspects of interacting spin-1 and spin-2 theory. We saw how very special these theories are even though there are also quite a few similarities. Many authors stress the similarities, and as we have noted several times, the spin-1 and spin-2 theories are indeed sources of ideas and methods to generalize to higher spin interactions. However, in my opinion, this might very well be just wishful thinking. That said, the situation is the following: Higher spin fields are in practice described by some kind of indexed fields, the indices referring to “internal” spaces and “space-time”. This is firmly grounded in our way of going from the abstract group theory to the concrete field theoretic realizations. To collect all the higher spin fields in one object, we contract the indices with variables or generators spanning some mathematical structure, in practice vector spaces with an algebraic structure on top. When the generators are commuting, multiplying such higher spin objects together (mimicking interactions) is trivial, but uninteresting. In order to construct products in some way capturing interactions, we would have to introduce some kind of “vertex operators” and perform relevant “integrations”. This is what has been tried in BRST-inspired approaches in Minkowski space-time, borrowing techniques from string field theory. When the generators are not commuting—satisfying instead a nontrivial algebra—the result of multiplying two higher spin objects is much more interesting and could potentially provide for interactions without (much) further work. The reason being that now one can perform nontrivial “simplifications” using the commutation relations of the algebra of the generators. This is basically the idea behind the Vasiliev approach. These two examples do not exhaust the approaches tried in higher spin theory, but they capture recurring ideas common to them all. Are these ideas powerful enough to eventually provide a positive solution to the higher spin problem? Or must they be surpassed? Not knowing the answer to that question, these circumstances turn out to guide much of the rest of the algebraic theory outlined in the present chapter.

Our first topic will be the Heisenberg and Weyl algebras. These two kinds of algebras are closely related. One may get away with the impression, by just looking at the formulas, that they are the same thing. But there is an important difference in perspective. The Heisenberg algebras are considered as Lie algebras, whereas the Weyl algebras are treated as associative algebras. Technically, a Weyl algebra is the universal enveloping algebra of the corresponding Heisenberg algebra. This we have to sort out in the present section. In the process, we will have to examine how we actually compute in classical and quantum mechanics.20 Brackets and commutators: a note on terminology There is an irritating issue of terminology having to do with notation such as [⋅ , ⋅]. When we want to discuss abstract Lie algebras, then the bracket [⋅ , ⋅], or its graded variant, is the fundamental product in the algebra. If, however, a Lie algebra derives from an associative algebra, then the bracket is indeed a derived object. The problem resides not so much in practical computations, but rather in speaking about them. I have tried

20 All this is well known to the experts, but I choose to recapitulate it here in the rethinking tradition that the book is written in. It may be useful to the newcomer.

92 � 3 Algebraic structures in higher spin theory

to write commutator when I intend [a, b] = ab − ba, and bracket (or Lie bracket) if no such equation is presupposed. No doubt though, I have failed to be consistent. See also the note below on Poisson algebras.

3.2.1 The Heisenberg algebra The Heisenberg algebras, as their name suggest, are based on the familiar commutator [q,̂ p]̂ = iℏ of basic quantum mechanics. In d dimensions, a Heisenberg algebra Hd is generated by elements q̂ ν and p̂ μ with ν, μ = 1, 2, . . . , d, satisfying the nontrivial brackets [q̂ ν , p̂ μ ] = Cδνμ

(3.25)

and the trivial ones [q̂ ν , q̂ μ ] = 0

[p̂ ν , p̂ μ ] = 0 [C, q̂ ν ] = 0

[C, p̂ ν ] = 0

(3.26)

We allow for a slight generalization with an arbitrary central element C rather than iℏ. This then also covers cases where the q and p are linearly recombined into, for instance, creation and annihilation oscillators. The dimension of the algebra is 2d + 1, i. e., the number of generators with C counted as one of them. Concrete representations can be obtained in the usual way by multiplicative q’s and the p’s as q derivatives, acting in appropriate function spaces. Now what kind of algebra is this? If we view the generators abstractly and the brackets [⋅ , ⋅] are considered as the products in the algebra, then the algebra is a Lie algebra, actually an associative Lie algebra, since the Jacobi identities are trivial. Indeed, for any ̂ c]̂ = [a,̂ [b,̂ c]] ̂ = 0. three generators a,̂ b̂ and ĉ we have [[a,̂ b], However, this cannot be the full story in quantum mechanics, since in practice we encounter higher powers of the generators and we have to—or want to—compute commutators between such. Clearly then, the relations (3.25) and (3.26) are not enough. The answer is that we use formulas of the type [a, bc] = [a, b]c + b[a, c]

(3.27)

This is all very well, but it should be pointed out that if we consider higher powers of the generators and supply the basic brackets of (3.25) and (3.26) with formulas of the type (3.27), then we have moved beyond Heisenberg algebras proper. Indeed, we have moved into the realm of Poisson algebras. A reminder of Poisson algebras Historically, it all emanates from classical mechanics (as most things in theoretical physics does). Remember Hamiltonian mechanics as we outlined it in Section 3.2 of our Volume 1. There, the Poisson brackets of a classical mechanical system were established to satisfy the following fundamental properties:

3.2 Higher spin related algebras

{a, b} = −{b, a}

antisymmetry

{a + b, c + d} = {a, c} + {a, d} + {b, c} + {b, d} {a, bc} = {a, b}c + b{a, c}

Leibniz rule

{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0 Jacobi identity

� 93

(3.28a) linearity

(3.28b) (3.28c) (3.28d)

The classical Poisson bracket of mechanics satisfy—as already noted—these equations by construction. Upon quantization, turning Poisson brackets {⋅ , ⋅} into quantum brackets [⋅ , ⋅] in the standard way, these same equations are required to be satisfied by the brackets [⋅ , ⋅]. They can be considered as axioms satisfied by the quantum bracket. Algebras satisfying these axioms are called Poisson algebras. They are associative multiplicative algebras that also have a Lie bracket that satisfy the Leibniz rule, i. e., the Lie bracket acts as a derivation with respect to the multiplication (here written simply as juxtaposition of elements). When the operators q̂ and p̂ of a one-dimensional quantum system are represented with a multiplicative coordinate q and a derivative −iℏd/dq, respectively, acting in an appropriate Hilbert space of functions, all the formulas (3.28) are satisfied. The quantum Lie bracket [q,̂ p]̂ is represented as a commutator q̂p̂ − p̂q̂ where the multiplication is the associative multiplication of the Poisson algebra.

The observation that we work with higher powers of operators in quantum mechanics leads us to the Weyl algebras. They can be introduced in a few related ways, but the intuition is that higher powers of the basic generators are among the elements of the algebra. Let us be simpleminded before introducing the more exact terminology. Consider again generators q̂ ν and p̂ μ , but initially not imposing any bracket relations, rather allowing for multiplying them together into monomials and adding to get arbitrary polynomials with real or complex coefficients. This essentially provides us with a free associative word concatenation algebra. Let us call it Wd . To get something more interesting, we impose the bracket relations (3.25) and (3.26), now thought of as commutators in terms of the fundamental multiplication. This allows for “simplifications”, making ostensibly different looking expressions actually being equal. A simple example in two dimensions is p̂ 1 x̂2 x̂1 p̂ 2 = iℏx̂2 p̂ 2 + x̂1 x̂2 p̂ 1 p̂ 2 . The two expressions of each side of the equality sign look different, but they are actually equal using the commutators. This is a signal of the presence of an equivalence relation.

3.2.2 The Weyl algebra We will now try to understand various aspects of Weyl algebras, but first some more background. A historical and conceptual digression Early on in higher spin gauge theory, it was noticed that the spin-2 infinite-dimensional diffeomorphism algebra could be captured, at least in a heuristic way, by commuting two objects ξ μ 𝜕μ and ην 𝜕ν to obtain

[ξ μ 𝜕μ , ην 𝜕ν ] = (ξ μ 𝜕μ ην − ημ 𝜕μ ξ ν )𝜕ν . For spin 1, one has the finite-dimensional Lie algebra [ξa T a , ηb T b ] =

94 � 3 Algebraic structures in higher spin theory μν

f abc ξa ηb T c . It was then quite natural to try commuting objects like ξa T a 𝜕μ 𝜕ν for spin 3 and so on for higher spin. This was implicit in the Fang and Fronsdal “deformation of gauge groups” paper [150] and was explicitly studied by the present author in [155] and with I. Bengtsson in [157]. Even in such a quite naive approach, it was clear that once one invokes spin 3, one must have an infinite tower of higher spin gauge fields, if one hopes for any success at all. This initial idea was not followed up at the time, but it is now clear that the proper setting is within the theory of higher-order differential operators. A higher-order differential operator, of order m, on the space Rn with coordinates x μ and partial derivatives 𝜕μ = 𝜕/𝜕x μ , can be written as m

D = ∑ Dμ1 ⋅⋅⋅μr 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μr r=0

(3.29)

where the coefficients Dμ1 ⋅⋅⋅μr are functions (possibly restricted to some functional class) of the coordinates. When the coefficients are restricted to polynomials in xμ , one gets a form of Weyl algebras. We will return to this general kind of operators in Section 3.6.

The Weyl algebra 𝒜d is defined as the associative algebra formed by arbitrary (real or complex) polynomials in the generators q̂ ν and p̂ ν with ν = 1, 2, . . . , d modulo the nontrivial relations [q̂ ν , p̂ μ ] − Cδνμ = 0

(3.30)

The trivial relations (3.26) could be included in the definition, but that would be a bit too pedantic. Working modulo variables that commute is indeed trivial to implement in calculations. Working modulo the nontrivial commutators is more interesting as it gives rise to nontrivial equivalence classes of algebra elements. For instance, in one dimension, the elements p̂ q̂ and q̂ p̂ − 1 are equivalent since we can go from the first to the second expression by adding 0 = [q,̂ p]̂ − 1. Indeed, all elements in the free-word concatenation algebra 𝒲d proportional to μ ̂ [qν , p̂ μ ] − Cδν forms a two-sided ideal ℐ such that ℐ𝒲d = 𝒲d ℐ ⊆ ℐ . The Weyl algebra 𝒜d then appears as the quotient algebra 𝒲d /ℐ of equivalence classes [a] defined by the equivalence relation a ∼ b when a = b + ι where ι ∈ ℐ . These deliberations lead to two related questions: (i) how do we define sensible representatives of the equivalence classes, and (ii) what happens upon multiplication? For an answer to the second question, on general grounds it should be clear that multiplying two elements a1 and a2 from the equivalence classes [a1 ] and [a2 ], always gives an element in the equivalence class [a1 a2 ] of the product a1 a2 . That is, however, not so useful in calculation, unless one can implement it in a practical algorithm. Pursuing the first question, it is related to the question of ordering the product. There are two natural choices: normal ordering having all the q̂ ν to the left, and Weyl ordering where the elements are completely symmetric sums of products of the basic operators q̂ ν and p̂ ν . Normal ordering, usually associated with oscillator variables, corresponds to the representation of polynomial higher-order differential operators with all the x to the left as in (3.29).

3.3 Star product

� 95

We will now choose Weyl ordering. That is, in each equivalence class we will pick the completely symmetric combination of operators as representative. This means that in a general algebra element, being a polynomial sum over monomials, each monomial is to be Weyl ordered. Then returning to the second question: what happens upon multiplication of two Weyl ordered elements? Well, the products will not in general be Weyl ordered without further ado. Let us take a look at how this works out in practice. Weyl ordering in practice In order to study the basic mechanism of Weyl ordering, it is convenient to work with oscillator-type generaμ tors âi with commutator [âi , âj ] = ϵij or [âi , âjν ] = ϵij ημν in higher than one dimension. Consider products of n such oscillators (in one dimension) in any conceivable order. Adding some such different products with, for instance, integer coefficients, the resulting polynomial will have no particular symmetry upon interchange of the oscillators in positions k and l say (1 ≤ k < l ≤ n). Weyl ordering means that one takes all such possible products and adds them with equal coefficients. One choice of coefficient would be 1 divided by the number of different orderings. An example with n = 4 and products of two α̂1 and two α̂2 is W (α̂1 , α̂1 , α̂2 , α̂2 ) =

1 (α̂ α̂ α̂ α̂ + α̂1 α̂2 α̂1 α̂2 + α̂1 α̂2 α̂2 α̂1 = α̂2 α̂2 α̂1 α̂1 + α̂2 α̂1 α̂2 α̂1 + α̂2 α̂1 α̂1 α̂2 ) 6 1 1 2 2

(3.31)

The symmetry upon interchanging any two oscillators is of course built into the expression. More than that, the expression for W (α̂1 , α̂1 , α̂2 , α̂2 ) is invariant under applying the nontrivial commutation relations to any pair of oscillators. This is the essence of choosing Weyl ordering for the representative of the equivalence classes. One may entertain oneself by computing some examples. Here is one. Compute the square of the Weyl ordered expression W (α̂1 , α̂2 ) = 21 (α̂1 α̂2 + α̂2 α̂1 ). The result is not Weyl-ordered, but can be made so using the commutation relations. One then gets W (α̂1 , α̂2 )W (α̂1 , α̂2 ) = W (α̂1 , α̂1 , α̂2 , α̂2 ) − 1

(3.32)

where the right-hand side belongs to the equivalence class [α̂1 α̂1 α̂2 α̂2 ].

When operators like p̂ are represented by derivatives, there enters the issue of “scoop”. In some contexts, it is practical to limit the scoop of a derivative to the object standing immediately to the right of it. In the present context though, the common convention is that derivatives act on everything to the right of it. This amounts to associativity.

3.3 Star product The so-called star product, also called the Moyal product, plays an important role in the Vasiliev higher spin theory.21 The basic idea is that we have a set of commuting variables 21 For an interesting history of these products and their role in quantum mechanics, see [179].

96 � 3 Algebraic structures in higher spin theory upon which we want to impose a new well-defined noncommutative product. The original example is the classical phase space variables q and p, which upon quantization enjoys the noncommutative product q̂ p̂ = p̂ q̂ + iℏ. Now, instead of promoting the coordinates and momenta into operators, we can let the noncommutativity be maintained by a new product ⋆, and write q ⋆ p = p ⋆ q + iℏ. This should be done so that arbitrary polynomials in the phase space variables can be multiplied in this sense. Let us begin by focusing what we want to achieve. First, we want to set up a correspondence between classical functions f (q, p) and quantum operators f ̂(q,̂ p)̂ such that f ̂ → f in the limit ℏ → 0. Any such correspondence is not unique due to operator ordering ambiguities in the sense that many different operator orderings correspond to any one classical monomial qm pn . We then choose, as already noted, Weyl ordering on the quantum side in order to set up a bijection between f ̂ and f . Second, we want the product ⋆ to respect this very bijection. There are therefore three parts to this mathematical technique: (i) the Weyl map from classical phase-space functions to the corresponding Weyl-ordered quantum operator functions, and (ii) the inverse Wigner map from quantum operators to classical variables and (iii) the star product itself. For ease of presentation, we will do the construction for a one-dimensional system. Generalizing to many dimensions is basically straightforward. Adaptations toward the notation used in the Vasiliev theory will be given below in Section 3.6.7 and in Chapter 7. But first, a few words of caution regarding a notion that will be used throughout. What is a symbol? The concept of a symbol of an operator, and the corresponding term seems to be used in slightly different ways in the literature (in mathematics and in physics). The concept is simple for the basic operators q̂ and p̂ themselves. The symbol of q̂ is q and the symbol of p̂ is p. One could say that the symbol of a higherorder quantum operator f ̂ is the corresponding classical expression f , but the ordering ambiguities must be decided on in some way. Reordering a quantum expression such as q̂3 p̂2 will produce lower-order terms. Should these be included in the corresponding symbol or not? The choice made here is to define the symbol as the leading order term. This way the Weyl map and taking the symbol become one-to-one invertible. We will have occasion to return to this discussion in Section 3.6.

3.3.1 The Weyl map The Weyl map is supposed to map classical phase functions f (q, p) to Weyl ordered quan̂ To be definite, we restrict ourselves to polynomials, or possibly tum operators f ̂(q,̂ p). power series, in order not to have to deal with deeper analysis.22 Introducing some notation, we want a map 22 Whether this is sufficient for the use in higher spin theory or not is a difficult question.

3.3 Star product

̂ (q,̂ p)̂ W : f (q, p) → fW

� 97

(3.33)

̂ in order to emphasize the parwhere we have decorated the quantum operator as fW ticular Weyl ordering that we choose: A monomial qm pn is mapped into a sum of all ̂ and n p’s, ̂ divided by the total number of orderdifferent orderings of products of m q’s ings (m + n)!/m!n!. It goes without saying that the Weyl map must act linearly on each term in a polynomial. Thus, the Weyl map is quite simple to perform by hand. There are also explicit formulas for it. Consider the following double Fourier transform: ̂ (q,̂ p)̂ = fW

1 ∬ dσdτ ∬ dqdpf (q, p) exp(iσ(q̂ − q) + iτ(p̂ − p)) (2π)2

(3.34)

Formally doing the q and p integrals give ̂ (q,̂ p)̂ = fW

1 ∬ dσdτ f ̃(σ, τ) exp(iσ q̂ + iτ p)̂ 2π

(3.35)

where f ̃(σ, τ) is the Fourier transform of the classical function f (q, p). Now performing the inverse transform indicated in (3.35) it should be clear that one gets back to the corresponding polynomial form, but now with operators q̂ and p̂ in place of q and p. The symmetric Weyl ordering comes about since the order of the operators must be kept. ̂ the term of order σ m τ n generates the sum of Indeed, in the expansion of exp(iσ q̂ + iτ p), ̂ and n p’s. ̂ all orderings of m q’s Formal derivation of the Weyl transformation formula The formula (3.35) may be derived through some steps of formal manipulations. Consider computing exp(q̂

󵄨󵄨 𝜕 𝜕 󵄨󵄨 + p̂ )f (q, p)󵄨󵄨󵄨 󵄨󵄨 𝜕q 𝜕p 󵄨p=q=0

(3.36)

̂ and n for f (q, p) the monomial qm pn . The result is the corresponding Weyl ordered sum of products of m q’s ̂ with equal weights (m + n)!/m!n!. So, this formula performs the Weyl map. p’s First, enforce the setting of q = p = 0 with an integral ∬ dqdpδ(q)δ(p) and then express the delta functionals with their Fourier integrals. Second, partial integrate the exponential in (3.36) from f (q, p) onto the Fourier integrals for the delta functionals. Third, use a simple instance of the Baker–Haussdorf–Campbell formula to arrive at (3.34). Note, by the way, that no ℏ factors are involved in the Weyl map.

The bijection between Weyl-ordered quantum operators and their symbols is quite simple at the vector space level, before any products are considered. It just amounts to an exchange q̂ ↔ q and p̂ ↔ p in the Weyl-ordered basis.23 The situation is, however, more 23 Of course, at the symbol side, any sum of Weyl ordered monomials collapses into just one term.

98 � 3 Algebraic structures in higher spin theory complicated at the algebra level, since the noncommutativity on the operator side has the effect that the product of two Weyl ordered polynomials is not Weyl ordered. The product can of course be Weyl ordered by using the basic commutators, but the process, in general, produces lower-order terms and the result does not correspond to the corresponding product of symbols. This is clear from the simplest example of multiplying the Weyl ordered operators q̂ and p.̂ The product can be Weyl ordered to 21 (q̂ p̂ + p̂ q)̂ + iℏ2 but this does not correspond to qp on the symbol side. It should be noted that we are thinking about an associative product here. One may imagine considering a commutator product instead, and hope that the quantum commutator on the operator side corresponds to the Poisson bracket on the symbol side. It is, however, not so simple, as will be clear below. Eventually, we will see that the proper setting for all this is universal enveloping algebras to which we turn in Section 3.4. Disambiguating Weyl map and Weyl ordering This may be a trivial note, but do not conflate “Weyl map” and “Weyl ordering”. The Weyl map is a map from symbols to operators. Weyl ordering is a map from operators to operators involving the basic commutators.

3.3.2 The Moyal product The Moyal product can be seen as a deformation of the ordinary associative product of classical phase space variables q and p. It is meant to mimic the noncommutativity of the corresponding quantum operators q̂ and p.̂ Let us jump right in quote the formula as it applies to this case. Consider f (q, p) and g(q, p) to be two-phase space polynomials. The star product is defined as f ⋆ g = f exp[

iℏ ← 󳨀→ 󳨀 ← 󳨀→ 󳨀 (𝜕 𝜕 − 𝜕p 𝜕q )]g 2 q p

(3.37)

The factor iℏ may be a bit awkward to carry along in computations, but some deformation parameter is convenient to keep. The two basic examples are q ⋆ p = qp +

iℏ 2

and

p ⋆ q = pq −

iℏ 2

(3.38)

Note that the variables are still commuting under the ordinary product so there is really no need for keeping track of orderings in the result of computing the star product. This is reflected in the following nice result upon computing the commutator of q and p under the star product: [q, p]⋆ = iℏ

(3.39)

3.3 Star product

� 99

Thus, the star commutator reproduces the corresponding quantum commutator. However, this is as far as one gets without considering ordering questions. Example 2 (Star product and ordering). Consider computing the star commutator [q2 , p2 ]⋆ . We get q2 ⋆ p2 = q2 p2 + 2iℏqp −

ℏ2 2

and

p2 ⋆ q2 = p2 q2 − 2iℏpq −

ℏ2 2

(3.40)

so that [q2 , p2 ]⋆ = 4iℏqp

(3.41)

However, computing the corresponding quantum commutator produces [q̂ 2 , p̂ 2 ] = 4iℏq̂ p̂ + 2ℏ2

(3.42)

What went wrong? Well, nothing actually went wrong. It is just ordering ambiguities that hit us. Implicitly, I have chosen normal ordering on the right-hand side of (3.42). With Weyl ordering, the right-hand side is 2iℏ(q̂ p̂ + p̂ q)̂ and the result has the same form as the right-hand side of the star product commutator in (3.41). This example focuses the fact that working with the star product we are still in the classical realm. It is the Weyl map that takes us into the quantum realm. Indeed, W ([q2 , p2 ]⋆ ) = 2iℏ(q̂ p̂ + p̂ q)̂

(3.43)

Once in the quantum realm, we may reorder our expressions at will, using the quantum commutators. 󳶣 Example 2 may prompt a few more questions. Do we not now have an algebra with two products, the usual one and the star one? Should one worry about associativity? Let us first state an important fact on how the Weyl map interacts with the star product. Then we will do some more examples that illustrate these questions.

3.3.3 The Weyl map and the Moyal product One may prove that the Moyal star product has the following nice property. Consider two phase space functions f and g. Then W (f ⋆ g) = W (f )W (g)

(3.44)

The formula may be readily checked in simple cases with f and g low order monomials in q and p. It will be proved below in Section 3.3.4.

100 � 3 Algebraic structures in higher spin theory Example 3 (Star product, associativity and Weyl ordering). Consider computing the following star product: qp ⋆ qp = q2 p2 +

ℏ2 4

(3.45)

Upon contemplating the left-hand side, one may find it a bit strange with a mixture of ordinary products and a star product, although of course, the star product is defined in that way in formula (3.37). But what happens if the ordinary products are replaced by star products, and one computes (q ⋆ p) ⋆ (q ⋆ p) = q2 p2 + iℏqp

(3.46)

The results are different, not even under the Weyl map can they be made to agree. Furthermore, in qp ⋆ qp one may contemplate moving the star product around in the expression. One then finds that qp ⋆ qp ≠ q ⋆ pqp = qpq ⋆ q. This shows that one should not make the mistake of thinking that the star product associates with the ordinary product. On the other hand, it may be entertaining to check the following equalities: W ((q ⋆ p) ⋆ (q ⋆ p)) = W (q ⋆ p)W (q ⋆ p) = W (q)W (p)W (q)W (p)

(3.47)

They should be true according to (3.44). The star product is associative, so it does not matter in what order the product q ⋆ p ⋆ q ⋆ p is computed. That is not the point. However, the calculation is anyway illuminating since neither W (q ⋆ p)W (q ⋆ p) nor W (q)W (p)W (q)W (p) come out Weyl ordered from the direct computation. Let us look at details. From (3.46), we get W ((q ⋆ p) ⋆ (q ⋆ p)) = W (q2 p2 ) + iℏW (qp)

(3.48)

while from (3.38) we have W (q ⋆ p)W (q ⋆ p) = W (qp)W (qp) + iℏW (qp) −

ℏ2 4

(3.49)

and finally W (q)W (p)W (q)W (p) = q̂ p̂ q̂ p̂

(3.50)

which clearly shows that we cannot expect Weyl ordering to be respected by the use of formula (3.44). However, the right-hand sides are indeed equal to Weyl-ordered expressions upon use of the basic commutation relations. 󳶣

3.3 Star product

� 101

A historical and conceptual digression The topics treated here properly belong to the conceptual and mathematical development of quantum mechanics and its relation to classical mechanics, the problem of “quantization” for short. As such, it is a huge subject that one should perhaps not write about as an amateur. I will limit myself to offering some first step guides to the review literature, and then state some of the facts, in the hope of precluding confusion. As for historical and review references, I will mention two books. First, [180] (there is a short extract [179]), which treats the subject “concisely” (as the title states) containing many further references. Second, there is the more detailed textbook [181]. Then there is the review article [182]. There are of course many more texts on this subject. As for some relevant facts, we have the Groenewald theorem that effectively invalidates “simple-minded quantization schemes”. The theorem says that there is, in general, no invertible linear map Q : f (q, p) → f (̂ q,̂ p)̂ from functions f (x, p) and g(x, p) in phase space to Hermitean operators f (̂ x,̂ p)̂ and g(̂ x,̂ p)̂ in Hilbert space such that the Poisson brackets are preserved in the sense of being mapped to quantum commutators Q({f , g}) =

1 [Q(f ), Q(g)] iℏ

(3.51)

Rather, the linear Weyl map W in (3.34) from phase-space functions to Weyl-ordered operators is such that the ⋆-product of (3.37) satisfies W (f ⋆ g) = W (f )W (g). Then the Moyal Bracket defined by {f , g}mb = [f , g]⋆

(3.52)

is mapped to quantum commutators W ({f , g}mb ) =

1 [W (f ), W (g)] iℏ

(3.53)

This means that the—often stated24 —quantization scheme that Poisson brackets {f , g} go to quantum brackets iℏ1 [f ,̂ g]̂ cannot be expected to work for arbitrary operators, not even for polynomial operators. It is, at beast, a working heuristic. These facts are indeed illustrated in the examples given above.

3.3.4 Integral form of the star product The star product can be expressed by an integral formula. We will derive it in a simple one-dimensional case. Consider a commutator [x̂i , x̂j ] = iϵij between operators x̂i with i = 1, 2 and ℏ = 1. The differential formula for the star product is i← 󳨀→ 󳨀 f ⋆ g = f exp[ 𝜕i 𝜕j ϵij ]g 2

(3.54)

with 𝜕i = 𝜕/𝜕x i . The corresponding integral formula is (f ⋆ g)(x) =

24 Also in our Volume 1.

1 ∫ d 2 sd 2 tf (x + s)g(x + t) exp[2isi tj ϵij ] (2π)2

(3.55)

102 � 3 Algebraic structures in higher spin theory The equivalence can be shown in the following way. Express f and g in (3.54) as Fourier transforms f (x) =

i 1 ∫ d 2 sf ̃(s)eix si 2π

g(x) =

1 ix i ti ̃ ∫ d 2 t g(t)e 2π

(3.56)

Then one gets (by expanding the middle exponential) i 1 i← 󳨀 → 󳨀 ix i ti ̃ ∫ d 2 sd 2 t f ̃(s)eix si exp[ 𝜕 m 𝜕 n ϵmn ]g(t)e 2 2 (2π) m i 1 ix (sm +tm ) ̃ exp[− ϵij si tj ] = ∫ d 2 sd 2 t f ̃(s)g(t)e 2 2 (2π)

f ⋆g =

(3.57)

On the other hand, doing the same thing in (3.55) yields 1 i(x m +sm )σm i(x n +t n )τn (2isi t j ϵij ) ̃ e e ∫ d 2 sd 2 td 2 σd 2 τ f ̃(σ)g(τ)e 4 (2π) 1 ix m (σm +τm ) it n τn ̃ = e ∫ d 2 σd 2 τd 2 t f ̃(σ)g(τ)e (2π)2 m n 1 × ∫ d 2 seis (σm +2ϵmn t ) 2 (2π) 1 ix m (σm +τm ) it n τn 2 ̃ = e δ (σm + 2ϵmn t n ) ∫ d 2 σd 2 τd 2 t f ̃(σ)g(τ)e 2 (2π) i 1 ix m (σm +τm ) ̃ exp [− σ n τn ] = ∫ d 2 σd 2 τ f ̃(σ)g(τ)e 2 2 (2π)

f ⋆g =

(3.58)

The equality of the last lines of the computations (3.57) and (3.58) shows the equivalence of the differential and integral representations of the star product. As a byproduct of the computation (3.57), we get a proof of the formula (3.44). Applying the Weyl map, we get 1 i ix̂ m (sm +tm ) ̃ exp[− ϵij si tj ] ∫ d 2 sd 2 t f ̃(s)g(t)e 2 2 (2π) 1 ̂ m sm ix̂ m tm 2 2 ̃ i x ̃ = e = W (f (x))W (g(x)) ∫ d sd t f (s)g(t)e (2π)2

W ((f ⋆ g)(x)) =

i

j

i

i

1

i

(3.59)

j

using the simple instance ex̂ si ex̂ tj = ex̂ si +x̂ ti e 2 [x̂ ti ,x̂ sj ] of the Baker–Campbell–Haussdorff formula.

3.4 Universal enveloping algebras From an associative algebra, one can always construct a Lie algebra. The same underlying vector space V is used and the Lie bracket between two vectors u and v is defined in

3.4 Universal enveloping algebras

� 103

terms of the associative product ⬦ as the commutator [u, v] = u ⬦ v − v ⬦ u. The Jacobi identity then holds identically due the associativity of the product ⬦. The reverse process—constructing an associative algebra from a Lie algebra—is more involved, but has a fairly natural solution in terms of universal enveloping algebras. There is an abstract definition of this concept, employing notions from category theory, but here we will prefer a more concrete definition. First, we have some intuition. The intuitive concept of a universal enveloping algebra The basic idea is to embed a Lie algebra g into an associative algebra 𝒜 such that the Lie bracket [u, v] in g is represented by the commutator u ⬦ v − v ⬦ u in 𝒜. How can this be done? It is important to disambiguate notation and terminology here in order to avoid confusion. We should be thinking of an “abstract” Lie algebra with basis elements Xi satisfying the Lie brackets [Xi , Xj ] = ∑nk=1 cijk Xk and the Jacobi identity. Nothing more is assumed about the basis elements. In particular, nothing is assumed to be known about products of elements in the Lie algebra, except the Leibniz rule for the “interaction” between products of elements and the Lie bracket. Thus, we really have a Poisson algebra structure. A natural first question to ask is: What is to be the product ⬦ in 𝒜? The answer is the simplest possible. It is just (word) concatenation which we write as uv. In that way, one ensures associativity (uv)w = u(vw) = uvw.25 This can be phrased as: 𝒜 is generated by the elements of g, meaning that one works with arbitrary products (concatenations) of elements of g. This means that 𝒜 is infinite-dimensional. To be a little bit more precise, the universal enveloping algebra 𝒰 (g) is generated by powers of elements xi (i = 1, 2, . . . , n) subject only to n

xi xj − xj xi = ∑ cijk xk k=1

(3.60)

where n is the dimension of g. A generic element of 𝒜 is linear combination of products of generators in any order. As an infinite-dimensional basis for 𝒰 (g) one may take 1, xi , xi xj , xi xj xk , . . . . Such a basis is, however, overcomplete due to ordering ambiguities since the relation (3.60) can be used to simplify the products into a complete set by making a choice on ordering. One such set is in terms of lexicographical ordered products k k k x1 1 x22 ⋅ ⋅ ⋅ xk n One may also check that the Leibniz rule holds for the commutator in 𝒰 (g) since [u, vw] = u(vw) − (vw)u = v(uw − wu) + (uv − vu)w = v[u, w] + [u, v]w using the associative product.

Let us now put all this on a little more firmer ground. We shall describe two ways of arriving at the universal enveloping algebra. Since a Lie algebra is also a vector space (by forgetting the Lie bracket structure), we start from this underlying vector space V and contemplate the freely generated tensor algebra ϒ(V ) ∞

ϒ(V ) = ⨁ ϒk (V ) k=0

(3.61)

25 This question is seldom asked in the literature. Perhaps because of the simplicity of the answer.

104 � 3 Algebraic structures in higher spin theory where ϒk (V ) is the tensor product ⊗ of k factors of V .26 This tensor algebra is associative under the multiplication map ϒm (V ) ⊗ ϒn (V ) → ϒm+n (V ). The goal is now to impose a Lie bracket structure on ϒ(V ). This is done through a process called “lifting”. First, the bilinear Lie bracket map g × g → g is lifted to a corresponding bilinear, skew-symmetric map V × V → V defined by [u, v]⊗ = u ⊗ v − v ⊗ u = [u, v]

(3.62)

where on the left-hand side we have used tensor product notation for clarity, and in the right-hand side we have the Lie algebra product, interpreting u and v as Lie algebra elements.27 The lifting to higher order is done recursively via Leibniz type rules [u ⊗ v, w]⊗ = u ⊗ [v, w] + [u, w] ⊗ v

[u, v ⊗ w]⊗ = v ⊗ [u, w]⊗ + [u, v]⊗ ⊗ w

(3.63a) (3.63b)

In this way, the commutator [⋅ , ⋅]⊗ can be lifted to arbitrary ϒk (V ) and thereby to the full tensor algebra ϒ(V ). By its construction, it is a Poisson algebra. In this way, ϒ(V ) is an associative enveloping algebra for g, but it is also a Lie algebra since the Lie bracket has been lifted to it. It should therefore rightly be designated by ϒ(g). In order to arrive at the universal enveloping algebra 𝒰 (g), one further step is required. Many elements in ϒ(g) are equivalent under the equivalence relation generated by (3.62). Indeed, the relation (3.62) splits up ϒ(g) into equivalence classes (cosets).28 Thus, 𝒰 (g) is the quotient space ϒ(g)/(u ⊗ v − v ⊗ u = [u, v]). A second way, closely related and equivalent to the above, is the following. Return to the tensor algebra ϒ(g) before the lifting process.29 All elements proportional to u⊗v− v ⊗ u − [u, v] (with u and v in g) form a two-sided ideal, call it I. The universal enveloping algebra 𝒰 (g) is defined as ϒ(g)/I. The description given here is clearly quite heuristic and in need of mathematical elaboration. The central theorem making all this consistent is the so-called Poincaré– Birkhoff–Witt theorem. It, in particular, shows that the lexicographical base mentioned in the box above exists and is unique in the sense that it does not matter in which order the application the basic commutator (3.62) is applied. A textbook reference is [183]. A physics inspired review paper is [173].

26 See Section 3.7.10 in our Volume 1. 27 So, on the left-hand side we have the commutator bracket over the tensor product, and on the righthand side the Lie algebra product. See the box [Brackets and commutators:...] at the beginning of Section 3.2 above. 28 See Section 3.9.2 in our Volume 1. 29 So, that Lie brackets between vectors u and v may still be computed, but we do not consider computing commutators in ϒ(g).

3.5 General transformations revisited

� 105

Looking back at the theory reviewed in Section 3.2, we can now see that the Weyl algebra is the universal enveloping algebra of the Heisenberg algebra. At the beginning of Section 3.2.2, we mentioned higher-order differential operators and their relation to the higher spin problem. How such operators fit into the picture painted here we will come to in Section 3.6. There we will also see how the Fradkin–Vasiliev higher spin algebras— computed using star products—relate to the Heisenberg and Weyl algebras.

3.5 General transformations revisited In Volume 1, Section 3.9, we discussed transformation groups, i. e., groups acting on sets, “moving” its elements around. We need to elaborate both in the direction of concreteness and in the direction of generality. In particular, our interest concerns gauge transformations. In general, the sets that the groups act on, has more structure, being vector spaces or manifolds. As we have alluded to in several places, the gauge transformations of massless higher spin field theory falls quite naturally into the structure of strongly homotopy Lie algebra, also denoted as L∞ algebras. These are algebras where the Jacobi identity fails in a certain well-defined way. Now, as we know, the lower-spin gauge theories are described by Lie algebras, so it becomes very interesting to understand how these mathematical structures are related in higher spin theory. We will come to L∞ algebras is Section 4.4.2. We will start by elaborating on a bit of well-known theory, namely the theory of Lie groups. In our Volume 1, Sections 3.9, 3.10 and 3.11, we briefly discussed transformation groups, differential manifolds and Lie groups, but we did not really make the relation between the three concepts concrete. To have something that further motivates the effort, we begin by introducing the Maurer–Cartan equation. We will present it as it is often done in short reviews. The confusion that may result, prompts us toward an understanding that will be useful when we have to deal with the generalization in Vasiliev theory. Maurer–Cartan theory is also central to the group manifold and free differential algebra approach to supergravity, and we will come to that in Section 4.1. 3.5.1 Maurer–Cartan theory This is a brief introduction to Maurer–Cartan theory.30 Consider a Lie algebra with generators Ta satisfying the usual commutation relations 30 One may introduce much more differential geometric notation for what follows here, but the idea should be clear enough from our mostly verbal description. We will instead focus on a potentially confusing point often glossed over in the physics literature. A thorough reference is [184], which although constituting a mathematical treatment, contains many useful conceptual comments.

106 � 3 Algebraic structures in higher spin theory [Ta , Tb ] = fab c Tc

(3.64)

We can think of the generators Ta as a basis in the Lie group manifold near the identity. i. e., as a basis in the tangent space Te G. Then consider the corresponding cotangent space spanned by a dual basis of 1-forms that we denote by Ωa . We then have, as usual in vector space theory, ⟨Ωa , Tb ⟩ = δba (see Section 3.7.5 in our Volume 1). The next step is to introduce an exterior derivative d = dx a 𝜕a and a wedge product ∧. In terms of these, one may define a curvature Ra through 1 Ra = d ∧ Ωa + fbc a Ωb ∧ Ωc 2

(3.65)

When Ra = 0, formula (3.65) define the Maurer–Cartan equations. They are differential equations on the Lie group manifold. Consistency requires dRa = 0, which upon computation and using Ra = 0 again, yields the Jacobi identities for the structure constants fab c . In this context, dRa = 0 is called an integrability condition, or sometimes a Frobenius integrability condition (of which it is an instance). It remains to convince one-self that the two equations (3.64) and (3.65) with R = 0 are equivalent expressions of the same Lie group geometry. But there is clearly something amiss here since Ωa is a 1-form while Ta denote a set matrices. The generators Ta must be promoted to vector fields in order for an equation like ⟨Ωa , Tb ⟩ = δba to make sense. To clarify, we must remind ourselves of a little more differential geometry, in particular, as applied to Lie groups.

3.5.2 Some more differential geometry recollected The concept of a differential manifold was introduced in Section 3.10 in our Volume 1. The very basics of the theory as it applies to Lie groups, fiber bundles and coordinate transformations were then outlined in Sections 3.11–3.13. Here, we will take a little closer look at the theory. To get going, let us redefine the concept of a tangent vector in a more abstract way that is useful in the present context.31 In all of this section, it is understood that we are working on a differentiable manifold M, so we do not have to repeat that in every instance. Functions are supposed to be infinitely differentiable, or C ∞ , smooth for short. A tangent vector at the point x, denoted by Vx , is a mapping from the space of differentiable functions f , g, . . . , on some neighborhood of x into R satisfying linearity and the Leibniz rule Vx (af + bg) = aVx (f ) + bVx (g)

a, b ∈ R

31 There are three standard ways to define tangent vectors. See Section III.B.1 in [185].

(3.66)

3.5 General transformations revisited

� 107

Vx (fg) = Vx (f )g + fVx (g)

(3.67)

Any such linear mapping satisfying the Leibniz rule defines a derivation. To define the tangent space Tx M at the point x, we need only consider the set of all tangent vectors at the point subject to linearity in the sense (aUx + bVx )(f ) = aUx (f ) + bVx (f )

(3.68) μ

In a local coordinate system x μ , a tangent vector may be represented as Vx = Vx 𝜕μ . Then the directional derivative of a function in the direction of the vector Vx (action of a vector μ on a function32 ) is Vx (f ) = Vx 𝜕μ f . To define a vector field, one may initially think of letting the point x vary in Vx and write it as V (x) or simply as V . As we discussed in Section 3.12 in Volume 1, a more sophisticated concept is often introduced. A vector field V is then a cross-section of the vector bundle TM.33 The set of all vector fields on the manifold will be denoted as vectM . The notation is meant to be suggestive of a Lie algebra, which it is. Consider vector fields V and U acting on functions f and g. Addition of vectors, multiplication of a vector by a function and multiplication of vectors are naturally defined as (U + V )f = Uf + Vf

(3.69a)

(gV )f = g(Vf )

(3.69b)

(UV )f = U(Vf )

(3.69c)

respectively. Multiplication of vectors is associative, but not closed since it produces higher-order derivatives. Working in local coordinates, we find34 (UV )f = U μ 𝜕μ (V ν 𝜕ν f ) = U μ V ν 𝜕μ 𝜕ν f + U μ 𝜕μ V ν 𝜕ν f

(3.70)

However, the commutator of two vectors, the Lie bracket [U, V ] = UV − VU, is a vector field. Indeed [U, V ]f = (U μ 𝜕μ V ν − V μ 𝜕μ U ν )𝜕ν f

(3.71)

32 In our Volume 1, I denoted this with V [x]f . Notation is not standardized here, and often it is convenient just to write Vx f for the action of a vector on a function, as we indeed do below.

33 Providing for more “interesting” geometry when the fibers at different points are not trivially related, in particular, when nontrivial topology needs to be taken into account. 34 Remember our convention that the scope of derivatives are always just reaching the first differentiable object to the right, unless otherwise indicated. Note also that, with f a (scalar) function, Vf is again a (scalar) function, so the definition (3.69c) does indeed make sense.

108 � 3 Algebraic structures in higher spin theory One may show that all axioms of Lie algebras are satisfied, in particular, the Jacobi identity. The set of all vector fields on a manifold is therefore a Lie algebra vectM . The importance of the Lie bracket [U, V ] of vector fields U and V stems from the fact that it is equal to (or may be defined as) the Lie derivative of V in the direction of U, LU V = [U, V ]

(3.72)

This form of the Lie derivative is another way of introducing the Lie derivative of q general relativity that produces general coordinate transformations35 of tensors Tp , q q 36 generically as δϵ Tp = −Lϵ Tp . We will not show that here, but rather just record a few more useful facts. The Lie bracket formula (3.72) may also be expressed as L[U,V ] = [LU , LV ]

(3.73)

implicitly acting on any tensor on both sides of the equation. Next, we need the action of the Lie derivative on a 1-form. 1-forms live in the cotangent vector space Tx⋆ dual to the tangent vector space Tx . Working in local coordinates we write an arbitrary 1-form ω as ω = ωμ dx μ . The bases in tangent and cotangent spaces are dual in the sense ⟨dx ν , 𝜕μ ⟩ = δμν

(3.74)

⟨ω, V ⟩ = ων V μ ⟨dx ν , 𝜕μ ⟩ = ωμ V μ

(3.75)

and we have the inner product

Note that very often one writes ω(V ) = ⟨ω, V ⟩ thinking of it as the action of the 1-form ω on the vector V . This is consistent with the action of df on a vector V being equal to the action of the vector V on f , i. e., in formulas ⟨df , V ⟩ = V (f ) = V μ 𝜕μ f

(3.76)

From the instance LU ⟨ω, V ⟩ = ⟨LU ω, V ⟩ + ⟨ω, LU V ⟩ of the Leibniz rule, one finds the action of a Lie derivative on a 1-form LU ω = (U ν 𝜕ν ωμ + ων 𝜕μ U ν )dx μ

(3.77)

Also, by direct computation in a coordinate system, one finds the formula V (ω(U)) − U(ω(V )) − ω([U, V ]) = dω(V , U)

(3.78)

35 Our Volume 1, Section 3.13.1. 36 The procedure is found in many books; see, for instance, Section 5.3 in [186] or Section II.C.2 in [185]. A particularly succinct account, at the level of rigor of our work, can be found in [187].

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which will enable us to see the equivalence of the Maurer–Cartan structure equation with (3.64). This was one goal with this short excursion into differential geometry. However, as a very important byproduct, what we have here from a theoretical physics perspective—particularly embodied in formula (3.71) is the Lie algebra of general coordinate transformations. This is an infinite-dimensional Lie algebra, in contrast to the finite-dimensional Lie algebras of Lie groups. The relation between these two concepts of Lie algebra is important in higher spin theory, at least as we know it at the present. We end this section by introducing one more important concept, that of the interior product. The interior product iV ω between a p-form ω and a vector V may be defined by the following three clauses: 1. iV is an antiderivation, i. e., it obeys “the Leibniz rule with signs”. 2. iV f = 0, i. e., it yields zero acting on functions. 3. iV dx μ = V μ , so that iV ω = ⟨ω, V ⟩ = ωμ V μ on a 1-form ω. It then follows, by direct calculation, that acting with iV on a p-form we get37 iV ω = =

p

1 ∑ (−1)n−1 V μn ωμ1 ⋅⋅⋅μn ⋅⋅⋅μp ∧ dx μ1 ∧ ⋅ ⋅ ⋅ ∧ dx μn ∧ ⋅ ⋅ ⋅ ∧ dx μp ?󳨀󳨀? p! n=1 1 V μ ωμμ2 ⋅⋅⋅μp dx μ1 ∧ ⋅ ⋅ ⋅ ∧ dx μp (p − 1)!

(3.79)

where in the first line, dx μn , signifies that the n-th differential shall be removed. ?󳨀󳨀? Using the interior product and the exterior derivative, the Lie derivative of a form can be written in a nice way LV ω = (diV + iV d)ω

(3.80)

On 1-forms, the formula follows from computing the right-hand side and comparing to formula (3.77).38 This formula, which may very well give the impression of being a pure, but convenient, technicality will later on cast some more light on the relation between local Poincaré translations and general coordinate transformations in gauge theory of gravity.

3.5.3 Lie algebras of Lie groups One way of thinking about Lie algebra is as follows. First, we have a transformation group G and we assume that the group elements g form a differential manifold in such

37 Our p-form conventions can be found in Section 3.10.2 of Volume 1. 38 For general p-forms see, for instance, Section 5.4.3 in [186].

110 � 3 Algebraic structures in higher spin theory a way that the group structure and manifold structure are compatible. This gives us a Lie group. Lie groups may then act as transformation groups on other structured sets. In particular, the group may act on itself. This will give us the Lie algebra g. The first part can be phrased as a definition: A Lie group G is a differential manifold maintaining a group structure such that the group operations of multiplication and inverse G×G→G:

(g1 , g1 ) 󳨃→ g1 g2

G→G:

g 󳨃→ g

(3.81a) (3.81b)

−1

are differentiable.39 For the second part, we will first digress somewhat. A few Lie algebra related concepts Vector fields on a manifold may be considered as generators of groups of transformations. Remember40 that one way of defining the concept of a tangent vector to a manifold at a point was to define the tangent vector Vx as the equivalence class of all curves tangent at, and passing through, the point x. Letting x vary we have a vector field. Then conversely, a vector field V (x) may define a curve σ(t) in the manifold through the differential equation dσ(t) = V (σ(t)) dt

or in local coordinates

x μ (t) = V μ (x(t)) dt

(3.82)

where we let x denote points in the manifold as well as the coordinates of the curve σ. The solution—called μ an integral curve—x μ (t, x0 ) depends on the initial condition x μ (0, x0 ) = x0 . Given the initial condition, the curve, which is defined for some real interval I(x0 ) around t = 0 and with σ(0, x0 ) = x0 , is unique.41 It also satisfies the nice property σ(t, σ(s, x)) = σ(t + s, x). The map I(x) × M → M : (x, t) → σ(x, t) is called the flow of the vector field V (x). Given certain regularity assumptions, a vector field generates a one-dimensional group of diffeomorphisms of the manifold. To focus this aspect, it is convenient to use the notation σt for the curve and think of it as a map M → M : x 󳨃→ σ(t, x) (for fixed t). In a language familiar to theoretical physics, one would say that the vector field is the infinitesimal generator of the infinitesimal transformation μ σϵ (x) = σ μ (ϵ, x) = x μ + ϵV μ (x). Next, consider the important case of a Lie group of transformations on the manifold, i. e., differentiable maps σ :G×M →M

where (g, x) 󳨃→ σ(g, x)

(3.83)

satisfying certain natural conditions that we will come to. The notation σ(g, x) anticipates the connection to the one-dimensional transformations of above. For a fixed (but arbitrary) group element g, we write σg (x) = σ(g, x) for the map σg : M → M. We can then think of σg(t) (x) as curves in the manifold through points

39 See, for instance, Section 1.2 in [188]. 40 See Section 3.10.1 in our Volume 1. 41 Precise statements can be found in [185], Section III.C.1.

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x, parametrized by group elements g(t) if we let the group elements themselves be parametrized by real parameters t. This leads to the concept of curves g(t) within the Lie group itself, defining a one parameter subgroup of a Lie group to be a differentiable map g(t) from R to G with the desirable properties g(t)g(s) = g(t + s)

and

g(0) = e

where e is the unit group element

(3.84)

Returning to the conditions to be attached to the maps (3.83), these are the group properties σgh = σg ∘ σh and σe = idM from which follows σg−1 = σg−1 . Other desirable properties for the Lie group actions are those of transitive, effective and free action (see Section 3.9.1 of our Volume 1).

We will not elaborate more on this,42 but rather return to the Lie group acting on itself, in particular, to the concepts of left- (and right-)invariant vector fields on the group manifold. This will lead to the Lie algebra. Consider a left translation Lh : G → G in a Lie group G, defined by Lh (g) = hg

(3.85)

where g is an arbitrary point in the group manifold, and h picks out a particular left translation. Such a map induces a corresponding linear map Lh⋆ on the tangent space Tg G at g, namely Lh⋆ : Tg G → Thg G

(3.86)

Push-forward and pull-back maps In general, if f is a map M → N between two manifolds, f⋆ is the corresponding linear approximation map Tx M → Tf (x) N between the corresponding tangent spaces, called a push forward. The pull back map f ⋆ goes in the other direction between the cotangent spaces Tf⋆(x) N → Tx⋆ M. The lower star in Lh⋆ in formula (3.86) is

associated with the map Lh (i. e., not with h).43 Let us record some facts about the f⋆ map. First, to define it properly, consider a map f : M → N between two manifolds. Focus on a fixed but arbitrary point p ∈ M to begin with. We want to define the map f⋆ : Tp M → Tf (p) N mapping tangent vectors Vp to specified tangent vectors Wf (p) . Then pick an arbitrary function g defined on the manifold N as a “carrier” and define Wf (p) (g) ≡ Vp (g ∘ f ). The specified tangent vector Wf (p) (g) is the image of Vp under the map f⋆ , i. e., we have Wf (p) = (f⋆ V )f (p) .

42 For the reader who looks for more details, good treatments can be found in Section 5.6 in [186] (on the level of sophistication chosen here) or in Section III.D in [185] (more sophisticated regarding regularity requirements). 43 As an aside, notation here is really awkward, in particular, the use of parentheses of various kinds. I have given up—as seems to be the common practice—and notation is very much tailored to the computation at hand, inserting parentheses where they may clarify what acts on what and in which way, while not making expressions unreadable, or ambiguous, by overuse of parentheses.

112 � 3 Algebraic structures in higher spin theory

Now, the resulting definition (f⋆ V )f (p) ≡ Vp (g ∘ f ) is a numerical equality since on the left-hand side we compute at the point f (p) and on the right-hand side at the point p. To make it a functional equality, we use the invertibility of f and finally arrive at (f⋆ V ) ≡ Vp (g ∘ f ) ∘ f −1

(3.87)

Consider now two vector fields U and V in vectM . By direct computation, it follows that f⋆ [U, V ] = [f⋆ U, f⋆ U]

(3.88)

This result will soon be useful. Turning to the pull-back map f ⋆ , it is defined by requiring ⟨f ⋆ ω, V ⟩ = ⟨ω, f⋆ V ⟩

(3.89)

A left-invariant vector field V is then defined as one for which Lh⋆ (V (g)) ≡ V (Lh (g)) = V (hg)

(3.90)

Now consider the set of tangent vectors at the identity e of the group, i. e., the tangent space Te G. We want to set up bijection between the tangent space at the identity and the set of left-invariant vector fields. This means that for each tangent vector T in Te G there should be a unique left-invariant vector field V (g), and conversely, any left-invariant vector field shall define a unique tangent vector at the identity. The second part is the easier. Choose h = g −1 in (3.86) and (3.90). The point g is then translated to the identity e where the left-invariant vector field V defines the tangent vector V (e) ∈ Te G. For the first part, which is a bit more tricky, define a left translation Lg ′ with some arbitrary g ′ ∈ G. Next, let T be a tangent vector in Te G. Nothing now prevents us from defining a vector field UT throughout the group manifold by UT (g ′ ) = Lg ′ T. This vector field is left invariant as the following computation of UT (hg ′ ) shows: UT (hg ′ ) = Lhg ′ ⋆ T = (Lh Lg ′ )⋆ T = Lh⋆ Lg ′ ⋆ T = Lh⋆ UT (g ′ )

(3.91)

where we have only used the definition of UT and composition of the linear map Lg⋆ . One may denote the isomorphic sets of tangent vectors at the identity and leftinvariant vector fields by g, but we do not yet have a Lie algebra. For that, it remains to show that the commutator of left-invariant vector fields satisfy the axioms of Lie algebras. We can do that, with a little good faith, in the following way. Since g is a set of vector fields, it must be a subset of the set vectG of all vector fields on the Lie group manifold. Therefore, the Lie bracket between vector fields, defined above in Section 3.5.2, is defined also on g. What we need in order to get a Lie subalgebra, is closure under left invariance. This follows from the general property (3.88) for vector fields in vectG .

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It remains to define the dual concept of left-invariant 1-forms. That is done via the pull-back map f ⋆ : Tf (x) N → Tx M. A 1-form ω is left invariant if L⋆h (ω(hg)) ≡ ω(g)

(3.92)

The dual nature of the definitions (3.90) and (3.92) is very clear if we study them near the identity by choosing g = e. Then we have Lh⋆ (V (e)) = V (h) L⋆h (ω(h))

= ω(e)

(3.93a) (3.93b)

It may be checked that these formulas are compatible with the correspondence between push-forward and pull-back maps of the definition (3.89).

3.5.4 The structure constants We may now begin to discern that something dramatic happens when we pass from the general coordinate transformations on a generic manifold M to the corresponding transformations on a Lie group manifold G. The infinite-dimensional Lie algebra of Lie brackets of vector fields gets trimmed down to a finite-dimensional Lie algebra g. Clearly, it must be the group structure of formulas (3.81) that is responsible for that. From infinite to finite dimension Consider the real Lie group GL(n) with Lie algebra gl(n). We write the matrix corresponding to the group element g as x ij |g . Actually, the x ij |g can be thought of as coordinates on the group manifold.44 A left translation hg is then given by matrix multiplication. A vector in Te GL(n) can be written as A(e) = Aij

𝜕 𝜕x ij |e

(3.94)

with matrix elements Aij . The corresponding left invariant vector field can be computed from Lg⋆ A(e) = A(ge) = Aij

𝜕 kl lm 𝜕 𝜕 𝜕 x |g x |e km = x ki |g Aij kj = (gA)kj kj 𝜕x ij |e 𝜕x |g 𝜕x |g 𝜕x |g

(3.95)

All repeated indices are summed over, regardless of whether they are upper or lower. The formalism is awkward, but the result is natural. It is now possible to commute two left-invariant vector fields A and B in this form to find

44 There is no deeper reason for writing x ij |g rather than x ij (g) other than typographical. The formalism of concrete differential geometry is full of forced compromises and any one symbol may serve several not entirely related purposes.

114 � 3 Algebraic structures in higher spin theory x ij ([A, B]jk )

𝜕 𝜕x ik |g

(3.96)

which is again a left-invariant vector field. In this formula, [A, B]jk are the matrix elements of the matrix commutator AB − BA.

Although this is all well known, it is a theoretical phenomena that is worth emphasizing in the context of higher spin gauge theory. Indeed, already in connection with the problems of gauge theories of gravity, we saw the different structures of the infinitedimensional diffeomorphism group and the finite-dimensional local Poincaré group (see our Volume 1, Section 4.6.3). For now, let us see how the structure constants emerge. Since Te G is an n-dimensional vector space, we can introduce a basis of n left-invariant vector fields. Conforming to common practice, we name them {X1 , X2 , . . . , Xn } using lower-covariant vector indices as in Xa . Since we already know that the commutator of two left-invariant vector fields is again a left-invariant vector field, it must be possible to expand any commutator [Xa , Xb ] in terms of the set {Xc }. We then arrive, almost by default, at the familiar formula (3.64) above, but now in terms of left-invariant vector fields rather than matrix generators. We will elaborate on this in the next section. So, this is the theory in a brief outline. Some fine mathematical points have certainly been glossed over, but nothing essential for a theoretical physicist has been left out. We can return to the Maurer–Cartan equation of Section 3.5.1. A few questions that the reader might have asked herself are: What kind of basis vectors Ta and Ωa do we really have here? And what are the derivatives 𝜕a ? Indeed, what is d?

3.5.5 Frames, coframes and the structure of Lie algebras From the discussion above, and in analogy to the theory of general relativity, we can now piece together answers to these questions. First, taking a clue from the box [From infinite to finite dimension] above, the group generators Ta are promoted to leftinvariant vector fields through Xa = x ik Takj 𝜕ij

(3.97)

where we think of x ij as local coordinates for the group elements g. In analogy with general relativity, one could ponder thinking of the vector fields Xa as a non-coordinate basis in the group manifold. Second, the Ωa are left-invariant 1-forms and may be written as Ωa = Ωa ij dx ij = (x −1 )ik Tkja dx ij

(3.98)

3.5 General transformations revisited

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where (x −1 )ij denotes the local coordinate representation for the group element g −1 . Now it follows by direct calculation that ⟨Ωa , Xb ⟩ = δba

(3.99)

using j

(3.100)

𝜕 𝜕x ij

(3.101)

⟨dx ij , 𝜕kl ⟩ = δki δl The exterior derivative d is given by d = dx ij 𝜕ij = dx ij

Finally, that the two ways of presenting the Lie algebra structure 1 d ∧ Ωa + fbc a Ωb ∧ Ωc = 0 2 [Xa , Xb ] = fab c Xc

(3.102a) (3.102b)

are equivalent, follows from the general formula (3.78). We see that it makes sense to consider the Xa as frames and the Ωa as coframes. One may convince one-self that the structure constants are indeed constants by using the fact that both sides of the two equations (3.102) are left invariant. We can now restart the story of Section 3.5.1 in a way that makes more sense. The left-invariant 1-forms are coframes, or vielbeins, on the group manifold whose coordinates we may denote by x μ . Explicitly, we have Ωa = Ω(x)a μ dx μ . These coframe fields have a fixed-coordinate dependence and can therefore not serve as dynamical variables, but rather as a background configuration, indeed one with zero curvature. Dual to the μ coframe fields, we have the frame fields Xa = X(x)a 𝜕μ , which are the left-invariant vector fields. From the duality formulas (3.99) and (3.100) for the bases, the latter of which μ μ we can now write as ⟨dx μ , 𝜕ν ⟩ = δν , we find that Ωa μ and Xa behave as inversely related vielbein fields μ

Ωa μ Xb = δba

and

Xa μ Ωa ν = δνμ

(3.103)

As in general relativity, the pair of dual bases (dx μ , 𝜕ν ) in cotangent and tangent space, are called coordinate or holonomic bases, while the pair (Ωa , Xb ) are called noncoordinate or anholonomic bases.45 We will continue this discussion in the next chapter in Section 4.1 on the group manifold approach to gravity. The relevance for the Vasiliev theory will become clear in Chapter 7, in particular in Section 7.3.5 on free differential algebras. 45 The corresponding discussion in general relativity can be found in Section 4.5.1 of our Volume 1.

116 � 3 Algebraic structures in higher spin theory

3.6 Higher-order differential operators That higher-order differential operators should be related to the higher spin problem is an old, and quite natural observation, dating back to the early days of massless higher spin field theory. Fronsdal’s 1979 conference paper [2] contains what is probably the first explicit attempt that goes beyond mere allusions to “deformed gauge groups”. The idea was taken up by I. Bengtsson and the present author in the mid-1980s papers [156] and [157], but we did not get very far. For various reasons—and perhaps not so surprisingly— the idea has turned out to be very difficult to implement. The initial, quite naive attempts can be referred to the Fronsdal (Minkowski spacetime) program. The Fradkin–Vasiliev “higher spin algebra” approach in AdS space-time, of which we will see more in Chapter 7, is arguably the most developed implementation. It is related both to the universal enveloping Weyl algebra of the Heisenberg algebra and to mathematical results on “higher-conformal symmetries of the Laplacian” of M. Eastwood [189]. Another related implementation is Y. A. Segal’s conformal higher spin theory [190]. There is also deeper precursor work by M. Flato and C. Fronsdal in [191] and follow up papers on singletons. Comments like these may not be very useful to the newcomer, but we will try to clarify the connections as we continue, even though the full picture is still not clear. There are indeed a few aspects of the subject of higher-order differential operators and their relation to the higher spin problem that one should be aware of from the outset.

3.6.1 Higher-order differential operators and tentative higher spin algebras One aspect is the counting of degrees of freedom carried by the fields. At its very simplest, consider a tower of symmetric tensors hμ1 ⋅⋅⋅μs collected into a single object 1 μ1 ⋅⋅⋅μs h (x)pμ1 ⋅ ⋅ ⋅ pμs s! s=0 ∞

h(x, p) = ∑

(3.104)

where, as the notation suggests, the pμ may be thought of as “momenta” or “differential operators”. We will come to interpretations of the pμ below. Likewise, one may consider the corresponding “gauge parameters” 1 ξ μ1 ⋅⋅⋅μs−1 (x)pμ1 ⋅ ⋅ ⋅ pμs−1 (s − 1)! s=0 ∞

ξ(x, p) = ∑

(3.105)

All this is clearly indicative of notation that we are already familiar with from higher spin theory. Assuming algebraic or/and differential properties of the pμ , one may start commuting objects like h(x, p) and ξ(x, p) and study the ensuing algebras to see if anything interesting comes out. At least we want to reproduce the lowest-order gauge transformations

3.6 Higher-order differential operators

δhμ1 ⋅⋅⋅μs = 𝜕(μ1 ξ μ1 ⋅⋅⋅μs−1 )

� 117

(3.106)

One may, however, first worry about the relation between the symmetric tensors hμ1 ⋅⋅⋅μs and massless higher spin particles, i. e., how the massless Poincaré representations sit in the hμ1 ⋅⋅⋅μs . Counting degrees of freedom off-shell and on-shell Remember that the number of field components in a symmetric Lorentz tensor field ϕμ1 ⋅⋅⋅μs in four dimensions is (s+3 ) (see Section 5.1 in our Volume 1). This number grows like s3 , so in order to bring down the number s of field components to match the Poincaré particle representations, we need conditions and invariances. In the massive case, the number of physical field components is 2s + 1 through the conditions of tracelessness and transversality (See Section 2.5 in our Volume 1). For the massless, gauge field case, the conditions of double tracelessness for the fields and tracelessness for the gauge parameters, bring down the number of physical components to 2 for all spin s. Then turn to the fields and parameters in (3.104) and (3.106). To move forward with such an approach, one needs a way to implement constraints on the components of the expansion. In this context, we can take the opportunity to comment on the usage of the terms “on-shell” and “offshell” in higher spin theory. The term on-shell always means—as in other contexts—that the equations of motion are imposed. However, whereas the common usage of off-shell in the meaning off-shell formulation, refers to the existence of a Lagrangian (so that one may quantize using path-integrals), in Vasiliev inspired higher spin theory it is not uncommon to use “off-shell” referring to the equations of motion not being imposed even in the case when no Lagrangian is known. That is, one has only the fields and their gauge transformations. Thus, in particular, in the Vasiliev theory, one should not read “off-shell formulation” as implying that there is an action for the theory (which there is not as of writing). Indeed, not imposing the equations of motion in the Vasiliev theory, means having just the fields, and possibly the gauge transformations.

Another aspect is the relation to the underlying manifold (space-time). We know that general coordinate transformations correspond, infinitesimally, to the Lie algebra of left-invariant vector fields on the manifold (see Section 3.5.2). These are first-order differential operators and their action integrates to the full group of general coordinate transformations. What could possibly by gained by looking at higher-order differential operators when the first order already provides it all?46 However, the question is not really “what could be gained”, but how to accommodate the problems that one might get into. Consider, intuitively, an infinitesimalcoordinate transformation generated by a spin-2 gauge transformation. Such a transformation is generated by a vector field on the manifold ξ μ 𝜕μ . By exponentiation, such a Lie algebra transformation can be integrated to a Lie group transformation, i. e., to a general coordinate transformation. Higher-orders s of derivatives are then suppressed

46 I. Bengtsson made this comment a long time ago when we tried to use higher-order differential operators for the higher spin gauge algebra problem. Unfortunately, at that time, we were unaware of the mathematical literature on higher-order differential operators that existed already then.

118 � 3 Algebraic structures in higher spin theory by higher powers of the infinitesimal parameter ξ μ . They are of 𝒪(ξ s ). However, higher spin infinitesimal transformations such as ξ μ(s) (𝜕μ )s are of the same order as ξ μ , unless one declares that the higher spin gauge parameters ξ μ(s) are of order (ξ μ )s . Therefore, exponentiation of higher spin infinitesimal transformations is prone to be problematic. We will return to this problem in Section 3.6.6. A third aspect is more related to the very starting point of the endeavor. In what we have referred to as the “gauging approach” (to which all the approaches mentioned above belongs) one starts with a gauge algebra—in practice one makes an informed guess about the structure of the higher spin algebra—and from there, one tries to construct the interacting theory, most likely involving a deformation of the algebra along the way. However, in a pure Noether procedure—Fronsdal program approach—one starts with a spectrum of free fields and tries to construct the interactions and the nonlinear gauge transformations ab initio with the idea that the gauge algebra will appear along the way. When this is done for spin 1 and spin 2, the correct gauge algebras appear early in process; so, early indeed, that the full Noether procedure can be “short-circuited”. Enough of these generalities, let us now see what can be said about higher-order differential operators. We will be following, some way, a recent paper by X. Bekaert that offers a developed exposition of this approach [192].47 Useful material can also be found in [193].

3.6.2 Two (or more) approaches to higher spin gauge algebras The intuition is to introduce some kind of “commutators” between objects like those in (3.104) and (3.105) so that a commutator between a field h and a parameter ξ corresponds to a transformation δξ h and a commutator between two ξ1 and ξ2 yields the gauge algebra. There are two quite natural ways to do that. The first one is to interpret the “momenta” pμ as differential operators 𝜕μ . To be definite, we take pμ → −il𝜕μ or, if one prefers, pμ → −il∇μ with ∇μ the covariant derivative of some maximally symmetric background (l is a Planck-like constant of mass dimension −1). The derivatives are then thought of as acting on the coordinates x μ upon which the fields and parameters depend. With this interpretation, the expansions (3.104) and (3.105) become (introducing new notation along the way) ∞ (−il)s μ1 ⋅⋅⋅μs Ĥ = ∑ h (x)𝜕μ1 ⋅ ⋅ ⋅ 𝜕μs + (ordering terms) s! s=0 ∞ (−il)(s−1) μ1 ⋅⋅⋅μs−1 Ξ̂ = ∑ ξ 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μs−1 + (ordering terms) (s − 1)! s=1

47 We will not follow through all details of Bekaert’s argument. It is quite technical.

(3.107a) (3.107b)

3.6 Higher-order differential operators

� 119

One may worry about the interpretation of these formulas with regard to the underlying manifold structure that the coordinates x μ and derivatives 𝜕μ refer to. It should be safe to think of a linearized approximation—around some fixed (symmetric) background— to an eventual full higher spin theory including gravity. Then the components may be treated as fiber tensors. Simpler still, one may take the space to be R(3,1) , i. e., Minkowski space-time. Since the differential operators have a nontrivial action on the fields and parameters, one could contemplate—as indicated in the formulas—to order the terms in some different way than the “normal order” of the explicitly written terms. By Weyl, or symmetric ordering, is meant the result of thinking of Ĥ as acting on a function f (x) and computing 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μs (hμ1 ⋅⋅⋅μs (x)f (x)) using the Leibniz rule. That is, the “ordering terms” in (3.107) stands for lower-order terms, in derivatives, resulting from the ordering prescription. In order to reproduce the free theory gauge transformations (3.106), a term proportional to 𝜕2 must be added to Ĥ [2]. This is natural, since at the spin-2 level we have the metric expressed as the sum of a background and a fluctuation according to gμ1 μ2 = ημ1 μ2 + hμ1 μ2 . Therefore, we take l2 Ĝ = Ĥ − 𝜕2 2

(3.108)

In this scheme, the non-Abelian deformation of the free theory Abelian gauge algebra would result from the commuting two parameters [Ξ̂ 1 , Ξ̂ 2 ]. A transformation of the field would result from the adjoint action of parameter on the field i ̂ δΞ̂ Ĥ = − [Ξ,̂ H] l

(3.109)

Since transformations like these are substitutions, the Jacobi identity must hold. Thus computing the commutator of two transformations yields 1 ̂ = − i [− i [Ξ̂ 1 , Ξ̂ 2 ], H] ̂ [δΞ̂ , δΞ̂ ]Ĥ = 2 [[Ξ̂ 1 , Ξ̂ 2 ], H] 1 2 l l l

(3.110)

Consequently, we get for the gauge algebra [δΞ̂ , δΞ̂ ]Ĥ = δΞ̂ Ĥ 1

2

3

i where Ξ̂ 3 = − [Ξ̂ 1 , Ξ̂ 2 ] l

(3.111)

Anyone doing this for the first time may be forgiven for exclaiming a “eureka” and rushing out on the town square to announce the discovery, in particular, since the low spin Maxwell and Einstein gauge algebras are contained. The Maxwell spin-1 Abelian gauge algebra is quite trivial, and not so exciting, but for spin 2 one indeed gets μ μ μ [ξ1 𝜕μ , ξ2ν 𝜕ν ] = (ξ1 𝜕μ ξ2ν − ξ2 𝜕μ ην1 )𝜕ν that, with some hand-waving, can be interpreted as the Lie algebra of local diffeomorphisms. But this is roughly as far as one gets before

120 � 3 Algebraic structures in higher spin theory trouble sets in. Before discussing that, let us look at the second natural approach to a tentative higher spin algebra. If we call the above the higher-order differential operator approach to higher spin gauge algebra, we may call the second the Hamiltonian cotangent approach. Then return to the expansions (3.104) and (3.105) and interpret the pμ as canonical momenta conjugate to the coordinates x μ . One may then think of the fields h(x, p) and parameters ξ(x, p) as functions on the cotangent bundle T ⋆ M of the underlying manifold M.48 Also, in this case one introduces a background metric term according to 1 g(x, p) = h(x, p) + p2 2

(3.112)

In this case, the algebraic structure is generated by the Poisson bracket between two functions ξ(x, p) and η(x, p), {ξ, η} =

𝜕ξ 𝜕η 𝜕ξ 𝜕η − 𝜕x μ 𝜕pμ 𝜕pμ 𝜕x μ

(3.113)

Then gauge transformations are given by δξ g(x, p) = {ξ(x, p), g(x, p)}

(3.114)

and the gauge algebra is computed from {ξ1 (x, p), ξ2 (x, p)} = ξ3 (x, p)

(3.115)

The resulting algebraic structure is not the same as the one ensuing from the higherorder differential operator approach. For instance, picking out two parameters ξ1 and ξ2 with n1 and n2 indices, respectively, one gets (μ1 ⋅⋅⋅μn1 μn1 +1 ⋅⋅⋅μn1 +n2 −1 )μ ξ2

{ξ1 , ξ2 }μ1 ⋅⋅⋅μn1 +n2 −1 = n2 𝜕μ ξ1

(μ1 ⋅⋅⋅μn2 μn2 +1 ⋅⋅⋅μn1 +n2 −1 )μ ξ1

− n1 𝜕μ ξ2

(3.116)

This is an example of a Schouten bracket, defined for symmetric, multivector fields. It generalizes the Lie bracket for vector fields. However, the bracket (3.111) between differential operator parameters produces “lower-order” terms with correspondingly more than one derivative acting on the parameters. These are not produced by the Poisson bracket. A third approach to the higher spin gauge algebra problem, or rather a variant of the first approach, starts from the observation that one would like to attach some kind of internal indices to at least the odd spin fields. One attempt in that direction was made in [157] where matrices T a were attached to the fields in a Chan–Paton fashion. We leave it to the reader to explore further. 48 Again in a linearized approximation.

3.6 Higher-order differential operators

� 121

A fourth, related approach, also tried in [157], is to work with frame-like higher spin fields rather than metric-like fields. The fields are then 1-forms eμ,a1 ...as−1 and ωμ,b,a1 ...as−1 indexed by base space index μ and tangent space indices ai and b subject to certain symmetry and trace conditions (see Section 5.7 of our Volume 1). This is clearly reminiscent of the Vasiliev construction, but not quite the same. An approach along these lines eases—at least initially—the interpretational problem of what space the higher spin indices live in. 3.6.3 Some general properties of higher-order differential operators Having so gained some intuition about the possible role of higher-order differential operators in higher spin theory, we can now record a few of their properties. Consider, as in (3.29), a differential operator of order m on Rn , m

D(m) = ∑ Dμ1 ⋅⋅⋅μr (x μ )𝜕μ1 ⋅ ⋅ ⋅ 𝜕μr

(3.117)

r=0

They are linear operators in the sense that they can be added freely with real or complex coefficients. The functions Dμ1 ⋅⋅⋅μr (x μ ) (belonging to some functional class) are simply called “coefficients”. They are symmetric tensors on Rn . The leading coefficient is sometimes called the symbol of D(m) .49 Two, or more, differential operators may be composed associatively in the obvious way as D(m) D(n) and the result is, by direct calculation, a new differential operator of order m + n that does not, however, have symmetric coefficients. It can be made symmetric by introducing a symmetrized product between coefficients. For two symbols μ1 ⋅⋅⋅μm2 μ1 ⋅⋅⋅μm1 D1 and D2 , we define (D1 ∨ D2 )μ1 ⋅⋅⋅μm1 +m2 = D1

(μ1 ⋅⋅⋅μm1

μm1 +1 ⋅⋅⋅μm1 +m2 )

D2

(3.118)

using unit weight symmetrization (as usual). An example is clarifying. Example 4 (Product of differential operators). For a simple example, multiply the two opμμ νν erators D1 1 2 𝜕μ1 𝜕μ2 and D2 1 2 𝜕ν1 𝜕ν2 with the result μμ

νν

D1 1 2 𝜕μ1 𝜕μ2 (D2 1 2 𝜕ν1 𝜕ν2 ) μμ

νν

μμ

νν

μμ

νν

= (D1 1 2 𝜕μ1 𝜕μ2 D2 1 2 )𝜕ν1 𝜕ν2 + 2(D1 1 2 𝜕μ2 D2 1 2 )𝜕μ1 𝜕ν1 𝜕ν2 + D1 1 2 D2 1 2 𝜕μ1 𝜕μ2 𝜕ν1 𝜕ν2

(3.119)

The first, lowest-order coefficient is symmetric, but the second and third coefficients need to be symmetrized. The meaning of this actually the following. Consider the last, 49 This a variant of the terminology of symbols in connection with the star product algebra; see Section 3.3.

122 � 3 Algebraic structures in higher spin theory highest-order term. Since the partial derivatives are totally symmetric, only the symμμ νν metric part of the coefficient D1 1 2 D2 1 2 will contribute to the differential operator. However, if one wants to consider the coefficients by themselves, one may want to assure their symmetry. This can be done by defining a symmetric product, in this case (D1 ∨ (μ μ μ μ ) D2 )μ1 μ2 μ3 μ4 = D1 1 2 D2 3 4 possibly with a combinatorial weight. Analogous symmetrizations can be done for the lower-order coefficients. 󳶣 Corresponding to the symmetrization (3.118), one can define a product ∘ between differential operators according to D1

(m1 )

∘ D2

(m2 )

= (D1 ∨ D2 )μ1 ⋅⋅⋅μm1 +m2 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μm +m + lower-order terms 1

2

(3.120)

Under this product, and linear addition, higher-order differential operators form an associative algebra. Higher spin adaptions? Most likely, as we have already seen, in a higher spin application, one would have to tailor definitions like these to the needs at hand, for instance, regarding symmetrization weights, and in particular, regarding trace constraints. In this context, one thing that is quite obvious, but anyway interesting to note, is the fact that higher-order (than first) differential operators do not obey the Leibniz rule, and are therefore not derivations. As pointed out already in [157], this may be related to the difficulties with higher spin interactions, since standard differential geometric techniques cannot be employed. However, higher-derivative operators, acting on themselves through a commutator bracket (adjoint action) do obey the Leibniz rule; see Section 3.6.6 below.

The need to define higher-order differential operators as polynomials in 𝜕μ as in (3.117) (and not just as monomials) is clearly an effect of their properties under composition. This property can be captured by the concept of filtered algebras (see Section 3.1.7). It remains, for us, to explore two further aspects of higher-order differential operators. The first one is the relation to the Weyl algebra, and the second is the relation to the Hamiltonian cotangent algebra.

3.6.4 The Weyl algebra and higher-order differential operators This is actually a point where we make contact with the Vasiliev higher spin theory, or at least move a bit closer to it. First, we want to establish a connection between higherorder differential operator algebras and Weyl algebras. That is quite easy given what we have already done. The Weyl algebra was introduced in Section 3.2.2. Mimicking what we wrote there, in the notation of the present section, the Weyl algebra 𝒜d is the associative algebra formed by arbitrary (real or complex) polynomials in the generators x̂ν and p̂ ν with

3.6 Higher-order differential operators

� 123

μ

n = 1, 2, . . . , d, modulo the nontrivial relations [x̂ν , p̂ μ ] = iℏδν of equation (3.30). Now, if we take p̂ μ = iℏ𝜕μ , this relation is “automatically” satisfied, so to speak, by the simple μ fact that iℏ𝜕μ (xν f (x)) = iℏδν f (x) + iℏxν 𝜕μ f (x). This means that replacing the coefficients Dμ1 ⋅⋅⋅μr (x) in the differential operator (3.117) with arbitrary polynomial coefficients, one does indeed get an element of a Weyl algebra, albeit in a “normal ordered” form (with all the coordinates to the left). To be a little bit more specific, consider one term in (3.117) with r = k with l factors of xμ . It can be written as f μ1 ⋅⋅⋅μk+l xμk+1 ⋅⋅⋅ xμk+l 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μk

(3.121)

where the coefficients Dμ1 ⋅⋅⋅μk (x) are given by the polynomials (actually homogenous polynomials) f μ1 ⋅⋅⋅μk+l xμk+1 ⋅⋅⋅ xμk+l of order l. Normal ordering is not mandatory, and we can just as well consider arbitrary polynomials in xμ and 𝜕μ . Thus, we do indeed get a Weyl algebra. The normal ordered form results from just letting all the derivatives act on all factors of x to the right. The coefficients f μ1 ⋅⋅⋅μk+l can be chosen real or complex.50

3.6.5 Hamiltonian cotangent operators and higher-order differential operators In Section 3.6.2, we outlined two examples of approaches to tentative higher spin algebras. How do they compare? We already noted above, after equation (3.116), that they do not yield the same deformation due to the different structure of action of the derivatives. Intuitively, the Hamiltonian cotangent approach is “classical” in that it works with Poisson brackets, while the higher-order differential approach is “quantum” since it works with space-time derivatives that do not commute with the coordinates. But can anything more specific be said? μ1 ⋅⋅⋅μn2 μ1 ⋅⋅⋅μn1 For that, pick two terms in (3.107b) with parameters ξ1 and ξ2 and compute the corresponding commutator. The derivatives will distribute according to the Leibniz rule, producing new generators with a “spectrum” of derivatives ranging from min(n1 , n2 ) to n1 + n2 − 1. For the leading term, one gets [ξ1 , ξ2 ]μ1 ⋅⋅⋅μn1 +n2 −1 = il

(−il)n1 +n2 −1 {ξ , ξ }μ1 ⋅⋅⋅μn1 +n2 −1 (n1 + n2 − 1)! 1 2

(3.122)

in terms of the Schouten bracket term produced in the Hamiltonian cotangent example (3.116), and in accordance with (3.111). Lower-order terms could be reproduced by replacing the Poisson bracket with the Moyal bracket.

50 In the Vasiliev theory application, they will actually be fields.

124 � 3 Algebraic structures in higher spin theory 3.6.6 A few steps into the theory of higher spin diffeomorphisms We will not attempt to review the theory of higher spin diffeomorphisms in any detail here.51 A inroad to that theory is [192], based on mathematical papers [194, 195]. But let us try to get some intuition on what a study might entail, as the theory is clearly relevant for further advances in higher spin theory. As outlined in [192], the theory can be thought of as running in parallel for the two examples quoted above: the higherorder differential operator approach and the Hamiltonian cotangent approach. Let us focus on the first. As already noted in Section 3.6.3, differential operators of finite order form an infinite-dimensional filtered associative algebra. The filtration is by order of derivatives. With some more mathematical underpinning, they can be thought of as linear operators acting on C ∞ functions on some manifold M, in short, “higher spin” operators like (3.107) with a finite number of derivatives. One may denote such algebras by 𝒟(M), and for elements of filter order m use the notation D(m) of (3.117). As we have seen above, the filtration is such that D(m) D(n) ⊆ D(m+n) . If a graded algebra gr(𝒟(M)) is desired, one may “divide out” order m − 1 from order m operators as in Section 3.1.7. A corresponding infinite-dimensional Lie algebra D(M) may defined as the commutator algebra in 𝒟(M). The higher spin gauge transformations of formula (3.107b) can then be interpreted as the adjoint action of the algebra D(M) (on itself, so to speak). As such, it is an inner derivation (see Section 3.1.8). Clearly, filtrations and gradations carry over to the Lie algebras. Now consider the commutator of two differential operators. From our discussion above, it is clear that [D(m) , D(n) ] ⊆ D(m+n−1)

(3.123)

From this equation, some simple conclusions can be drawn. For m = n = 1, we have a closed algebra, and this is the algebra of spin-2 infinitesimal diffeomorphisms. It may be integrated to the full group of general coordinate transformations. First-order differential operators may be interpreted as vector fields on the manifold. In analogy with the definition of a Lie derivative LU V along a vector field U in equation (3.72), we can write LD(1) D(n) = [D(1) , D(n) ]. Since the filtration is preserved by such an operation, it is an infinitesimal automorphism. That is, however, not true for the higher spin analogue. Defining a higher spin Lie derivative in the direction of a differential operator D(m) as the adjoint action of D(m) , LD(m) D(n) = [D(m) , D(n) ] 51 It requires quite a lot of machinery to be set up.

(3.124)

3.6 Higher-order differential operators

� 125

we find that is does not preserve the filtration, instead it increases the order by m − 1. Infinitesimally, this may not be seen as any big problem. We know that as soon as we pass the spin-2 border, we will activate the full tower of higher spin transformations. A new problem comes when one contemplates an attempt to integrate the infinitesimal Lie algebra transformations to finite Lie group actions via some kind of one-parameter exponentiation of the infinitesimal transformations. Consider, heuristically, acting with exp(ξD(m) ) by iterating (3.124) in a Taylor expansion of the exponential around ξ. At order r, the filtration degree would increase by m(r − 1). This is much more serious than the problem at the Lie algebra level where one at least stays within a finite increase of order of derivatives (it is just a potential infinity). The exponential, as r → ∞, pushes the increase in filter degree to an actual infinity, going beyond the realm of countable number of derivatives. This may be seen as another manifestation of the gauge algebra problem that we alluded to at the beginning of Section 3.6.1. It can also be seen as another aspect of the now well-recognized, nonlocality problem of higher spin interactions at the quartic level. This, admittedly, very hand-waving argument, can be made exact by proper definitions and analysis; see [194, 195] cited above. A way around this “no-go” result is argued in [192]. 3.6.7 Toward the Fradkin–Vasiliev higher spin algebras To move a little bit toward the implementation of these ideas to the Vasiliev theory, let us modify the notation to conform to the literature. Consider generators X̂ A and P̂ B subject to the only nontrivial relations (ℏ taken to 1) [X̂ A , P̂ B ] = iηAB

(3.125)

The indices A, B, C, . . . run from 0 to d and we take a metric ηAB with signature −++ ⋅ ⋅ ⋅+−. The generators X̂ A and P̂ B are considered to be Hermitian under Hermitian conjugation (in the standard fashion). We can then build the (real or complex) Weyl algebra 𝒜d+1 as the universal enveloping algebra of the Heisenberg algebra given by (3.125). We also introduce the corresponding commutator Lie algebra [𝒜d+1 ] obtained from −i[⋅ , ⋅] in the Weyl algebra, consistent with (3.125). With the chosen metric, we can realize the AdS Lie algebra o(d − 1, 2) through the generators M̂ AB = X̂ A P̂ B − X̂ B P̂ A

(3.126)

in the standard way.52 This is clearly a subalgebra of the algebra [𝒜d+1 ]. Another Lie subalgebra is spanned by the three generators 52 See Section 3.4.2 in our Volume 1.

126 � 3 Algebraic structures in higher spin theory X̂ A X̂ A

1 ̂A ̂ (X PA + P̂ A X̂ A ) 2

P̂ A P̂ A

(3.127)

These generators span the Lie algebra sp(2). The generators of sp(2) commute with the generators of o(d − 1, 2) and together they form a so-called Howe dual pair. To get to the Vasiliev higher spin algebras, two ideas are put together. One is the gauging idea, the idea of “localizing”—so to speak—a global symmetry algebra. To do that for spin 2, the most effective way is to follow the approach of Kibble in gauging the Poincaré group.53 In this approach, the gravitational fields are described by two 1-forms e and ω with indices in the tangent space. These tangent space indices are contracted into abstract generators of the Poincaré group algebra. This can also be done for the AdS space-time symmetry algebra (see Section 7.1.6). If one wants to extend this to a theory containing an infinite tower of higher spin fields, the idea is to try to extend the Poincaré or AdS algebra to an infinite-dimensional algebra containing these algebras as subalgebras. Knowing what we know now, the Weyl algebra is a natural candidate.54 However, the Weyl algebra, taken off the shelf, contains too many generators, built as it is from arbitrary polynomials in the basic generators X̂ A and P̂ B . We need some organizing principle to bring down the set of generators to something that matches the free higher spin theory.55 To do that, one can argue in the following way. In Section 5.7.3 in our Volume 1, we described spin s with a collection of extended frame-like fields ωμ,a(s−1),b(t)

with 0 ≤ t ≤ s − 1

(3.128)

The transformations are given by δωμ,a(s−1),b(t) = 𝜕μ ξa(s−1),b(t) + hμc ξa(s−1),cb(t)

(3.129)

The idea is to collect the fields into Weyl algebra generators, and likewise for the parameters. To do that, there must be some translation between the A, B indices and the a, b 53 For the streamlined approach, see Section 4.6.3 of our Volume 1. For the history, see Section 2.5 of the present volume, where also the contributions of pioneering work of Utiyama and clarifying work of Sciama are outlined. 54 I. Bengtsson and the present author played with this idea in 1983–1985, but we did not know about Weyl algebras, so we did not get far. In discussions at that time, R. Marnelius pointed out that there were no conserved quantities associated with the would be higher spin gauge parameters. This confused me to the point of not pursuing the idea further. It still confuses me, to be honest. He was right, in a way. I think what he wanted to say was that there are no global, measurable charges connected to the higher spin symmetries. That is what the Weinberg soft emission theorem and the Coleman–Mandula theorem say, in short. 55 E. S. Fradkin and M. A. Vasiliev did that in certain way in the paper [196]. Some details of this can be found in Section 8.3.

3.6 Higher-order differential operators

� 127

indices. Putting d = 4, we see that the operators M̂ AB generate the AdS algebra o(3, 2) in a 5-dimensional notation. The indices a, b are, however, 4-dimensional. Remembering the translation M̂ AB → {M̂ ab , M̂ 4a } gives the clue.56 For spin 2, the fields ωμ,a and ωμ,a,b are collected into on 1-form field ωμ,A,B , anti-symmetric in A, B. This generalizes to higher spin in that the full collection (3.128) of fields for spin s can be represented by ωμ,A(s−1),B(s−1) = ωμ,|A1 B1 |⋅⋅⋅|As−1 Bs−1 |

(3.130)

where the first notation highlights the separate symmetry in Ai and Bi indices and the second notation highlights the antisymmetry within each pair Ai Bi . The fields are traceless in both index sets Ai and Bi . In terms of Young tableaux, we have k k

→

k k

⊕

k k−1

⊕ ⋅⋅⋅ ⊕

1

k

⊕

k

(3.131)

where the left-hand side corresponds to the fields in (3.130) and the right-hand side to the fields in (3.128). This is an example of a branching rule, in this case for so(d + 1) to so(d). We will not prove it here, but the intuition is to generalize the index split M̂ AB → {M̂ ab , M̂ 4a } to higher-order tensors. The next step in understanding of the relation between the Vasiliev fields and the Weyl algebra is to note that the fields in (3.130) correspond to special forms of coefficients for Weyl algebra generators; namely, those that are completely symmetric, and traceless, separately in the indices Ai and Bi , and antisymmetric in the exchange of any Ai with any Bj . This is precisely the symmetry of the Young tableaux on the left-hand side of (3.131). What has to be done is to study the subalgebra of the Weyl algebra given by the elements that commute with sp(2). That yields a subalgebra of elements corresponding to all tracefull coefficients of the form of the left-hand side of (3.131). Finally, dividing out a certain ideal, one obtains a subalgebra corresponding to all traceless coefficients. It turns out, a bit surprisingly, that the AdS higher spin algebra is generated by powers of the basic generators M̂ AB . This is indeed the simplest guess one could have made. An higher spin algebra from the Weyl algebra Return to the sp(2) subalgebra of 𝒜d+1 with generators given in (3.127). One may convince oneself that there is an associative subalgebra of 𝒜d+1 of all elements that commute with sp(2). This subalgebra is a centralizer 57 and is denoted by 𝒞𝒜d+1 (sp(2)). There is a corresponding commutator Lie algebra [𝒞𝒜d+1 (sp(2))],

56 For instance with A, B, . . . ranging over 0, 1, 2, 3, 4 and a, b, . . . over 0, 1, 2, 3. For details, see Section 7.1.2. 57 See Section 4.8 in the textbook [174].

128 � 3 Algebraic structures in higher spin theory

and real and complex forms can be considered. The real form of this algebra is called the off-shell higher spin algebra with varying notation (see [197] for one variant). To get a grip on the elements of 𝒞𝒜d+1 (sp(2)) let us first Weyl-order them and consider polynomials ̂ The corresponding symbol is denoted by G(X, P) and can be represented in the notation of (3.130), GW (X,̂ P). G(X, P) = ∑ G|A1 B1 |⋅⋅⋅|At Bt |At+1 ⋅⋅⋅Ar PA1 PA2 ⋅ ⋅ ⋅ PAr X B1 X B2 ⋅ ⋅ ⋅ X Bt 1≤t≤r

(3.132)

The “asymmetry” in the expression between P’s and X’s is deliberate, and is meant to correspond to two-row Young diagrams. The choice of assigning the P’s to the upper row is conventional, and that “asymmetry” may be avoided by working with oscillators (see the next box). The commutativity of polynomials GW (X,̂ P)̂ with elements of sp(2) translates in symbol language to the three equations: XA

𝜕G =0 𝜕PA

XA

𝜕G 𝜕G = PA A 𝜕X A 𝜕P

PA

𝜕G =0 𝜕X A

(3.133)

As they come from a non-Abelian Lie algebra, these equations are not independent of each other. The effect of either the first, or the third equation, is antisymmetry in each index pair |Ai Bi |. The effect of the second equation is r = t. All in all, the coefficient tensors in (3.132) are described by two-row rectangular Young tableaux.58 We now begin to see the relation to the higher spin fields gauge fields in (3.130). However, it remains to impose tracelessness. Here, again, sp(2) plays a role. From the representation (3.127) of sp(2), we see that elements of the algebra that contain a factor sp(2) correspond to traces. For instance, a factor of P̂ A P̂A corresponds to factor of ηAi Aj with i ≠ j in the tensor G|A1 B1 |⋅⋅⋅|Ai Bi |⋅⋅⋅|Aj Bj |⋅⋅⋅|At Bt | . To make this more exact, note that there are two ideals in the centralizer algebra 𝒞𝒜d+1 (sp(2)), namely those consisting of elements that can be written as sums of products of elements in sp(2) and 𝒜d+1 either as sp(2)𝒜d+1 or 𝒜d+1 sp(2). These elements must be divided out. The result is called is called the on-shell higher spin algebra. To clinch the arguments made here, note that when working with the symbol formulation of the algebras one should employ the Moyal star product.

Now all this may seem quite abstract, but there is a remarkable fact, alluded to above, that brings it all down to earth again. The centralizer 𝒞𝒜d+1 (sp(2)) is isomorphic to the complex associative algebra spanned by all Weyl-ordered powers of the o(d − 1, 2) generators of (3.126). This may be shown by direct computation. For the reader who wants to delve deeper into these topics, we recommend the excellent article [193] from which we have drawn inspiration. In the application to the Vasiliev theory, a more symmetric formulation in terms of oscillators is used. Oscillator and star product realization The operators X ̂ A and P̂ A may be linearly recombined into creation and annihilation operators in the standard way Y1̂ A =

58 Tensors of gl(d + 1).

1 (X ̂ A + i P̂ A ) √2

Y2̂ A =

1 (X ̂ A − i P̂ A ) √2

(3.134)

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Then we have [YiA , YjB ]⋆ = ϵij ηAB

(3.135)

proceeding at once to a symbol and Moyal star-product formulation. The multidimensional, differential and integral forms of the star product can be worked out along the lines of Section 3.3.4 above. While ϵij is always the two-dimensional antisymmetric symbol raising and lowering indices in the standard fashion Y i = ϵ ij Yj and Y i = Yj ϵji , the “metric” may be any o(n, m) symmetric form with n + m = d + 1. The Weyl algebra 𝒜n+m is the vector space of all polynomials in the variables (YiA , YjB ) with product the Moyal star product taken modulo the star commutator (3.135). That is, it is the universal enveloping algebra of the Heisenberg algebra given by (3.134) (and the trivial commutators). A generic element is A

A

B

B

f (Y ) = ∑ fA1 ⋅⋅⋅An ,B1 ⋅⋅⋅Bm Y1 1 ⋅ ⋅ ⋅ Y1 n Y2 1 ⋅ ⋅ ⋅ Y2 m

(3.136)

n,m

All the theory reviewed above can be restated. In particular, the sp(2) generators are given by the symmetric elements mij = YiA YjA and the antisymmetric o(n, m) elements by M AB = 21 Y iA Yib . Gauge fields are ∞

A

A

B

B

ωμ (Y ; x) = ∑ ωμ;A1 ⋅⋅⋅Ak ,B1 ⋅⋅⋅Bk Y1 1 ⋅ ⋅ ⋅ Y1 k Y2 2 ⋅ ⋅ ⋅ Y2 k k=0

(3.137)

where the spin s is related to the summation index as s = k +1 (compare to (3.130)). The Vasiliev d-dimensional theory on AdSd may be developed on this ground as done in [197].

4 General theory of interactions In this chapter, we will discuss general approaches to the problem of constructing interactions in field theory. In the corresponding Chapter 4 in our Volume 1, we discussed the well-established methods to construct interactions for lower spin massless fields. These methods, and developments from them, have naturally been a source of concepts, methods and inspiration for attempts at the higher spin problem. We will start with the group manifold approach that was invented in connection to supergravity theory in the 1980s. Further on, in Chapter 7, we come to the closely related free differential algebra approach, which is employed in the Vasiliev theory together with the unfolding method. This can be seen as following the “gauging” road to interactions, that we wrote about in the introductory Chapter 1, although with an increasingly stronger reliance of differential geometric concepts and methods, and in general to the need for deformation. Then we will turn to the “classic” Fronsdal program as it was developed by Burgers, Berends and van Dam (BBvD) in connection to the spin-3 interaction problem. The actual concrete results obtained within this approach will be reviewed in Chapter 5.1 The Fronsdal–BBvD method is a Noether coupling technique. It is “minimal” in the sense that it does not introduce any more concepts or formalism than is absolutely necessary to get started on the higher spin interaction problem in a covariant Minkowski context. But, as the approach itself showed, the problem calls for more elaborate concepts and more general formalism and techniques.2 This leads over to the BRST based “antifield” method, also called the BRST-BV method or the antifield/antibracket method. This approach was largely developed from the BRST and BRST-BV methods for gauge theory quantization by G. Barnich, M. Henneaux and F. Brandt, and adapted to the problem of constructing interactions in field theory with higher spin as one major example. References will be provided as we continue.

4.1 The group manifold approach The group manifold approach—to gravity and supergravity in particular—is closely related to gauge theory of gravity, and to the MacDowell–Mansouri/Stelle–West approach [102, 103]. This is not always apparent in the literature from the heydays in the early 1980s, but it was made explicit already in the paper by Y. Ne’eman and T. Regge [198] (letter version [199]) that initiated the approach. Perhaps it was too obvious to note in the 1 The review of the general Fronsdal–BBvD analysis of interactions starts in Section 2.6. 2 Unless perhaps if one chooses to work within the Dirac light-front program, which makes do with quite simple tools. The light-front approach—part of the Dirac program—is not treated in the present chapter as we will treat it in detail in Chapter 6. https://doi.org/10.1515/9783110675528-004

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later literature. There are now some very good reviews in [200, 201, 202], and inspired by these, we will here approach the subject with higher spin applications in mind. The early “self-contained presentation” [203] is also well worth reading as well as the more detailed and mathematically more stringent first reference [198] on the subject. The group manifold theory itself is applicable to any Lie group G, but in the application to gravity and supergravity, it is the Poincaré group and its neighbor the AdS group and the corresponding supergroups that are relevant. It is then quite interesting to compare the result of such an application to the ordinary Kibble-style gauging of the spacetime groups. One item in particular, that we will focus on, is the difference between diffeomorphisms in the group manifold and gauge transformations, an instance of which we have seen in connection with the gauge theory approach to gravity reviewed in Section 4.6 of our Volume 1. Here, we will have the opportunity to clarify this in a more general context. It should also be noted that, as the approach was applied to the construction of extended supergravity theories and to 11-dimensional supergravity, it was developed into what was initially called Cartan Integrable Systems (CIS). In these, higher-order forms were included, something that was needed in order to accommodate extended and higher-dimensional supergravity theories. The approach was later renamed Free Differential Algebra (FDA), a concept then already known and studied in the mathematics literature [204]. It is the FDA methodology that was adapted—in a certain way—by Vasiliev to AdS higher spin theory. 4.1.1 Maurer–Cartan theory To get going, we return to the Maurer–Cartan theory developed in Section 3.5. We will first change notation for the group indices in order to conform to common practice in the literature. Thus, lower Latin case letters a, b, c, etc. will be replaced by uppercase letters A, B, C, etc.3 Let us therefore record the two equivalent ways of capturing the Lie algebra structure from formulas (3.102), which now take the form 1 d ∧ ΩA + f ABC ΩB ∧ ΩC = 0 2 [XA , XB ] = f CAB XC

(4.1a) (4.1b)

As before, the 1-forms ΩA and vectors XA are left invariant and fixed by the group geometry, and are therefore inadequate to describe any dynamics. We can think of ΩA and XA as defining a background, for instance, a Poincaré invariant background. In the Vasiliev theory, an equation of the type (4.1a), defines the AdS background. 3 This also frees the lowercase Latin letters for use as tangent space indices for the space-time groups as has been our convention in Volume 1.

132 � 4 General theory of interactions The fields depend on the group manifold coordinates that we will denote by z𝒜 and if we need to emphasize the coordinate basis of the 1-forms and vectors, we can write ΩA = ΩAℬ dzℬ and XA ℬ 𝜕ℬ , respectively. Clearly, we can think of these fields as coframe and frame fields, or as a pair of inversely related vielbein fields, still however, with a fixed functional dependence on z. To highlight that aspect, we can write ΩAℬ = eAℬ and XA ℬ = eAℬ . The discussion and formulas from Section 3.5.5 is entirely applicable here as we have only changed notation.4 Introduce also the notation G for the general—background geometry—group considered and ISO(3, 1) for the example of the four-dimensional Poincaré group. A direct way to derive the Maurer–Cartan formula from the Lie-algebra and vice versa In Section 3.5.5, we mentioned a way to show the equivalence of the two representations of the Lie algebra structure given in equation (4.1). There is a more direct way to derive the Maurer–Cartan equation from the Lie algebra commutator. First, use the representation XA = eA 𝒜 𝜕𝒜 for the Lie algebra generators to derive the following differential conditions on the vielbeins: e[A𝒜 eB]ℬ 𝜕𝒜 eℬC = −f CAB

(4.2)

Then by computing dΩA from ΩA = eA ℬ dzℬ , one indeed finds the Maurer–Cartan formula. To add a little bit more context to this computation, one could note that in terms of local coordinates z𝒜 on the group manifold, the generators XA are indeed a basis of tangent vectors in tangent space, as the 1-forms ΩA are a basis in cotangent space. It may be interesting to invert the computation, deriving the Lie algebra of the tangent vectors from the Maurer–Cartan equations.

In order to accommodate dynamics, we think of deforming around the background G, and consequently, introduce dynamical 1-form fields σ A and vector fields ϒA . Instead of the formulas (4.1), we will then have 1 RA = d ∧ σ A + f ABC σ B ∧ σ C ≠ 0 2 [ϒA , ϒB ] = f CAB ϒC − RC (ϒA , ϒB )

(4.3a) (4.3b)

where the deviation from the background is given by curvatures RA . These two equations are again equivalent expressions for the same underlying geometry, and normally one concentrates on the first equation (4.3a) for the 1-forms.5 4 One item of notation to note is that duality brackets such as ⟨ωA , XB ⟩ are often written as ΩA (XB ) in the literature and thought of as “evaluating along” the generators XB .

5 The notation in the second term on the right-hand side of (4.3b) signifies contracting the vectors into the 2-form curvature, or phrased according to the previous footnote: “evaluating the curvature along the vectors”.

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Bianchi identities for the curvatures RA are derived by computing d ∧ RA from (4.3a) using d 2 = 0 and the Jacobi identities for the structure constants. The result is d ∧ RA + f ABC σ B ∧ RC = (∇R)A = 0

(4.4)

where we have used the definition of the covariant derivative ∇ from formula (4.9) below. Is the Lie group manifold flat or curved? Intuitively, one may not think of a Lie group manifold as flat, but if the curvature—as defined here—is taken as a measure of flatness, the manifold is flat. Perhaps “rigid” would be a better designation. The situation is clarified upon considering the group manifolds corresponding to the groups SO(3, 2) or SO(4, 1). After the processes of “factorization” and “horizontality” (see below) has been enforced, either by hand or through appropriate field equations following from a “gravitational” action, the conventional curvature defined on the relevant coset spaces will be constant.

We can now apply this formalism to the Poincaré group. The group coordinates z𝒜 are specialized to x μ and yμν where x are for the translation subgroup and y for the Lorentz subgroup SO(3, 1). Correspondingly, the tangent space indices and 1-form indices split up as σ A → {ea , ωab }. Formally, plugging the structure constants for the Poincaré algebra into (4.3a), one gets T a = d ∧ e a + ωa b ∧ e b

Rab = d ∧ ωab + ωac ∧ ωcb

(4.5) (4.6)

where we have used notation suggestive of torsion T a and Lorentz curvature Rab both of which are 2-forms on the group manifold. These two formulas have the precise outward appearance of the formulas for the torsion and curvature that appear upon following the Kibble gauging of the Poincaré symmetry of Minkowski space-time. We reviewed this in Section 4.6 in our Volume 1, and the formulas (4.5)–(4.6) and their derivation can be found there in Section 4.6.3.6 But the similarity of the formulas is somewhat deceptive, because the formulas (4.5) and (4.6) really live in the full 10-dimensional Poincaré group manifold, rather than the 4-dimensional space-time. To make this explicit, we can split the group manifold coordinates z𝒜 into components along the coordinates x μ and yμν of the translation and Lorentz subgroups. In detail, we have σ Aℬ dzℬ = eaμ dx μ + ωab μ dx μ + eaμν dyμν + ωab μν dyμν where all the fields depend on x μ and yμν . 6 Details on Kibble’s original paper can be found in the historical Section 2.5.2 above.

(4.7)

134 � 4 General theory of interactions A drastic way of reducing this abundance in field components and coordinate dependence would be to just declare yμν = 0 and dyμν = 0, but then not very much would have been gained. Therefore, in the group manifold approach a more sophisticated embedding of a four-dimensional space-time manifold into the 10-dimensional Poincaré group manifold was devised. In particular, when it came to higher dimensional, extended and gauged supergravities, the method offered a structured framework within which to approach the problems of constructing the theories. As this is not our main focus here, we refer the reader to the review literature [200, 201, 202] that applies the method to supergravity. Instead, we will discuss the relationship between gauge transformations and general coordinate transformations within this formalism. We will return to the question of getting back to ordinary space-time in Section 4.1.3 below. 4.1.2 Gauge vs. general coordinate transformations The setup described here is reminiscent of Yang–Mills gauge theory. Indeed, ordinary spin-1 Yang–Mills theory is perhaps the simplest instance of this scheme. Here, we will first discuss what would naturally be considered as gauge transformations in the present context, and then how such relate to coordinate transformations in the group manifold. This, in its turn, hinges on the conceptual status that can be attached to the 1-form fields σ A . Let us continue the story from where we left it around formulas (4.3) above. The field σ can be considered as a G-valued 1-form σ = σ A XA , and as already noted, it will have a nonvanishing curvature R(σ) = d∧σ +σ ∧σ ≠ 0, which is just formula (4.3a) again. This formula certainly reminds us of a Yang–Mills field-strength on the one hand—if we think of the σ as gauge fields, but also on the other hand, of a gravitational curvature— if we think of σ as a connection. This latter interpretation is supported by the Poincaré example given above. But as we saw in the example, the situation is not that simple. At the present stage of generality, the field σ can be considered as a “generalized connection”,7 but not on a fiber bundle, but rather on the group manifold G. If we write out the formula σ = σ A XA making the group manifold coordinate differentials explicit, we get σ = σ A XA = σ Aℬ XA dzℬ

(4.8)

This way of writing may support the interpretation of the field σ as a connection if we think of the index ℬ as referring to a curved deformation of the group manifold G, sometimes denoted by G̃ in the original literature, and the index A as referring to the corresponding tangent space G. The same reasoning may also support the

7 It was designated a “pseudo-connection” in some of the original literature; see [203], page 7.

4.1 The group manifold approach � 135

interpretation of the field σ as a “vielbein” or “coframe”. Thus, at this level of generality, it seems that the field σ straddles both interpretations, as is also clear from the Poincaré example above.8 However, one should perhaps not press these interpretations too far; components of the field σ take on the proper roles of coframe or connection only after the processes of factorization and horizontality has taken place. In order to define the concept of a gauge transformation, we first define a covariant derivative. Consider a p-form field ψa with an index a in some representation vector space of the group G and the corresponding representation matrices Dab (XA ). Then the covariant derivative is defined, as usual, as ∇ψa = dψa + σ A Dab (XA ) ∧ ψb . In the adjoint representation, which we are primarily interested in here, we have ∇ψA = dψA + f ABC σ B ∧ ψC

(4.9)

Based on this covariant derivative, we can define an infinitesimal gauge transformation of the coframe field σ A with parameter ξ A as δσ A = ∇ξ A

(4.10)

The definition is reasonable, because entirely standard computations yield the gauge covariance of the curvature δRA = f ABC RB ξ C

(4.11)

Again, everything look very much like ordinary Yang–Mills gauge theory, which is of course intentional, and will guarantee that both Yang–Mills theories and versions of gravity and supergravity—more precisely, gauge versions of the latter theories—can be derived within this methodological framework. Let us next consider infinitesimal coordinate transformations. This should be a reparametrization of the group coordinates, which we write as δzℬ = ζ ℬ . The coframe field then transforms as covariant vector (with respect to the base manifold G)̃ δσ A = (δσ A𝒩 )dz𝒩 = −(ζ ℳ 𝜕ℳ σ A𝒩 + (𝜕𝒩 ζ ℳ )σ Aℳ )dz𝒩

(4.12)

This formula may be rewritten in terms of the anholonomic transformation parameter ζ A = σ A𝒩 ζ 𝒩 (such a parameter can be compared to a gauge parameter) δσ A = −(𝜕𝒩 ζ A + ζ ℳ (𝜕ℳ σ A𝒩 − 𝜕𝒩 σ Aℳ ))dz𝒩

(4.13)

8 This discussion may also explain why the original literature sometimes argues in terms of a “deformation” of the initial group manifold when introducing the σ field, but sometimes just introduces it as a 1-form with nonzero curvature. This is clearly an important conceptual issue that one should think through. It is related to the practical question of whether deformation is done perturbatively or not.

136 � 4 General theory of interactions The first term is just dζ A . To further rewrite the second term, consider the 2-form dσ A = 𝜕ℳ σ A𝒩 dyℳ ∧ dy𝒩 . Contracting with two ϒ vectors, we get9 1 dσ AFG ≡ dσ A (ϒF , ϒG ) = iϒ (iϒ (dσ A )) = ϒF ℳ ϒG𝒩 (𝜕ℳ σ A𝒩 − 𝜕𝒩 σ Aℳ ) 2 1 = σF ℳ σG𝒩 (𝜕ℳ σ A𝒩 − 𝜕𝒩 σ Aℳ ) 2

(4.14)

where we have used the fact that the vectors ϒ considered as frame fields, are the inverses of the coframe fields σ, i. e., ϒF ℳ = σF ℳ . Next, compared to (4.13), we get δσ A = −dζ A − 2ζ F σ G dσ AFG

(4.15)

This formula is an instance of the general formula (3.80), which can be seen by making the identifications diζ σ A = dz𝒩 𝜕𝒩 ζ A and iζ dσ A = 2dz𝒩 ζ F σ G𝒩 dσ AFG while noting that δσ A = −Lζ σ A in our conventions from our Volume 1.10 Then finally adding and subtracting the term f AFG σ F ∧ ζ G in order to fill out the first term to a covariant derivative and the second term to a full curvature, we get δσ A = −∇ζ A + 2RAFG σ F ∧ ζ G = −∇ζ A − iζ RA

(4.16)

We have already seen an instance of this formula in connection with the Kibblestyle gauging of the Poincaré algebra leading to the Riemann–Cartan space-time U4 with torsion. This was discussed at some length in Section 4.6 in Volume 1. The exact correspondence between what was discussed there, and how to interpret formula (4.16) in that connection, we will return to after a few more items have been clarified. Suffice it to note here, that what we have done in formula (4.16) is to rewrite a diffeomorphism in the manifold in a way that separates out a term that can be quite naturally interpreted as a gauge transformation, and a curvature term, precisely because of the extra structure afforded by the Lie group structure. When the curvature term iζ RA vanishes, we see that the diffeomorphism reduces to a gauge transformation. In this case, one says that the curvature is horizontal along the directions ϒA in ζ A ϒA . 4.1.3 Horizontality and factorization The main application of the group manifold approach was to the construction of gauged extended supergravity theories through gauging a supersymmetry group. However, as seen already in the simple case of gauging the Poincaré group in this way, one gets “too many” curvatures and “too large” a functional dependence. Eventually one wants to get

9 The inner product was introduced in Section 3.5.2. 10 Which differ by an overall sign from what is common in the group manifold literature.

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back to an ordinary space-time field theory, with something gained, namely the action, field equations and symmetries for a new model. In the case of the Poincaré group, the bottom line will be that “horizontality” removes the corresponding 1-form directions dyμν , then “factorization” follows and takes care of dropping the dependence on the Lorentz coordinates yμν . Underlying these phenomena is the semidirect product nature of the Poincaré group ISO(3, 1) and its Lie algebra, which makes it natural to realize it as a transitive action on the quotient coset space ISO(3, 1)/SO(3, 1) (see Section 3.9.3 in our Volume 1). But the procedure is applicable also to semisimple groups such as the AdS group, so we will discuss it in some generality. In such a general setting, the idea is to first construct a model on the group manifold G and then—preferably through field equations derived from an action—reduce the model down to a fiber bundle (G/H, H) with base space the quotient G/H and fiber H, where H is a suitable subgroup of G. It would take us to far to carry through this program in any detail, but we will outline the symmetry aspects of it, stopping short of the action and field equations. Consider equation (4.16) that describes all symmetry transformations of the theory. Split the indices so that A, B, . . . run over the full group G, while H, H ′ , . . . run over a subgroup H (SO(3, 1) in the Poincaré or AdS cases) and K, K ′ . . . run over the coset space G/H, i. e., Minkowski or AdS space-time in the Poincaré or AdS case. The curvature 2-from is RA = RABC σ B ∧ σ C . When RABH = 0, the curvature is said to be horizontal on H. Then it follows from (4.16) that diffeomorphisms along the H directions reduce to gauge transformations since the curvature term is zero. With parameters ζ H , we get δσ H = −dζ H − f HH ′ H ′′ σ H ∧ ζ H ′

δσ K = −f KK ′ H σ K ∧ ζ H ′

′′

(4.17a) (4.17b)

A term of the form f HKH ′ σ K ∧ ζ H in the first equation can be dropped since for semisimple Lie algebras the structure constants f HKH ′ can be set to zero through an appropriate choice of basis. For the Poincaré algebra, there is no such structure constant, and neither for the AdS algebra in the standard form. ′

Specializing to the Poincaré group and the AdS group Particularizing the structure constants f ABC with indices A, B, C running over H (indices H) and G/H (indices K ) one may specialize to the Poincaré case. Then indices H may be written as antisymmetrized pairs ab (the corresponding 1-form fields being spin connections ωab ) while the indices K are a (the corresponding 1-form fields being vierbeins ea ). The equations one finds are precisely those of Poincaré gauge theory reviewed in Section 4.6.3 of our Volume 1. To put the ordinary torsion T a 2-form to zero, an action is needed. The AdS case works similarly.

138 � 4 General theory of interactions For the gravity and supergravity, actions can be constructed so that the field equations lead to horizontality and factorization. All this is reviewed in references [200, 201, 202].

4.2 The Fronsdal–BBvD general theory of deformation We will here continue the exposition of the BBvD general analysis from Chapter 2, as it is more technical than historical, and its subsequent elaboration by mathematicians. 4.2.1 The BBvD general analysis analyzed Let us now continue the analysis of the contents of the BBvD-G paper from where we left it in Section 2.6.2. There we saw that the formalism set up by BBvD is different from the general gauge theory formalism that we reviewed in Section 3.1 of our Volume 1.11 Comparing the formalism and the concepts involved will be illuminating for the understanding of the higher spin interaction problem. The general analysis performed in BBvD-G is, on the face of it, in a single spin formalism. A gauge field is denoted by ϕ. It can carry an internal index so that a multiplet of spin s fields is possible, as in Yang–Mills theory. It is not stated explicitly whether ϕ is to be thought of a as a single spin s field or if it can stand for a collection of fields with different spin. The paper says that ϕ can represent a “family of fields”, but whether these may have different spin or not, is not clear. However, it is clear that it can be done in concrete cases. Worked out examples for all possible cubic interactions s1 -s2 -s3 for spins si ≤ 3, are listed in the Burgers thesis BBvD-T and in BBvD-G. It should also be said that the higher spin fields are in all cases taken as symmetric tensor and tensor–spinor fields. In any way, the BBvD formalism is certainly powerful enough to capture the inherent difficulties of the higher spin interaction problem. We know that any consistent higher spin gauge theory must contain an infinite tower of higher spin fields.12 Indeed one result of the BBvD-G analysis is that a pure spin-3 gauge theory is not possible. We have already seen examples of formalisms capable of maintaining infinite towers of higher spin fields, and we will return to a discussion of this in Section 4.4.5 below on the mathematical elaboration of the BBvD theory. As may have become clear from the work we have already discussed, the interaction problem can be approached from two sides. It can be approached via an iterative Noether coupling procedure, as in Section 3 of the BBvD-G paper, and it can be approached by studying possible higher spin gauge algebras ab initio. Eventually, the 11 Where we were following the treatment of Henneaux and Teitelboim in [175]. 12 This insight is sometimes referred to the Vasiliev approach to higher spin, but it really goes back to Fronsdal’s work in the second half of the 1970s and was clear by 1984.

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two approaches must be merged and consistent with each other. Indeed, for low spin, the cubic order Noether procedure allows us to guess and check the full gauge algebra in one step. Therefore, in the fourth section of BBvD-G, the authors perform a general analysis of the kind of gauge algebras that one may encounter for higher spin gauge fields. The BBvD analysis of general gauge algebras Consider the gauge transformations δξ ϕ = T (ϕ, ξ) of (2.132) where T is linear in the parameter but generally not in the field. Such transformations are substitutions and are therefore associative so that the Jacobi identity ∑ [δξ1 , [δξ2 , δξ3 ]]ϕ = 0

cyclic

(4.18)

holds for infinitesimal transformations. BBvD also assumes that the algebra is closed [δη , δξ ]ϕ = δα ϕ

(4.19)

This means that the transformation on the right-hand side is “of the same kind”, loosely speaking, as those commuted on the left-hand side. As the authors discuss, to achieve this one may sometimes have to enlarge the set of transformations. This is also thoroughly discussed in [175], Section 3.1. See also Section 3.1 in our Volume 1. According to BBvD, that complication does not appear for higher spin. Now two cases present themselves: (i) α does not depend on ϕ, and (ii) α does depend on ϕ. In the first case, it makes no sense to assume ϕ dependence for the parameters on the left-hand side of (4.19) either. The algebra is field independent, and one may define [ξ1 , ξ2 ] through [δξ1 , δξ2 ]ϕ = δ[ξ1 ,ξ2 ] ϕ

(4.20)

∑ [ξ3 , [ξ2 , ξ1 ]]ϕ = 0

(4.21)

The Jacobi identity (4.18) then becomes cyclic

The parameters form a Lie algebra that is independent of the fields. For the second case, one could write instead of (4.20) [δξ1 , δξ2 ]ϕ = δC(ϕ,ξ1 ,ξ2 ) ϕ

(4.22)

But BBvD initially generalize this by assuming that also the parameters on the left-hand side are field dependent, even though they are not so in the higher spin free field theory that one is deforming. To capture this, the transformations are written as δπ ϕ = T (ϕ, π(x, ϕ(x)))

(4.23)

It then makes sense to define the commutator [π1 , π2 ] of two such field dependent parameters through [δπ1 , δπ2 ]ϕ = δ[π1 ,π2 ] ϕ

(4.24)

in analogy with (4.20). Computing the left-hand side of (4.24), one may extract an expression for [π1 , π2 ],

140 � 4 General theory of interactions

[π1 , π2 ] = Dϕ (π1 |T (ϕ, π2 )) − Dϕ (π2 |T (ϕ, π1 )) + C(ϕ, π1 , π2 )

(4.25)

where the first term is the derivative of the first parameter π1 in the direction of change in ϕ induced by the second parameter, and vice versa for the second term. This directional derivative is defined by Dϕ (π(ϕ)|T (ϕ, ω)) = lim

λ→0

π(ϕ + λT (ϕ, ω)) − π(ϕ) λ

(4.26)

In the general case, the Jacobi identity ∑cyclic [π, [π2 , π3 ]]ϕ = 0 gets considerably more complicated, but for higher spin, where the parameters π1 , π2 , π3 are taken as field independent, so that only C(ϕ, π2 , π1 ) occurs on the right-hand side of (4.24), one gets ∑ {C(ϕ, π3 , C(ϕ, π2 , π1 )) − Dϕ (C(ϕ, π2 , π1 )|T (ϕ, π3 ))} = 0

cyclic

(4.27)

This, already quite complex, instance of the Jacobi identity, reflects the fact that although the parameters are taken as field independent, the structure “constants” of the algebra are not so.13

After this general analysis, BBvD turns to the actual task of extracting algebraic information from the first-order (in the field) gauge transformations of formula (2.132) with field independent ξ’s. Therefore, only C(ϕ, π1 , π2 ) appears in the right hand side of (4.24). The parameter C(ϕ, ξ1 , ξ2 ) is itself expanded in powers of g, C(ϕ, ξ1 , ξ2 ) = gC0 (ξ1 , ξ2 ) + g 2 C1 (ϕ, ξ1 , ξ2 ) + ⋅ ⋅ ⋅

(4.28)

and the lowest order of the Jacobi identity (4.27) becomes g 2 ∑ {C1 (δζ , η, ξ) + C0 (C0 (η, ξ), ζ )} = 0 cyclic η,ξ,ζ

(4.29)

One then assumes that one has found a first-order deformation L1 of the Lagrangian, unique up to field redefinitions. From such an L1 , one can derive a first-order deformation T1 to the gauge transformation.14 Such a T1 depends on the field redefinitions, and thus may look different depending on how L1 has been chosen. Furthermore, T1 also depends on gauge parameter redefinitions ξ → ξ + gΞ(ξ, ϕ). Field and parameter redefinitions made concrete Perhaps a closer look at what kinds of field redefinitions are possible will be useful at this stage. Generically, one writes field and parameter redefinitions to first order in g as

13 The appearance of a directional derivative Dϕ in the Jacobi identity, although there is no such in the commutator (4.24) in this case, is due to the commutator being used twice in computing the identity. 14 See Section 2.6.2. For details in the Yang–Mills case, see Section 4.4 in our Volume 1.

4.2 The Fronsdal–BBvD general theory of deformation

ϕ → ϕ + gΦ(ϕ, ϕ)

and

ξ → ξ + gΞ(ξ, ϕ)

� 141

(4.30)

respectively. This can be made a bit more concrete by dimensional analysis for particular spin. For spin 1, this means that Φ cannot be more than linear in ϕ, and that Ξ is linear in ξ and independent on ϕ.15 For spin 2, there are nontrivial field redefinitions quadratic in ϕ and parameter redefinitions linear in ξ and ϕ. Any such field redefinition applied to the free spin-2 action produces fake cubic interactions with two derivatives. For spin 3, there must be a derivative in both the field and parameter redefinitions. One could write ϕ → ϕ + gΦ(ϕ, ϕ, 𝜕) and ξ → ξ + gΞ(ξ, ϕ, 𝜕). It is quite obvious how the pattern goes on for higher spin. In all cases, such field redefinitions applied to the free actions produce fake cubic interactions with the correct number of derivatives. A dimensional correct, cubic level, generic formula for a spin s field redefinition may be written as ϕ′s = gs f (ϕs , ϕs , 𝜕n ) where n = s − 2

(4.31)

The derivatives are distributed over the two fields, and there are in total 2s+n = 3s−2 indices to be contracted down to s indices, requiring s − 1 contractions to be done.16 For a theory with a tower of higher spin fields, the formula must be appropriately generalized. Corresponding deliberations may be conducted for parameter redefinitions.

Given such a first-order deformation of the gauge transformations, one checks if the gauge algebra (4.19) can be satisfied with α given by α = gC0 (η, ξ) + g 2 C1 (ϕ, η, ξ) + ⋅ ⋅ ⋅

(4.32)

according to (4.28) and (4.24) in the case at hand with the Dϕ terms zero. To do this, one computes [δη , δξ ]ϕ using the gauge transformations (2.132) up to second order and compares to δα ϕ. This results in an equation to order g, 𝜕C0 (η, ξ) = T1 (𝜕ξ, η) − T1 (𝜕η, ξ)

(4.33)

and another equation to order g 2 , 𝜕C1 (ϕ, η, ξ) = T1 (T1 (ϕ, ξ), η) − T1 (T1 (ϕ, η), ξ) − T1 (ϕ, C0 (η, ξ))

+ T2 (𝜕ξ, ϕ, η) − T2 (𝜕η, ϕ, ξ) + T2 (ϕ, 𝜕ξ, η) − T2 (ϕ, 𝜕η, ξ)

(4.34)

For spin 1 and spin 2, these two equations can be satisfied by a C0 and a T1 and all higher Cn and Tn equal to zero.

15 No inverse powers of derivatives are allowed in redefinitions. 16 The number of derivatives is given by dimensional analysis with gs the spin s coupling constant of mass dimension 1 − s.

142 � 4 General theory of interactions Spin-1 and spin-2 gauge algebras a For spin 1, we have δϕaμ = 𝜕μ ξ a + gf abc ϕbμ ξ c (see formula (4.20) in our Volume 1) and can so read off T1μ =

f abc ϕbμ ξ c . For C0 , one can compute [ηa t a , ξ b t b ] using the algebra of generators t a to find C0a = f abc ηb ξ c . Alternatively, one may compute [δη , δξ ]ϕ to order g to find the same expression for C0a . Computing [δη , δξ ]ϕ

to order g2 amounts to checking (4.33) and involves the Jacobi identities for the structure constants f abc . Then checking (4.34) (with T2 = 0) is just the Jacobi identity again. Thus, the general analysis applied to spin 1 reproduces the familiar Yang–Mills results—with a field independent gauge algebra—and the full theory can be derived. The analogous analysis can be done for spin 2 with similar results. For hints on how to do it, see Section 5 of BBvD-G, reference [151].

We will continue with the spin 3 case in the next chapter in Sections 5.1 and 5.4.

4.3 BRST-BV antifield theory The BRST approach to field theory, and its development into the Batalin–Vilkovisky antifield method, has its origin in the quantization and renormalization problems of Yang– Mills theory.17 The method, however, has applications beyond the original, very important questions. Our focus is in the construction of new interacting field theories for higher spin fields. As we saw above in our review of the Fronsdal/BBvD Noether coupling approach to higher spin interactions, there are several problems to solve. What are the field equations and the gauge invariant Lagrangian? What are the corresponding gauge transformations and what algebra do they obey? A third problem is to sort out trivial interactions from the nontrivial ones. A fourth problem is to implement spacetime locality. It turns out that the BRST-BV antifield method treats all these aspects of the problem in one comprehensive algebraic scheme. The generalities of the approach will be outlined in the present section, while the concrete applications to higher spin fields will be treated in Chapter 5.

4.3.1 Gauge fields, antifields and transformations For the general formalism, we will adopt a notation close to the textbook [175] and used in many of the papers on the subject, in particular, in the very useful review [207]. Other standard review literature are [208, 209]. The formalism is in its turn based on the con17 We will not attempt any review of this huge subject. The references cited below contain such material, and further references. The method itself appears under several names given by combinations of the words “BRST”, “BV”, “antifield” and “antibracket”, to name the most common parts. A good access point to the original literature is [205] as well as [206].

4.3 BRST-BV antifield theory

� 143

densed notation of DeWitt [210].18 We outlined this general gauge theory formalism in Sections 3.1 and 3.14 in our Volume 1. Thus, our gauge fields will be φi subject to the gauge transformations δξ φi = Ria ξ a where Ria is a differential operator (acting on the parameter) depending on the fields. Let us take this as the starting point. One step in the standard BRST formalism for Yang–Mills gauge theory is to replace the gauge parameters ξ a with odd ghost fields C a , so that we now have BRST transformations δϵ φi = ϵRia C a with some infinitesimal, odd parameter ϵ.19 Rather than trying to motivate the formalism more than this, we will jump right in, and see, as we proceed, how it subsumes many, if not all, features of gauge theory. We will use almost standard notation, thus the physical fields φi and the ghosts C α (and possibly ghosts for ghosts, etc.) are collectively denoted by ΦA ⊃ {φi , C α , . . .} and the corresponding antifields by Ψ⧖A ⊃ {φ⧖i , Cα⧖ , . . .}.20 We will only write formulas for the irreducible case—where the gauge transformations are independent—which is the case of higher spin gauge theory.21 For any functions X and Y of the fields/antifields, the antibracket is defined as (X, Y ) =

δ X δl Y δr X δl Y − r δΨA δΨ⧖A δΨ⧖A δΨA

(4.35)

in terms of right and left functional derivatives. Under this bracket, the fields/antifields are conjugate according to (ΨA , Ψ⧖B ) = δBA

(4.36)

Right and left functional derivatives This is one place where something basically simple may become confusing. Does “right derivative” mean “acting to the right” or “acting from the right”? My inclination is to think “to the right” but it seems that the literature prefers “from the right”. Adopting the majority view, one could then write for the opposite “left derivative” → 󳨀 δl F δF = δϕ(x) δϕ(x)

(4.37)

18 All space-time and internal indices as well as space-time dependence is collected in one abstract index i. Sums as well as integrals are subsumed under the Einstein summation convention. Due to the abstractedness of the formalism, all questions about metric signatures, upper and lower indices, etc., can and must be dealt with as soon as concrete and particular theories are considered. 19 The parameter ϵ has no space-time dependence, so BRST transformations are often termed “global”. Instead, the ghost field C a is a local field. 20 The standard notation for antifields is Ψ∗ with some variant of “asterisk”. We choose another notation ⧖ (a vertical bow tie) in order to disambiguate with respect to other usages of ∗ such as complex conjugate and dual tensors. 21 The gauge parameters of the Fronsdal theory are normally considered to be independent, although they are constrained to be traceless.

144 � 4 General theory of interactions

while for the right derivative, we could write ← 󳨀 δr F Fδ = δϕ(x) δϕ(x)

(4.38)

in the notation introduced in connection with the Moyal product in Section 3.3.2. The right and left functional derivatives in the antibracket are defined to take the Grassmann properties of the functionals and fields into account. Consider computing the variation δF of the functional F with the functional derivative acting from the left. Then we would have δF = ∫ d 4 xδϕ(x)

δl F δF = ∫ d 4 x r δϕ(x) δϕ(x) δϕ(x)

(4.39)

where the first equality can be considered to be a natural definition, while the second is more of a requirement. Taking the Grassmann parities into account, one finds δl F δF = (−1)π(ϕ)(π(F)+1) r δϕ(x) δϕ(x)

(4.40)

Incidentally then, up to Grassmann signs, the definition of the antibracket is in accordance with the linear term in the Moyal bracket of formula (3.37).

In terms of the field theory BRST symmetry operator s, the BRST transformations are written δs X = sX

(4.41)

where we have suppressed the infinitesimal parameter ϵ that plays no role in the subsequent algebraic treatment. The idea is to implement s as a “canonical” transformation, in the space of fields and antifields, using the field/antifield bracket sX = (X, S)

(4.42)

In this formula, S is an enhancement of the classical action of the theory with terms involving the antifields in such a way as to produce the equations of motion, the gauge transformations and indeed the full gauge structure of the theory. But before continuing the discussion of the BRST-BV generator S, we collect properties of the relevant formalism in the box below.22 Properties of the antibracket For definiteness, we work in a graded vector space (really a supercommutative algebra) of fields Ψ ⊃ ⧖ {. . . , Cα , φi , φ⧖ i , Cα , . . .} defined on a space-time manifold M. The antibracket, defined in (4.35), has the following properties for generic objects X, Y , Z in the algebra:

22 The generator S does not seem to have a well-established name in the literature. One may read “master action”, “action functional”, “BRST generator”, “solution to the master equation”, to name a few.

4.3 BRST-BV antifield theory

Grassman parity: π(X, Y ) = π(X) + π(Y ) + 1 Field ghost number:

� 145

(mod 2)

(4.43a)

ghf (X, Y ) = ghf (X) + ghf (Y ) + 1

(4.43b)

Symmetry:

(X, Y ) = −(−1)(πX +1)(πY +1) (Y , X)

(4.43c)

Derivation:

πY (πX +1)

Y (X, Z)

(4.43d)

(XY , Z) = X(Y , Z) + (−1)

πY (πZ +1)

(X, Z)Y

(4.43e)

(X, (Y , Z)) = ((X, Y ), Z) + (−1)

(πX +1)(πY +1)

Derivation: Jacobi identity:

(X, YZ) = (X, Y )Z + (−1)

(Y , (X, Z))

(4.43f)

Note that the bracket (⋅ , ⋅) itself is odd. This algebraic structure, (⋅ , ⋅) : Ψ ⊗ Ψ → Ψ, is sometimes called a Gerstenhaber algebra in the mathematics literature. The field ghost number assignment in (4.43b) is clarified in Table 4.1 below.

The BRST operator s is an odd nilpotent derivation with the following properties: π(sX) = π(X) + 1

(mod 2) πY

s(XY ) = X(sY ) + (−1) (sX)Y ss = 0

(4.44a) (4.44b) (4.44c)

The form of the Leibniz rule (4.44b), with “right action”, is consistent with the antibracket implementation (4.42) and the rules of (4.43). The reader may now ask: what is S and what is it supposed to do? Let us give a straight initial answer. The master action S is a sum of terms, possibly of infinite number in perturbation theory, the first two of which can be immediately written down as soon as one has decided on which theory to work with. The first term is the classical action, generating the field equations, and the second term generates the BRST-transformations. Thus, we have S = Scl + ϕ⧖i Ria C a + ⋅ ⋅ ⋅

(4.45)

where, as said, Scl is the classical action. Here, Scl and Ria may encode the full interacting action and gauge transformations, as in Yang–Mills, where it is known, or just the free theory, in cases when the formalism is intended to be used for a perturbative approach to the full interacting theory. For Yang–Mills, the gauge algebra structure appears at the next antighost level. In any case, the full algebraic structure of the theory is captured by the master equation (S, S) = 0

(4.46)

expressing the nilpotency of S, and consequently of s. A note on reducible gauge theories A gauge theory, or its gauge transformations, are said to be reducible when the gauge transformations are not independent. A simple example is the Abelian 2-form theory with antisymmetric field Bμν = −Bνμ and field

146 � 4 General theory of interactions

strength Hμνρ = 𝜕μ Bνρ + 𝜕ν Bρμ + 𝜕ρ Bμν . The Lagrangian, proportional to Hμνρ H μνρ , is invariant under gauge transformations δBμν = 𝜕μ Λν −𝜕ν Λμ . These gauge transformations are, however, not independent in that they are zero for parameters of the form Λμ = 𝜕μ ξ. In the present formalism, the nonindependence of the gauge transformations can be expressed as equations Zαa Rai = 0 for some set of functions Zαa . The effect of this is the appearance of “ghosts for ghosts” C α and the corresponding antifields Cα⧖ . The various ghost number assignments are given in Table 4.1 below. If the functions Zαa themselves are independent, we have a “first-order reducible theory”. The level of dependence may however go to higher orders, with the appearance of “ghosts for ghosts for ghosts”. In the case of a first-order reducible theory, there will be a term Ca⧖ Zαa C a in S (2) (see (4.48)).

To continue, we have to understand the structure of S in more detail. A key to this is the various ghost numbers relevant to the theory. There are three of them, two of which are independent. The pure ghost number ghp is nonzero for the ordinary BRST ghost fields; 1 for the ghosts, 2 for the ghosts for ghosts, etc. The antighost number gha is nonzero for the antifields; −1 for the antifields ϕ⧖i , −2 for the antighosts Ca⧖ , etc. Finally, what I prefer to call the field ghost number ghf is given by23 ghf = ghp − gha

(4.47)

The spectrum of ghost numbers is illustrated in Table 4.1 where we also note the Grassmann parity for the fields. Table 4.1: Ghost number and Grassmann parity assignments. Ghost number/Field

⋅⋅⋅

Cα⧖

Ca⧖

ϕ⧖ i

ϕi

Ca

Cα

⋅⋅⋅

ghp gha ghf Grassmann parity π

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

� � −� �

� � −� �

� � −� �

� � � �

� � � �

� � � �

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

From the assignments of Table 4.1, we see that the antibracket has Grassmann parity and field ghost number as given in formulas (4.43a) and (4.43b). It has no definite antighost number. We also see that S has field ghost number zero and is Grassmann even. More interestingly, though, is the fact that the first two terms of the expansion of S as in equation (4.45) have antighost numbers 0 and 1, respectively. Anticipating a continued expansion in antighost number, we can write S = S (0) + S (1) + ⋅ ⋅ ⋅ = ∑ S (n) n≥0

(4.48)

23 Often called just “ghost number” in this context. The field ghost number ghf is related to the mechanics BRST theory ghost number ghm (defined in Section 5.4.2 of our Volume 1) by ghf = −(ghm +1/2).

4.3 BRST-BV antifield theory

� 147

where n is the antighost number. The expansion is also an expansion in pure ghost number. The sum might be finite or infinite. 4.3.2 Solution of the master equation Inserting the first two terms of the expansion (4.45) of S into the master equation (4.46), it is clear that it cannot in general be satisfied without adding higher-level terms in the antighost number. “Solving the master equation” then becomes a matter of determining such higher-level terms, if they exist. Since the antibracket has no definite antighost number, we will devise a notation—not standard in the literature—to keep track of that. According to the definition (4.35), the bracket may reduce the antighost number by 1 or 2 in an irreducible theory or higher in a reducible theory. To denote the reduction in antighost number by the bracket, we will write (X, Y )−n . We can then write the first few terms of the master equation sorted according to antighost number. The first term (S (0) , S (0) ) = 0 is trivial since S (0) is the classical action and there is nothing to compute. Continuing, we get for the first few levels gha = 0: (S (0) , S (1) )−1 = 0

(4.49a)

gha = 1: 2(S (0) , S (2) )−1 + 2(S (1) , S (2) )−2 + (S (1) , S (1) )−1 = 0

gha = 2: 2(S , S )−1 + 2(S , S )−1 + (S , S )−2 = 0 (0)

(3)

(1)

(2)

(2)

(2)

(4.49b) (4.49c)

The first equation expresses gauge invariance, or rather BRST invariance, of the classical action. When the formalism is applied to a known theory, free or interacting, that equation should also be satisfied by construction.24 Next, computing (S (1) , S (1) ) ≡ D(1) in the second equation, we find j

δl Rkb

δl Rka a b )C C δϕk δϕk ij δS = −ϕ⧖i (Cabc Ric + Mab clj )C a C b δϕ

D(1) = (ϕ⧖i Ria C a , ϕ⧖j Rb C b ) = −ϕ⧖i (Rka

− Rkb

(4.50)

where the second line can be read off from formula (3.19) in Section 3.1.4 of our Volume 1. The term captures the structure of the gauge algebra of the theory. For a free theory, it is zero, and the expansion of the master action stops. For an interacting theory, however, one must now ascertain a solution to the homological equation (4.49b) that now reads 2(S (0) , S (2) )−1 + 2(S (1) , S (2) )−2 + D(1) = 0. This is a good point to break off and look a little closer at the inner workings of the theory. 24 In the original literature on the subject, this is sometimes expressed as S (0) and S (1) (and a part of S (2) related to reducibility of the gauge transformations) being fixed by “boundary conditions”. See, for instance, [207], Section 7.4.

148 � 4 General theory of interactions 4.3.3 The Koszul–Tate and longitudinal differentials and homology What we have done above is the “top-down” approach to BRST-(co)homology. We will now look inside the formalism and see what it contains in terms of the so-called Koszul– Tate differential δ and the longitudinal differential γ.25 For that, we return to the realization of the BRST symmetry operator s as sX = (X, S). Inserting the expansion (4.48), one will get an expansion over antighost number also for sX. This expansion may be written s = δ + γ + s(1) + ⋅ ⋅ ⋅

(4.51)

with gha (δ) = −1 and gha (γ) = 0. Since the antibracket itself has no definite antighost number, the terms in (4.51) may receive contributions from more than one term in (4.48). So, for instance, will the Koszul–Tate differential receive contributions from both S (0) and S (1) for an irreducible theory.26 To be explicit, let us compute the antibracket with the fields and antifields. The result is as follows: (ϕi , S (0) + S (1) ) = 0 + Ria C a a

(C , S (Ca⧖ , S (0) (ϕ⧖i , S (0) + S (1) ) =

(0)

+S )=0+0 (1)

+ S ) = 0 + ϕ⧖i Ria j δS δRa a C − cli + ϕ⧖j δϕ δϕi (1)

⇒

δϕi = 0, γϕi = Ria C a

(4.52a)

⇒

δC = 0, γC = 0

(4.52b)

⇒

δCa⧖

(4.52c)

⇒

δϕ⧖i = −

a

a

= 0,

δCa⧖

=

ϕ⧖i Ria

j

δScl δRa a , γϕ⧖i = ϕ⧖j C δϕi δϕi

(4.52d)

We have interpreted the result of the computation to the left in terms of the differentials δ and γ. The third line sticks out with the antibracket (Ca⧖ , S (1) ) = ϕ⧖i Ria interpreted as δCa⧖ rather than γCa⧖ . This interpretation can be seen as dictated by the antighost number counting. This “phenomenon”, δ receiving contributions not just from S (0) but also from S (1) , continues in higher orders and is the concrete manifestation of the antibracket having no definite antighost number. It is interesting to continue the computation up to antighost number 1, equation (4.49b), in general terms. With the first two terms of the equation interpreted in terms of the Koszul–Tate differential, the equation reads 2δS (2) + D(1) = 0

(4.53)

25 Many expositions work “bottom-up” starting with the constituent differentials to arrive at the antifield/antibracket formalism. The reader may ask for deeper motivations for the choices made in building up this formalism, and there are certainly varying points of view taken in the literature. Not wanting to enter into that, let us just state: it works and does the job. 26 For a first-order reducible theory (only ghosts for ghosts, but not higher), also S (2) will contribute to δ.

4.3 BRST-BV antifield theory

� 149

If the equation can be solved, then D(1) must be δ closed (i. e., δD(1) = 0). This follows immediately from the action of δ given in formulas (4.52) applied to (4.50). Next, one uses the acyclicity of δ, i. e., the fact that Hk (δ) = 0 for k > 0 (see next section). This is also clear from the action of δ as given in formulas (4.52). This means that D(1) is exact and we have D(1) = δS (2) for some S (2) . A Grassmann even ansatz for S (2) with antighost number 2 and field ghost number 0 is given by ij

S (2) = Ca⧖ k abc C b C c + ϕ⧖i ϕ⧖j kbc C b C c

(4.54)

ij

for some functions k abc and kbc depending only on the fields ϕi . Manifest space-time invariance is also assumed. Then computing 2(S (0) , S (2) )−1 + 2(S (1) , S (2) )−2 and compared to D(1) in (4.50), we can solve the master equation to this order by making the identifications 1 k abc = − Cbca 2

1 ij ij and kab = − Mab 4

(4.55) ij

In Yang–Mills theory, the gauge algebra is closed, and we have Mab = 0 and Cbca are the Lie algebra structure constants. The results obtained so far means that for a general irreducible gauge theory we have a solution for the master equation up to S (2) . The remaining terms in the expansion of S can be analyzed recursively in the following way. Assume that S has been constructed to order n − 1 with n ≥ 3. Consider then the antighost number expansion of S in (4.48) up to gha = n − 1, R(n−1) = ∑ S (i) i≤n−1

(4.56)

Upon computing (R(n) , R(n) ) at antighost number n−1, one finds two sets of terms.27 First, those that can be written as 2δS (n) coming from (S (0) , S (n) )−1 and (S (1) , S (n) )−2 , and second, terms D(n−1) coming from (R(n−1) , R(n−1) ) with antighost number n−1 and depending only on function S (i) with i ≤ n − 1. The master equation thus reads 2δS (n) + D(n−1) = 0

(4.57)

For this equation to have a solution, it is necessary that D(n−1) is δ-closed. Here, this follows from the Jacobi identity ((D(n−1) , D(n−1) ), D(n−1) ) = 0 at antighost number n − 2. Furthermore, acyclicity of δ at positive antighost number implies that D(n−1) is δ-exact, so there must indeed be some S (n) satisfying equation (4.57).

27 There are also higher-order terms in the antighost number. The lower-order terms sum to zero due to already solved S (i) .

150 � 4 General theory of interactions 4.3.4 Homological perturbation theory—Lagrangian version Seeing all this for the first time, one may get the impression that it is too good to be true. But one must then realize that the formalism is constructed to work in this way. How do we know that the Koszul–Tate differential δ is acyclic in positive antighost degree? The answer is that it is constructed to be so. We have not actually said much about what the Koszul–Tate differential is supposed to do, except stating the two equations δϕ⧖i = −

δScl δϕi

and

δCa⧖ = ϕ⧖i Ria

(4.58)

that follows from the first two terms S (0) + S (1) in S. Let us now supply a more “physical” context for the construction. We are considering a generic field theory where by I we denote the infinite-dimensional space of all field histories. The theory may be, or may not be, a gauge theory. In this space, the equations of motion δScl /δϕi = 0 define a subspace Σ called the stationary surface. Observables are defined as smooth functions on Σ, i. e., elements of C ∞ (Σ). One may also consider smooth functions on I belonging to C ∞ (I). These spaces are vector spaces and algebras under ordinary pointwise multiplication. A few words of caution The “functions” we are considering here can be thought of as ordinary functions when the indices on the fields are ranging over finite sets, as in mechanics. When doing field theory, where the indices range also over continuous sets, the situation is clearly much more subtle. We have functions on field histories or on field configurations. However, as stressed in many of the original papers on the subject, the formalism set up here is concerned with the algebraic aspects of gauge theory where the functional analysis aspects of the theory may be downplayed. If one is interested in deeper aspects of the theory, one should consult mathematical papers on the subject.

The idea is now to implement the restriction from I to Σ through a differential δ acting in some extended space of fields. In more detail, δ shall “compute C ∞ (Σ) through its homology”, as it is phrased. This means that H0 (δ) =

Ker(δ) 󵄨󵄨󵄨󵄨 ∞ 󵄨 = C (Σ) Im(δ) 󵄨󵄨󵄨0

(4.59)

and, furthermore, that Hk (δ) = 0

for k ≠ 0

(4.60)

so that δ is acyclic in degree ≠ 0. The grading here is the antighost number. That C ∞ (Σ) can be represented by a quotient, follows from the following argument.

4.3 BRST-BV antifield theory

� 151

Consider the subspace 𝒩 of C ∞ (I) of all functions that vanish on Σ. This subspace forms an ideal in C ∞ (I). Functions in the quotient C ∞ (I)/𝒩 are therefore considered equivalent if they differ by a function in 𝒩 , and consequently only the “values” on Σ matters. On the other hand, any function on Σ may be extended off Σ to a function on I. Two such extensions must differ by a function in 𝒩 , and thus a function in C ∞ (Σ) represents an element in C ∞ (I)/𝒩 . To implement (4.59) and (4.60), consider first a theory without gauge invariance. One may then take Ker(δ)|0 = C ∞ (I)

and

Im(δ)|0 = 𝒩

(4.61)

The first part of (4.61) is achieved by setting δϕi = 0

(4.62)

By the Leibniz rule, this implies δF(ϕi ) = 0 for all functions on ℐ and, therefore, the kernel of δ is all of C ∞ (I). To achieve the second part of (4.61), one assumption is needed as well as the introduction of one new set of objects. It is assumed that the elements of 𝒩 can be written as combinations of the field equations G(ϕi ) ∈ 𝒩 ⇔ G(ϕi ) = λj (ϕi )

δS0 δϕj

(4.63)

This quite natural assumption is called regularity and is discussed at length in [211, 208].28 At this stage, new variables (really fields, often called “generators”) are needed. For a function G to belong to Im(δ)|0 = 𝒩 , it must be δ on something. One therefore introduces as many new fields ϕ⧖i as there are field equations, and sets δϕ⧖i = −

δS0 δϕj

(4.64)

Now we have G = δ(−λi ϕ⧖i ) as required to satisfy (4.63). The Koszul–Tate differential is taken as a boundary operator, so it lowers the antighost number by one unit. Therefore, the antifield ϕ⧖i must have an antighost number equal to 1. It then remains to show that Hk (δ) = 0 for k ≠ 0 so that δ provides a resolution of C ∞ (I)/𝒩 . Since the number of independent equations of motion are equal to the number of new generators, one may used the method of contracting homotopies to show that H1 (δ) = 0 (see Section 3.1.9). When there is gauge invariance, this is no longer true. The field equations are no longer independent, but satisfy the Noether identities δS0 i R =0 δϕi a 28 A concise discussion can be found in Chapter 17, Section 17.1.5 in [175].

(4.65)

152 � 4 General theory of interactions There are therefore δ-closed functions in antighost degree 1, namely Ria ϕ⧖i . These are nonexact, and thus H1 (δ) ≠ 0. To recover H1 (δ) = 0, one adds a new set of antighost fields Ca⧖ , one for each cycle in Ria ϕ⧖i with the action of δ defined as δCa⧖ = Ria ϕ⧖i

(4.66)

with properties for Ca⧖ as given in Table 4.1. Extending the action of δ to any polynomial in the fields and antifields by the Leibniz rule, one recovers δ2 = 0. Again, one may show, for an irreducible theory, that Hk (δ) = 0 for k > 0. For a reducible theory, the process of adding new antifields in order to cancel higher-order homology has to be continued.29 The process itself is called resolution and the objective is to concentrate the homology to H0 (δ). 4.3.5 Solution of the master equation: summary We can now summarize the parallel development of the last few sections. The BRST differential s is implemented as sX = (X, S) in equation (4.42). The nilpotency of s is then equivalent to the master equation (4.46). The master action S is expanded in antighost number as in formula (4.48), the first few terms of which should generate the differentials δ and γ.30 Then the master equation is solved perturbatively, order by order in antighost number, subject to the boundary conditions S (0) = Scl

(4.67a)

S (1) = ϕ⧖i Ria C a

S

(2)

=

Ca⧖ Zαa C α

(4.67b) + ⋅⋅⋅

(4.67c)

where S (2) may contain further terms (signified by the dots) coming from the solution of the master equation. What is obtained then is a so-called proper solution. Satisfying the boundary conditions excludes taking the trivial solution S = Scl . The proper solution corresponds to the resolution of the homology algebra, i. e., obtaining Hk (δ) = 0 for k > 0. 4.3.6 Space-time locality of the BRST-BV formalism For a field theory developed in the BRST-BV antifield formalism, not just the classical action, but all functions and functionals must be local, in the sense of depending only 29 See [207] Section 5, or [175], Chapter 17 for review type expositions. 30 Depending on the order of reducibility of the theory.

4.3 BRST-BV antifield theory

� 153

on the fields and a finite order of derivatives of fields. The concept of locality is defined in detail below in Section 4.4.3. In particular, the master action S must be local. Starting with the classical action Scl , it must be ascertained that all further terms in the expansion of S are local functionals. This was an issue when the BRST-BV theory was developed, and according to [212], first simply assumed [213], but then proved in [212] (see also [214, 211, 215] and the review [208]). When reading the original papers pertaining to locality, from a higher spin perspective, it is useful to keep in mind that there are actually two aspects of locality involved. The first one, where locality turned out to be guaranteed, is when the antifield formalism is applied to already known interacting field theories such as Yang–Mills with matter. As we have remarked, the BRST-BV formalism can be seen as built “on the outside” of an already existing theory with local interactions. The BRST-BV antifield formalism does not introduce any new nonlocalities. The second aspect is when the formalism is applied to the problem of deforming a free field theory into an interacting theory, as is certainly the case for higher spin gauge theory. Then locality turns out to be a problem. The nonlocalities encountered upon trying to introduce higher-order interactions, can be detected by the BRST-BV method, and are then signaled by what is called obstructions to the deformation. We will see examples of such obstructions in higher spin gauge theory in Section 5.4. Looking back at the proof that the master equation can be solved, i. e., that there is a solution S = S (0) + S (1) + S (2) + ⋅ ⋅ ⋅ we see that it hinges on the acyclicity of the Koszul–Tate differential δ. The rest of the argument is algebra that does not depend on the locality requirement. The existence of S is equivalent to the existence of the BRST differential s = δ + γ + s(1) + ⋅ ⋅ ⋅ . So, in order to extend the existence theorem to be true when locality is demanded, acyclicity for δ must be proved for local functions. As it is phrased in [212], even though D(n−1) in (4.57) is a local functional as soon as the lower-order terms S (k) for k ≤ n − 1 are local, it is not clear that the solution S (n) is also a local functional. The locality requirement is expressed in terms of the concepts outlined in Section 4.4.3. That is, a function and, therefore, a local functional before evaluation, depends on the fields and all their multiple partial derivatives up to some finite order. Up to now, the development has been phrased in terms of functionals such as S (n) . In order to explicitly discuss locality, it is also convenient to transfer the discussion to functions. Since when working with functionals, one can make partial integrations at the price of surface terms, it is clear that the corresponding functions are nonunique up to divergences. Such divergences do not contribute to the field equations.

4.3.7 Deforming the master equation and locality of interactions The development so far can be seen as collecting all interesting gauge structure of an already existing theory into one formalism; the master action satisfying the master equation. Now we will see how the formalism can be applied to the problem of deforming a

154 � 4 General theory of interactions free field theory into an interacting field theory, in the process capturing all the gauge structure. The key paper here is [216] (see also the review paper [217]). Both the aim of the paper—that of reanalyzing the problem of constructing consistent interactions among field with gauge freedom in the antibracket formalism—and its results, are clearly stated in the Introduction, from which we quote. [. . .] using the properties of the antibracket, [. . .] there is no obstruction to constructing interactions that consistently preserve the gauge symmetries of the free theory if one allows the interactions to be nonlocal [removed footnote]. Obstructions arise only if one insists on locality.

Let us start by fixing the notation for the master action. The focus is now shifted from the expansion over antighost number to interaction order, but still, at any interaction order (including the free level) there are terms in S with a different antighost number. To have a handle on this, we write Sk for the master action at interaction order k, where k = 0 signifies the free theory, k = 1 the cubic interaction order, and so on. We continue to write S (i) for the terms in the master action of antighost number i. With these conventions, Sk(i) , denotes terms at interaction order k and antighost number i.31 Focusing on interaction order, the master action is expanded according to S = S0 + gS1 + g 2 S2 + ⋅ ⋅ ⋅

(4.68)

Here, S0 = S0(0) + S0(1) with S0(0) the free classical action and S0(1) = ϕ⧖i Ria C a with Ria given by the free theory gauge transformations (see (4.67b)).32 Expanding the master equation according to interaction order, we get g0 :

(S0 , S0 ) = 0

(4.69a)

g :

(S0 , S1 ) = 0

(4.69b)

g :

2(S0 , S2 ) + (S1 , S1 ) = 0

(4.69c)

g :

(S0 , S3 ) + (S1 , S2 ) = 0

(4.69d)

1

2

3

⋅⋅⋅ The recursive nature of the equations is obvious. If lower-order terms Sk are obtained, the question is to determine the next Sk+1 . The first equation (4.69a) is true by construction, since we assume that the free theory is gauge invariant. The second equation (4.69a), can be solved for higher spin (and lower spin such as Yang–Mills and gravity). The crucial test comes with the third equation. Now, let us see what the paper [216] has to say about this. 31 These conventions differ from the original paper [216] where S (k) (or rather tion order g k . This is a bit confusing since expansion.

(k) S

(k) ) S

denotes interac-

is in other contexts used to indicate antighost number

32 S0 may contain a term at antighost number 2 if the theory is reducible (see formula (4.67c)), but we do not consider that here.

4.3 BRST-BV antifield theory

� 155

Consider the BRST cohomology H ∗ (s) generated by the BRST differential s through the antibracket sA = (A, S) as in (4.42), where A is an element (functional) of the cohomology. Using the Jacobi identity (4.43f), one may show that the antibracket induces a well-defined map H p (s) × H q (s) → H p+q+1 (s) : ([A], [B]) = [(A, B)]

(4.70)

where [A] denotes the cohomological class of the BRST-closed element A. This simply means that the bracket of closed elements maps to closed elements. What is not trivial is the following theorem. The antibracket map is trivial: the antibracket of two BRST-closed functionals is BRST-exact.

The proof is only sketched in the paper and we will not try to reproduce it here. Let us instead see what the theorem implies for the deformation procedure. Consider the third equation (4.69c) of the deformation equations (4.69). Since the second equation (4.69b) means that S1 is BRST-closed, the theorem says that (S1 , S1 ) is BRST-exact, i. e., there is some A such that (S1 , S1 ) = (S0 , A). Therefore, the third equation is automatically satisfied and the deformation S2 exists. In a similar fashion, it follows that the higher-order terms S3 and so on, also exist. However, as the paper states [. . .] while (S (1) , S (1) ) is always cohomologically trivial, it is not true, in general, that it is the BRST variation of a local functional.

It may be, but there is no guarantee, and it must be checked in the concrete case at hand. A possible source of a nonlocality is the presence of S (0) on the left-hand side of the deformation equation (4.69c). In field theories, S (0) contain the free field equations that generically sport quadratic derivatives such as ◻. In solving the equation, one may have to resort to naively inverting ◻, producing nonlocalities of the 1/◻ type. In higher spin theories, S (1) contains terms with a countable infinity of derivatives. This makes (S (1) , S (1) ) itself nonlocal, albeit local for any particular set of higher spin fields, since then only a finite number of derivatives enter. These questions have to be dealt with in particular models. When taking space-time locality into account in practical computations, it is convenient to work with functions rather than functionals. To retain the information in surface terms—that may result from partial integration—i. e., integrals of total derivatives, cocycle and coboundary conditions are modified. Denoting by a the integrand (function) corresponding to the functional A, we have cocycle condition: sA = s ∫ a → sa + dk = 0 coboundary condition: A = sB = s ∫ b → a = sb + dl The integrals are over top form degree, say n. Then k and l are (n − 1)-forms.

(4.71)

156 � 4 General theory of interactions Following the paper, we write S (k) = ∫ ℒ(k) requiring that ℒ(k) be a top-form depending on the variables/fields and a finite number of their space-time derivatives. For the antibracket, we have (A, B) = ∫(a, b) where (a, b) is defined up to d-exact terms. The first two nontrivial deformation equations can now be written 2s(0) ℒ(1) = dj(1) 2s ℒ

(0) (2)

+ (ℒ , ℒ ) = dj (1)

(1)

(2)

(4.72a) (4.72b)

In Section 5.4, we will review examples of this method applied to higher spin gauge theory in within the Fronsdal–BBvD approach. This kind of local cohomology is denoted by H p (s(0) |d), to be understood as the relative cohomology for s(0) modulo d. The map H p (s(0) |d) × H q (s(0) |d) → H p+q+1 (s(0) |d) : ([A], [B]) = [(A, B)]

(4.73)

is nontrivial in general, and there is no guarantee that the deformation equations can be solved. Yang–Mills and gravity provide examples of nontrivial, local solutions.

4.3.8 The algebraic Poincaré lemma and the descent equations To see how the method works in practice, not the least somewhat subtle interplay between the ghost number expansions and the interaction order expansion, it is interesting to look at Yang–Mills theory. But before that some more general theory is needed connected to the space-time differential d. The Poincaré lemma is a result in de Rahm cohomology. Let the space-time manifold be Rn (nontrivial topology will not be considered). The Poincaré lemma states that all closed forms with p > 0 are exact H p (d) = 0 p

for p > 0

H (d) ∼ R for p = 0

(4.74)

This can be proved by the method of contracting homotopies; see Section 3.1.9. The algebraic Poincaré lemma is a generalization where the p-forms are local functions f (x, [φ]) (see Section 4.4.3) depending on x and a finite number of derivatives of the fields (denoted by [φ]). It is well known that if a term f in the Lagrangian is a total derivative, then it does not contribute to the Euler–Lagrange equations. The converse is also true: if the Euler– Lagrange equations computed for a certain term f vanishes, then the term is a total derivative. Thus, we have f = 𝜕μ jμ ⇔

δf =0 δφi

for all fields φi

(4.75)

4.3 BRST-BV antifield theory

� 157

Consider now a local n-form w = fdx n . It is closed by construction. Then taking jμ in (4.75) as an (n − 1)-form, we find that w is exact if and only if the function f is a total derivative, i. e., w = dk ⇔ f = 𝜕μ jμ . Now we can formulate the algebraic Poincaré lemma. The cohomology of d in the algebra of local forms is given by H p (d) ≠ 0 p

H (d) = 0 p

for p = n

for n > p > 0

H (d) ∼ R for p = 0

(4.76)

The new feature here is the nontrivial cohomology at top form p = n. For the reader who wants a fuller exposition of this theory with proofs and further references, there is the original paper [211] and the review [208] to consult. We will now continue to an example computation. The descent equations and the Yang–Mills example In our Volume 1, Chapter 4, we outlined three, more or less different, ways of constructing the Yang–Mills Lagrangian. The BRST-antifield method provides a fourth way [218]. We will work with nonintegrated local densities ℒ in four dimensions according to S = ∫ ℒd 4 x. The starting point is the free theory master “Lagrangian” given by 1 aμν (1) a ℒ0 = ℒ(0) Faμν + A⧖μ a 𝜕μ C 0 + ℒ0 = − 4 F

(4.77)

The corresponding BRST differential, computed according to (4.42), becomes s0 = 𝜕ρ F aρσ

𝜕 𝜕 𝜕 − 𝜕ρ A⧖ρ + 𝜕μ C a μ a 𝜕A⧖σ 𝜕Ca⧖ 𝜕Aa a

(4.78)

The first nontrivial equation to solve, in order to find cubic interactions, is (4.72a), which takes the form s0 ℒ1 + da[3] = 0

(4.79)

where ℒ1 denotes the sought for cubic local interaction master Lagrangian, and a[3] is a local 3-form. The attempted solution is by a process of “descent” in form degree, resulting in the so-called “descent equations”. Acting with s0 on equation (4.79) and using the fact that s0 commutes with d, one arrives at d(s0 a[3] = 0). Then, due to the algebraic Poincaré lemma, any closed form of degree not of 4 or 0 must be exact. Therefore, one may write s0 a[3] = −da[2] for some local 2-form a[2] . Iterating this argument, one gets the descent equations s0 ℒ1 + da[3] = 0

s0 a[3] + da[2] = 0 .. .

s0 a[m] = 0

(4.80)

158 � 4 General theory of interactions

The descent cannot go beyond a 0-form, but it may stop at some m-form with m > 0 for other reasons related to general and particular required properties of the theory. Clearly, all local forms a[i] must be cubic in the fields and their derivatives (including antifields and ghosts). Furthermore, since s0 has field ghost number 1, it follows that the a[i] must have field ghost number 4 − i. Since s0 and d carry no dimension, all a[i] are of mass dimension 0 (just as ℒ1 in form language). General requirements can be found in Section 5.4.1. Having determined possible lowest form degree candidates for a[m] , one attempts to work up the descent by working out a[m+1] and so on all the way up to ℒ1 . In this “lifting” process, BRST trivial forms—of the type a[i] ∼ a[i] + s0 b[i] + dc[i−1] for some local forms b[i−1] and c[i] —should be discarded. It is in the lifting that obstructions may occur. Barring auxiliary fields, that do not change the degree of freedom count, the requirement of consistency implies that no other fields than those appearing in the free theory are allowed. Then it is clear that no candidate cubic 0-form a[0] of field ghost number 4 is possible. Moving up one step, it is also clear that no candidate cubic 1-form a[1] of field ghost number 3 is possible. At the next level up, there are, however, two candidates, both containing the ghost field C a , a b c μ ν a[2] = kfa|bc Fμν C C dx dx

and

a

′ a[2] = k ′ fa|bc ∗ F μν C b C c dx μ dx ν

(4.81)

with k and k ′ constants to be chosen conveniently and fa|bc antisymmetric in ab. In the second candidate, is the dual field strength. Let us work with a[2] first, which will turn out not to lead to any nontrivial interactions. To perform the attempted lift to a[3] , first compute da[2] with the result ∗ a F μν

a da[2] = 2kfa|bc Fμν 𝜕ρ C b C c dx μ dx ν dx ρ

(4.82)

since on the 2-form F a we have dF a = 0. This should be matched to −s0 a[3] , leading to a b c μ ν ρ a[3] = −2kfa|bc Fμν Aρ C dx dx dx

(4.83)

Next, computing da[3] , we will find one term (among two terms) that can be written as a b c μ ν ρ σ Fρσ C dx dx dx dx . This term cannot be gotten from any −s0 ℒ1 . It must therefore be zero, which it kfa|bc Fμν is with a totally antisymmetric fabc . Then, however, a[2] is BRST-exact, since a[2] = s0 (2kfabc Aaμ Abν C c dx μ dx ν ) + d(2kfabc Aaμ C b C c dx μ dx ν )

(4.84)

′ Continuing instead with the candidate a[2] , we get a

(4.85)

a

(4.86)

′ da[2] = k ′ fa|bc (ϵμνρσ (𝜕 ⋅ F)aσ C b C c + 2∗ F μν 𝜕ρ C b C c )dx μ dx ν dx ρ ′ This should now be matched to −s0 a[3] , leading to ′ a[3] = −k ′ fa|bc (ϵμνρσ A⧖aσ C b C c + 2∗ F μν Abρ C c )dx μ dx ν dx ρ ′ Next, computing da[3] , we get five terms ′ da[3] = −k ′ fa|bc (−𝜕 ⋅ A⧖a C b C c + 2A⧖aρ 𝜕ρ C b C c + 2𝜕μ F aμν Abν C c + 2F aμν 𝜕μ Abν C c + 2F aμν Abν 𝜕μ C c )d 4 x

(4.87)

′ Finally, matching to −s0 a[4] , we will find ′ a[4] ∼ fabc (F aμν Abμ Acν − 2A⧖a ⋅ Ab C c − C ⧖a C b C c )

(4.88)

4.4 Some further notes on Noether coupling techniques

� 159

4.4 Some further notes on Noether coupling techniques The BBvD approach reviewed in Section 4.2 is indeed a general Noether coupling technique. The BRST-BV antifield formulation, developed in the previous section is a way of systematically working with the action, the field equations and the gauge transformations in one comprehensive formalism. We will now mention a few related complementary or mathematically advanced approaches that has been proposed in the literature, and has indeed found their home in the BRST-BV approach to higher spin interactions. Or perhaps one should say that the BRST-BV framework, supplied with the jet bundle formulation to be reviewed below, is capable of encompassing them all.

4.4.1 The Stasheff synthesis of ideas J. Stasheff wrote, during the 1990s, a number of highly interesting research and review papers on the general field theory interaction problem, in the process realizing—as judged from comments made in the papers—the role played “infinity algebras”, in particular strongly homotopy Lie, or Lie∞ algebras, but also Ass∞ associative algebras. This was connected with B. Zwiebach’s work on interacting closed string field theory [219] and with the development of the BRST-BV antifield method by M. Henneaux and collaborators, and the similarities with the mathematical theory of homological perturbation theory as applied to constrained Hamiltonian systems. We see here the confluence of several powerful concepts, methods and concrete models. One article, that is well worth studying, is the highly interesting and comprehensive review article [220] that brings together jet bundle Lagrangian field theory, BRST-BV antifield theory, higher homotopy algebra and closed string field theory. The synthesis is not complete—perhaps not to be expected—but highly suggestive. Given that the higher spin interaction problem is a deformation theory problem, what tools can one bring to the workplace? It is not entirely clear how to approach this circle of ideas and where to start the exposition, short as it will be. We will, however, begin with describing the concept of strongly homotopy Lie algebras as abstract mathematical objects. Then we will focus on Lagrangian field theory as formulated on jet spaces, relevant as it is in higher derivative higher spin theory and in order to control questions of locality. In this context, we must mention a quite “minimalistic” approach to the interaction problem due to S. Anco [221]. It is an mathematical elaborate version of a perturbative Noether coupling technique, but it introduces no superstructure of any form, except what is absolutely necessary in the calculus of variations. The paper develops two “variational identities”, one for arbitrary local variations of the Lagrangian and one for arbitrary local variations of the Euler–Lagrange equations of motion. This is applied to arbitrary infinitesimal local symmetries to obtain a general perturbative scheme for determining locally symmetric Lagrangians and equations of motion as power series

160 � 4 General theory of interactions based on the linear theories. The paper also contains a thorough, and stringent, discussion about locality in Lagrangian field theory. A feature, common to all perturbative—power series—approaches, appears with bearing on the locality issues in higher spin field theory. Namely, the recursive nature of the resulting equations, where zero-order infinitesimal variations δ(0) L(k) of the order k Lagrangian is expressed recursively in terms of lower-order Lagrangians and variations. This pattern appears in all perturbative approaches as we have seen. See Section 3.14.6 in our Volume 1, the antifield/antibracket equations (4.69) above or the lightfront formulas (6.31) to be discussed in Chapter 6. On the theme of mathematically elaborate approaches to Noether coupling, there is a further development of variational calculus that goes under the name of variational bicomplex that studies the subject in terms of the cohomology of two [222] “perpendicular” differentials, a horizontal dH and a vertical dV related to the total derivatives and Euler–Lagrange derivatives of the fields (see also [223]). Next, to get an idea of how and why, the same pattern of recursive equations tend to come up in all perturbative approaches to field theory interactions, and indeed fall into the pattern of Lie∞ algebra; a simple “syntactical” analysis will be offered. We close this section, perhaps appropriately, with the Fulp–Lada–Stasheff elaboration of the BBvD theory. Some further papers, specifically targeted on the BRST approach to higher spin interactions, but general in their content, will be taken up in Section 5.2.4 in the next chapter. Some inroad references to infinity algebras and their relation to neighboring subjects Infinity algebras arose in mathematics (mid-20th century) in the context of algebraic topology. There is a intricate dichotomy in topology in that the subject concerns distinguishing properties of spaces that cannot be deformed into one another. Thus, there are discrete algebraic aspects of the subject as well as infinitesimally deformation theoretic aspects. In general terms, one can say that infinity algebras are connected with deformation problems of algebraic structures. The literature is extensive and any list of references will be hopelessly inadequate. But for the benefit of the reader who wants an inroad to the subject, here are a few, quite random choices. On the subject of deformation theory, there is the early paper [224] by M. Gerstenhaber on deformations of rings and algebras. For some history of deformation theory, see [225] in the very useful volume [226]. A classic reference in the theory of deformations is the Kontsevich paper [227] related to deformation quantization. Gerstenhaber algebras are discussed in [228] in the context of topological field theory. Reference [229] offers a first introduction to homological perturbation theory—the Hamiltonian counterpart to Lagrangian BRST-BV antifield/antibracket theory reviewed above. Further down that road there is [230] that takes us closer to field theory. Closer to higher spin field theory, we have, for instance, [231, 232] that explore connections to the Vasiliev higher spin theory and the technique of “unfolding”. A later paper in that research program is [233]. A recent paper on Lie∞ algebras and field theory is [234].

4.4 Some further notes on Noether coupling techniques

� 161

4.4.2 Strongly homotopy Lie algebras There are a few variants of the basic definition of strongly homotopy Lie algebras in the literature (see, for instance [235, 236]), but the following, mildly technical definition, is sufficient for our purpose of application to higher spin gauge interactions. Consider a Z2 graded vector space V = V0 ⊕ V1 over some number field, and denote the elements by x. The grading is given by ϱ with ϱ(x) = 0 if x ∈ V0 and ϱ(x) = 1 if x ∈ V1 . V is supposed to carry a sequence of n-linear products denoted by brackets. The graded n-linearity is expressed by the following two equations: [x1 , . . . , xn , xn+1 , . . . , xm ] = (−1)ϱ(xn )ϱ(xn+1 ) [x1 , . . . , xn+1 , xn , . . . , xm ] [x1 , . . . , an xn + bn xn′ . . . , xm ] = an (−1)ι(an ,n) [x1 , . . . , xn , . . . , xm ]

+ bn (−1)ι(bn ,n) [x1 , . . . , xn′ , . . . , xm ]

(4.89a) (4.89b)

where the signs picked up during the performed operations is accounted for by ι(an , n) = ϱ(an )(ϱ(x1 ) + ⋅ ⋅ ⋅ + ϱ(xn−1 )). The defining equations—bracket identities—for the algebra are, for all n ∈ N, k+l=n

∑ k=0 l=0

∑ ϵ(π(k, l))[[xπ(1) , . . . , xπ(k) ], xπ(k+1) , . . . , xπ(k+l) ] = 0

π(k,l)

(4.90)

where π(k, l) stands for (k, l)-unshuffles. A (k, l)-unshuffle is a permutation π of the indices 1, 2, . . . , k + l such that π(1) < ⋅ ⋅ ⋅ < π(k) and π(k + 1) < ⋅ ⋅ ⋅ < π(k + l). The factor ϵ(π(k, l)) is the sign picked up during the unshuffle as the points xi with indices 0 ≤ i ≤ k are taken through the points xj with indices k+1 ≤ j ≤ l. This is just the normal procedure in superalgebras. The low index, n = 0 and n = 1 brackets are treated separately, thus [⋅] = 0

and

[x] = Dx

where D is a derivation

(4.91)

4.4.3 Jet spaces, Lagrangian field theory and locality From a theoretical physics perspective, jet spaces can be seen as a generalization of the idea of treating generalized coordinates qi (t) and velocities q̇ i (t) as independent variables in Lagrangian mechanics, to systems where higher-time derivatives such as q̈ i (t) occur. In field theory, this means treating the fields ϕ and their space-time derivatives 𝜕μ ϕ, 𝜕μ 𝜕ν ϕ, . . . as independent variables. Clearly, such a formalism may be useful in the higher spin context. It was, and still is, used in the development of the Vasiliev AdS theory, albeit not explicitly phrased as a jet space formulation. The Vasiliev theory contains infinite sequences of variables being higher and higher-order derivatives of the basic fields of the theory. In that context, the procedure is called “unfolding”.

162 � 4 General theory of interactions To make the discussion a bit more definite, consider a field theory where the Lagrangian ℒ depends on the fields ϕi and their derivatives 𝜕μ1 ϕi , 𝜕μ1 𝜕μ2 ϕi , . . . , 𝜕μ1 . . . 𝜕μk ϕi up to some finite order k. These derivatives are considered as independent variables, just as one considers ϕi and 𝜕μ ϕi independent when varying the action in an ordinary two-derivative theory. A shorthand notation is often used, writing ϕi(μ1 ⋅⋅⋅μk ) ≡ 𝜕μ1 . . . 𝜕μk ϕi (it goes without saying that the partial derivatives are symmetric). To denote that the Lagrangian depends on this set of independent variables, it is common to write ℒ[ϕ]. More generally, we may have to consider functions f of the fields and their derivatives as well as of the space-time coordinate xμ itself. Then we write f (x, [ϕ]). To further develop the theory, one then thinks of the set {ϕi , ϕμ1 , . . . , ϕi(μ1 ⋅⋅⋅μk ) } as constituting a k + 1 dimensional vector space Vk . Next, thinking of Vk as fibers over Minkowski space-time M, we have the jet space M × Vk of order k. This opens the possibility of considering nontrivial jet bundles J k (E) over space-time M with nontrivial projections π : E → M.33 Since, as in higher spin gauge theory, the order of derivatives may be unbounded, or not known a priori, one can introduce the infinite jet bundle π ∞ : J ∞ (E) = M ×V ∞ → M. However, actually having derivatives of infinite order is a clear sign of nonlocality. By a local function f , we intend in this context a function f (x, [ϕ]) depending only on a finite order of derivatives. In the Lagrangian field theory context, local functions can be assumed to be polynomial in the derivatives of the fields. The zero-order jet space J 0 (E) have coordinates (x, ϕ). As in fiber bundle theory, a field ϕ : M → V 0 can be represented by a section (or a “graph”) grϕ : M → M ×V 0 , which evaluates as grϕ (x) = (x, ϕ(x)). Such a section induces a section of J k (E) through evalu-

ation of the partial derivatives ϕi(μ1 ⋅⋅⋅μk ) |grϕ = 𝜕μ1 . . . 𝜕μk ϕi . Evaluating a local function at a section yields a space-time function, i. e., f (x, [ϕ])|grϕ = f (x, ϕ(x)). We can now generalize the Lagrangian field theory that we outlined in Section 3.14 in our Volume 1 to the case of higher-derivative theories. It is basically straightforward. Treating all derivatives of the field ϕi as independent variables, varying the action and computing the Euler–Lagrange derivative, amounts to the formula 𝜕ℒ δℒ δℒ δℒ δℒ = i − 𝜕μ1 i + 𝜕μ1 𝜕μ2 i − ⋅ ⋅ ⋅ + (−1)k 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μk i i δϕ 𝜕ϕ δϕμ1 δϕ(μ μ ) δϕ(μ ⋅⋅⋅μ k

= ∑ (−1)n 𝜕μ1 ⋅ ⋅ ⋅ 𝜕μn n=0

1 2

δℒ δϕi(μ ⋅⋅⋅μ 1

1

k)

(4.92) n)

One slight subtlety here is when computing the multiple partial derivatives, they must be interpreted as total partial derivatives according to

33 The theory of jet bundles is developed in detail in [237]. Fiber bundle theory is briefly reviewed in Section 3.12 in our Volume 1.

4.4 Some further notes on Noether coupling techniques

𝜕μ =

� 163

𝜕 𝜕 𝜕 𝜕 + ϕiμ i + ϕi(μν1 ) i + ⋅ ⋅ ⋅ + ϕi(μν1 ...νk ) i μ 𝜕x 𝜕ϕ 𝜕ϕν 𝜕ϕ(ν ...ν 1

1

(4.93) k)

as one indeed does already for the simple case of k = 1.34 A historical digression with hindsight In the early days of massless higher spin interactions, the generic higher-derivative nature of the interactions was revealed with the construction of light-front cubic vertices in 1983 and the covariant spin-3 vertex in 1984. Such a higher-derivative property may even have been suspected on general grounds, without any detailed calculations. One instance is Fronsdal’s attempt at a higher spin gauge algebra involving fields of all integer spin (see Section 2.6.1). Another is the Weinberg low energy theorem for the S-matrix (see Section 2.4.3), based on power-counting for cubic vertices for higher helicity. A more subtle suspicion for higher derivatives comes from properties of the Poincaré group [238] as mentioned in [152]. The nature of quartic and higher-order interactions was obscure in the early days, and remains so today, if not indeed nonexistent within conventional field theory due to the nonlocality problems. For any finite value of the spin s, one could make a power-counting ansatz for a pure quartic vertex. For a pure spin s vertex of order v, the power-counting number of derivatives is 2 + (v − 2)(s − 2). However, as we will see, no such pure spin s vertices exist by themselves. Thus, the higher spin, higher-derivative Lagrangian field theory that we set up a formalism for here, may be nonexistent as conventional field theory.

4.4.4 Syntactic–semantic approach to Lie∞ algebras in interacting field theory Toward the very end of the conference paper [220], Stasheff writes three formulas from Zwiebach’s closed string theory [219]; for the action, the field equations and the gauge transformations. The first is the action ∞ n−2 κ 1 {Ψ ⋅ ⋅ ⋅ Ψ} S(Ψ) = ⟨Ψ, QΨ⟩ + ∑ 2 n! n=3

(4.94)

where the first term is the free kinetic action with Q the BRST operator, |Ψ⟩ a string field and ⟨⋅ , ⋅⟩ an inner product. The expression {Ψ ⋅ ⋅ ⋅ Ψ} ≡ {Ψn } contains n terms and is given in terms of the basic brackets [Ψ, . . . , Ψ] by ⟨Ψ, [Ψ, . . . , Ψ]⟩. The particular case {Ψ2 } for n = 2 is given by the kinetic term. The basic brackets, in their turn, are some kind of products between string fields. The field equations follow by varying the action κn−2 {δΨ, Ψn−1 } n! n=2 ∞

δS = ∑

(4.95)

34 One could introduce a special notation such as 𝜕μt for the total derivative when writing general expressions. However, in practical calculations, and in particular when the Lagrangian depends polynomially on the field and its derivatives, this never causes any problems. One just proceeds with the derivations according to the chain rule and/or Leibniz rule.

164 � 4 General theory of interactions Likewise, gauge transformations are given by κn n [Ψ , Λ] n! n=0 ∞

δΛ Ψ = ∑

(4.96)

Contemplating this, one may be struck by the thought that these formulas are nothing particular to closed string theory. Abstracting away from any special interpretation of the ingredients, these are just formulas—written in one fell swoop, so to speak—that apply to any gauge theory. Guided by that thought, one may embark upon an attempt to derive them in the simplest possible way with minimal requirements. That is done in the paper [239]. The paper was inspired by computer science thinking; apart from abstracting away from the particulars of the problem, there is an explicit splitting of the problem into a syntactical part and a semantical part.35 To make this explicit, the syntax was chosen in a deliberately nonphysics looking language. What is to be the higher spin gauge fields (in the semantics) is represented by variables Φ of a data type ℋ, formally written as Φ :: ℋ. To connect this to semantics, there should be some “function call” to extract a particular spin s field ϕs , tentatively written as get(Φ, s) → ϕs . In order to distinguish instances of the variables Φ, we will write Φ(σ) letting σ standing for all properties maintained by the data type ℋ. The syntax has to make room for properties of real field theories. A first requirement is that the type ℋ supports a graded vector space structure. A second, that a kinetic action can be represented. For this, the paper introduced an inner product in and a kinetic operator K, in(⋅ , ⋅) :: ℋ2 → C and K :: ℋ → ℋ

(4.97)

The syntax for the free field theory action is A(Φ) = in(Φ, KΦ)

(4.98)

To be able to model interactions, the syntax has to support products pr :: ℋn → ℋ

with Φ(σn+1 ) = pr(Φ(σ1 ), Φ(σ2 ), . . . , Φ(σn ))

(4.99)

A priori, this product has no symmetries, an issue to which we will return below. A shorthand notation is useful when the field arguments are not needed pr(Φn ) ≡ pr(Φ(σ1 ), Φ(σ2 ), . . . , Φ(σn )) ≡ pr(Φ1 , . . . , Φn )

(4.100)

We will also need expressions like pr(Φk , Ψl ), which are naturally expanded as need be pr(Φk , Ψl ) = pr(Φ1 , . . . , Φk , Ψ1 , . . . , Ψl )

(4.101)

35 It can be argued that the split is not absolutely maintained in the paper, and that there is actually quite some unprincipled mix of syntax and semantics. Be that as it may, the idea is to separate the language to describe the theory from its eventual concrete implementation.

4.4 Some further notes on Noether coupling techniques

� 165

The product is supposed to be multilinear as defined by (4.89) replacing brackets [ ] by products pr( ) and xi by Φi .36 In order for this product to serve as a basis for introducing interactions, it must have further properties. The bare minimum of such properties will be derived from the requirement of gauge invariance of the action. The low indices n = 0 and n = 1 merit simplified notation. Thus, we define pr(Φ0 ) ≡ pr( ) = 0

and

pr(Φ) = KΦ

(4.102)

We now have the abstract tools for writing interactions. By taking the product between n − 1 HS fields and the inner product with an nth field, a candidate for an n-field interaction term can be written as a multilinear map vx :: ℋ⊗n → C :

vx(Φ1 , Φ2 , . . . , Φn ) = in(Φn , pr(Φ1 , Φ2 , . . . , Φn−1 ))

(4.103)

Just as the product (4.99), vx has no a priori symmetries. However, if this is to be useful in an interaction term, it must at least be cyclic symmetric in the field.37 This finishes setting up of the basic syntax of the theory (there are more details in the paper). The paper then defines an action ∞

A(Φ) = ∑ i=2

g i−2 ∑ vx(Φ1 , Φ2 , . . . , Φi ) i! π[i]

(4.104)

where the sum ∑π[i] means the sum over all permutations. A new summation symbol ∞ ∑∞ π(i=m) ≡ ∑i=m ∑π[i] will be used. The action can be written, highlighting the kinetic term explicitly and expanding vx, in the form g i−2 in(Φi , pr(Φ1 , Φ2 , . . . , Φi−1 i! π(i=3) ∞

A(Φ) = in(Φ, KΦ) + ∑

g i−2 in(Φ, pr(Φi−1 )) i! π(i=3) ∞

= in(Φ, KΦ) + ∑

(4.105)

where the kinetic term is 1 in(Φ, KΦ) = (in(Φ1 , K2 Φ2 ) + in(Φ2 , K1 Φ1 )) 2

(4.106)

36 This does not make the argument circular, it is just a refinement of the vector space requirement. What we want to derive is the formula (4.90). 37 This can be fixed by explicitly summing over all permutations of the fields. Alternatively, one could just sum over all cyclic permutations. It turns out, though, that summing over all permutations leads to simpler formulas. Computationally, it is inefficient to sum over all permutations. However, in actual implementations, the permutation symmetries will be explicit, and the combinatorial sums collapse into at most 𝒪(n2 ) terms.

166 � 4 General theory of interactions The gauge transformation will also be written as a formal power series. At this stage, we have a choice either to introduce a new abstract product prg (renaming the previously introduced product pra ), or use the same product as the one used for the action. This is one of the points where one is confronted with a dilemma as to the generality of the ansatz. We will be conservative here and use the same product. The gauge transformation then reads ∞ gi gi pr(Φi , Ξ) = KΞ + ∑ pr(Φi , Ξ) i! i! π(i=0) π(i=1) ∞

δΞ Φ = ∑

(4.107)

Gauge invariance of the action to all orders of interaction amounts to δΞ A(Φ) = 0.

(4.108)

To compute this, one has to apply δΞ to the action (4.104) ∞

δΞ A(Φ) = ∑ i=2

g i−2 δ ∑ vx(Φi ) i! Ξ π[i]

(4.109)

We immediately run into the problem of how to perform this operation within the weak structure we have introduced. However, by analyzing how the operation of varying the action is normally done in standard field theories, we see that this can be done by just textually substituting δΞ Φ for all the occurrences of Φ one at a turn. This is a very weak form of equational reasoning that we certainly want to do in any formalism, but strictly speaking, rewrite rules, belong to semantics. Then by demanding (4.109) to vanish, one can derive the requisite demands on the maps in, pr and vx. The paper goes through this calculation meticulously noting all steps where required properties of the maps has to recorded. By doing that, one can target precisely where syntax no longer suffices and “real” semantics—beyond vector space equational reasoning—is demanded. For the actual detailed computations, we refer to the paper. Here, we will only record the results.38 The variation of the action (4.109) computes to k+l=n

δΞ A(Φ) = ∑

π(k=0) π(l=0)

g n−1 in(Ξ, pr(Φk , pr(Φl ))) k! l!

(4.110)

Invariance should not depend on the gauge parameter Ξ, so in order for this sum to vanish, we must require for all n ∈ N, k+l=n

∑

π(k=0) π(l=0)

1 pr(Φk , pr(Φl )) = 0 k! l!

(4.111)

38 In view of the preceding discussion of field theory within the antifield/antibracket formalism, it should be clear that locality cannot be treated syntactically or within weak semantics.

4.4 Some further notes on Noether coupling techniques

� 167

This is a nontrivial demand on the map pr. The other demands can be considered as part of the syntax, but this one definitely involves the semantics of the theory. We will refer to this requirement as the product identity. Low level special cases of the product identity It is informative to write down the first few low-order cases of the product identity. When n = 0, the identity trivializes to pr(pr()) = pr(0) = 0

(4.112)

The case n = 1, expressing gauge invariance for the free theory, becomes pr(Φ, pr()) + pr(pr(Φ)) = KK Φ = 0

(4.113)

When n = 2, taking permutations into account and noting that pr(Φ1 , Φ2 ) is actually symmetric, we get pr(pr(Φ1 , Φ2 )) + pr(Φ1 , pr(Φ2 )) + pr(Φ2 , pr(Φ1 )) = 0

(4.114)

K pr(Φ1 , Φ2 ) + pr(Φ1 , K Φ2 ) + pr(K Φ1 , Φ2 ) = 0

(4.115)

or

This equation expresses gauge invariance of the cubic interaction term. Granting that we already know how to implement the n = 1 equation in terms of an appropriate field and a kinetic operator, the n = 2 equation is the first nontrivial equation. It involves the two-product pr(⋅ , ⋅), which so far is undefined. The next level, n = 3 involves the quartic interaction term K pr(Φ1 , Φ2 , Φ3 ) + pr(K Φ1 , Φ2 , Φ3 ) + pr(Φ1 , K Φ2 , Φ3 ) + pr(Φ1 , Φ2 , K Φ3 ) + pr(Φ1 , pr(Φ2 , Φ3 )) + pr(Φ2 , pr(Φ3 , Φ1 )) + pr(Φ3 , pr(Φ1 , Φ2 )) = 0

(4.116)

This equation expresses gauge invariance up to the quartic level. In order to solve it, the full two-product pr(⋅ , ⋅) must have been obtained first. Clearly, it can then be seen as a “differential equation” for the threeproduct pr(⋅ , ⋅ , ⋅) with K acting as differential operator. It follows that a necessary condition for the interaction to be cubic is that the two-product satisfies a Jacobi identity. In that case, the three-product pr(⋅ , ⋅ , ⋅) is zero, and the first four terms in (4.116) must be zero. Therefore, the last line of equation (4.116) is sometimes called the Jacobiator. It being nonzero signals the presence of a Lie∞ algebra.

The product identity will have to be generalized in order to close the gauge algebra. But first, let us record the field equations that are straightforward to compute by varying the action and demanding δΦ A(Φ) = 0. The result is g i−1 pr(Φi ) = 0 i! π(i=1) ∞

W(Φ) = ∑

(4.117)

expressing the basic intuition that the product captures the interactions. Up until now, we have not had to analyze the nature of the gauge parameters and their relation to the fields. There has only been one parameter Ξ around, and it has been

168 � 4 General theory of interactions tacitly assumed that it commutes with the fields Φi . When studying the gauge algebra, and perhaps higher-order structures, there will be at least two gauge parameters in the computations, and there is a choice as to how they commute or anticommute. In the antifield/antibracket BRST-BV formalism, the gauge parameters are Grassmann odd while the fields are Grassmann even. However, in the BBvD analysis, the parameters are even. Presumably—but I do not know—this is a technical choice with no fundamental bearing on the existence of interactions. In the paper [239], the parameters were chosen to be Grassmann odd in order to pave the way for a semantic map to a BRST-BV formalism. This said, to examine the demands of the gauge algebra, one shall have to compute— as always—the commutator [δΞ1 , δΞ2 ]Φ = δΞ1 (δΞ2 Φ) − δΞ2 (δΞ1 Φ). This is done in the paper. The computation results in an equation with terms that bear resemblance to the product identity, but now with a mixture of fields to powers Φk and Φl and with gauge parameters Ξ1 and Ξ2 . This forces a reexamination of the product identity in the case where the fields and parameters may carry a Z2 grading. Such an analysis leads to the bracket identities (4.90) of a strongly homotopy algebra, again with brackets [ ] replaced by products pr( ) and the graded elements xi represented by fields Φi and parameters Ξj . With such a generalized product identity, the commutator of gauge transformations become gk pr(Φk , Ξ1 , Ξ2 , W (Φ)) k! π(k=0) ∞

[δΞ1 , δΞ2 ]Φ = δΞ(Φ,Ξ1 ,Ξ2 ) Φ − g 2 ∑

(4.118)

where the new gauge parameter, in the first term, is g l+1 pr(Ξ1 , Ξ2 , Φl ) l! π(l=0) ∞

Ξ(Φ, Ξ1 , Ξ2 ) = − ∑

(4.119)

The first term can be recognized as a field dependent gauge transformation and the second as being proportional to the field equations, thus conforming to expectations. We can now record the required properties of the product. Apart from being able to make substitutions, add and compare equal terms, i. e., use standard equational reasoning with a pinch of graded vector space structure, the strongly homotopy Lie algebra product identity is the only nontrivial demand on pr. Let us, however, add a few more comments. Further comments on the syntactic–semantic approach First, field redefinitions should be taken into account. Given the free theory equation of motion in equation (4.113), this leads to a cohomological problem. Second, referring back to BBvD analysis of the higher spin gauge algebra, it is clear that the above formalism does not take field dependence of the gauge transformations into account fully. The commutator of two gauge transformations, produce a field dependent gauge transformation according to (4.118).

4.4 Some further notes on Noether coupling techniques

� 169

Therefore, as in BBvD, the commutator should be reexamined with field dependent parameters also on the left-hand side. Given that such a field dependence can be assumed to be polynomial, that should be a tractable computation. Third, gauge parameter redefinitions affect the structure of the gauge algebra [153]. Such parameter redefinitions make the parameters field dependent, thus interplaying with the second item above. It is not clear if this has been studied in the literature. Fourth, given that Lie∞ algebras are, in some sense, generalizations of Lie algebras, what about the general statement that infinitesimal gauge transformations form a Lie algebra and that the finite gauge transformations form a Lie group? As phrased in [175], Section 3.1.4: “There is no escape to that result because invertible transformations leaving something (here the action) invariant always obey the group axioms.”. At least two comments may clarify. The gauge algebra of all gauge transformations may be very large and infinite-dimensional and not at all reducible to a finite-dimensional matrix Lie algebra. Indeed, the commutator of two gauge transformations that we found in formula (4.118) is squarely within the context of [175]. Furthermore, it can be checked that Jacobi identity of three gauge transformations δΞ1 , δΞ2 and δΞ3 is satisfied in Lie∞ algebras [234] (see also the next section). Thus, we are not going outside the framework of Lie algebras. Rather, the Lie∞ structure of the n-ary products of the perturbative approach to interacting gauge theories, probes the underlying substructure of the gauge theory.

4.4.5 The Lie∞ mathematical elaboration of the BBvD theory The paper [240] by Fulp, Lada and Stasheff (FLS) is a mathematical elaboration of the BBvD theory. The paper shows that if the gauge algebra of the Lagrangian higher spin field theory of Berends, Burgers and van Dam [151] exists, then it has the structure of a strongly homotopy Lie algebra. In Section 4.2.1 above, we discussed the generality of the BBvD approach. The FLS paper interprets the BBvD analysis as pertaining to arbitrary field theories. The FLS paper formulates what they call the BBvD hypothesis: “[. . .] the requirement that the commutator of two gauge symmetries be another gauge symmetry whose gauge parameter is possibly field dependent.”. The authors note that BBvD “[. . .] do not require an a priori given Lie structure to induce the algebraic structure of the symmetry algebra.” (in contract to other referred works). Instead, as we have also seen, the algebraic structure should come out of the construction of the interactions. The paper, and the proof, is phrased in a coalgebraic language—rather than in an algebraic language—which is beyond our limits here. So, we will content ourselves with a brief restating of the results only.39 We might as well continue to let the authors speak for themselves. From the Introduction, we quote the following:

39 The “colanguage” derives from category theory where one studies morphisms between categories of mathematical objects such as algebras. Going from “algebra” to “coalgebra” means inverting all the morphisms. For instance, rather than studying products of two elements a and b computing to a new element c, one studies coproducts taking elements c in the underlying vector space V and “splitting” them up into elements (a, b) of V ⊗ V . A nice introduction to this, and related algebraic and geometric topics, is the master thesis [241].

170 � 4 General theory of interactions

When the BBvD hypothesis is satisfied, we show that the gauge symmetry algebra of a large class of field theories is an sh-Lie algebra.

There is a comment about when this reduces to the familiar Lie structures (Yang–Mills and gravity). There is also a comment about “open algebras” that close only on-shell. The paper discusses also these two cases. We formulate the relevant structures in BBvD’s theory in terms of linear maps from a certain coalgebra Λ∗ Φ [One may think of this object as the vectors space Φ ⊗ ⋅ ⋅ ⋅ ⊗ Φ.] into the respective vector spaces Φ of fields and Ξ of gauge parameters [. . .]. It turns out that the space Ξ of gauge parameters has, in general, no natural Lie structure, but the space of linear maps from Λ∗ Φ into Ξ is a Lie algebra under certain mild assumptions along with the BBvD hypothesis. [Compare to our fourth comment in the box above.] Our main result is [under the same assumptions] [. . .] the fields and parameters combine to form an sh-Lie algebra.

The analysis of the FLS paper can be seen as a mathematically sophisticated support for our syntactical–semantical approach in Section 4.4.4 and the results found there. Unfortunately, however, the BBvD hypothesis turns out to be false. As we will review in Section 5.4, the spin-3 gauge algebra does not close in the sense of being consistent at the g 2 order, not even by introducing fields of spin higher than 3. This was contrary to belief at the time.

5 Covariant approaches in Minkowski space-time At the time of this writing in early 2023, the covariant Minkowski approach to interacting higher spin gauge theory—the Minkowski branch of the Fronsdal program—finds itself in an impasse. Although cubic interactions are quite well understood, serious consistency problems arise at the next to lowest level, the quartic level of the gauge algebra closure. That said, the complete set of cubic interactions are known within the Noether coupling scheme [242]. The overall result agrees with the light-front result and classification [160] and with the BRST vertex [159]. Comparisons have been made with limits of string theory [243] and with cubic vertices from the Vasiliev theory [166]. More references will be given below as we proceed. From what we have learned in the previous chapters, it is indeed possible to set up general schemes, flexible enough to accommodate interactions between more or less arbitrary spectra of higher spin gauge fields, and powerful enough to analyze general properties. Such schemes range from the BBvD Noether coupling analysis, over the Fulp–Lada–Stasheff elaboration of the BBvD analysis, the BRST vertex operator approach to the full antifield/antibracket BRST-BV formulation as we have reviewed in the previous chapter. Although such general schemes need not be tied to just a Minkowski space-time background per se, we will—in this chapter—have that case in our minds. We will now approach the problem of actually implementing the general methods in practice.

5.1 Spin-3 cubic covariant interactions—BBvD Covariant spin-3 field theory was, for obvious reasons, the first case beyond spin 2 that was seriously explored in a covariant formulation. We saw in Section 4.4 of our Volume 1 that spin 1 is very nice and easy, whereas already spin 2 involves quite a bit of work. That is apparent from the Fang–Fronsdal 1979 paper [150] on deformations of gauge groups (see Section 2.8 in our Volume where also references to earlier work can be found). The spin-3 covariant cubic interaction was found by F. A. Berends, G. H. J. Burgers and H. van Dam in the 1984 paper [152] (BBvD-3).

5.1.1 The spin-3 cubic self-interaction The approach that was used was based on the method outlined in the Fang–Fronsdal paper—and applied there to spin 2—but, as BBvD writes “[. . .] it has never been applied successfully to a case, where the theory was not known beforehand.”. The method is summarized as follows: 1. Find the free Lagrangian ℒ0 with its gauge invariance and source constraint. https://doi.org/10.1515/9783110675528-005

172 � 5 Covariant approaches in Minkowski space-time 2. 3. 4. 5.

Construct an interaction Lagrangian g ℒ1 , trilinear in the fields, satisfying the source constraint to order g. Extend the gauge invariance of ℒ0 to a gauge invariance of ℒ0 + g ℒ1 up to order g. Find the commutator of two (new) gauge transformations of ℒ0 + g ℒ1 . Hopefully, the full gauge algebra can be guessed from this. When the gauge algebra is known, the complete Lagrangian could be constructed.

In the paper, this program was applied to spin 3 and steps 1–4 were carried out. However, as soon became clear, in step 4, the full gauge algebra cannot be guessed. This is in contrast to spin 2 where the full Lie algebra of infinitesimal coordinate transformations can be guessed from the order g commutator as was shown by Fang–Fronsdal. Even worse, BBvD soon realized that the gauge algebra does not even exist. There is no gauge theory of pure spin-3 fields. This was shown in the follow up paper [151] (BBvD-G) that we have discussed at length in Sections 2.6.2 and 4.2 (see also [155, 157]). But that was not yet known, so let us continue to review the spin-3 paper (BBvD-3). In Section 2 of the paper, the cubic spin-2 interaction is discussed in detail as a preparation for the much more complex spin-3 calculation. Field redefinitions and gauge parameter redefinitions are discussed. After that, Sections 3 and 4 on spin 3 are quite short. We will review the discussion as it applies to spin 3. The free Lagrangian ℒ0 , the field equations written as ℒ0,ϕαβγ and the free theory gauge transformations δ0 ϕαβγ of the field ϕαβγ in terms of the traceless gauge parameter ξαβ , are given as in the Fronsdal theory. Gauge invariance of the action leads to an operator constraint 1 𝜕γ (ℒ0,ϕαβγ − ηαβ ℒ0,ϕ δ ) = 0 γδ 4

(5.1)

If the free field theory is coupled to a source Tαβγ (coming for instance from selfinteraction), then there is a corresponding source constraint 1 𝜕γ (Tαβγ − ηαβ T ρ ργ ) = 0 4

(5.2)

For the cubic interaction term ℒ1 , the possibility of a set of spin-3 fields ϕaαβγ is allowed for. It is noted that no “suitable” interaction with one derivative is found. The next case is three derivatives, distributed over the fields in any way. It is written g ℒ1 = gf abc ℒ(ϕa , ϕb , ϕc ), summed over a, b, c from 1 to N (no symmetry properties for the f abc are assumed). The coupling constant g has dimension inverse mass squared. There are 48 different terms in ℒ1 (for spin 2 the number is 14). They are not listed in the paper. Next, paralleling the discussion for spin 2, there are nonlinear gauge transformations ϕaαβγ = δ0 ϕaαβγ + gδ1 ϕaαβγ + ⋅ ⋅ ⋅

(5.3)

5.1 Spin-3 cubic covariant interactions—BBvD

� 173

No ansatz for δ1 ϕaαβγ in terms of fields and parameters is given. At this stage, it is enough to note that δ1 ϕaαβγ is linear in the gauge parameter. Therefore, after varying the action ℒ0 + g ℒ1 with δ0 ϕ + gδ1 ϕ and collecting terms of order g, all derivatives acting on the gauge parameter can be shifted off the parameter through partial integrations. The result of this operation is what the paper calls “the first order operator constraint” 1 abαβ (ηα ρ ηβ σ δab 𝜕τ − ηαβ ηρσ δab 𝜕τ )ℒ1,ϕbρστ = −H1 ρστ ℒ0,ϕbρστ 4 abαβ

(5.4) abαβ

The paper notes that H1 ρστ is symmetric and traceless in αβ. It is for the H1 ρστ and ℒ1 that an ansatz will be needed. The right-hand side of (5.4) is zero when the free field equations are satisfied. Then the source constraint for ℒ1 remains. The solution then proceeds in two steps. First, a solution for ℒ1 is found satisfying the source constraint (for on-shell fields). Field redefinitions (there are 18 of them) of the free Lagrangian, producing fake interactions, are sorted out. This yields a solution for ℒ1 with 25 terms. Second, taking this ℒ1 for off-shell abαβ fields and inserting it in (5.4), a solution for the H1 ρστ can be obtained. Here, there is a abαβ

new subtlety in that the solution for H1 ρστ is not unique, gauge parameter redefinitions must be taken into account. There are 12 of them. The order g gauge transformations abαβ gδ1 ϕaαβγ themselves can be obtained from the H1 ρστ by “reversing” the partial integrations, so to speak. This two-step procedure to solve (5.4) is generic for the Noether procedure, as we have noted in several places (see, for instance, Section 3.14.6 in our Volume 1). It is illustrated for the simple spin-1 case in Section 4.4 in Volume 1. Clearly, however, the complexity increases dramatically with spin. No discernible structure can be seen from the explicit solution obtained in the paper, and perhaps, nor should one be expected. Even the spin-2 cubic interaction looks quite unremarkable. Only for spin-1 Yang–Mills, with its single cubic term, do we have something simple to contemplate. The paper ends with a computation of the commutator of two-order g gauge transformations. The algebra closes, the paper states.1 As for the Jacobi identity, the freedom of parameter redefinitions must be used. This is left to the next paper. The paper notes the need for fully antisymmetric coefficients f abc and for three derivatives. This is conjectured to generalize to higher spin in the now well-known way. 5.1.2 Gauge algebra problems In the follow up paper [151] (BBvD-G), after the general discussion that we have reviewed at some length, the authors turn to the concrete gauge algebras for spin 1, 2 and 3 (see Section 5 of the paper, it is well worth reading). The all-important Jacobi identities are studied. 1 Probably prematurely, it is contradicted in the follow-up paper to be reviewed in Section 5.1.2.

174 � 5 Covariant approaches in Minkowski space-time For spin 1, the analysis shows the well-known fact that the gauge algebra is independent of the fields. Indeed, it reduces to an ordinary matrix Lie algebra. This can be seen as the underlying reason for why the full Yang–Mills theory can be derived by a—now standard text-book—gauging procedure. Spin 2 is a bit more involved, but the result is the expected. The spin-2 gauge transformations to order g (derived ab initio, by deforming the free spin-2 theory, not from the Einstein theory) closes on field independent parameters. Furthermore, the Jacobi identity is satisfied. The result is the Lie algebra of infinitesimal general coordinate transformations to first nontrivial order. From there on, the full Einstein theory can be obtained. At spin 3, the pattern breaks down. When computing the commutator of two firstorder gauge transformations, one is really probing the theory at second order. This is analyzed in detail in the paper (see Section 4, reviewed in Section 4.2.1 here). It is useful to look back at the lower spin cases. Doing that one sees that for spin 1, closing the gauge algebra in the sense of the commutator producing a gauge transformation of the same form, already requires the Jacobi identity. The same phenomena occurs for spin 2. In both cases, the Jacobi identity holds. But not so for spin 3. Let us see if we can disentangle the problem. The spin-3 gauge algebra breakdown The gist of the argument, which is quite involved, may be summarized as follows. There is some choice in the order g spin-3 gauge transformation, but there is always a term c S a (ϕ, ξ) = gf abc ϕbρστ,αβ ξαβ

(5.5)

that cannot be removed by parameter redefinitions. The paper makes a certain choice and computes the commutator of two transformations δξ and δη . The order g terms then give C0 (η, ξ) in the commutator (see formulas (4.22) and (4.28)). Now, as reviewed in Section 4.2.1, equation (4.34) at order g2 must be satisfied. Spin 1 and spin 2 works without invoking order g2 terms g2 T2 (ϕ, ϕ, ξ) in the gauge transformations (see (2.132)) or in the field dependent g2 C1 (ϕ, ξ1 , ξ2 ) parameters (see (4.32)). The paper shows that this is not possible for spin 3. The first three terms in (4.34) does not vanish (as they do for low spin). This can be seen from the unavoidable presence of the S a (ϕ, ξ) term in the transformation. Commuting S a (ϕ, ξ) with S a (ϕ, η) (corresponding to the first two terms in (4.34)) produces e c S(S(ϕ, ξ), η) − S(S(ϕ, η), ξ) = g2 (f abc f bde − f abe f bdc )ϕdρστ,αβγδ ξαβ ηγδ

(5.6)

with a free index a on the left hand side. It is a term with four derivatives acting on the field. It cannot be canceled by the third T1 (ϕ, C0 (η, ξ)) term in (4.34), since this term cannot have four derivatives on the field. Therefore, one has to consider 𝜕C1 terms and perhaps T2 terms. However, 𝜕C1 (ϕ, η, ξ) is ruled out since in (5.6) all free indices are on the field, whereas in 𝜕C1 at least on index is on a derivative. The term (5.6) cannot be canceled. In a similar way, T2 terms are ruled out. At least one derivative in the T2 terms in (4.34) is used on a parameter ξ or η.

The situation is summarized in the paper. A unique nontrivial cubic vertex for selfinteracting spin-3 fields exists. A corresponding gauge transformation can be derived.

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However, the commutator of two such gauge transformations does not close on a similar transformation, not even utilizing possible generalizations going beyond what is needed for the low spin gauge fields.2 However, the hope remained: could the theory be saved by adding in higher spin fields, of spins 4 and 5, and eventually of all integer spin? This had been suggested already by Fronsdal. However, to investigate that, much more powerful methods had to be applied, and they were not yet available.

5.2 The covariant BRST approach A covariant BRST approach to higher spin gauge fields, that produced the correct cubic Yang–Mills vertex, was initiated by the present author in 1988 in the paper [159]. The underlying free field theory was thoroughly reviewed in Section 5.4 in our Volume 1. The method employed, based on vertex operators, was inspired by the Witten bosonic string cubic vertex [244] in the formulation of the Gross–Jevicki papers [245, 246, 248].3 In surveying the research into this approach, it seems that it has not come to a clear conclusion, and there may be more serious work to do.

5.2.1 Vertex operator approach to interactions—a reminder For a short reminder about the vertex operator approach to interactions, consider generalized fields |Φ(x)⟩ living within the Fock space of any number—finite or infinite—of μ† physical oscillators αn and their ghost companions βn† . This means the Φ’s are appropriate expansions over the oscillators. The free theory Lagrangian is given in terms of the BRST operator as ⟨Φ|Q|Φ⟩. For the cubic interaction, it is convenient to work in momentum space. The three fields, shortly written as |Φi ⟩ then comes with their own momenta and lives in their own Fock spaces. A vertex operator ⟨V123 |—to be determined—encodes the details of the interactions between the component space-time fields of the |Φi ⟩. The Lagrangian, to cubic order, is then given by 1 1 L = ⟨Φ|Q|Φ⟩ + g⟨V123 |Φ1 ⟩|Φ2 ⟩|Φ3 ⟩ 2 3

(5.7)

The gauge transformations are also given by the same vertex operator, roughly as δΦ3 ∼ ⟨V123 (|Φ1 ⟩|Ξ2 ⟩+⋅ ⋅ ⋅) with appropriately symmetrized and weighted terms. The fields may 2 The negative conclusion regarding the gauge algebra for a pure spin-3 theory was also reached in the papers [155, 157] using slightly different methods, but within the same kind of formalism. 3 Thanks to the Queen Mary College senior common room pub, where the papers were read and pondered over lunch time pints of Fuller’s London Pride (not more than one a day, though).

176 � 5 Covariant approaches in Minkowski space-time very well carry Chan–Paton-type internal indices of some simple gauge group, and in that case there are appropriate traces over these indices assumed in the expression for the action and the corresponding gauge transformation. Furthermore, there are implicit space-time integrals or momentum space integrals understood in (5.7). We will provide details when treating concrete models. 5.2.2 The first mid-1980s BRST attempt at interactions I. G. Koh and S. Ouvry were quick to approach the higher spin interaction problem within the BRST formalism, building on the Ouvry and Stern paper [247] on the free field theory. As we know, the BRST approach to free higher spin gauge fields was modeled on free string field theory,4 so it was natural to look at string theory also for the interaction problem. String field theory interactions was a very active research area at this time. In the paper [158], Koh and Ouvry works with a higher spin BRST model with a countable infinite set of oscillators inherited from the bosonic string. The first class constraints are those of massless higher spin theory [247, 35]. Their cubic interaction Lagrangian is that of formula (5.7) with the fields denoted by |χ⟩. For the gauge transformations, they take5 δ⟨χ3 | =

1 ⟨Λ |Q + ⟨V123 |((α2 /α1 )|χ1 ⟩|Λ2 ⟩ − (α1 /α3 )|Λ1 ⟩|χ2 ⟩) g 3 3

(5.8)

with |Λ⟩ for the gauge parameters. In this formula, the αi are “length” parameters. The authors motivate their construction in the following way. In our case the basic field |χ⟩ [reference to [247]], even though it contains at the free level the same local fields as a string field, is not a string-like object. However, when interactions are turned on, we may assume that a sort of string field picture arises, as a result of the interactions between the local fields. Thus, in the same way as a “length” parameter was introduced in the string context as an ingredient for the construction of the interaction, we assign to |χ⟩ a “length” parameter α, such that when the interaction takes place between the fields |χ1 ⟩, |χ2 ⟩ and |χ3 ⟩, the algebraic length α1 + α2 + α3 = 0 is conserved (in what follows we take α1 > 0, α2 > 0 and α3 < 0). We then show that it is possible to find a cubic vertex which ensures per se the nilpotency of the corresponding BRST operator; this in any space-time dimension.

The authors comment that the vertex is different from the string vertex, referring to papers by Neveu and West and by Hata, Itoh, Kugo, Kunimoto and Ogawa, two groups of researches then active in string field theory. The authors also write that “[they] have 4 But it could have been set up independently, had the historical circumstances been different. The method itself is more general than string theory. See our Volume 1, Section 3.3.3. 5 The paper just writes V for ⟨V123 |.

5.2 The covariant BRST approach � 177

no arguments regarding the uniqueness or the fact that a Higgs mechanism could take place: quartic interactions may still be needed in that respect.”.6 Up to some minor notational changes, the free field theory that the paper starts from is, as already noted, the one of [247] (described in Section 2.11.2 in our Volume 1). Thus there is an infinite number of space-time oscillators αnr with n = −∞, . . . , −1, 1, . . . , ∞ and r = 1, 2, 3 and where α0r denotes the momentum variable of the fields. The negative n-indices signify creation operators. The ghost oscillators are correspondingly denoted by (βn , β̄ n ) with (β0 , β̄ 0 ) the ghost zero mode. The gauge invariance of the Lagrangian, to first order in the coupling, under the gauge transformations (5.8) translates into the equation 3

Qt |V ⟩ = 0 α 123 t=1 t

(5.9)

∑

The authors then consider an ansatz for |V123 ⟩ in the form of a product of bosonic vertex VB and a fermionic (ghost) vertex VF VB = ⟨0123 | exp( ∑ Nnrs αnr ⋅ α0s ) n>0 r,s=1,2,3

r ̄s VF = ⟨0123 | exp( ∑ Rrs n βn β0 ) n>0 r,s=1,2,3

(5.10)

The ansatz is clearly inspired by string field theory with the coefficients Nnrs in the place rs where the string theory Neumann functions Nn0 would be. Note that there are no coeffirs cients corresponding to the Neumann functions Nnm with n, m ≠ 0 in the ansatz. Therefore, there are no cross couplings between the oscillators, and one could well consider a theory with just one oscillator. The authors give no explanation for not including such terms in the ansatz.7 It is known, however, from the analysis of [159] and the light-front theory, that in a one-oscillator theory, any coefficients N11rs are diagonal in field labels, i. e., r = s, thus not contributing to interactions. In string field theory, the Yang–Mills interaction comes from combinations of prodtu r ucts N11rs N10 α1 ⋅ α1s α1t ⋅ α0u where it is essential that r can be different from s. One may therefore suspect that a vertex of the form (5.10) cannot produce the Yang–Mills coupling for spin-1 gauge fields. This is also confirmed by the solution found in the paper, and explicitly stated in the text where the authors write “Finally, as for the component

6 Note that this does not refer to the Witten theory of interacting bosonic strings—the “Witten” vertex— but to the variant based on covariantizing the light-front string field theory vertex. The Witten theory is cubic and requires no quartic interactions. In a certain sense it is “Chern–Simons”-like. There is also no length parameter in the Witten theory. The length parameter in the non-Witten approaches derived form the light-front α, which is actually the p+ momentum of the string fields. 7 That is, apart from noting their absence in connection to a discussion of the string theory “horn diagrams” of the string theory papers referred to above. Physically, cross couplings between the oscillators would correspond to string vibrations. Here, we have no tension.

178 � 5 Covariant approaches in Minkowski space-time expansion, our gauge invariant interaction is quite different from the string’s one. For example the spin-one local field is shown to couple to other fields with higher derivatives but does not have the AApA coupling of Yang–Mills theories.”. 5.2.3 The 1988 Yang–Mills producing higher spin vertex In the winter and early spring of 1987, I started to work on a covariant higher spin vertex in a BRST formulation, inspired by the new Witten theory to interacting bosonic strings [244], in particular in the operator formulation of Gross and Jevicki [245, 246, 248]. From the work with I. Bengtsson and N. Linden, the previous autumn on the light-front cubic vertices—to be extensively reviewed in Chapter 6—I knew that it would not work with an exponential of operators bilinear in oscillators and momenta. This was also confirmed by the Koh and Ouvry paper reviewed above; at least operators with three oscillators and one momenta were necessary. A supposition that was also born out by the actual computations made in the covariant formulation. The theory [159] was based on the BRST formulation of the free field theory of [35]. Here, we will translate the theory into the equivalent notation of Section 5.4 of our Volume 1 and make a few aesthetic adjustments in the formulas.8 The action to cubic order, in arbitrary space time dimension D, is taken as in formula (5.7) with the vertex written as |V123 ⟩ = C1 eΔ |−⟩123

(5.11)

with C1 a numerical constant to be chosen and the vacuum |−⟩123 = |−⟩1 |−⟩2 |−⟩3 for the three interacting fields |Φr ⟩. More precisely, in D = 4 the interaction term should be written g Tr ∫⟨V123 |Φ1 ⟩|Φ2 ⟩|Φ3 ⟩δ4 (p1 + p2 + p3 )d 4 p1 d 4 p2 d 4 p3 κ

(5.12)

Here, κ is a constant of mass dimension −1 while g is a dimensionless coupling constant. The trace operation Tr is over the Chan–Paton-type matrices thought to be attached to the fields. Since momentum space fields have mass dimension −3, this expression is dimensionless, as it should be. In the sequel, and in the paper itself, the momentum space delta function and integrals, and the trace operation, will not be written out explicitly. They may be considered as included in the Fock space contractions ⟨⋅||⋅⟩. This also goes for the gauge transformation formula (5.15) below. The problem is to determine the operator Δ and for this we need an ansatz. It must be built out of operators that are Lorentz scalar, bosonic, have ghost number zero, not 8 Essentially, this amounts to changing notation for the ghosts according to β0 → c0 , β̄ 0 → b0 , β → c− , β̄ → b+ , β† → c+ and β̄ † → b− .

5.2 The covariant BRST approach

� 179

annihilating the vacuum and be of mass dimension zero. Then we have the following components with the required properties αr† ⋅ αs†

καr† ⋅ ps

cr+ b−s

κcr+ b0s

κ2 pr ⋅ ps

(5.13)

where r, s = 1, 2, 3. From our light-front deliberations,9 we knew that Δ could be expected to be built from sums of powers of these components. This is in contrast to the bosonic string, where only linear terms in the operators of (5.13) occur. But as many authors have noted over the years, the exponential form exp Δ for the vertex operator is an inheritance from the string, and once higher powers of the basic bilinears of (5.13) are introduced, the exponential form exp Δ may be unnecessary or even wrong. For the string, the exponential form derives from the three-string overlap conditions encoding coordinate continuity, translating into a linear partial differential equation for the vertex operator.10 Anyway, in the notation of the paper, the ansatz for the first two powers in an expansion Δ = Δ2 + Δ4 + ⋅ ⋅ ⋅ were taken as Δ2 = X rs cr+ b−s + Y rs αr† ⋅ αs† + Z rs αr† ⋅ ps + W rs cr+ b0s

Δ4 =

X rstu cr+ b−s ct+ b0u

+

Y rstu αr†

⋅

αs† αt†

⋅ pu +

Z rstu cr+ b−s αt†

(5.14a) ⋅ pu +

W rstu αr†

⋅

αs† ct+ b0u

(5.14b)

In Δ2 , we have included all bilinears except the last one. Powers of κ are absorbed into the coefficients. In Δ4 , we have only written out contributions with coefficients proportional to κ. The paper lists further terms proportional to κ0 , κ2 and κ3 . However, the terms in (5.14b) are the ones that can be expected to contribute to the Yang–Mills cubic coupling. Since they are proportional to κ, they will produce spin-1 cubic interactions proportional to the coupling g. The nonlinear gauge transformation corresponding to the cubic interaction is g δ|Φ3 ⟩ = C2 (⟨Φ1 |⟨Ξ2 | − ⟨Ξ1 |⟨Φ2 |)|V123⟩ κ

(5.15)

with momentum space delta function and integrals understood as noted above. Gauge invariance to cubic order is now equivalent to 3

∑ Qr |V123 ⟩ = 0

r=1

(5.16)

with C2 = 43 C1 . 9 Private discussions with I. Bengtsson and the paper [160] with N. Linden. 10 To this day, no such simple geometric, or mechanical, picture for the higher spin cubic vertex exists. The present author have made some attempts, and discussions can be found in the papers [249, 250, 251, 252]. See also Section 5.4.1 in our Volume 1, and Section 2.2 in the present Volume, in particular, 2.2.3.

180 � 5 Covariant approaches in Minkowski space-time In the paper, two global symmetries of the free theory were discussed, and it was argued that they should carry over to the interacting theory. One of them is given by the Hermitian (or anti-Hermitian) operator ℛ=R±R

where R = α ⋅ p − 2b0 c−

†

(5.17)

This operator resembles the list of operators Kn = Ln −(−1)n L−n that generate symmetries in covariant string field theory [244]. The Hermitian alternative of (5.17) was, and will be here, taken as a symmetry also of the cubic theory. Since it is supposed not to get modified by interactions, the condition for invariance is 3

∑ ℛr |V123 ⟩ = 0

(5.18)

r=1

The second global symmetry will not be discussed here, instead see discussion in the original paper and in reference [260]. Finally, let it be noted that the theory works with the full triplet of (reducible) fields for all spins encoded in |Φ⟩. Thus, no trace conditions are imposed on component fields or gauge parameters. Now, let us proceed to the solution. Since the two equations (5.16) and (5.18) are “linear” in their action on Δ, the solution for Δ4 will be independent from Δ2 . For Δ2 , we will just quote the result of the computation. Confirming the light-front result and the result of [247] (see Section 5.2.2) we get X rs = 2Y rs = −δrs

(5.19)

while all components of the symmetric part of Z rs are equal to a constant ZS and all components of the antisymmetric part of Z rs is equal to another constant ZA and W rs = 2ZArs . In order to get cubic interactions for spin 1, we have to expand exp Δ2 to second order, and it is clear that the Yang–Mills interaction is not reproduced. Turning to Δ4 , the equations coming from BRST invariance (5.16) are

(Z

rtsu

b0t pμu

+

X rstu b0s cs+ cr+ ct+ b0u = 0

(5.20a)

=0

(5.20b)

=0

(5.20c)

=0

(5.20d)

=0

(5.20e)

(W rstu p2u + Y rstu pt ⋅ pu )αr† ⋅ αs† ct+ (X rstu p2u + Z rstu pt ⋅ pu )cr+ b−s ct+ 2W rstu b0u pμr − X rstu b0u pμs )cr+ ct+ αsμ† (2Y rstu pμr pνu − Z rstu pμs pνu )cr+ αsμ† αtν†

All repeated r, s, t, u indices are summed over 1, 2, 3. Note that in some places, an index may occur more than once in the lower position. Note also the deviant index order for the Z term in equation (5.20d). Some signs and factors differ from the paper [159] due to the different conventions for Q used here.

5.2 The covariant BRST approach

� 181

Likewise, there are equations coming from the global symmetry (5.18) X rstu b0s cr+ ct+ b0u = 0

(5.21a)

=0

(5.21b)

=0

(5.21c)

Y rstu αr† ⋅ ps αt† ⋅ pu Y rstu αr† ⋅ αs† pt ⋅ pu Z rtsu cr+ b−s pt ⋅ pu (W rstu − Z turs )αr† ⋅ ps ct+ b0u

=0

(5.21d)

=0

(5.21e)

These two sets of equations are actually quite weak, and they do not suffice to determine the vertex functions completely. Counting the number of coefficients, we get 18 Y rstu , 18 W rstu , 27 Z rstu and 9 X rstu for a total number of 72. For the equations, on the other hand, there are 44 independent Q-equations and 38 independent R-equations. However, not all of the R equations are independent of the Q-equations. The total number of independent equations is 56, leaving 16 free coefficients. The nonuniqueness of the solution is related to field redefinitions, an issue that was discussed but not explicitly addressed in the paper. The solution given in the paper was therefore not unique, but it reproduced the Yang–Mills cubic interaction, so that parts of the field redefinition freedom was fixed implicitly. The solution in the 1988 paper was computed by hand. Running the equations through a modern mathematics software package, one may obtain a parametrized solution in terms of 16 undetermined constants. Then field redefinitions reduce this number down to 1 undetermined constant which we chose to be Y 1231 . The solution is given in Table 5.1. Table 5.1: Solution for the coefficients. The rest of the coefficients follows from cyclic permutation 1 → 2 → 3 → 1. All coefficients are expressed through Y1234 = 1 and − stands for nonexistent entries. For the Y and W , the entries are redundant due to the extra symmetry in the first two indices. Index

Y

W

Z

X

Index

Y

W

Z

X

Index

Y

W

Z

X

1231 1232 1233 1213 1223 1212 1221 1211 1222

1 −1 0 −1 1 −1 1 0 0

1 −1 0 0 0 0 0 −1 1

0 2 2 2 2 2 0 0 2

−1 1 −1 – 1 – 1 – −1

2131 2132 2133 2113 2123 2112 2121 2111 2122

1 −1 0 −1 1 −1 1 0 0

1 −1 0 0 0 0 0 −1 1

−2 0 −2 −2 −2 0 −2 −2 0

−1 1 1 −1 – −1 – 1 –

1123 1113 1112 1132 1131 1121 1122 1133 1111

3 1 −1 −3 −1 1 0 0 0

1 1 −1 −1 1 −1 2 −2 0

2 2 −2 −2 0 0 −2 2 0

1 – – −1 1 −1 1 −1 –

182 � 5 Covariant approaches in Minkowski space-time Field redefinitions For our purpose, it will be enough to study spin-1 field redefinitions. Now, referring back to the generic formula (4.31), it would seem that there are no nontrivial spin s field redefinitions. Consider, however, the tripletBRST action for the free spin-1 field11 1 ∫ d 4 x(φ◻φ − H 2 + 2H𝜕 ⋅ φ) 2

(5.22)

The auxiliary field H is subject to a field redefinition H → H + gφ ⋅ φ that clearly produce fake cubic and quartic interactions. We will now explore the consequences of this observation. Introducing a field redefinition vertex |F123 ⟩, a field redefinition in the BRST approach can be written |Φ3 ⟩ → |Φ3 ⟩ +

g ⟨Φ |⟨Φ |F ⟩ κ 1 2 123

(5.23)

Counting mechanical ghost number ghm , |F123 ⟩ must have ghm = 1/2, since the fields have ghm = −1/2. Performing this redefinition in the free BRST action, we get 3 g 1 1 ⟨Φ3 |Q3 |Φ3 ⟩ → ⟨Φ3 |Q3 |Φ3 ⟩ + ⟨Φ1 |⟨Φ2 |⟨Φ3 | ∑ Qr |F123 ⟩ 2 2 3κ r=1

(5.24)

using symmetry in the field labels 1, 2, 3. We see that ∑r Qr |F123 ⟩ are fake cubic interactions on the same level as |V123 ⟩. Indeed, the invariance requirement (5.16) becomes an identity 3

3

r=1

s=1

∑ Qr ∑ Qs |F123 ⟩ = 0

(5.25)

due to the nilpotency of the BRST operators ∑r Qr ∑s Qs = 0. Therefore, we have a cohomological problem: we are looking for BRST-closed solutions to equation (5.16) modulo BRST-exact solutions |D123 ⟩ = ∑r Qr |F123 ⟩. We can now proceed to details. We need to lift the field redefinition H → H + gφ ⋅ φ to BRST formalism. From Section 5.4.2. (Volume 1), we take the formula |Φ⟩ = (φ + Cc + b− + Hb− c 0 )|+⟩ for a spin s field. For spin 1, there is no C field (no trace). Some experimentation hints at the following tentative formula for the field redefinition of H H3 b−3 c30 |+⟩3 =

g ⟨ϕ ⋅ α |⟨ϕ ⋅ α |F rst αr† ⋅ αs† b−t |−⟩123 κ 1 1 2 2

(5.26)

By symmetry, there are 6 free parameters F rst . Consider next the BRST operator Q = −c 0 p2 + c + α ⋅ p + c − α† ⋅ p + c + c − b0

(5.27)

Then consider acting with ∑i Qi on F rst αr† ⋅ αs† b−t |−⟩123 . Short computations yield ∑ ci− αi† ⋅ pi F rst αr† ⋅ αs† b−t |−⟩123 → F rst αr† ⋅ αs† αt† ⋅ pt |−⟩123 i

∑ ci+ αi ⋅ pi F rst αr† ⋅ αs† b−t |−⟩123 → F rst αr† ⋅ ps cs+ b−t |−⟩123 i

11 See our Volume 1, Section 5.4.3.

mixing with Y rstu terms mixing with Z rstu terms

(5.28a) (5.28b)

5.2 The covariant BRST approach � 183

∑ ci+ ci− b0i F rst αr† ⋅ αs† b−t |−⟩123 → F rst αr† ⋅ αs† ct+ b−t |−⟩123 i

mixing with W rstu terms

(5.28c)

Using the freedom offered by these field redefinitions, it is possible to reduce the number of undetermined vertex coefficients from 16 to 1.

It is possible to compile the concrete numerical coefficients of Table 5.1 into a vertex operator Δ4 as in the ansatz (5.14b). We will not do it here, but just note that the leading term for the Y coefficients yield Δ4,leading = Yα1† ⋅ α2† α3† ⋅ (p1 − p2 ) + cycl. perm. + ⋅ ⋅ ⋅

(5.29)

where the dots stand in for the corresponding leading W , Z and X terms; by “leading” terms is meant terms where the oscillators and ghosts are labeled by three different field labels 1, 2, 3. This very same pattern appears both in the covariant cubic vertex [253] that we will review in Section 5.3, and in the light-front vertex. This leads to a question, that as far as I know, has not been fully investigated. In the light-front approach, the spin s cubic self-interaction is given by simply taking the spin-1 cubic vertex and raising it to the power s [160, 254]. Such factorization may be expected in a formalism that only employs physical states,12 but does it hold in a covariant formalism, or a BRST enhanced formalism? This is clearly related to the question of writing a “generating function” exp(Δ4 ) for all cubic higher spin interactions, or some other function f (Δ4 ) that can be expanded in powers. At least two questions arise: (i) are the individual spin s cubic interactions correctly reproduced for such a function, and (ii) are the relative coefficients such that the vertex can be continued to the quartic level, or as in the light-front approach, such that the theory can be made to stop at the cubic level. In the light-front approach, this latter phenomenon actually happens, as will be reviewed in Section 6.5.

5.2.4 General analysis of the BRST approach A few papers, from the late 1980s and early 1990s era, that can be classified as general analysis of the BRST approach, should be mentioned. They are interesting and important in this respect, but they did not lead to new progress in the actual construction of higher spin vertices. Although some of the authors were quite active in the 1980s higher spin gauge theory research, the papers have largely remained outside the standard stock of higher spin references.

12 See the review [255] for a discussion and references to the underlying “scattering amplitudes” research as it applies to higher spin. See also the textbook [134].

184 � 5 Covariant approaches in Minkowski space-time The first paper is [162], where criteria for a purely cubic BRST-type theory are analyzed. It is argued that the necessary and sufficient conditions for the existence of a purely cubic interaction take the form of Lie algebra on which the free theory BRST operator acts as an exterior derivative (much as in a generalized Maurer–Cartan form of a Lie algebra). Much of the discussion in the paper runs in parallel with, and draws inspiration from, string field theory, in particular, the Witten cubic open string theory. But as the authors note, the massless higher spin case is much more difficult due to the lack of an underlying geometrical picture. It is not uncommon in the higher spin gauge theory literature from this era to model interactions on the Witten theory. Unfortunately, such approaches do not seem to have gotten off the ground. In a second paper [163], the cubic restriction is lifted, and a formulation very close to the BRST-BV approach is arrived at, however, without explicitly introducing the antifield language.13 The formalism is based on a bracket [A, B](Γ) defined in a space of functionals A(Γ) of fields Γ defined on a manifold ℳ̄ including ghost coordinates. The classical action is Q(Γ) and its free part is Q0 (Γ). Gauge invariance is expressed through the master equation [Q, Q](Γ) = 0

(5.30)

The interactions are introduced perturbatively as Q(Γ) = ∑ Qn (Γ) = Q0 (Γ) + Qint (Γ) n≥0

(5.31)

Due to the nilpotency of the free Q0 , the master equation becomes 1 [Q0 , Qint ] = − [Qint , Qint ] 2

(5.32)

Writing out this equation, order by order, one gets 1 n−1 [Q0 , Qn ] = − ∑ [Qp , Qn−p ] 2 p=1

(5.33)

The paper then proceeds to a formal solution of this equation, and instances of this kind of theory are discussed at length, but let us stop here. The formula, in one form or another, is what one is almost inevitably bound to arrive at in any perturbative approach to the higher spin field theory. The standard Noether procedure takes this form. We will see that the light-front perturbative approach takes this form. We saw this pattern in Section 4.3 when we discussed the BRST-BV approach (see equations (4.69)). When the interaction terms are expressed as n-ary products, or brackets, the structure of Lie∞ 13 There is, however, reference to Batalin–Vilkovisky.

5.2 The covariant BRST approach � 185

theory emerges. However, as we have discussed in Section 4.3.7, a formal solution is not of much worth. What we need is a local solution.

5.2.5 Follow up papers on the BRST approach Starting in the early 2000s, there was a revival—or rather a rediscovery by a few groups of authors—of the BRST approach to higher spin massless fields. First of the free field theory (see Section 2.11.6 in our Volume 1) and then toward the mid-2000s, of the cubic interacting theory. Here, we will mention one such paper [256] in which the vertex of Section 5.2.3 is reproduced. The paper contains references back to earlier work within this series papers. The Koh–Ouvry vertex, reviewed in Section 5.2.2, is also discussed in some of the papers. There is also a paper by Metsaev [257] on the BRST approach to cubic interactions for massless and massive higher spin fields. The paper is very systematic, but unfortunately quite short on details. It treats a cubic vertex of the type of Section 5.2.3 in the case of three massless fields of different spin. From the appearance of the formulas, it seems that the massless cubic vertex corresponds to what the authors of the papers reviewed in Section 5.3 call “leading terms”. Roughly speaking, without explicit traces and divergences of the higher spin fields. However, the formulation is within the triplet version of the free field theory (as is the vertex of Section 5.2.3), hence the situation deserves further study.

5.2.6 Mechanics BRST vs. field theory BV formalism If there is a mechanical model—or geometrical as in string theory—underlying a certain field theory, then one might suspect that there is some relation between the mechanics BRST operator and the field theory BRST transformations. Indeed there is, as was pointed out by Siegel in the context of string field theory [258, 259]. Using the language of the Batalin–Vilkovisky field/antifield formalism [205], the correspondence is very close. The BRST operator of the underlying mechanical model gives the free field theory part of master action, not just the classical action, but also the BRST-transformation generating term. The question of the existence of an underlying mechanical model behind higher spin gauge interactions remains elusive. It has not been very much researched.14 In the first analysis, one could argue that the BRST theory for free higher spin gauge fields reviewed in Section 5.4 in our Volume 1, is as far as one would hope to get. It indeed

14 The Vasiliev theory does not require any such model and does not seem to say anything about the question.

186 � 5 Covariant approaches in Minkowski space-time furnishes a supporting structure for the Fronsdal theory of symmetric higher spin gauge fields of any spin. With more than one oscillator, one gets a theory for Fronsdal fields of any spin and symmetry, also in any dimension as well, if that is desired. Still, one may feel that it should be possible to get deeper. In Section 2.2, we have reviewed, at some length, historically interesting mechanical models for hadrons and nonhadrons, in the latter case also with cubic interactions. Attempts in this direction has not met with success so far. The Fronsdal reproducing BRST theory seems to be too “singular” to be interpreted “geometrically” or even in terms of some two-particle “action-at-a-distance” model. Discussions about underlying mechanical models are scarce in the literature, apart from what the present author has written, but one exception is in the Introduction to [162]. The present author have discussed mechanical models in [250, 251]. It is indeed possible to transfer the BRST vertex operator approach of Section 5.2.3 to a full BRST-BV antifield/antibracket formalism. This was done by the present author in [260]. We will not go through all the details here, but record the overall structure. The fields |Φ⟩ are augmented to a full graded vector space of fields/antifields and ghosts/antighosts according to |Ψ⟩ = |𝒞 ⟩ ⊕ |Φ⟩ ⊕ |Φ⧖ ⟩ ⊕ |𝒞 ⧖ ⟩

(5.34)

The antifield/antibracket apparatus of Section 4.3 can be taken as it stands. For instance, the free field theory master action now becomes the standard sum of two terms 1 1 S0 = S0(0) + S0(1) = ⟨Φ|Q|Φ⟩ + ⟨Φ⧖ |Q|𝒞 ⟩ = ⟨Ψ|Q|Ψ⟩|ghm =0 2 2

(5.35)

where the decoration “(i) ” denotes antighost number gha . These are the only two bilinear terms of ghost number zero that can be formed out of fields/ghosts and antifields/antighosts and the BRST operator Q.15 The first term is the classical action, while the second term encodes the free theory BRST transformations. A sample calculation suffices to bring this out. Consider 1 δs |Φ⟩ = (|Φ⟩, ⟨Φ|Q|Φ⟩ + ⟨Φ⧖ |Q|𝒞 ⟩) = (|Φ⟩, ⟨Φ⧖ |)Q|𝒞 ⟩ = Q|𝒞 ⟩, 2

(5.36)

since the only nonzero bracket is (|Φ⟩, ⟨Φ# |) = 1. One should note that the mechanics bra and ket structure plays no role here. The antibracket (⋅ , ⋅) is completely indifferent to the Fock complex canonical structure of mechanical oscillators and mechanical ghosts. This also makes sense from the point of view that the abstract operation ⧖ of forming antifield and antighosts has nothing to do with neither complex nor Hermitian conjugation. The rest of the BRST-transformations of can be derived similarly. 15 Note that, due to the interplay between various ghost numbers, ghm = 0 corresponds to ghf = 0 in the standard antifield/antibracket formalism.

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Therefore, it is possible to lift the (mechanics BRST-based) gauge invariant ⟨Φ|Q|Φ⟩ free field theory into a (still mechanics BRST-based) field theory BRST-BV-invariant ⟨Ψ|Q|Ψ⟩ free theory in the antifield/antibracket formalism. The free field theory operator s is exactly represented by the mechanics operator Q. Thus, Q, as indeed should be anticipated, encodes both the δ and γ differentials. A perturbative ansatz for the interacting theory can now be written as ∞ 1 󵄨 󵄨 S = ⟨Ψ|Q|Ψ⟩󵄨󵄨󵄨gh =0 + ∑ g n−2 ⟨Ψ|⊗n 𝒱n |−⟩1⋅⋅⋅n 󵄨󵄨󵄨gh =0 m m 2 n=3

(5.37)

with “polymorphic” vertex operators 𝒱n encoding all gauge structure of the theory.

5.3 Non-BRST approaches—post millennium investigations At the cubic level, one may, as we know, consider the interaction problem without invoking a tower of higher spin gauge fields. This may even be convenient, since the superstructure needed to maintain a full spectrum of fields—although devised to control the inherent complexity of the problem—clearly comes with its own price of formalism. We will first look at an approach to cubic self-interactions s-s-s and three different spin interactions s1 -s2 -s3 , mainly developed in a series of papers by R. Manvelyan, K. Mkrtchyan and W. Rühl. This work itself involves some complexity controlling notational and logistical developments of standard Minkowski space higher spin formalism. We will call it the MKR-series of papers. Before reviewing the actual results, it might be good to have a scale on which to gauge them. From the light-front classification, to be reviewed in the next chapter, we borrow the following. Light-front benchmark The number of space-time derivatives in a higher spin cubic interaction s1 -s2 -s3 can be classified as follows [160] (see Section 6.1.1). There are four ways to symmetrically add and subtract three integers s1 , s2 and s3 . Correspondingly, there are four types of cubic interactions found in the light-front approach. (i) s1 = s2 = s3 = 0: scalar cubic interaction, no derivatives. (ii) s1 < s2 + s3 : s2 + s3 − s1 derivatives. (iii) s1 > s2 + s3 : s1 − s2 − s3 derivatives. (iv) No restriction: s1 + s2 + s3 derivatives. The cubic self-interactions s-s-s with the minimal number of derivatives belong to case (ii). To this case also belongs higher spin gravitational interactions s-2-s with two derivatives. Case (iii) contains no self-interactions. Case (iv) contains higher-order interactions corresponding to products of higher spin curvature tensors.

188 � 5 Covariant approaches in Minkowski space-time The first paper in the series concerns some particular trilinear higher spin interactions [261]. It was based on earlier work on scalar field conformal interactions with higher spin fields in AdS. The interactions considered here are squarely within the Fronsdal Minkowski program and can be seen as a direct continuation of the BBvD work.16 The completely symmetric, Lorentz tensor higher spin fields hμ(s)1 ...μi ...μs (x) are collected into a formal expansion over vectors aμ , s

h(s) (x, a) = ∑(∏ aμi )hμ(s)1 ...μi ...μs (x) μi

i=1

(5.38)

Then, denoting space-time derivatives with ∇μ and derivatives with respect to the vectors aμ by 𝜕a (with an implicit μ index), three standard operations may be defined Grad :

(Grad h)(s+1) (x, a) = (a ⋅ ∇)h(s) (x, a)

1 ◻ h(s) (x, a) Tr : (Tr h)(s−2) (x, a) = s(s + 1) a 1 Div : (Div h)s−1 (x, a) = (∇ ⋅ 𝜕a )h(s) (x, a) s

(5.39a) (5.39b) (5.39c)

with ◻a = 𝜕a ⋅ 𝜕a . Conforming to the notation of the papers, μ contraction dots will be left out from now on, instead indicated by parentheses. In this formalism, a gauge transformation then looks as δh(s) (x, a) = s(a∇)ξ (s−1) (x, a)

(5.40)

and the trace constraints take the form ◻a ξ (s−1) (x, a) = 0

(◻a )2 h(s) (x, a) = 0

(5.41)

Contractions between tensors with equal number of indices can be expressed through a ∗ product ∗a =

1 s ←μ󳨀i 󳨀→ ∏ 𝜕 𝜕a (s!)2 i=1 a μi

(5.42)

The Fronsdal tensor becomes 1 F (s) (x, a) = ◻h(s) (x, a) − (a∇)(∇𝜕a )h(s) (x, a) + (a∇)2 ◻a h(s) (x, a) 2

(5.43)

in terms of which the free Lagrangian reads17 16 We will stay close to the notation of the originals, but make minor adjustments to conform to the present book. 17 Note the unconventional sign. It affects the formula (5.50) below.

1 2

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� 189

1 ◻ h(s) (a) ∗a ◻a F (s) (a) 8s(s − 1) a

(5.44)

ℒ0 (h (a)) = − h (a) ∗a F (s)

(s)

(s)

(a) +

The authors make no distinction between the Lagrangian and the action, freely allowing them to formally partial integrate over space-time coordinates (hence retaining only the variables a in action/Lagrangian level expressions). Also, introduce the traceless de Donder tensor G(s−1) , which in this notation takes the form G(h)(s−1) = (Div h)(s−1) −

s−1 1 1 (Grad Tr h)(s−1) = ((∇𝜕a ) − (a∇)◻a )h(s) 2 s 2

(5.45)

In terms of the de Donder tensor, the Fronsdal tensor is written as F (s) (x, a) = ◻h(s) (x, a) − s(a∇)G(s−1) (x, a)

(5.46)

This equation will be used in cubic computation to express ◻h(s) (x, a) in terms of the Fronsdal tensor and the de Donder tensor. As there will also occur terms with ◻ξ (s−1) (x, a), we note the equation ◻ξ (s−1) = δG(s−1) (x, a)

(5.47)

The de Wit–Freedman higher spin Christoffel symbols and curvatures Γ(s) are in(n) troduced. They can be compactly denoted by introducing one further formal vector bμ over which space-time derivatives may be expanded. Here, we note the homogeneity properties of the symbols n s (s) Γ(s) (n) (x, βb, αa) = β α Γ(n) (x, b, a)

(5.48)

that indicates the number of derivatives. The gauge invariant higher spin curvatures (s) are Γ(s) (x, b, a) = Γ(s) (x, b, a), i. e., the Christoffel symbols with n = s. The definition is s

(−1)k (b∇)s−k (a∇)s (b𝜕a )k h(s) (x, a) k! k=0

Γ(s) (x, b, a) = ∑

(5.49)

All this is just a compact way of expressing the free field theory that we reviewed in Sections 5.1 and 5.2 in our Volume 1. The Noether coupling method, at first order in interaction, reads in the notation of the papers δ0 ℒ1 (h(s) (a)) = −δ1 ℒ0 (h(s) (a)) = (F (s) (a) −

a2 ◻ F (s) (a)) ∗a δ1 h(s) (a) 4 a

(5.50)

where on the right-hand side, the free field equations appear as they come out of varying the action (5.44). As usual, a trace operation must be performed in order to arrive at the Fronsdal wave equations F (s) (a) = 0. Taking into account the fact that the variation

190 � 5 Covariant approaches in Minkowski space-time δ1 h(s) (a) must be double traceless, the variation can be shifted by a trace term so that Noether equation takes the simplified form δ0 ℒ1 (h(s) (a)) = F (s) (a) ∗a δ1 h(s) (a)

(5.51)

As for the results of the paper, after a few lower spin exercises, it employs the Noether coupling procedure to derive an expression for the 2s-s-s cubic interactions 1 (2s) h (x, b) ∗b Ψ(2s) Γ (x, b) where 2s b2 μ (s) (s) Ψ(2s) (x, b) = Γ (b, a) ∗ Γ (b, a) − 𝜕b Γ(s) (b, a) ∗a 𝜕b Γ(s) (b, a) b Γ 2(s + 1) μ

ℒ(1) (h (a), h (s)

(2s)

(b)) =

(5.52)

When reading formulas such as the one above, one should note that the introduction of the formal expansion vectors a and b allows for compact expressions involving quite complicated contractions between indices on space-time derivatives and indices on the higher spin fields. To interpret the result, note that the objects Ψ(2s) Γ (x, b) may be thought of as generalized Bell–Robinson tensors, already noted and discussed by BBvD. They are essentially given by sums of contractions over bilinear terms in higher spin curvatures, or higher spin Weyl, tensors. Compared to the light-front benchmark, these interactions belong to case (ii). The number of derivatives is 2s + s − s = 2s. The second paper in the series [262]18 discusses the cubic self-interaction and presents a solution for spin-4 fields. It does what BBvD did for spin 3. The complexity increases, quite dramatically. Anyone who has ever contemplated setting up an ansatz for a general higher spin cubic interaction will have been confronted with a barrage of terms. Organizing principles are needed, and several such are used in the paper. Focus first on cubic self-interactions s-s-s with the minimal number of s of derivatives. There will be 4s indices in all to contract, amounting to 2s contractions. Letting f , g and h stand for the three participating fields, there will be contractions of the following, not exclusive, generic types: 1. fgμ hμ 1’. fgh′ 2. f ∇μ g∇μ h 2’. fg◻h 3. fgμ ∇μ h 3’. f ∇ ⋅ gh and all combinations thereof, so that all indices are contracted. The first pair of cases indicates contractions of indices on the fields. All three fields may occur with their traces. Luckily, double traces (on the same field) will not occur. 18 Published after the third paper [242]. The papers were simultaneous.

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The second pair of cases shows contractions between derivatives. Such terms mix under partial integration. Factors ◻h may be traded for a sum of the Fronsdal tensor and the de Donder tensor through formula (5.46). This becomes important upon processing the left-hand side of the Noether equation (5.51). To understand why, consider the last pair of cases. Clearly, they mix under partial integration. Consider, however, terms of the form 3. with no traces. MKR call them “leading terms”, being of the simplest type with no traces, no divergences and no contracted derivatives. The solution process can be started with these terms. The processing the left-hand side of the Noether equations will entail partial integrations, producing terms with divergences as factors. Such terms can be traded for new terms with trace and de Donder factors using the definition of the de Donder tensor (5.45). Thus, an organizing principle for the computation can be to treat the terms in the ansatz for the action according to the numbers of traces and de Donder tensors they contain. Terms with one more trace or one more de Donder tensor that occur at a certain stage is then moved up to a next stage. Returning to the second pair of cases, the ones that entailed contracted derivatives, the terms produced there will either move up one stage (de Donder tensor) or contribute to the right-hand side of the Noether equations (Fronsdal tensor). Indeed, the whole point of the process is to vary the ansatz for ℒ1 and then process the terms in order to get zero, a field redefinition of the free action, or a nontrivial contribution F (s) (a) ∗a δ1 h(s) (a) to the right-hand side. Returning to the “leading terms”, MKR argues for the following form for them:19 (0,0)

ℒ1

∼

s! ∫ dx1 dx2 dx3 δ(x − x1 )δ(x − x2 )δ(x − x3 ) n !n n1 +n2 +n3 =s 1 2 !n3 ! ∑

× [(𝜕b 𝜕c )n1 (𝜕a ∇2 )n1 (𝜕c 𝜕a )n2 (𝜕b ∇3 )n2 (𝜕a 𝜕b )n3 (𝜕c ∇1 )n3 ]h(x1 , a)h(x2 , b)h(x3 , c) (5.53)

It is quite clear that there is a certain resemblance to both the light-front structure, to be reviewed in the next chapter, and to the BRST structure reviewed above. The notation ℒ(0,0) refers to the number of trace and de Donder factors. 1 The actual computation leading to the spin-4 cubic self-interaction is not described in any detail in the paper. The result is quoted in terms of types of Lagrangians ℒ(i,j) where i stands for the number of de Donder factors and j for the number of trace factors, where (i, j) = (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1). All in all, there are 63 terms. Needless to say, the result is not particularly illuminating. This abundance of terms can be compared to the light-front result, looking forward to formula (6.1) in the next chapter. For the cubic self-interaction for spin s there are just s + 1 terms, or noting that interaction is a binomial expansion, just one terms for any spin. So, the physics of cubic higher spin interactions is very simple. We seem to pay a high price for manifest Lorentz invariance, which in its turn requires gauge invariance.

19 The notation is adjusted somewhat to be closer to the next paper in the series. Note also that in places the notation a = a1 , b = a2 and c = a3 is used.

192 � 5 Covariant approaches in Minkowski space-time However, more structure may appear as we move from cubic interactions for any particular spin to general trilinear interactions. We now turn to the third paper in the series [242] where one may see the algorithm in action in some more detail. The following ansatz is made for the leading order s1 -s2 -s3 Lagrangian, where the spins are supposed to be ordered s1 ≥ s2 ≥ s3 , (0,0)

ℒ1

= ∑ Cns11,s,n22,s,n33 ∫ dx1 dx2 dx3 δ(x3 − x1 )δ(x2 − x1 ) ni

̂ 12 , Q23 , Q31 |n1 , n2 , n3 )h(s1 ) (x1 , a)h(s2 ) (x2 , b)h(s3 ) (x3 , c) × T(Q

(5.54)

with the operator T̂ given by ̂ 12 , Q23 , Q31 |n1 , n2 , n3 ) = (𝜕b 𝜕c )Q23 (𝜕a ∇2 )n1 (𝜕c 𝜕a )Q31 (𝜕b ∇3 )n2 (𝜕a 𝜕b )Q12 (𝜕c ∇1 )n3 T(Q

(5.55)

The total number of derivatives is Δ = n1 + n2 + n3 . As there are s1 indices on the field hs1 (x1 , a) and correspondingly for the other fields, one gets the equations n1 + Q12 + Q31 = s1

n2 + Q23 + Q12 = s2

n3 + Q31 + Q23 = s3

(5.56)

These equations can be solved by Q12 = n3 − ν3

Q23 = n1 − ν1

Q31 = n2 − ν2

(5.57)

where 1 νi = (Δ + si − sj − sk ) with i, j, k all different 2

(5.58)

For all spin equal, the ansatz reduces to (5.53). With the ordering s1 ≥ s2 ≥ s3 , the minimal number of derivatives is given by s1 + s2 − s3 and the numbers νi become ν1 = s1 − s3 , ν2 = s2 − s3 and ν3 = 0. The coefficients in the ansatz then turn out to be C{ni } = Cns11,s,n22,s,n33 ∼ {s } i

s3 ! (n1 − s1 + s3 )!(n2 − s2 + s3 )!n3 !

(5.59)

This is proved by computing three currents J (si ) (xi , ai ) from the ansatz (5.54) by varying with respect to h(si ) and demanding that their divergences vanish on-shell. This is actually the first stage in the Noether procedure. The Noether equation takes the form, relying on the discussion above 3

∑ δ0i ℒ1 (h(s1 ) (a), h(s2 ) (b), h(s3 ) (c)) = 0 + 𝒪(F (si ) (ai )) i=1

Explicitly, it reads

(5.60)

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� 193

̂ ij |nk )[(a∇1 )ξ (s1 −1) h(s2 ) h(s3 ) + h(s1 ) (b∇2 )ξ (s2 −1) h(s3 ) + h(s1 ) h(s2 ) (c∇3 )ξ (s3 −1) ] C{ni } T(Q {s } i

= 0 + 𝒪(F (si ) (ai ), G(si −1) , ◻ai h(si ) (ai ))

(5.61)

where on the right-hand side, it is indicated what kinds of terms that may appear at nonzero order. The processing of this equation is simplified by noting that ̂ ij |nk )(ai ∇i )ξ (si −1) (ai ) = [T(Q ̂ ij |nk ), (ai ∇i )]ξ (si −1) (ai ) T(Q

(5.62)

̂ ij |nk ) on ξ (s1 −1) (a1 ) say, there are Q12 + This is true simply because when acting with T(Q Q31 + n1 derivatives 𝜕a1 but only s1 − 1 factors of a1 in ξ (s1 −1) (a1 ). This one factor too few according to (5.56), and the result is zero. The commutators are then evaluated to ̂ 12 , Q23 , Q31 − 1|n1 , n2 , n3 + 1) ̂ ij |nk ), (a∇1 )] = Q31 T(Q [T(Q ̂ 12 , Q23 , Q31 |n1 − 1, n2 , n3 )(∇1 ∇2 ) × n1 T(Q

̂ 12 − 1, Q23 , Q31 |n1 , n2 + 1, n3 ) − Q12 T(Q ̂ 12 − 1, Q23 , Q31 |n1 , n2 , n3 )(𝜕b ∇2 ) − Q12 T(Q

(5.63)

The origins of these terms are as follows. The first results from the commutator with (𝜕c 𝜕a )Q31 . The second results from the commutator with (𝜕a ∇2 ). The last two terms result from the commutator with (𝜕a 𝜕b )Q12 upon using ∇1 + ∇2 + ∇3 = 0, which is an instance of partial integration. The combinations of derivatives in the second term can be rewritten as ∇1 ∇2 = 21 (◻3 − ◻2 − ◻1 ). Then there are corresponding equations for (b∇2 ) and (c∇3 ). Remember now, that these terms sit within the left-hand side of the Noether equation (5.61). Then, as the authors point out, one may note that the first and third term in (5.63) remain within the confines of “leading terms” and, therefore, just reproduce the computations that yielded the ansatz (5.54) in the first place. The second and fourth term, however, contribute terms containing to terms of the type ◻h and ◻ξ and to divergences. In processing them, there will be further commutators to compute, and instances to use equations (5.46) and (5.47). Indeed, the only way to cancel terms containing factors of ◻1 ξ (s1 −1) say, is to have contributions to the Lagrangian containing G(s1 −1) h(s2 ) h(s3 ) . Thus, there are contributions ℒ(1,0) in the notation of the paper. On the other hand, terms containing factors of ◻1 h(s1 ) can be traded for two types of terms: (i) terms with the Fronsdal tensor F (s) (x1 , a1 ), these being put aside for the computation of the lowest order nonlinear gauge transformations, and (ii) terms with (a1 ∇1 )G(s−1) (x1 , a1 ), leading to further commutators. As for the terms with divergences, these are traded for terms with de Donder tensors and traces. We may thus begin to see how the process of solving the Noether equation will require the introduction of new terms ℒ(i,j) in the Lagrangian where i stands for the power de Donder tensors and j for the power of traces with i + j = 3. Some details of the computation are given in an Appendix to the paper. It is clearly a long and complex computation and the result is again quite complicated. But it is interesting that there is enough structure to make the computation systematic and feasible.

194 � 5 Covariant approaches in Minkowski space-time In the last paper of the series [253], the authors propose a generating function for the higher spin cubic interactions. It was inspired by a contemporary work by A. Sagnotti and M. Taronna on cubic higher spin interactions derived as limits of string theory amplitudes (see comments on that work in Section 5.4.5). The generating function for cubic interactions may be written in a few different forms, and we will quote one of them. The starting point is the Lagrangian for the leading terms of the previous paper, in a notation close to that of the fourth paper (0,0)

ℒs1 s2 s3 =

1 ∫ dx1 dx2 dx3 δ(x − x1 )δ(x − x2 )δ(x − x3 ) α!β!γ! α+β+γ=n ∑

̂ a , 𝜕b , 𝜕c , ∇1 , ∇2 , ∇3 )h(s1 ) (x1 , a)h(s2 ) (x2 , b)h(s3 ) (x3 , c) × T(𝜕

(5.64)

where the operator T̂ (a variant of the operator T̂ from the previous paper) may be written in two equivalent ways T̂ = (𝜕a 𝜕b )γ (𝜕b 𝜕c )α (𝜕c 𝜕a )β (𝜕c ∇1 )s3 −n+γ (𝜕a ∇2 )s1 −n+α (𝜕b ∇3 )s2 −n+β

= (𝜕a 𝜕b )γ (𝜕b 𝜕c )α (𝜕c 𝜕a )β (𝜕c ∇12 )s3 −n+γ (𝜕a ∇23 )s1 −n+α (𝜕b ∇31 )s2 −n+β

(5.65)

where ∇12 = ∇1 − ∇2 , ∇23 = ∇2 − ∇3 and ∇31 = ∇3 − ∇1 . The equivalence follows from using ∇1 + ∇2 + ∇3 = 0 and dropping divergences upon partial integration. Next, collecting all higher spin fields into one object 1 (s) h (x, a) s! s=0 ∞

Φ(x, a) = ∑

(5.66)

all leading order interactions can be expressed as (a1 = a, a2 = b, a3 = c), 𝒜

(0,0)

= ∫ dx1 dx2 dx3 δ(x − x1 )δ(x − x2 )δ(x − x3 ) exp Ŵ × Φ(x1 , a1 )Φ(x2 , a2 )Φ(x3 , a3 )|a1 =a2 =a3 =0

(5.67)

where Ŵ is the operator20 λ2 Ŵ = [(𝜕a1 𝜕a2 )(𝜕a3 ∇12 ) + (𝜕a2 𝜕a3 )(𝜕a1 ∇23 ) + (𝜕a3 𝜕a1 )(𝜕a2 ∇31 )] 2 1 + [(𝜕a3 ∇12 ) + (𝜕a1 ∇23 ) + (𝜕a2 ∇31 )] 2

(5.68)

One may convince one self that expanding exp Ŵ and acting on the three fields in (5.67), evaluating the result at a1 = a2 = a3 = 0, results in interactions of the form (5.64). The rest of the Lagrangian terms containing trace and de Donder operators, can be derived 20 The authors work with a “scaling dimension” for the vectors ai . The factor of λ2 compensates for that.

5.4 Obstructions to the Fronsdal–BBvD program

� 195

by performing a zero-order gauge transformation in (5.67). As in the previous paper, the nonzero terms, and the terms containing the Fronsdal tensor, so produced, will have to be compensated by additional contributions to the Lagrangian involving trace and de Donder operators. For details on the computation, we refer to the paper itself. The paper also introduces a quite compact formula for the complete cubic interaction vertex operator Ŵ based on three pairs of Grassmann variables (ηai , η̄ ai ). Taking certain bilinears in them, they can be used to maintain trace and de Donder operators on the same level as the (𝜕ai 𝜕aj ) and (𝜕ai ∇k ). Again, we refer the reader to the paper itself. Let us instead comment on the form of the operator Ŵ . In retrospect, it looks very natural, and very simple. Much of the complexity of the full cubic interaction comes from the factors of traces and divergences. But these must be computed in order to extract the first order of the nonlinear gauge transformations. Even so, it is striking how close Ŵ mimics the cubic light-front vertex operator that we will discuss in the next chapter. Furthermore, the overall structure resembles the BRST vertex of Section 5.2.3. A direct comparisson still remains to be done, however.

5.4 Obstructions to the Fronsdal–BBvD program In this section, we will review a series of papers from the early 2000s by X. Bekaert and N. Boulanger, with collaborators S. Leclercq, S. Cnockaert and P. Sundell in various configurations, that may be viewed as a direct continuation of the BBvD–Fronsdal program. Remember that the BBvD spin-3 paper ended with the statement that the found first order in g spin-3 gauge transformations closed. What actually was checked was the closure at the order g level. In the next paper, BBvD-G, containing the first general analysis of the higher spin interaction problem, it was shown that the gauge algebra did not close at the order g 2 level. In other words, the Jacobi identity could not be satisfied. Even so, the hope remained, that the inclusion—very natural anyway—of still higher spin fields might cure the inconsistency. But the theory lay dormant until the early 2000s, most likely because of the intractability of the problem. In the meantime, the antifield/antibracket technology had been developed, and it was proposed how it could be applied to the higher spin problem. This we reviewed in Section 4.3.7. The hope for a s > 3 cure for the s = 3 inconsistency, however, was in vain, as was proved by Bekaert, Boulanger and Leclercq in [263]. To try to understand how this came about, is the main objective of the present section.

5.4.1 Some general properties of interactions Let us first list some general properties (as noted in some of the papers) required of the nonlinear deformation of the free theory.

196 � 5 Covariant approaches in Minkowski space-time Poincaré invariance and parity: The deformed Lagrangian should be invariant under the Poincaré group. There should be no explicit dependence on the space-time coordinates. It should also be invariant under parity transformations. All indices are contracted by the Minkowski metric only (no antisymmetric tensor). Nontriviality: Trivial deformations, arising from field redefinitions, are rejected. Consistency: The deformed theory must possess the same number of (deformed) independent gauge symmetries, reducibility identities, etc., as the free theory. That is, the number of physical degrees of freedom should be unchanged. Locality: The deformed action must be a local functional. The deformed gauge transformations and the field redefinitions are local functions. The actual implementation is a direct application of the antifield/antibracket theory that we have reviewed in Section 4.3. Higher spin gauge theory is irreducible so it suffices to introduce ghosts and antifields to accompany the higher spin fields. These are taken as constrained totally symmetric tensors ϕ(s) = ϕaμ1 ...μs in the standard way, where the index a denotes some kind of “internal” index that we know occur for odd spin fields. Replacing, as usual, the gauge parameters by the ghost fields, we have the following fields of theory: Cμ⧖a 1 ...μs−1

ϕ⧖a μ1 ...μs

ϕaμ1 ...μs

Cμa1 ...μs−1

(5.69)

with precisely the ghost number assignments of Table 4.1 (the middle four columns). Due to irreducibility, the free BRST differential is given by s = δ + γ. 5.4.2 Gauge invariants The first paper in the series is [264] that concerns a question in the free field theory of higher spin gauge fields. Its results are used in the subsequent papers on interactions. Remember the two covariant approaches to free higher spin gauge theory: (i) the constrained Fronsdal approach with double traceless fields and traceless gauge parameters, and (ii) the unconstrained (but nonlocal) theory of Francia and Sagnotti based on de Wit–Freedman curvature tensors.21 As the paper writes, it is generally believed that in the unconstrained formulation, a local function of the totally symmetric gauge field is gauge invariant if and only if it depends on the field only through the de Wit–Freedman curvature tensor. Likewise, in the constrained formulation (i. e., with the trace and double trace constraints enforced) a local function of the totally symmetric gauge field is gauge invariant if and only if it depends on the field only through the de Wit–Freedman curvature and on the Fronsdal tensor. 21 See Chapter 5 of our Volume 1 for theory and references.

5.4 Obstructions to the Fronsdal–BBvD program � 197

The paper sets out to put this on “firm mathematical grounds”. The techniques used are the jet space formulation of field theory and the anti-field BRST formalism. The paper also concerns Killing tensor fields on Minkowski space-time, i. e., symmetric tensors ϵμ1 ...μs−1 that obey the “Killing-like” equation 𝜕(μ1 ϵμ2 ...μs−1 ) = 0. This is obviously interesting for Minkowski higher spin theory and lead naturally over to questions regarding Minkowski higher spin algebras that we have briefly touched on in Section 3.6, but we will not have more to say about this. Let us instead focus on one of the results from the paper that is subsequently used, namely the cohomology the γ differential for the constrained theory. We will state the theorem as it was “streamlined” in the paper [263]. The local functions in the cohomology H ∗ (γ) for a spin s Fronsdal theory depend only on: 1. The Fronsdal tensor ℱμa1 ...μs and all its higher-order derivatives. 2.

3. 4.

a The de Wit–Freedman curvature tensor K|μ in 1 ν1 |...|μs νs |

s s

Young symmetry and all its

higher-order derivatives. ⧖a The antifields ϕ⧖a μ1 ...μs and antighosts Cμ1 ...μs−1 and all their higher-order derivatives. (i)a Certain non-γ exact ghost tensors U|μ ν

1 1 |...|μi νi |νi+1 ...νs−1

with i < s. These are the traceless part of

the i-times antisymmetrized ith derivatives of the ghosts Cμa1 ...μs−1 .

The new gauge invariant objects here are items 3 and 4 that can only be seen in the BRST antifield formulation.

5.4.3 Spin 3 revisited The next paper in the series is [265] with S. Cnockaert. The spin-3 theory of BBvD is revisited with the new stronger methods. These are described in detail in the paper in Section 4 where the BRST method is applied to spin 3, and in Section 5, where the actual deformations are computed. Here, we will just state the results. The results concern cubic interaction terms (order g) and gauge algebra closure at order g (corresponding to the commutator) and at order g 2 (corresponding to the Jacobi identity) in dimensions n ≥ 4. The results are stated in two technical theorems, but we will rephrase them more leisurely. There are two types of cubic interactions: first, the BBvD cubic interaction with three derivatives, second, a new cubic interaction term with five derivatives in dimensions n ≥ 5. At order g, cubic interactions exist with three and five derivatives, and the corresponding gauge transformations (with two and four derivatives, respectively) close off-shell. At order g 2 , the gauge transformations with four derivatives close off-shell, but not so for the BBvD gauge transformations with two derivatives that suffers an obstruction. The n ≥ 5 gauge transformations does not suffer any obstruction.

Therefore, in four dimensions, the BBvD results are confirmed.

198 � 5 Covariant approaches in Minkowski space-time 5.4.4 Strong obstruction of the BBvD vertex In the paper [263] with S. Leclercq, it is proved that the gauge algebra nonclosure at the second order cannot be remedied by the inclusion of fields with higher spin, or lower spin, than 3. The BBvD spin-3 cubic interaction is “strongly obstructed”. Thus a longstanding conjecture, hypothesis or hope, was proved to be wrong. The proof is highly nontrivial, and we will not review its details, instead trying to understand its overall logic. For the review of this paper, let us first translate the formalism of Section 4.3.7 into the notation of the paper. The paper writes W for S and the terms gSi in the expansion (4.68) as g i W (i) . The nonintegrated function w correspond to the action W according to ∫ w = W . The free theory BRST generator is s. Furthermore, the first-order deformation w(1) is written simply as a. In this notation, the first-order deformation equation (W (0) , W (1) ) = sW (1) = 0, for nonintegrated functions, become sa + db = 0

(5.70)

Such deformations belong to the cohomology class H 0,n (s|n). The next order in deformation, where obstructions might occur, become 1 (w(1) , w(1) ) = − sw(2) + de 2

(5.71)

The cohomology class here is H 1,n (s|n). At the first stage (5.70), finding the cubic deformation a, the generic form of the possible terms can be sorted out by counting. We need to look for cubic terms of the form (C ⧖ )k (ϕ⧖ )l (ϕ)m (C)n . Clearly, k + l + m + n = 3. Since a must have ghf = 0, we also have the relation −2k −l +n = 0. Thus, k = m+2n−3 ≥ 0 with l free. The only possibilities are a0 ∼ ϕϕϕ with antifield number gha = 0, a1 ∼ ϕ⧖ ϕC with antifield number gha = 1 and a2 ∼ C ⧖ CC with antifield number gha = 2. The equation (5.70) can now be split up into components with fixed antighost number using gha (δ) = −1 and gha (γ) = 0, γa2 + db2 = 0

δa2 + γa1 + db1 = 0

(5.72)

δa1 + γa0 + db0 = 0

It then becomes clear the importance of the theorem on the cohomology of γ from Section 5.4.2. The next question is what concrete ansatz to put for the γ-invariant a2 ∼ C ⧖ CC?22 It is quite interesting to see how subtle the formalism is. 22 A possible term ϕ⧖ ϕ⋈ CC is ruled out since—apart from it being quartic—it corresponds to an open algebra, and the transformations are known [266] to close off-shell in lowest order.

5.4 Obstructions to the Fronsdal–BBvD program

� 199

The terms of a2 must in principle contain the antighost Cμ⧖a and derivatives of 1 ...μs−1

(i)a it and two factors of the tensor U|μ . However, since a2 is defined modulo 1 ν1 |...|μi νi |νi+1 ...νs−1 d, any derivatives acting on the antighost may be thought as having been partially integrated onto the U tensors. An ansatz for a2 can be written a a2 = fbc|(i)(j) Ca⧖ U (i)b U (j)c

(5.73)

a where fbc|(i)(j) are structure constants. All indices, internal as well as space-time, are of course contracted (i and j stands for space-time indices on derivatives). Since a2 does not involve the physical fields, this shows that the gauge algebra must close off-shell at order g. Then, coming to the second order equation (5.71), its maximal antighost number component reads

(a2 , a2 ) = γc2 + de2

(5.74)

This equations encodes the Jacobi identity for the gauge algebra at lowest order g 2 . After this, the rest is very technical, regarding the possible ways to contract the tensors in (5.73) and bounds relating spins and number of derivatives. Based on this, the BBvD three derivative spin-3 vertex can be derived, as well as the new five derivative vertex. This was the result of the “spin 3 revisited” paper where it was also shown that (a2,BBvD , a2,BBvD ) produced an obstruction with terms that are not γ-exact modulo d and, therefore, cannot be compensated within the pure spin 3 theory. This is in four spacetime dimensions. In the three dimensions, there is no obstruction. The new five derivative vertex, vanishes in three dimensions, is Abelian in four and non-Abelian in five and more dimensions. Then, finally, comes the question of whether the obstruction can be lifted by the inclusion fields of other spin. Since (a2 , a2 ) is quartic in the spin-3 fields, new a2 vertices must be of the form s-3-3. The analysis of bounds on the spin values, performed in the technical part of the paper, yields 1 ≤ s ≤ 5. The cases 1-3-3 and 2-3-3 were studied in [266]. These candidates both yield (a2 , a2 ) = 0. It remains to study the cases 4-3-3 and 5-3-3. These candidates also fail to lift the obstruction. This result is a serious blow to the Fronsdal–BBvD program. Incidentally, it proves that the “BBvD hypothesis” of the Fulp–Lada–Stasheff elaboration of the BBvD theory, does not hold (see Section 4.4.5). This means that the it cannot be proved that the BBvD theory is represented by a Lie∞ algebra. 5.4.5 A few more references We will end this chapter with a few further references to cubic interactions in Minkowski space-time.

200 � 5 Covariant approaches in Minkowski space-time (i) In [267], the authors investigate couplings between spin-2, spin-3 fields. They find a nonminimal 2-3-3 vertex with four derivatives. They also find the 3-2-2 vertex with three derivatives corresponding to spin 2 in a spin-3 background [151]. These results are then further studied in [266] where the 2-3-3 vertex is shown to be unique and consistent to the next order. (ii) Furthermore, in [266] the vertices of [267] are compared to limits of the Vasiliev cubic interactions [268, 269]. The vertex 2-3-3 is the leading term in the flat limit. However, there are also subleading terms, one of which corresponds to a two-derivative minimal gravitational coupling of spin-3 fields, an interaction that is ruled out by flat space no-go theorems, but are present in the light-front formulation. (iii) The results of (i) and (ii) are corroborated by the string theory analysis of [243]. Studying the first Regge trajectory, a sequence of 2-3-3 vertices with powers of momentum factors 0, 2, 4, 6, 8 is found. Since this analysis is based on string theory there naturally appears a variable ξ related to the string theory oscillators. In terms of such variables, the interactions that survive in the massless limit are essentially generated by differential operators of the form 𝒢 = √α′ /2[(𝜕ξ1 ⋅ 𝜕ξ2 )(𝜕ξ3 ⋅ p12 ) + (𝜕ξ2 ⋅ 𝜕ξ3 )(𝜕ξ1 ⋅ p23 ) + (𝜕ξ3 ⋅ 𝜕ξ1 )(𝜕ξ2 ⋅ p31 )]

(5.75)

where 1, 2, 3 indexes external states and where p12 = p1 − p2 , etc., in terms of external momenta. This is interesting because this form resembles the light-front vertex operators to be studied in the next chapter, and in a covariant framework to the vertex operators computed in [242, 253] (see Section 5.2.3). (iv) In [270], M. Taronna tries to synthesize much of the accumulated knowledge of cubic (and quartic) interactions into a comprehensive scheme based on a general construction of FDA’s as we have reviewed in Section 4.1 and L∞ methods in the spirit of [220, 235, 240, 219]. See also [239, 260] for the present author’s attempts along such lines.

5.4.6 Chapter 5 epilogue The Minkowski–Fronsdal program indeed finds itself in serious troubles. This, however, is the normal situation in higher spin gauge theory. Judging from the experience of 50 years of stubborn work against all odds, we have not seen the end of the story yet.

6 Light-front interactions In our Volume 1, we developed the free field theory of of light-front higher spin fields. Here, we will continue with the interacting theory. In retrospect, it can be seen as a certain implementation of the Dirac program—the program of finding all nonlinear realizations of the Poincaré group—in field theory and for zero mass and arbitrary helicity. This chapter is a direct continuation of the corresponding Chapter 6 in Volume 1, to which we refer the reader for details of notation. Over the years, there have accumulated a body of literature on cubic interactions on the light-front, in various dimensions and for various types of fields, massless as well as massive, most of this work done by R. R. Metsaev, and we will return to a short description and list of references toward the end of this chapter. For the quartic level, the published record is much more scarce. There are the pioneering papers by Metsaev in 1991, and the revisiting paper by Ponomarev and Skvortsov. These papers, are of crucial importance for the subject, and we will review their contents. What we will do in this chapter is quite elementary as compared to recent, and somewhat unexpected, developments. We will derive all cubic interactions for massless higher spin fields on the light-front in a formalism that is capable to address also the questions of higher-order interactions, but we will stop short of doing that. As for the recent developments, they are of two kinds and related to the existence of a purely cubic higher spin theory, often called the chiral higher spin gravity. This theory is “chiral”—and nonunitary—in that only half of the cubic interactions are kept, and that the light-front Poincaré algebra closes on this subtheory without any higher-order interactions. This was implicit in work by Metsaev in [271, 272], but went unnoticed for 25 years until it was explicated by D. Ponomarev and E. Skvortsov in [273]. It is a “higher spin gravity” since the higher spin fields enjoy minimal gravitational interactions, as shown in [160], a fact that also went unnoticed for a long time until it was pointed out in [252]. One recent development is the exceptional good quantum properties of the theory. Another ongoing line of research is attempts at a covariant formulation of the theory in twistor language. We will return to a brief review at the end of the chapter.

6.1 The cubic interactions: review A characteristic feature of the light-front cubic interactions for massless higher-helicity fields ϕ is the simple binomial expansion form they take λ λ 𝜕 λ ∫ d 4 x ∑ (−1)n ( )(𝜕+ ) ϕ [ + ] n 𝜕 n=0

(λ−n)

n

𝜕 ϕ̄ [ + ] ϕ̄ + c. c. 𝜕

(6.1)

where λ is the helicity. This expression, from the original paper [274], with gauge group structure constants (antisymmetrization) understood for odd helicity, appeared somehttps://doi.org/10.1515/9783110675528-006

202 � 6 Light-front interactions what mysteriously in 1983 out of the deformed Poincaré algebra. Its structure becomes more clear when the interaction terms are reformulated in momentum space in terms of vertex operators as was done a few years later in reference [160]. In that formulation, the momentum structure for the helicity λ cubic interaction is essentially given by ℙλ where ℙ is defined by1 1 3 ℙ = − ∑ γ̃r pr 3 r=1

with γ̃r = γr+1 − γr+2 (cyclic) and γ = p+

(6.2)

A little more concretely, one may introduce the combinations ℙij = γi pj − γj pi

with i ≠ j = 1, 2, 3

(6.3)

At the cubic order, there are three such combinations, but due to momentum conservation, there is only one linearly independent combination. We therefore write it as ℙ with the understanding ℙ = ℙ12 = ℙ23 = ℙ31 . Similar formulas hold for the complex conjugate ℙ.̄

The purpose of the present section is to clarify the structure of the cubic vertices and derive the full solution to that order in the Poincaré algebra. It will be observed that the kinematics of the vertex—in a certain sense to be clarified—suffice to determine it. Particular cubic couplings involving different higher helicities will be exhibited. In this connection, a common misconception—that higher spin fields cannot interact gravitationally with spin 2, will be dispelled. There indeed exist such interactions on the lightfront [160]. In understanding the structure of the cubic interactions, and how they may generalize to higher orders, the momentum space representation is more convenient than working in configuration space with derivatives on fields. The fields can then essentially be dropped from consideration since all interaction data must be encoded in functions of the momenta. We will work through the cubic interactions in detail, but only review what is known about the quartic order, which as is common, displays features of nonlocality. Rationale for momentum space Begin by transforming (6.1) from configuration space to momentum space using ϕ(x + , x − , x, x)̄ =

+ − 1 ̄ + + ̄ ̄ i(−p x +px+̄ px) ∫ dp+ dpd pϕ(x , p , p, p)e (2π)3/2

or in shorthand

1 The practical ℙ variables originated in light-front superstring theory; see [275].

(6.4)

6.1 The cubic interactions: review �

ϕ(x) =

1 ∫ d 3 pϕ(p)eip⋅x (2π)3/2

203

(6.5)

where it should be understood that both x-space and p-space fields depend on the light-front time x + . Note ̄ Evolution in x + is genthat in the shorthand of (6.5), and in the computations below, p stands for (p+ , p, p). erated by the light-front Hamiltonian h = p− . Now inserting (6.5) into (6.1) yields (λ−n)

3 λ λ ip 1 λ ∫ dx + d 3 x ∫ ∏ d 3 pr eipr ⋅x ∑ (−1)n ( )(ip+1 ) ϕ(p1 )[ +2 ] 9/6 n ip2 (2π) r=1 n=0

n

̄ )[ ip3 ] ϕ(p ̄ ) + c. c. ϕ(p 2 3 ip+3 n

(λ−n)

=

3 3 λ λ p 1 λ ∫ dx + ∏ d 3 pr δ 3 (∑ pt )i 2λ ∑ ( )(p+1 ) [ +2 ] n p2 (2π)3/2 r=1 t=1 n=0

=

3 3 λ p p (−1)λ ̄ )ϕ(p ̄ ) + c. c. ∫ dx + ∏ d 3 pr δ 3 (∑ pt )(p+1 ) [ +2 − 3+ ] ϕ(p1 )ϕ(p 2 3 3/2 p p3 (2π) r=1 t=1 2

p3 ̄ )ϕ(p ̄ ) + c. c. ] ϕ(p1 )ϕ(p 2 3 p+3

[−

λ

(6.6)

Using cyclic symmetry in field labels 1, 2, 3 and momentum conservation, we get for the momentum structure, using (6.3), λ

(−1)λ (p+1 ) [

λ

λ

p γ p2 − 3+ ] = ( 1 ℙ) p+2 p3 γ2 γ3

(6.7)

which is the precise form of what we claimed above. For instance, the Yang–Mills cubic momentum vertex comes out directly for λ = 1. The gravity cubic vertex comes out as the square of the Yang–Mills cubic vertex. It is clear why the configuration space cubic interactions come out as binomial expansions.

6.1.1 All possible cubic interactions in four dimensions Let us for completeness list all possible cubic interaction terms in four dimensions. The list was first published in [160]. We will use a shorthand notation where ∫ stands for the momentum integrations and momentum conservation delta functions as in the last line of the computation (6.6) and ϕλ (p) for a helicity λ field. The complex conjugated field ϕ̄ λ (p) carries helicity −λ. We use the convention that λ is nonnegative, while s can be any integer s = ±λ. There are four possible types of interactions, namely (i) ∫ ϕ̄ 0 (p1 )ϕ0 (p2 )ϕ0 (p3 ) + c. c. (ii) ∫ (iii) ∫ (iv) ∫

λ

γ1 1

λ

λ

λ

λ

γ2 2 γ33 γ2 2 γ33 λ γ1 1

1

(6.8a)

(λ +λ −λ ) ℙ̄ 2 3 1 ϕ̄ λ1 (p1 )ϕλ2 (p2 )ϕλ3 (p3 ) + c. c.

with λ2 + λ3 > λ1

(6.8b)

(λ −λ −λ ) ℙ̄ 1 2 3 ϕλ1 (p1 )ϕ̄ λ2 (p2 )ϕ̄ λ3 (p3 ) + c. c.

with λ2 + λ3 < λ1

(6.8c)

λ λ λ γ1 1 γ2 2 γ33

(λ +λ +λ ) ℙ̄ 1 2 3 ϕλ1 (p1 )ϕλ2 (p2 )ϕλ3 (p3 ) + c. c.

(6.8d)

204 � 6 Light-front interactions The list includes all possibilities for cubic interactions excluding field redefinitions of the free field theory. Such field redefinitions involve powers of ℙ ℙ̄ and they may correspond to higher-order counterterms [276]. The first type (i) is just scalar ϕ3 . The second type (ii) is the one that contains Yang– Mills and Einstein three-point couplings and their higher spin generalizations. It also contains all sorts of cubic interactions between different helicities, most notably the gravitational interaction for arbitrary integer spin with two transverse derivatives. This type of interaction was long thought to be impossible in Minkowski space-time due to various covariant no-go theorems.2 The third type (iii) is a bit odd, it does not contain any self-interactions. The fourth type (iv) finally corresponds to models obtained by taking powers of field strengths in the covariant formulation. Possible counterterms can be found here. Gravitational interactions of higher spin massless fields Specializing type (ii) interactions to the case λ1 = λ3 = s arbitrary and λ2 = 2 we get ∫

γ1s

γ22 γ3s

2 ℙ̄ ϕ̄ s (p1 )ϕ2 (p2 )ϕs (p3 ) + c. c.

(6.9)

The momentum factor is rewritten using (6.3), γ1s

ℙ̄ 2 = s

γ22 γ3

γ13

( s−2

γ3

2

2

γ13 p̄2 p̄3 ℙ̄ ) = s−2 ( − ) γ2 γ3 γ2 γ3 γ3

(6.10)

The momentum space interaction may then be Fourier transformed into the configuration space form 2−n

2 𝜕̄ n 2 + s ∫ d 3 x ∑ (−1) ( )(𝜕 ) ϕ̄ s [ + ] n 𝜕 n=0

n

ϕ2 [

2−s 𝜕̄ ] (𝜕+ ) ϕs + c. c. 𝜕+

(6.11)

This is the basic spin s gravitational minimal interaction with two transverse derivatives [160]. For odd spin s, we know that the fields must carry an internal index, to be antisymmetrized over in the self-interaction terms. Here, we see that “color”, in that sense, is conserved in the gravitational interaction.

For other concrete examples of cubic interactions, the reader may consult [252]. We will now introduce a vertex operator formalism that can harbor all possible cubic interactions among arbitrary helicity fields, even connecting three different helicities as in the

2 The most often cited no-go results of this type are [277, 100, 99, 101], but see our Section 2.3 for some details. It is a curious twist of history that the existence of the minimal gravitational coupling of higherhelicity fields was not explicitly noted in 1987, neither by the authors of [160] nor by anyone else until it was unearthed in [252] by the present author. It is now well recognized within the research community.

6.1 The cubic interactions: review �

205

list (6.8) above. The formalism was initially inspired by string field theory, and was introduced in the paper [160] in 1987. It was reviewed and made more systematic in [278]. Here, the formalism will be further improved.

6.1.2 Vertex operator formalism for cubic vertices For each higher-helicity field |Φr ⟩, where r = 1, 2, 3, entering a cubic interaction, there is an associated two-dimensional transverse Fock space that carries the spectrum of helicities. In computing the effect of two consecutive quadratic transformations (generated by cubic vertex operators) that we would have to do in order to study the quartic level, we need to work efficiently with both bra and ket spaces. A slight change of notation will enhance the computational efficiency. By transferring the bra and ket indication from the field symbol Φ to the index symbol r, we write Φ|r⟩ and Φ⟨r|

instead of |Φr ⟩ and ⟨Φr |

(6.12)

We will mostly use this new notation when working with interactions, but keep the old notation occasionally. The fields are momentum space fields as given in formulas (6.57) and (6.58) of Section 6.3 of our Volume 1 (where the detailed formalism for higher-helicity fields is developed). Then the cubic interaction Hamiltonian is written as H(1) =

3 1 3 1 ∫ ∏ γr dγr dpr d p̄ r Φ⟨r| V |123⟩ = ∏ Φ⟨r| V |123⟩ 3 r=1 3 r=1

(6.13)

where, in the third expression, the momentum integrations ∫ γr dγr dpr d p̄ r are included in the Fock spaces inner product. Also, |123⟩ is short for |1⟩|2⟩|3⟩. The cubic order vertex is ̄ V |123⟩ = gκ−1 Δ123 Γ−1 3 δ(∑γr )δ(∑pr )δ(∑pr )|0123 ⟩ = Δ123 ⌀ |123⟩ r

r

r

(6.14)

In this formula, ⌀ |123⟩ is defined by ̄ ⌀ |123⟩ = gκ−1 Γ−1 3 δ(∑γr )δ(∑pr )δ(∑pr )|0123 ⟩ r

r

r

(6.15)

where Γ3 = γ1 γ2 γ3 and |0123 ⟩ is the vacuum for all three Fock spaces. All non-trivial interaction data is maintained by the operator Δ123 . To find all possible cubic interactions is equivalent to finding all possible forms for Δ123 , also denoted simply by Δ3 . This variant of bra–ket notation makes it easy work with vertices like V |1⟩⟨2||3⟩ with mixed bra and ket spaces. The formulas (6.13) and (6.14) generalize naturally to higher orders.

206 � 6 Light-front interactions Note also that the operator factor of the vertex is represented as Δ and not as exp Δ. The latter form is used in string field theory, but there is nothing at the cubic level that picks out any particular function of Δ in higher spin theory.3 Indeed, at the cubic level working with Δ or exp Δ makes no essential difference at all. The solution for the cubic Δ is an infinite sum of terms generating all possible cubic interactions between higherand lower-integer spin fields. We will comment more on this in Section 6.3.12. The dynamical cubic Lorentz generators are represented as a sum of an orbital part and a spin part J3− =

3 1 3 1 3 ∏ Φ⟨r| (∑ ai xi V |123⟩ + S |123⟩ ) = ∏ Φ⟨r| J −|123⟩ 3 r=1 3 r=1 i=1

(6.16a)

J3−̄ =

3 1 3 1 3 ∏ Φ⟨r| (∑ ai x̄i V |123⟩ + S̄ |123⟩ ) = ∏ Φ⟨r| J −̄|123⟩ 3 r=1 3 r=1 i=1

(6.16b)

where the ai are numbers to be determined, and where in analogy with (6.14) the spin vertex operators are S |123⟩ = χ3 ⌀|123⟩ S̄ |123⟩ = χ3̄ ⌀|123⟩

(6.17a) (6.17b)

This representation is very much in the spirit of the original paper [274]. The operators χ3 and χ3̄ turn out to be closely related to the Δ3 -operators and contain no independent interaction data. A short note on the momentum conservation delta-functions In the calculations to follow, it must be remembered that operators such as x, x ̄ and

𝜕 𝜕γ

act as derivatives on

the momentum conservation delta functions in ⌀|123⟩ . This must be taken into account using ∫ f (x)δ ′ (x)dx = − ∫ f ′ (x)δ(x)dx. In most cases, this is innocuous, since what occurs as f (x) are constants or x. It is not explicitly pointed out in every case below, but watch out when computing with the Lorentz generators and take a second look at formulas (6.16) above.

Historical detour with a lesson One major motivation behind the work that led to the paper [160] was the thought that perhaps string field theory formalism could throw light on the light-cone cubic interactions discovered some years earlier. This indeed turned out to be the case. It was found that the cubic interactions from [274] could be derived from the Hamiltonian

3 As pointed out by many workers.

6.1 The cubic interactions: review �

H(1) =

3 1 ∫ ∏ γr dγr dpr d p̄r ⟨Φr |V123 ⟩ 3 r=1

207

(6.18)

where the cubic vertex operator |V123 ⟩ is defined by

|V123 ⟩ =

g exp Δ|0123 ⟩Γ−1 3 δ(∑γr )δ(∑pr )δ(∑p̄ r ) κ r r r

Δ = κ ∑ Y rst (αr† αs† ᾱt† ℙ̄ + ᾱr† ᾱs† αt† ℙ)

(6.19)

r,s,t

Here, g is a dimensionless coupling constant and κ is of dimension −1. For the cubic interactions, g becomes the spin-1 coupling and gκ becomes the spin-2 coupling. The higher spin coupling constants gλ come out as gκ λ−1 . The Y rst are rational functions of γ that are determined by the Poincaré algebra. Γ3 is γ1 γ2 γ3 and it compensates the integration measure factor. The dynamical Lorentz generators acquire the following cubic deformations: − J(1) = −̄ J(1)

3 1 ̂ 123 ⟩ ∫ ∏ γ dγ dp d p̄ ⟨Φ |x|V 3 r=1 r r r r r

3 1 ̂̄ = ∫ ∏ γr dγr dpr d p̄r ⟨Φr |x|V 123 ⟩ 3 r=1

(6.20)

The prefactors x ̂ are (the formalism and notation of [160] is used) x̂ =

1 3 2i ∑ x + κ ∑ Y rst (̃γr + ̃γs − ̃γt )αr† αs† ᾱt† 3 r=1 r 3 r,s,t

(6.21)

with x ̂̄ the c. c. of x ̂ (the second term of these prefactors are replaced by the spin operators of (6.16a) and (6.16b) in the new improved formalism). These interaction terms deform the Poincaré algebra to first order, provided that Y rst =

γt γr γs

(6.22)

It is now clear how the momentum structure of (6.7) appears. One discovery of the 1987 paper was that the vertex operator Δ needed terms with three oscillators and one momenta in order to reproduce the known cubic interaction terms (see (6.19)). Guided by string theory, it would have been thought that quadratic forms in oscillators and momenta would suffice, but that is not the case. As shown in [160], a string-like ansatz of the form Δ = ∑ Y rs αr† ᾱs† + κ ∑ Y r (αr† ℙ̄ + ᾱr† ℙ) r,s

r

(6.23)

leads to the solution for the vertex functions Y rs and Y r Y rs = δrs

Yr =

1 γr

(6.24)

Such a vertex is incapable of reproducing neither the Yang–Mills, nor the gravity, cubic couplings. Now, if there is some depth to this vertex operator-oscillator formalism rather than just being a convenient book keeping

208 � 6 Light-front interactions

device, it indicates that there might be a profound difference between strings and higher spin models.4 As we have already seen in Sections 5.2.3 and 5.3, the same kind of structure as in the Δ of (6.19), later showed up in covariant Minkowski approaches.

6.2 Nonlinear realization of the Poincaré algebra In Section 6.3.2 in our Volume 1, the Poincaré algebra was split into the subalgebra 𝒦 of kinematic generators and the subalgebra 𝒟 of dynamical generators. The 21 commutators between kinematic generators are satisfied by the free field theory generators by construction. The form of the nonlinear Poincaré generators are then highly constrained by the (also 21 in number) commutators of the 𝒦 − 𝒟 part of the algebra. We will, however, start by first studying the structure of the 𝒟 − 𝒟 subalgebra, containing the remaining 3 commutators of the full algebra.

6.2.1 Transformations and Fock field commutators The dynamical generators are given by ∞

H = H(0) + ∑ H(ν−2) ν=3

(6.25)

and similar expressions for the dynamical Lorentz generators J − and J −̄ . The particular form of generators will be specified later. We have an option either to work directly in terms of the commutators such as [H, J − ] or in terms of transformations such as [δH , δJ − ]|Φ⟩. We will choose the second option although they are simply related through [δH , δJ − ]|Φ⟩ = −δ[H, J − ] |Φ⟩. In both options though, we have to sort out Fock space equal time commutators such as [Φ|r⟩ , Φ|s⟩ ]. A generic transformation is given by δG |Φ⟩ = [|Φ⟩, G]

(6.26)

δG |Φ⟩ = [|Φ⟩, G(μ) ]

(6.27)

or (μ)

4 Some other authors tried, or had already tried, string-like vertex operators for higher spin in a covariant formulation (see Section 5.2.2), and found the same result corresponding to (6.24), but did not proceed to higher order in oscillators. On a personal note, I was very confused by the notorious fact that a string-like vertex did not work. Going to higher order in oscillators was a bit of an act of desperation. In retrospect, the solution is quite obvious, but it shows the “singular” nature of higher spin interactions as compared to the “smooth” string-like interactions that can be captured in an exponential of a quadratic form of momenta and oscillators.

6.2 Nonlinear realization of the Poincaré algebra � 209

if a particular order of interaction μ is focused. What we need are commutators of the form [Φ|r⟩ , Φ|s⟩ ] = I|rs⟩

(6.28)

[Φ⟨r| , Φ|s⟩ ] = I⟨rs⟩

[Φ⟨r| , Φ⟨s| ] = I⟨rs|

where we think of the I|rs⟩ , . . . as delta vertices identifying momenta and Fock spaces labeled by r and s. They are supposed to work in the following way: Φ⟨r| I|rs⟩ = Φ|s⟩

(6.29)

where we understand that momenta are identified through pr + ps = 0 and the bra Fock space labeled by r is turned into the ket Fock space labeled by s. The rest of the commutators of (6.28) are interpreted correspondingly. Granted the existence of these delta vertices (they are constructed in Section 9.6 of [278] based on [245]), we can derive the structure of the algebra.

6.2.2 The 𝒟 − 𝒟 algebra For two generic dynamical generators A and B and a field |Φχ ⟩, we have [δA , δB ]|Φχ ⟩ = 0

(6.30)

since the dynamical subalgebra is Abelian. Expanding this equation a few orders in the interaction, we get Free:

[δA(0) , δB(0) ]|Φχ ⟩ = 0

Cubic: ([δA(0) , δB(1) ] + [δA(1) , δB(0) ])|Φχ ⟩ = 0

(6.31)

Quartic: ([δA(0) , δB(2) ] + [δA(2) , δB(0) ])|Φχ ⟩ = −[δA(1) , δB(1) ]|Φχ ⟩ Quintic: ([δA(0) , δB(3) ] + [δA(3) , δB(0) ])|Φχ ⟩ = −[δA(1) , δB(2) ] + [δA(2) , δB(1) ]|Φχ ⟩

The recursive nature of the equations is apparent where, for instance, in the third line, the cubic transformations act as “sources” to the quartic and so on. The general form of these equations can be written as ν−1

([δA(0) , δB(ν) ] + [δA(ν) , δB(0) ])|Φχ ⟩ = − ∑ ([δA , δB μ=1

(μ)

(ν−μ)

(ν−μ)

] + [δA

, δB ])|Φχ ⟩ (μ)

(6.32)

The left-hand sides of these equations will be called the differential commutator and the right-hand side the source commutator.

210 � 6 Light-front interactions

6.3 Computing the cubic interactions Having so established a notation for interactions in general, we now turn to the cubic interactions. The first step is to implement all restrictions on the interactions derivable from the 𝒦 − 𝒟 part of the algebra. For ease of reference, we repeat the commutators in the next section, grouped in a way that facilitates extracting their consequences for the interactions.

6.3.1 The 𝒦 − 𝒟 algebra The 21 commutators of this type have the following algebraic structure: (i)

[𝒦, 𝒟] ⊂ 𝒦

#6

(ii) [𝒦, 𝒟] = 0

#8

(iii)

#7

[𝒦, 𝒟] ⊂ 𝒟

The commutators of the first subtype are [ p+ , j− ] = −ip [ j , p ] = ip [ j+ , j−̄ ] = ij+− − j +

−

[ p+ , j−̄ ] = −ip̄ [ j+̄ , p− ] = ip̄ [ j , j ] = ij + ̄ −

+−

(𝒦𝒟.1) (𝒦𝒟.2) +j

(𝒦𝒟.3)

These commutators tell us that the kinematic transformations commute with the nonlinear part of the dynamic transformations. In practice, they therefore form a set of zero commutators together with the second subtype. These are [ p, p− ] = 0 [ p, j ] = 0 −

[ p,̄ p− ] = 0 [ p,̄ j−̄ ] = 0

[ p+ , p− ] = 0

(𝒦𝒟.4) (𝒦𝒟.5)

[j ,j ] = 0 [j ,j ] = 0

(𝒦𝒟.6)

[ j, p ] = 0

(𝒦𝒟.7)

+ ̄ −̄

+ − −

Together these will fix some of the structure of the interaction terms. The nonzero commutators of the third subtype are [j+− , p− ] = −ip− [ p,̄ j ] = −ip −

[ j , j ] = −ij +− −

[ j, j ] = j −

−

−

−

(𝒦𝒟.8) [ p, j ] = −ip [ j+− , j−̄ ] = −ij−̄

(𝒦𝒟.10)

[ j, j ] = −j

(𝒦𝒟.11)

− ̄

− ̄

−

− ̄

(𝒦𝒟.9)

6.3 Computing the cubic interactions �

211

These work order by order in the interaction and fix still more of the structure. Taken together, the 𝒦 − 𝒟 commutators determine the general form of the interactions up to p+ -structure. 6.3.2 Computation of commutators to cubic order We want to translate the Poincaré commutators of generators to a concrete computational form. Let k be a kinematical generator and d a dynamical generator with corresponding free theory operators k̂ and d̂ and cubic vertex |D⟩. Then we have on a certain field Φ|3⟩ δk Φ|3⟩ = k̂3 Φ|3⟩

(6.33)

δd Φ|3⟩ = d̂3 Φ|3⟩ + Φ⟨1| Φ⟨2| D |123⟩

(6.34)

Then consider the commutator [k, d] = g with g zero, kinematical or dynamical. In terms of transformations, we then have [δk , δd ]Φ|3⟩ = −δg Φ|3⟩

(6.35)

where we have remembered δ[k,d] Φ|3⟩ = −[δk , δd ]Φ|3⟩ . The left-hand side of the commutator becomes [δk , δd ]Φ|3⟩ = δk (Φ⟨1| Φ⟨2| D |123⟩ ) − δd (k̂3 Φ|3⟩ ) = δk Φ⟨1| Φ⟨2| D |123⟩ + Φ⟨1| δk Φ⟨2| D |123⟩ − k̂3 (⟨Φ1 |Φ⟨2| D |123⟩ ) = −Φ⟨1| Φ⟨2| (k̂1 + k̂2 + k̂3 )D |123⟩

(6.36)

Here, we have used the rule δk Φ⟨r| = −k̂r Φ⟨r|

(6.37)

Though natural, its precise derivation (with the minus sign) involves quite a few steps, using the free theory Poincaré generators of Section 6.3.3 in our Volume 1. Using this, we have the following cases. The commutator [k, d] = g becomes 3

∑ k̂r D |123⟩ = 0 for g zero or kinematical

r=1 3

∑ k̂r D |123⟩ = G |123⟩

r=1

for g dynamical

(6.38) (6.39)

Let us also include the commutator [d, g] between two dynamical commutators from the 𝒟 − 𝒟 subalgebra. The commutator becomes to cubic order 3

3

r=1

r=1

∑ d̂r G |123⟩ − ∑ ĝr D |123⟩ = 0

(6.40)

212 � 6 Light-front interactions 6.3.3 Immediate restrictions from the 𝒦 − 𝒟 algebra: cubic and arbitrary order We will now investigate how much of the vertex operators that are fixed by the kinematical part of the Poincaré algebra. We may just as well phrase this discussion for an arbitrary order ν vertex since the right-hand sides of the 𝒦 − 𝒟 commutators are either zero or works order-by-order in the interaction with no mixing of vertex operators of different orders. Indeed, the arguments that lead to formulas (6.38) and (6.39) generalize immediately to ν

[δk(0) , δd(ν) ]|Φ⟩ = 0

⇒

[δk(0) , δd(ν) ]|Φ⟩ = δg(ν) |Φ⟩

⇒

∑ k̂r D |1⋅⋅⋅ν⟩ = 0

(6.41)

∑ k̂r D |1⋅⋅⋅ν⟩ = G |1⋅⋅⋅ν⟩

(6.42)

r=1 ν r=1

A few of the 𝒦 − 𝒟 commutators can now be taken care off at once, namely the ones with zero or linear right-hand side, corresponding to the case of (6.41). – The commutators of (𝒦𝒟.4) tell us that there are no explicit occurrences of x − , x or x̄ in V|1⋅⋅⋅ν⟩ . The zero sums of the γ, p, p̄ momenta are guaranteed by the delta functions. – The commutators of (𝒦𝒟.1) tell us that the dynamical angular momentum vertices J −|123⟩ and J −̄|123⟩ do not contain x − . – The commutators of (𝒦𝒟.5) and (𝒦𝒟.9) tell us that the vertex J −|123⟩ contain the coordinate x linearly (but not x)̄ and correspondingly that the vertex J −̄|123⟩ contain the coordinate x̄ linearly (but not x). These results should be intuitively clear from the construction of the light-front frame. We have actually already built these restrictions into the expressions (6.16a) and (6.16b) for the dynamical angular momentum vertices J −|123⟩ and J −̄|123⟩ . For instance, the commutator [p,̄ j− ] = −ih implies 3

− = −iV|123⟩, ∑ p̄ r J|123⟩

r=1

(6.43)

which in its turn implies 3

∑ ai = 1 i=1

(6.44)

Symmetry among the field labels 1, 2, 3 would then yield all ai equal to 1/3. This will later follow from the 𝒟 − 𝒟 algebra. Sharper restrictions on the vertices will follow from the (𝒦𝒟.2), (𝒦𝒟.7) and (𝒦𝒟.8) commutators. But first we need an ansatz for the vertex functions, and before that a reminder on our notation for oscillators.

6.3 Computing the cubic interactions

� 213

6.3.4 Reminder on Fock space fields For ease of reference, we repeat the formulas for the Fock space higher-helicity fields here: ∞ 1 λ λ 󵄨󵄨 (ϕλ (p)(ᾱ + ) + ϕ̄ λ (p)(α+ ) )|0⟩ 󵄨󵄨Φ(p)⟩ = ∑ √ λ=0 λ!

(6.45)

In formulas like this, p is short for p, p̄ and γ = p+ . The conjugated Fock space field is given by ∞ 1 † 󵄨󵄨 󵄨 ⟨0|(ϕ̄ λ (−p)αλ + ϕλ (−p)ᾱ λ ) 󵄨󵄨Φ(p)⟩ = ⟨Φ(−p)󵄨󵄨󵄨 = ∑ √ λ=0 λ!

(6.46)

The fields ϕ and ϕ̄ being functions of x + , p+ , p, p.̄ There is a change of notation for the oscillators as compared to the corresponding formulas (6.57) and (6.58) in our Volume 1. This is explained in the box below. The light-front internal Fock-space I have been vacillating over how to denote the internal creation and annihilation operators used to build the Fock space of higher-helicity fields. When complexifying the transverse d. o. f. one gets a notational conflict between Hermitian conjugation, denoted by † , and the also often used notation † for creation operators. Fundamentally, the problem goes back to the need to distinguish between an operation and its effect.5 The choice to be used from now on is to use + for designating creation operators, having no operational meaning, and † for the operation Hermitian conjugation. To maintain fields of all helicities, we introduce a two-dimensional complex internal Fock space spanned by oscillator pairs (α, ᾱ + ) and (α,̄ α+ ) with [α, ᾱ + ] = [α,̄ α+ ] = 1

(6.47)

thus taking α+ and ᾱ + as creation operators and α and ᾱ as annihilation operators. The complex pairs of oscillators (α, ᾱ + ) and (α,̄ α+ ) can be defined in terms of a pair of oscillators (αi , αi+ ), i = 1, 2, with the usual commutators [αi , αj+ ] = δij . 1 (α + iα2 ) √2 1 1 α+ = (α+ + iα2+ ) √2 1 α=

1 (α − iα2 ) √2 1 1 ᾱ + = (α+ − iα2+ ) √2 1 ᾱ =

(6.48) (6.49)

The commutation relations of (6.47) are satisfied. The notation is now fully consistent in that one may, for instance, compute α† and get ᾱ + , which may look odd, but is correct. This is a change of notation in reference to the notation of my papers and of my Volume 1. In my papers, I used † for + and having no operational

5 See comments in our Volume 1, Section 1.4.

214 � 6 Light-front interactions

meaning for † in this context. There is no inconsistency, but perhaps a cause for confusion, and formulas from the papers translate to the present notation by just replacing † →+ in this context. The choice I made in Volume 1 (see Section 6.3.1) is also correct, but a bit unconventional and unaesthetic, and prone to misreading. If one only remembers not to confuse the plus + with a light-front direction, the present convention is preferable.

6.3.5 Preliminary ansatz for the vertex functions Much of the simplicity of the cubic vertex is due to the nice factorization of Δ3 into the factor Y rst αr+ αs+ ᾱ t+ and the special transverse momentum factor ℙ.̄ As we will see, this form is fixed by the 𝒦 − 𝒟 commutators. Suppose we were not quite sure about the factorization property of Δ3 . A slightly more general expression would be Δ3 = Y rstu αr+ αs+ ᾱ t+ p̄ u + c. c.

(6.50)

where sums over the labels are understood. Apart from the balance between barred and unbarred oscillators and momenta, this is almost as general as it can be, all γ dependence maintained by the Y . It might seem that we are excluding possibilities such as Y rstuv αr+ αs+ ᾱ t+ p̄ u γv + c. c.

(6.51)

However, such terms can taken care of by a redefinition of the Y -functions according to Y rstu = Y rstuv γv . Based on these observations, we try contributions to Δ3 of the form Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl t1 ⋅⋅⋅tm u1 ⋅⋅⋅un αr+1 ⋅ ⋅ ⋅ αr+k ᾱ s+1 ⋅ ⋅ ⋅ ᾱ s+l pt1 ⋅ ⋅ ⋅ ptm p̄ u1 ⋅ ⋅ ⋅ p̄ un

(6.52)

where summations are understood and the c. c. term should be added. The Y are symmetric in the r, s, t, u labels separately. We abbreviate it to6 k

l

Y (k)(l)(m)(n) (α+ ) (ᾱ + ) pm p̄ n

(6.53)

only showing explicit labels when needed. For any such term in Δ3 , there is a complex conjugated mirror term. A note on higher orders Clearly, nothing in the ansatz (6.52), except the summation ranges, is special for the cubic vertex. Thus, we can use the ansatz (6.52) for the Y -functions of an order ν vertex operator Δν by letting field label indices run from 1 to ν. However, we will soon find out that the cubic vertices are much more conveniently expressed in terms of

6 Dimensional factors of κ are suppressed.

6.3 Computing the cubic interactions

� 215

the momentum functions ℙ rather that the momenta pt themselves. Although this will generalize to higher orders, the superficial resemblance of higher-order functions to cubic functions will be lost. Nevertheless, useful insights into the higher-order structure can be gained from the expressions (6.52). In Sections 6.3.6 and 6.3.7 below, we will extract this general information along with the particular cubic implementation.

6.3.6 Further restrictions from the 𝒦 − 𝒟 algebra We start with the commutator of (𝒦𝒟.7) and work for arbitrary vertex order ν: ν

ν

∑ jr |Vν ⟩ = ∑ (xr p̄ r − x̄r pr − i(αr+ ᾱ r − ᾱ r+ αr ))Δν |⌀1⋅⋅⋅ν ⟩ r=1

r=1

= (i(n − m) − i(k − l))Δν |⌀1⋅⋅⋅ν ⟩ = 0

(6.54)

To satisfy this, we require (n − m) = (k − l), which can be interpreted as a balance between transverse orbital angular momentum and helicity. Written as k + m = l + n, the requirement also means that the overall number of unbarred and barred α’s and p’s should be the same. We then turn to the commutators of (𝒦𝒟.2). Here, we will do it first for the cubic case 3

3

r=1

r=1

∑ j+ |V3 ⟩ = ∑ xr γr Δ3 |⌀123 ⟩ 3

= Δ3 ∑ xr γr Y rstu αr+ αs+ ᾱ t+ p̄ u |⌀123 ⟩ r=1

= iΔ3 Y rstu αr+ αs+ ᾱ t+ γu |⌀123 ⟩ = 0

(6.55)

To satisfy this equation, we need 3

∑ Y rstu γu = 0

u=1

for all combinations of r, s, t

(6.56)

For the cubic vertex, there is a nice resummation formula.7 It states that for any indeterminate quantities cr , 3 3 3 p 1 3 1 ∑ cr pr = ( ∑ cr γr )(∑ s ) + ( ∑ cr γr γ̃r )ℙ 3 r=1 γ 3Γ r=1 r=1 s=1 s

(6.57)

where Γ = γ1 γ2 γ3 . It can be proved by direct calculation using momentum conservation and cyclic symmetry in the labels. 7 The original reference is unknown. The author learned it from N. Linden.

216 � 6 Light-front interactions If the resummation formula is applied to Y rstu p̄ u , i. e., with cu = Y rstu we get 3 3 3 p̄ 1 1 3 ∑ Y rstu p̄ u = ( ∑ Y rstu γu )(∑ s ) + ( ∑ Y rstu γu γ̃u )ℙ̄ 3 u=1 γ 3Γ u=1 u=1 s=1 s

(6.58)

The first term on the right is zero by (6.56) and the factor in front of ℙ̄ amounts to a redefinition Y rst = 1/(3Γ) ∑3u=1 Y rstu γu γ̃u .8 This is the way the form of the cubic Δ of equation (6.19) emerges. In analogy to (6.56), when applying j+ and j+̄ to an arbitrary order vertex V|1⋅⋅⋅ν⟩ , we get the following restrictions on the Y -functions (6.52): Y (k)(l)t1 ⋅⋅⋅ti ⋅⋅⋅tm (n) γti = 0

and

Y (k)(l)(m)u1 ⋅⋅⋅ui ⋅⋅⋅un γui = 0

(6.59)

meaning that replacing any one transverse momentum p or p̄ with γ and summing over labels yield zero (irrespective of all the other labels). The resummation formula (6.57) can be generalized to arbitrary vertex order [279], and it again yields that the expressions (6.52) for the contributions to Δν can be resummed into sums of products of Y -functions of γr and special momenta ℙrs and ℙ̄ rs factors where r, s run over 1 ⋅ ⋅ ⋅ ν. Next, we impose the commutator of (𝒦𝒟.8). Commutators involving the operator +− j require some care. The computational form of the commutator [j+− , h] = −ih, i. e., [δj+− , δh(ν) ]|Φ⟩ = iδh(ν) |Φ⟩ is ν

− ∑ (iγr r=1

𝜕 + i)|Vν ⟩ = i|Vν ⟩ 𝜕γr

(6.60)

The γ dependence of the vertex sits in three places: Δν , Γ−1 ν and the γ conservation delta function. For the latter two, we have ν

∑ γr

r=1 ν

∑ γr

r=1

𝜕 −1 Γ = −νΓ−1 ν 𝜕γr ν

ν ν 𝜕 δ(∑ γs ) = −δ(∑ γs ) 𝜕γr s=1 s=1

(6.61) (6.62)

Therefore, equation (6.60) becomes ν

− ( ∑ iγr r=1

i. e., 8 ̃γ is defined in (6.2).

𝜕Δν + iν − iν − i)Δν |⌀1⋅⋅⋅ν ⟩ = i|Vν ⟩ 𝜕γr

(6.63)

6.3 Computing the cubic interactions � ν

∑ γr

r=1

𝜕Δν =0 𝜕γr

217

(6.64)

telling us that the functions Y making up Δν are homogeneous functions of the γr of degree zero. Cubic kinematics The kinematics of the interactions are such that for an order ν interaction we have momentum conservation ν

ν

ν

r=1

r=1

r=1

∑ pr = ∑ p̄r = ∑ γr = 0

(6.65)

It is convenient to write the transverse momentum dependence in terms of the combinations ℙij = γi pj − γj pi

and

ℙ̄ ij = γi p̄j − γj p̄i

(6.66)

The number of ℙij for an order ν vertex is n = ν(ν − 1)/2. Due to momentum conservation, only n − 2 of those are linearly independent. For the cubic, ν = 3, this means that there is only one ℙ = ℙ12 = ℙ23 = ℙ31 and similarly for ℙ.̄ Based on this, it is possible to derive linear recombination formulas. Let cr be arbitrary variables, then the cubic resummation formula (6.58) can be written in a way that generalizes to higher order 3

∑ cr pr =

r=1

3 p 1 3 1 3 (∑ cr γr )(∑ s ) − (∑ Sr cr ) ℙ 3 r=1 3 r=1 s=1 γs

(6.67)

where Sr = 1/γr+1 − 1/γr+2 . In this formula, the objects 3

∑

s=1

ps γs

and

ℙ

(6.68)

are independent basis vectors in the two-dimensional transverse momentum space {p1 , p2 , p3 } constrained by momentum conservation. A useful instance of the formula results from taking all cr = 0 except ca = 1/γa or cr = δra /γa , pa 1 3 p 1 Sa = ∑ s − ℙ γa 3 s=1 γs 3 γa

(6.69)

The generalization to arbitrary vertex order ν can be fould in reference [279].

6.3.7 Restrictions on the dynamical Lorentz generators A few more restrictions on the dynamical Lorentz generators follow from the remaining 𝒦 − 𝒟 commutators involving two Lorentz operators. Let us first remind ourselves of the overall form of the generators as given by formulas (6.16). The χ and χ̄ operators of the spin parts |S⟩ and |S⟩̄ of the dynamical Lorentz generators will have structure reminiscent of the Y -functions of the the Δ operators of the interacting Hamiltonian |V ⟩.

218 � 6 Light-front interactions Thus, we may expect χ and χ̄ to be a linear combinations of expressions of the general form X r1 ⋅⋅⋅rk′ s1 ⋅⋅⋅sl′ t1 ⋅⋅⋅tm′ u1 ⋅⋅⋅un′ αr+1 ⋅ ⋅ ⋅ αr+ ′ ᾱ s+1 ⋅ ⋅ ⋅ ᾱ s+′ pt1 ⋅ ⋅ ⋅ pt ′ p̄ u1 ⋅ ⋅ ⋅ p̄ u ′ k

m

l

(6.70)

n

X̄ r1 ⋅⋅⋅rk′′ s1 ⋅⋅⋅sl′′ t1 ⋅⋅⋅tm′′ u1 ⋅⋅⋅un′′ ᾱ r+1 ⋅ ⋅ ⋅ ᾱ r+ ′′ αs+1 ⋅ ⋅ ⋅ αs+′′ p̄ t1 ⋅ ⋅ ⋅ p̄ t ′′ pu1 ⋅ ⋅ ⋅ pu ′′ , k

l

m

n

(6.71)

respectively. The notation is chosen to indicate an expected complex conjugation relationship between χ and χ.̄ We will sharpen this shortly. The further restrictions from the 𝒦 − 𝒟 commutators are as follows: – The first commutator [j, j− ] = j− , of (𝒦𝒟.11), fixes the balance between unbarred and barred momenta and oscillators for the X functions. A calculation similar to (6.54) yields (k ′ − l′ ) − (n′ − m′ ) = 1 where k ′ , l′ , m′ and n′ are the numbers of α+ , ᾱ + , p and p,̄ respectively, as given in (6.70). Likewise, the second commutator [j, j−̄ ] = −j−̄ , of (𝒦𝒟.11) implies the balance equation (k ′′ − l′′ ) − (n′′ − m′′ ) = −1. The balance condition (k ′ − l′ ) − (n′ − m′ ) = 1 is entirely natural, it just says that the second term must have the same total transverse orbital and spin angular momentum as the coordinate. This balance condition can be satisfied by k ′ = k, l′ = l, m′ = m and n′ = n − 1 and this is indeed what follows from the dynamical part of the algebra. – The commutators of (𝒦𝒟.3) and (𝒦𝒟.6) tell us that when any one transverse momentum in X is replaced by a γ the result is zero. This is the same kind of condition as (6.59) on the Y . Using the resummation formulas, this results in the X functions being sums of products of functions of the γr and factors of ℙrs and ℙ̄ rs . – The commutator of (𝒦𝒟.10) fixes the γ homogeneity of the X and X̄ functions to zero. The computational form of the commutator [j+− , j− ] = −ij− to order ν is ν

− ∑ (iγr r=1

𝜕 + i)|J − ⟩ = i|J − ⟩ 𝜕γr

(6.72)

Then repeating the steps leading from equation (6.60) to (6.63), we find ν

∑ γr

r=1

𝜕χν =0 𝜕γr

(6.73)

and similarly for χν̄ . With this, we have now extracted all information from the 𝒦 − 𝒟 commutators. It will summarized in the next section. Summary of kinematical constraints Most of the kinematic constraints are quite trivial and just exclude or fix certain dependencies as we’ve seen in Sections 6.3.3 through 6.3.7. There are, however, three requirements that come out of the analysis that are crucial. They are as follows:

6.3 Computing the cubic interactions

� 219

γ-homogeneity This comes from the commutators with j +− . Y , X and X ̄ are all homogeneous functions of γ of degree 0. γ-replacement property This comes from the commutators with j + and j +̄ . Replacing a transverse momentum in Y , X and X ̄ with γ gives zero. For the cubic, this is implemented by the construction of the ℙ and ℙ.̄ Angular momentum balance This comes from the commutators with j. The number of unbarred and barred creators and transverse momenta in Y -functions must be the same. ̄ The number of unbarred and barred creators and transverse momenta in X- and X-functions must differ by ±1, respectively.

There is one further point that can be simplified for a generic vertex operator. The oscillator basis in formulas such as (6.52) or (6.53) always take the same form apart form the range of the field label sums {1, 2, . . . , ν}. A shorthand notation, recording the numbers of different oscillators, is convenient A+k l ̄ = αr+1 ⋅ ⋅ ⋅ αr+k ᾱ s+1 ⋅ ⋅ ⋅ ᾱ s+l

(6.74)

(A+k l ̄) = ᾱ r+1 ⋅ ⋅ ⋅ ᾱ r+k αs+1 ⋅ ⋅ ⋅ αs+l = A+kl̄ = A+lk̄

(6.75)

with complex conjugates ∗

There is of course full permutational symmetry in the field labels ri and sj in A+k l ̄ and A+kl̄ . 6.3.8 Definite ansatz for the cubic vertex functions Based on the preceding analysis, we can now write a general ansatz for the cubic vertex functions n m Δ3 = Y (k)(l)mn (A+k l ̄ ℙm ℙ̄ +A+kl̄ ℙ̄ ℙn )

(6.76)

where Y (k)(l)mn are real rational functions of the γr to be determined. The notation (k) and (l) serves as a reminder that the Y -functions have full permutational symmetry in the field labels ri and sj , respectively. The formula should be interpreted such that k and l are summed k = 1, 2, . . . and l ≤ k and the explicit form is n Δ3 = ∑ Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl mn αr+1 ⋅ ⋅ ⋅ αr+k ᾱ s+1 ⋅ ⋅ ⋅ ᾱ s+l ℙm ℙ̄ +c. c. r1 ⋅⋅⋅rk

(6.77)

s1 ⋅⋅⋅sl

Note that k − l = n − m from j-rotational invariance in the transverse space. The ansatz is such that Δ3 is real. For m = n (and therefore k = l), the two terms in the ansatz are equal. Such terms turn out not to produce any interactions so we might just as well take l < k. The ansatz is redundant as it stands since terms with ℙ ℙ̄ corresponds to field

220 � 6 Light-front interactions redefinitions of the free Hamiltonian. They can be removed by keeping only terms with either m = 0 or n = 0. Choosing m = 0, we get the ansatz n

Δ3 = Y (k)(l)n (A+k l ̄ ℙ̄ +A+kl̄ ℙn )

(6.78)

n = k − l. Going ahead with this ansatz, it will eventually turn out that we do reproduce the nontrivial cases (ii) and (iv) in Section 6.1.1, but not case (iii). To get case (iii), one must make the choice n = 0. We will only treat the case m = 0. See further comments in Section 6.3.12 below where we summarize and comment on the results. The ansatz for the spin X-function is such that there is one term for each term in Δ3 but with one factor less of ℙ.̄ Correspondingly, the ansatz for the spin X̄ is such that there is one term for each term in Δ3 but with one factor less of ℙ. Thus, n−1 χ3 = X (k)(l)n A+k l ̄ ℙ̄

and

χ3̄ = X̄ (k)(l)n A+kl̄ ℙn−1

(6.79)

6.3.9 Derivation of all cubic vertices in four dimensions The computational form (6.40) of the 𝒟 − 𝒟 commutators [h, j− ] = 0, [h, j−̄ ] = 0 and [j, j−̄ ] = 0 are 3

3

3

r=1

r=1

i

3

3

3

r=1

r=1

i

∑ jr− V|123⟩ − ∑ hr (∑ ai xi V|123⟩ + S|123⟩ ) = 0

∑ jr−̄ |V123 ⟩ − ∑ hr (∑ ai x̄i V|123⟩ + S̄|123⟩ ) = 0

3

3

3

3

r=1

i

r=1

i

∑ jr−̄ (∑ ai xi V|123⟩ + S|123⟩ ) − ∑ jr− (∑ ai x̄i V|123⟩ + S̄|123⟩ ) = 0

(6.80a) (6.80b) (6.80c)

The free theory generators are ̄ h = pp/γ j− = xh + ip

i 𝜕 − Mp 𝜕γ γ

(6.81) with M = α+ ᾱ − ᾱ + α

(6.82)

The 𝒟 − 𝒟 part of the Poincaré algebra leads to differential equations for the Δ3 , χ3 and χ3̄ operators. It turns out to be enough to compute the first commutator (6.80a). There are five types of terms contributing to the commutator; three from j− acting on |V ⟩ and two from h acting on |J − ⟩. We list them one by one. Notation will be shortened during the calculation, by dropping the field number indices on V|123⟩ , S|123⟩ and Δ3 .

6.3 Computing the cubic interactions

� 221

Terms from xh: These are ∑ xr hr |V ⟩ = ∑(hr xr + [xr , hr ])|V ⟩ r

r

= ∑ hr ([xr , Δ] + Δxr )|⌀⟩ + i ∑ r

r

pr |V ⟩ γr

= ∑ hr [xr , Δ]|⌀⟩ + ∑ Δhr xr |⌀⟩ + i ∑ r

r

r

pr |V ⟩ γr

(6.83)

In the last line, the second and the third terms will cancel contributions from the other parts of the commutator. To compute the first term, note that hr [xr , ⋅] will act on each and every ℙ and ℙ̄ in Δ with the result 3

∑ hr [xr , ℙ] = 0

r=1

and

3 3 3 p p̄ i ∑ hr [xr , ℙ]̄ = (ℙ ∑ r + ℙ̄ ∑ r ) 3 γ γ r=1 r r=1 r r=1

(6.84)

The first term on the last line of (6.83) therefore becomes 3 3 p̄ p in (k)(l)n n−1 Y (ℙ ∑ r + ℙ̄ ∑ r ) ℙ̄ A+k l ̄|⌀⟩ 3 γ γ r=1 r r=1 r

(6.85)

𝜕 Terms from ip 𝜕γ : These terms are computed using 3

∑ pr

r=1

𝜕 ℙ=0 𝜕γr

and

3

∑ pr

r=1

3 p̄ 𝜕 ̄ 1 ̄ 3 pr −ℙ∑ r) ℙ = (ℙ ∑ 𝜕γr 3 γ γ r=1 r r=1 r

(6.86)

and noting the action on the |⌀⟩ vacuum ∑ pr r

p 𝜕 −1 Γ δ(∑γr ) = −Γ−1 ∑ r δ(∑γr ) 𝜕γr r r γr r

(6.87)

Using this, we get i ∑ pr r

𝜕Δ 𝜕 𝜕 Δ|⌀⟩ = i ∑(pr )|⌀⟩ + i ∑ Δpr |⌀⟩ 𝜕γr 𝜕γr 𝜕γr r r = i ∑ pr r

+

𝜕Y (k)(l)n ̄ n + 𝜕Y (k)(l)n n + ℙ Ak l ̄|⌀⟩ + i ∑ pr ℙ Akl̄ |⌀⟩ 𝜕γr 𝜕γr r

3 p̄ p in (k)(l)n ̄ 3 pr n−1 Y (ℙ ∑ − ℙ ∑ r ) ℙ̄ A+k l ̄|⌀⟩ − i ∑ r |⌀⟩ 3 γ γ r γr r=1 r r=1 r

(6.88)

where the last term cancels the third term from the last line of (6.83). The next to last term will combine with the term (6.85).

222 � 6 Light-front interactions Terms from − γi Mp: The annihilators in M act on the creators in Δ inserting a term p/γ for every α and a term −p/γ for every α.̄ The result is (here we need explicit indices) − iY r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n (

pr1 γr1

+ iY r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n (

+ ⋅⋅⋅ +

pr1 γr1

prk γrk

+ ⋅⋅⋅ +

−

prk γrk

ps1 γs1 −

− ⋅⋅⋅ −

ps1 γs1

psl γsl

− ⋅⋅⋅ −

n

) ℙ̄ A+k l ̄|⌀⟩ psl γsl

) ℙn A+kl̄ |⌀⟩

(6.89)

To interpret these expressions correctly, note that there is no summation convention implied by the repeated indices. Instead, the proper interpretation is to think of the Y -functions of (6.77) as being replaced by Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n (

pr1 γr1

+ ⋅⋅⋅ +

prk γrk

−

ps1 γs1

− ⋅⋅⋅ −

psl γsl

(6.90)

)

for each and every index combination. Then there remains to compute the second term in (6.80a). We do it first for the orbital part, then for the spin part. Terms from ai xi : The computation runs as follows: − ∑ ht (∑ ar xr )|V ⟩ = − ∑ ht ∑ ar [xr , Δ]|⌀⟩ − ∑ Δht ∑ ar xr |⌀⟩ t

r

t

r

t

r

(6.91)

The second term cancels the second term of (6.83) if we choose all ar = 1/3. This is because all xr are equal on the vacuum, a consequence of momentum conservation, or locality in transverse directions. We are left with the first term. It is zero since ∑3r=1 ar [xr , ℙ]̄ ∼ ∑3r=1 γ̃r = 0 when all ar are equal. Terms from χ: The free Hamiltonian commutes with everything in the vertex |S⟩ and so just becomes a multiplication. The contribution is n−1 − X (k)(l)n ℙ̄ ∑ hr A+k l ̄|⌀⟩ r

(6.92)

where ∑r hr = − ℙ ℙ̄ /Γ. 6.3.10 The cubic differential equations Collecting the noncanceling terms from (6.85), (6.88), (6.89) and (6.92), we get two equations, one with oscillator basis A+k l ̄|⌀⟩,

6.3 Computing the cubic interactions

p 𝜕Y (k)(l)n ̄ n 2in (k)(l)n ̄ n n ℙ + Y ℙ ∑ r + Γ−1 X (k)(l)n ℙ ℙ̄ 𝜕γ 3 γ r r r r ps p p p r r s n − iY r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n ( 1 + ⋅ ⋅ ⋅ + k − 1 − ⋅ ⋅ ⋅ − l ) ℙ̄ ]A+k l ̄|⌀⟩ = 0 γr1 γrk γs1 γsl

� 223

[i ∑ pr

(6.93)

and one with oscillator basis A+kl̄ |⌀⟩, [i ∑ pr r

ps pr ps pr 𝜕Y (k)(l)n n ℙ +iY r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl ( 1 + ⋅ ⋅ ⋅ + k − 1 − ⋅ ⋅ ⋅ − l ) ℙn ]A+kl̄ |⌀⟩ = 0 𝜕γr γr1 γrk γs1 γsl (6.94)

Expanding out these equations, we get two equations for each concrete index combination r1 ⋅ ⋅ ⋅ rk s1 ⋅ ⋅ ⋅ sl . Since there is no source for the cubic differential, factors of ℙ and ℙ̄ can now be factored out. This gives p 𝜕Y (k)(l)n 2in (k)(l)n + Y ∑ r + Γ−1 X (k)(l)n ℙ 𝜕γ 3 r r r γr ps pr ps pr − iY r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n ( 1 + ⋅ ⋅ ⋅ + k − 1 − ⋅ ⋅ ⋅ − l ) = 0 γr1 γrk γs1 γsl

i ∑ pr

(6.95)

and i ∑ pr r

ps pr pr ps 𝜕Y (k)(l)n + iY r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n ( 1 + ⋅ ⋅ ⋅ + k − 1 − ⋅ ⋅ ⋅ − l ) = 0 𝜕γr γr1 γrk γs1 γsl

(6.96)

Both of these differential equations must be satisfied for any cubic vertex. We note that they are linear in transverse momentum p. As they stand, they can be solved by adding and subtracting them and using the recombination formula (6.67). In the next section, the equations will be solved in a way amenable to generalization to higher orders. 6.3.11 Solution of the cubic differential equations We apply the recombination formula (6.67) to the derivative term to obtain ∑ pr r

𝜕Y (k)(l)n 1 𝜕Y (k)(l)n 3 ps 1 𝜕Y (k)(l)n = ∑ γr − ∑ Sr ℙ ∑ 𝜕γr 3 r 𝜕γr s=1 γs 3 r 𝜕γr

and to the M-rotational term Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n (

pr1 γr1

+ ⋅⋅⋅ +

prk γrk

−

ps1 γs1

− ⋅⋅⋅ −

psl γsl

)

3 Sr Sr p 1 = Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n (k ∑ s − [ 1 + ⋅ ⋅ ⋅ + k ] ℙ) 3 γ γ γrk r1 s=1 s

(6.97)

224 � 6 Light-front interactions 3 Ss Ss p 1 − Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n (l ∑ s − [ 1 + ⋅ ⋅ ⋅ + l ] ℙ) 3 γ γs1 γsl s=1 s

=

k − l r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n 3 ps 1 Y − (SY )r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n ℙ ∑ 3 γ 3 s s=1

(6.98)

Here, we have introduced the convenient abbreviation (SY )r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n = Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n ([

Sr1 γr1

+ ⋅⋅⋅ +

Srk γrk

]−[

Ss1 γs1

+ ⋅⋅⋅ +

Ssl γsl

(6.99)

])

Even shorter, we will write (SY )(k)(l)n for this expression in analogy with Y (k)(l)n . Inserting these formulas into (6.95) and (6.96), we get p i 𝜕Y (k)(l)n 2in (k)(l)n i ( ∑ γr + Y − (k − l)Y (k)(l)n ) ∑ r 3 r 𝜕γr 3 3 r γr i 𝜕Y (k)(l)n i + (− ∑ Sr + (SY )(k)(l)n + Γ−1 X (k)(l)n ) ℙ = 0 3 r 𝜕γr 3

(6.100)

and p 𝜕Y (k)(l)n i i + (k − l)Y (k)(l)n ) ∑ r ( ∑ γr 3 r 𝜕γr 3 r γr i 𝜕Y (k)(l)n i + (− ∑ Sr − (SY )(k)(l)n ) ℙ = 0 3 r 𝜕γr 3

(6.101)

Now we can extract linearly independent parts of the differential equations. First, adding and subtracting the equations along the direction ∑ ps /γs yield ∑ γr r

𝜕Y (k)(l)n + nY (k)(l)n = 0 𝜕γr

(6.102a)

−(k − l) + n = 0

(6.102b)

This is at first a surprising result. The first equation is precisely the same equation that follows from the j+− , γ-homogeneity constraint. However, this is the light-front reflection of the fact that cubic amplitudes are determined by little group scaling (see Section 2.6 in [134]). We will return to a discussion of this in Section 6.4. The second equation is the helicity balance equation coming from j-invariance. Combining we get ∑ γr r

𝜕Y (k)(l)n = (l − k)Y (k)(l)n 𝜕γr

(6.103)

We then recover the fundamental solutions generating all higher spin cubic interactions

6.3 Computing the cubic interactions

Y r1 ⋅⋅⋅rk s1 ⋅⋅⋅sl n =

γs1 ⋅ ⋅ ⋅ γsl

γr1 ⋅ ⋅ ⋅ γrk

� 225

(6.104)

As a check, we see that these functions solve the differential equation (6.96). Next, we extract the equations along the direction ℙ from (6.100), ∑ Sr r

𝜕Y (k)(l)n − (SY )(k)(l)n + 3iΓ−1 X (k)(l)n = 0 𝜕γr

(6.105)

𝜕Y (k)(l)n + (SY )(k)(l)n = 0 𝜕γr

(6.106)

and from (6.101), ∑ Sr r

This last equation is a new differential equation for the Y -functions. Provided it can be satisfied, we get algebraic equations for the X-functions 2i X (k)(l)n = − Γ(SY )(k)(l)n 3

(6.107)

Thus, we see that the X-functions are determined by the Y -functions. One may convince oneself that the Y -functions in (6.104) indeed solves also the equations (6.106). This completes then the derivation of all cubic interaction vertex operators.

6.3.12 Summary of cubic interactions and the exp Δ question We may now summarize. The vertex functions for the Hamiltonian and the dynamical Lorentz generators take the following form: n Δ3 = Y (k)(l)n (A+k l ̄ ℙ̄ +A+kl̄ ℙn )

(6.108a)

χ3 =

(6.108b)

n−1 X (k)(l)n A+k l ̄ ℙ̄

More appropriately, we should write the general vertex operator as a sum over all contributions n

Δcubic = ∑ gk,l,n Y (k)(l)n (A+k l ̄ ℙ̄ +A+kl̄ ℙn ) k>l n=k−l

(6.109)

with coupling factors gk,l,n = gkl that cannot be determined from the cubic algebra. The first two terms are Δcubic′ = g10 Y (1)(0)1 A+10̄ ℙ̄ +g21 Y (2)(1)1 A+21̄ ℙ̄ +c. c. which, more explicitly, can be written as in (6.19) and (6.23),

(6.110)

226 � 6 Light-front interactions Δcubic′ = g10 ∑ Y r (αr† ℙ̄ + ᾱ r† ℙ) + g21 ∑ Y rst (αr† αs† ᾱ t† ℙ̄ + ᾱ r† ᾱ s† αt† ℙ) r

r,s,t

(6.111)

Now, as pointed out already in [160], all cubic self-interactions for arbitrary spin s can be obtained by simply taking the sth power of the last term in (6.111). More than that, taking arbitrary powers of Δcubic′ all interaction terms belonging to the classification cases (ii) and (iv), can be obtained.9 This raises the question of whether the vertex should be written as an exponential of Δcubic′ , in analogy with the cubic covariant vertex as reviewed above in Section 5.3, or as some other function f (Δcubic′ ). This question cannot be answered without going to the quartic order. See further comments at the end of the chapter, Section 6.6. 6.3.13 Comparision to the formalism of Metsaev and Ponomarev–Skvortsov Different formalisms have different merits, and in order to review the Metsaev quartic analysis, we will here outline how the present vertex operator formalism compares to the Metsaev formalism. We follow the elaboration of the Metsaev formalism done in [273] and write an order ν interaction as (the notation is somewhat informal, but hopefully clear enough) ∫ d 3ν pδ3ν (∑ pi )hλ1 ⋅⋅⋅λν (p1 , . . . , pν )Φλp11 ⋅ ⋅ ⋅ Φλpνν

(6.112)

where the interaction data is captured by the hλ1 ⋅⋅⋅λν functions of the light-front threeλ

momenta pi and helicities λi of the external fields Φpii . Note that the λi here take positive and negative integer values. Thus, for a certain spin |λ| ≥ 0, the helicities are ±|λ| and the two field components are Φ|λ| and Φ−|λ| . In the vertex operator formalism, for a higher-helicity field (containing all helicities) we have the Fock space expansion as in formula (6.45). Therefore, up to possible numerical factors, we have the correspondences Φ|λ| ↔ ϕ|λ| and Φ−|λ| ↔ ϕ̄ |λ| . The interaction Hamiltonian of order ν is written as H(ν−2) =

ν 1 1 ν ∫ ∏ γr dγr dpr d p̄ r Φ⟨r| V |1⟩⋅⋅⋅|ν⟩ = ∏ Φ⟨r| V |1⟩⋅⋅⋅|ν⟩ ν r=1 ν r=1

(6.113)

where, in the third expression, the momentum integrations ∫ γr dγr dpr d p̄ r are included in the Fock space inner products. The νth order vertex is 4−ν

V |1⟩⋅⋅⋅|ν⟩ = (gκ−1 )

̄ Δ |1⟩⋅⋅⋅|ν⟩ Γ−1 ν δ(∑γr )δ(∑pr )δ(∑pr ) r

r

= Δ |1⟩⋅⋅⋅|ν⟩ ⌀|1⋅⋅⋅ν⟩ 9 We did not choose to derive the vertex operator yielding the case (iii).

r

(6.114)

6.3 Computing the cubic interactions �

227

̄ where ⌀|1⋅⋅⋅ν⟩ is short for (gκ−1 )4−ν Γ−1 ν δ(∑r γr )δ(∑r pr )δ(∑r pr )|01⋅⋅⋅ν ⟩ and all non-trivial interaction data is maintained by the Δ’s. The order ν interaction is written in (6.113) in terms of the vertex operators (6.114). The essential part of V |1⟩⋅⋅⋅|ν⟩ , maintaining the interaction data, is the operator Δν , which generically take the form ̄ † ̄ = ℋνr1 ⋅⋅⋅rk s1 ⋅⋅⋅sl αr† ⋅ ⋅ ⋅ αr† ᾱ s† ⋅ ⋅ ⋅ ᾱ s† + c. c. Δν = ℋν (γ, ℙ, ℙ)A 1 k 1 l kl

(6.115)

r ⋅⋅⋅r s ⋅⋅⋅s where the ℋν1 k 1 l are functions of the transverse momentum variables ℙij and ℙ̄ ij and the γ. Coupling constants are included in ℋ as appropriate. Among the field labels r1 ≤ ri ≤ rk and s1 ≤ si ≤ sl , ranging over the set {1, 2, . . . , ν}, we denote by ki the number of occurrences of field label i among the ri indices, and likewise for li . Thus, we have, by definition, two ways to represent the ℋ functions r ⋅⋅⋅rk s1 ⋅⋅⋅sl † αr1

ℋν1

k

k

l

l

⋅ ⋅ ⋅ αr†k ᾱ s†1 ⋅ ⋅ ⋅ ᾱ s†l = ℋνk1 ⋅⋅⋅kν l1 ⋅⋅⋅lν (α1† ) 1 ⋅ ⋅ ⋅ (αν† ) ν (ᾱ 1† ) 1 ⋅ ⋅ ⋅ (ᾱ ν† ) ν

(6.116)

where then: k1 + k2 + ⋅ ⋅ ⋅ + kν = k and l1 + l2 + ⋅ ⋅ ⋅ + lν = l. Say that we want to extract a certain order ν interaction between external fields of helicities λ1 , . . . , λν , of which we can (by rearrangement) choose λ1 , . . . , λK to be positive and λK+1 , . . . , λν to be negative. Thus, we take λi = si for 1 ≤ i ≤ K and λi = −si for K + 1 ≤ i ≤ ν. In the product over bra-fields Φ⟨r| , we then pick out the particular combination 1 s s sK+1 ⟨0|ϕs1 ⋅ ⋅ ⋅ ϕsK ϕ̄ sK+1 ⋅ ⋅ ⋅ ϕ̄ sν ᾱ 11 ⋅ ⋅ ⋅ ᾱ KK αK+1 ⋅ ⋅ ⋅ ανsν √s1 ! ⋅ ⋅ ⋅ sν !

(6.117)

and compute its matrix element (6.113). This yields s1 ! ⋅ ⋅ ⋅ sν ! s1 ⋅⋅⋅sK sK+1 ⋅⋅⋅sν ℋ ϕs1 ⋅ ⋅ ⋅ ϕsK ϕ̄ sK+1 ⋅ ⋅ ⋅ ϕ̄ sν √s1 ! ⋅ ⋅ ⋅ sν ! ν = √s1 ! ⋅ ⋅ ⋅ sν ! ℋνs1 ⋅⋅⋅sK sK+1 ⋅⋅⋅sν Φs1 ⋅ ⋅ ⋅ ΦsK Φ−sK+1 ⋅ ⋅ ⋅ Φ−sν = √|λ1 !| ⋅ ⋅ ⋅ |λν !| ℋνλ1 ⋅⋅⋅λK ,−λK+1 ⋅⋅⋅,−λν Φλ1 ⋅ ⋅ ⋅ ΦλK ΦλK+1 ⋅ ⋅ ⋅ Φλν

(6.118)

We see that modulo ordering of the external fields and combinatorial factors the funcλ ⋅⋅⋅λ ,λ ...,λ tions hλ1 ⋅⋅⋅λν and ℋν1 K K+1 ν are essentially the same. Thus, the term-by-term comparison to Metsaev’s notation turn out to be fairly direct. For positive helicity, we have the identification λi = ki and for negative helicity λi = −li , the numbers ki and li being nonnegative. It is convenient to write in general ki − li = λi .The vertex operator formalism is capable to describe fields with mixed excitations with lower helicity (not subject to light-front tracelessness), and the vertex couples such fields also. The helicity of component bra-fields is, in that case, given by the difference between the power of unbarred oscillators α and the power of barred oscillators α.̄

228 � 6 Light-front interactions The explicit form of the cubic interactions become in this formalism [273] hλ1 ⋅⋅⋅λν = C λ1 ,λ2 ,λ3

λ +λ +λ ℙ̄ 1 2 3 λ

λ

λ

γ1 1 γ2 2 γ33

ℙ−λ1 −λ2 −λ3 + C̄ −λ1 ,−λ2 ,−λ3 −λ −λ −λ γ1 1 γ2 2 γ3 3

(6.119)

This covers all cases (i)–(iv). The complete cubic interaction Lagrangian is then given by the (6.112) with (6.119) inserted and summed over all helicities (λ1 , λ2 , λ3 ). Returning to the question of Section 6.3.12 whether the cubic interactions can be written in terms of some generating function, it depends on the coefficients C and cannot be decided at the cubic level of analysis.

6.4 Some deeper properties of the Poincaré algebra In the derivation of the cubic interactions, we have seen that the detailed dynamic calculations did not seem to impose any stricter requirements on the cubics than those following from the kinematics. This fact may create the impression that the cubics are in some sense trivial, an impression that may be strengthened by the often repeated statement that the cubics really do not probe the full underlying gauge algebra. Behind this statement there is, however, the unstated point of view that a covariant gauge invariant formulation is more fundamental than the non-covariant physical formulation. More true to the context would be to note that the cubic Poincaré algebra does not probe the full nonlinear Poincaré algebra. There is also a very close similarity between the form of the three-particle amplitudes for massless fields, expressed in the spinor helicity language, and the light-front cubic Hamiltonian interactions, as noted by S. Ananth in [254]. See also reference [280]. The cubic massless three-particle amplitudes can be derived using a technique called little group scaling.10 This should ring a bell, because we know the important role played by the little group in the Wigner classification11 of the unitary irreducible representations of the Poincaré group. It should therefore not be unexpected that the little group plays an important role also for the interactions. The question is: in what way? In order to understand this, we have to look deeper into the structure of the Poincaré group as it manifests itself on the light-front. Much of this section is inspired by a 1978 paper by H. Leutwyler and J. Stern [281] that stresses the fact that the little group is in fact the “core” of the Poincaré group. That paper concerns itself with massive representations and is partly placed within the research program of direct particle interaction theory and phenomenology that we reviewed above in Section 2.2, but we will try to transfer the main ideas to field theory. 10 See the textbook [134] both for little group scaling and a well-written overview of the modern amplitude techniques in scattering theory. 11 See Section 2.3 in our Volume 1.

6.4 Some deeper properties of the Poincaré algebra

� 229

6.4.1 “Goodness” split of the Poincaré algebra The light-front Poincaré algebra may be split in a few, interesting and related ways. One is the split into kinematical and dynamical generators 𝒦 = {p = γ, p, p}̄ ∪ {j, j +

,j ,j }

+− + + ̄

𝒟 = {p = h} ∪ {j , j } − − ̄

−

(6.120a) (6.120b)

that we have employed above in the derivation of the cubic interactions. The commutators between j+− and the rest of the Poincaré generators offer a way to split the algebra into three subalgebras. Any operator A that satisfies [j+− , A] = igA

(6.121)

is said to have goodness g. Then referring back to the list of commutators in Section 6.3.2 in our Volume 1, we can read of the following goodness classification12 of the Poincaré generators: 𝒢+ = {γ, j , j ̄ }

with g = +1

(6.122)

+− 𝒢0 = {p, p,̄ j, j } with g = 0 − − 𝒢− = {h, j , j ̄ } with g = −1

(6.123)

+ +

(6.124)

These are three subalgebras of the Poincaré algebra, and 𝒢+ and 𝒢− are Abelian. We also recognize the set of kinematical generators as 𝒦 = 𝒢+ ∪ 𝒢0 and the dynamical generators as 𝒢− , the Hamiltonians in Dirac’s terminology. Again, looking back at the commutators, it is clear that the algebra of kinematical generators is a semi-direct product of the algebra of goodness 0 generators with those of goodness +1, i. e., 𝒦 = 𝒢+ ⋊ 𝒢0

(6.125)

Indeed, since [𝒢0 , 𝒢+ ] ⊂ 𝒢+ , we see that 𝒢+ is an invariant subalgebra of 𝒦 (but not of the full Poincaré algebra).13 The goodness split is actually also a triangular decomposition of the light-front Poincaré algebra. Therefore, we have [𝒢0 , 𝒢± ] ⊆ 𝒢±

[𝒢+ , 𝒢− ] ⊆ 𝒢0

(6.126)

12 The terminology of “goodness” derives from the infinite momentum approach to quantum field theory. 13 Compare how the translations form an invariant Abelian subalgebra of the full Poincaré algebra, leading to the semidirect product structure of translations with Lorentz transformations.

230 � 6 Light-front interactions 6.4.2 The little group and the Hamiltonians Since 𝒦 leaves the null-plane x + = 0 invariant, it generates the stability group of the null-plane. Its commutators with the three Hamiltonians are given by the formulas (𝒦𝒟.1)–(𝒦𝒟.11) in Section 6.3.1 above. The Hamiltonians are closely related to the little group of the Poincaré group. This might be suspected since j− and j−̄ together with j are the only generators that involve spin angular momentum. However, the little group does not coincide with the dynamical generators because j belongs to the kinematical generators, while it is needed to close the little group. This we will now look deeper into. The free field theory Hamiltonians h, j− and j−̄ , that we have used up until now are particular to the massless, noncontinuous spin, representations of the Poincaré group. The general form for these generators, appropriate for massive, massless and continuous spin representations are given by pp̄ + m2 γ 𝜕 i − j = xh + ip − (Mp − μη) 𝜕γ γ i 𝜕 ̄ + ip̄ + (M p̄ − μη)̄ j−̄ = xh 𝜕γ γ h=

(6.127) (6.128) (6.129)

Here, M, η and η̄ are operators—to be related to the massive and massless little groups— while we may take m and μ as parameters of mass dimension. The ordinary massless representation is then obtained by choosing m = 0 and μ = 0 with M the spin part of j, where, in all representations ̄ +M j = i(x p̄ − xp)

(6.130)

To understand the massive and continuous spin representations, we work out the algebra between j, j− and j−̄ . Regardless of representation, they have to satisfy the following commutators: [j, j− ] = j−

[j, j−̄ ] = −j−̄

[j− , j−̄ ] = 0

(6.131)

Working out the first commutator, not assuming anything about the algebra satisfied by M, η and η,̄ we can satisfy it by demanding [M, η] = η. Likewise, the second commutator can be satisfied be demanding [M, η]̄ = −η.̄ Then the third commutator demands μ2 [η, η]̄ − 2m2 M. For the massive representation, we have to choose m ≠ 0. The parameter μ can be absorbed into the generators η and η̄ if one wishes, but one can also put μ = √2m. Then the demand from the third commutator of (6.131) becomes [η, η]̄ − M = 0. Thus, in the

6.5 Short orientation on recent developments � 231

massive case, the part (6.131) of the Poincaré algebra can be satisfied if (and only if) the operators M, η and η̄ satisfy the following algebra: [M, η] = η

[M, η]̄ = −η̄ [η, η]̄ = M

(6.132)

This is precisely the Lie algebra of the massive little group so(3). With the standard oscillator realization M = α+ ᾱ − ᾱ + α that we have used in the massless case, the other two generators are given by η = α+ α and η̄ = ᾱ + α.̄ Finally, we come to the continuous spin representations. Then we have m = 0 but μ ≠ 0. The first two commutators of (6.131) still require [M, η] = η and [M, η]̄ = −η̄ while the third commutator requires μ2 [η, η]̄ = 0. Thus, we get the Lie algebra iso(2) [M, η] = η

[M, η]̄ = −η̄

[η, η]̄ = 0,

(6.133)

which is the massless little group algebra of translations and rotations in a twodimensional plane. Ordinarily, one realizes η and η̄ trivially as being zero, resulting in the standard massless higher spin case. For continuous spin, one may take M = α+ ᾱ − ᾱ + α as usual, and η = α + α+

η̄ = ᾱ + ᾱ +

(6.134)

It is clear that the Lorentz algebra will transform between states of different spin.

6.5 Short orientation on recent developments As mentioned in the Introduction to the present chapter, the last couple of years have seen a quite dramatic renaissance for the light-front formulation of massless higher spin theory. There are some reasons for this, one of them being the realization—long due— that there existed a closed, local, massless higher spin theory with interactions, including gravitational. Another reason was the long dominance of the Vasiliev AdS theory, that after having attracted new researches, turned out to have its own problems, for instance, with locality. Add to this the inherent complexity of the theory with its multiplicity of infinite sets of auxiliary fields. Furthermore, interest had gradually started to shift to the covariant Fronsdal theory in Minkowski space-time. As we have noted in several places, AdS space-time is not necessary for higher spin gauge theory. To start the story of the light-front “enlightenment”, we have to go back to the papers by R. Metsaev on quartic interactions. 6.5.1 The Metseav quartic analysis A next step after finding the cubic interactions would be to move on to the question of quartic interactions. From our general analysis of the dynamical part of the Poincaré al-

232 � 6 Light-front interactions gebra in Section 6.2.2, we certainly have to expect the need for higher-order interactions to close the algebra. The first worker to publish substantive results on the quartic level of light-front massless higher spin theory was R. R. Metsaev with two papers [271, 272] in 1991.14 Before entering into details, let us briefly comment on an issue that has been discussed in the higher spin literature, namely the question about what function classes interactions may belong to (see [282] for a discussion and further references). This is closely related to the question of locality. For the known cubic interactions, it is clear that the interactions are given by polynomials (positive integer powers) in the transverse momentum variables ℙ and ℙ̄ times rational functions of the + direction momentum γ. If these restrictions can be sustained in higher orders of interaction is not yet known. What is clear is that already at the quartic level the (current exchange) interactions involve infinite sums of positive powers of transverse momenta [271, 272, 273]. This can be seen from the quartic level equation of (6.31). Commuting two cubic level transformations (on the right-hand side) to get the source for the quartic level (on the left-hand side), it is clear that one gets contributions with an infinite number of momentum factors. This can be interpreted as some sort of weak nonlocality, although each term in the expansion is local, as pointed out in [273]. The crucial question is whether the quartic level equation can be solved without bringing in inverse powers of transverse momenta coming from inverting ℙ ℙ̄ in the free level transformation. We will discuss this issue below in Section 6.5.2. As for the Metsaev papers, the first one concerns a quartic analysis for fields of even spin. The paper is a bit sketchy, providing few details. The paper does not state it explicitly, but it can be read out that the light-front Poincaré algebra can be closed at the cubic level, without quartic-order interactions, if the coefficients in (6.119) are chosen in a particular way C

λ1 ,λ2 ,λ3

λ +λ2 +λ3 −1

=

lp1

Γ(λ1 + λ2 + λ3 )

(6.135)

with lp a constant of dimension length. The important piece is the factorial (λ1 +λ2 +λ3 −1)! in the denominator. Cubic closure demands that one drops the second term in (6.119), effectively making the Hamiltonian non-Hermitian. This makes the quantum theory nonunitary. It also makes the theory chiral. The paper also writes a formal solution to the quartic deformation equation. We will explain it in some detail below. The second paper treats also odd spin fields by introducing internal indices. It provides some more details on the quartic computations, and contains a discussion of whether the quartic nonlocality can be avoided, or not. Rather than reviewing the paper,

14 These papers were preceded by letter with E. S. Fradkin treating cubic interactions in arbitrary dimensions.

6.5 Short orientation on recent developments

� 233

we will rephrase the discussion with the help of the “renaissance” paper of Ponomarev and Skvortsov that elicited the cubic closure of the chiral theory.

6.5.2 The Ponomarev–Skvortsov elaboration The Ponomarev–Skvortsov paper [273] reviews and reworks the old millennium lightfront theory and provides useful details of computation. It contains a derivation (quite tricky and referred to an Appendix) of the particular form of the coupling factors (6.135) that yields the cubic chiral theory. The discussion of the quartic deformation equation is made more explicit. Here we will focus on some general considerations regarding the quartic level. We will not embark upon detailed computations at the quartic level, but just record a few formal aspects of the analysis. Referring back to the order by order expansion of the Poincaré algebra in (6.31), we must analyze ([δA(0) , δB(2) ] + [δA(2) , δB(0) ])|Φχ ⟩ + [δA(1) , δB(1) ]|Φχ ⟩ = 0

(6.136)

for A and B chosen among H, J − and J −̄ corresponding to the commutators [H, J − ], [H, J −̄ ] and [J − , J −̄ ] expanded to the quartic level. First, and this is true to any order, if [H, J − ] = 0 has been verified, then [H, J −̄ ] = 0 also holds due to complex conjugation. Second, the commutator [J − , J −̄ ] = 0 yields no new restrictions, but follows from [H, J − ] = 0 and [H, J −̄ ] = 0 as was noted in the original work on cubic interactions. The result holds in general as noted in [273]. A proof runs as follows, starting at the cubic level. We want to prove [J2− , J3−̄ ] + [J3− , J2−̄ ] = 0

(6.137)

given that [H2 , J3− ] + [H3 , J2− ] = 0

and

[H2 , J3−̄ ] + [H3 , J2−̄ ] = 0

(6.138)

hold as well as the free level commutators. One then notes that H2 is an algebraic operator and, therefore, invertible of the mass-shell. We may then equivalently try to prove [H2 , [J2− , J3−̄ ]] + [H2 , [J3− , J2−̄ ]] = 0. Using the first the Jacobi indentity, then (6.138) and then the Jacobi identity again, we get [H2 , [J2− , J3−̄ ]] + [H2 , [J3− , J2−̄ ]] = −[J − , [J −̄ , H2 ]] − [J −̄ , [H2 , J − ]] − [J − , [J −̄ , H2 ]] − [J −̄ , [H2 , J − ]] 2

3

3

2

3

2

= −[J2− , [H3 , J2−̄ ]] + [J2−̄ , [H3 , J2− ]] = [H3 , [J −̄ , J − ]] + [J −̄ , [J − , H3 ]] + [J −̄ , [H3 , J − ]] = 0 2

2

2

2

2

2

2

3

(6.139)

234 � 6 Light-front interactions The proof at the quartic level proceeds in a similar way, starting with rewriting [H2 , [J2− , J4−̄ ]] + [H2 , [J4− , J2−̄ ]] + [H2 , [J3− , J3−̄ ]]

(6.140)

by judiciously using the Jacobi identity and already proved results at the cubic and free level, to show that it computes to zero. It is then clear that the result can be proved to arbitrary order by induction. Thus, the commutator [J − J −̄ ] need not be computed, unless as an explicit check on the calculations. This fact is a manifestation of the properties of the nonlinear Poincaré algebra. Next, we come to a somewhat more subtle observation relating to the possibility of solving the quartic deformation equation. It was first noted implicitly by Metsaev in [271], more explicitly in [272] and then elaborated in [273]. Let us first record the computational form of the quartic deformation equation 4

4

4

r=1

r=1

i

∑ jr− V|1⋅⋅⋅4⟩ − ∑ hr (∑ ai xi V|1⋅⋅⋅4⟩ + S|1⋅⋅⋅4⟩ ) = SC

(6.141)

where SC stands for the relevant source commutator built from the cubic vertices. One may now, once again, make the observation that h(4) = ∑4r=1 hr can be viewed as an algebraic, multiplicative operator, and that it has no further operational effect on what it acts on unless that which it acts on contains coordinates. Off the mass-shell, it is therefore invertible. Based on this observation, one may rearrange equation (6.141) into 4

4

r=1

i

h(4) S|1⋅⋅⋅4⟩ = ( ∑ jr− − h(4) ∑ ai xi )V|1⋅⋅⋅4⟩ − SC

(6.142)

Then, using the fact that the ai coefficients are equal to 1/4 and that the xi acts as derivatives i𝜕/𝜕p̄ i on the vertex, we can write 4 i 4 𝜕 )V|1⋅⋅⋅4⟩ − SC h(4) S|1⋅⋅⋅4⟩ = ( ∑ jr− − h(4) ∑ 4 r 𝜕p̄ r r=1

(6.143)

The authors of [273] introduce a new notation j̃r− for the operator acting on the vertex V|1⋅⋅⋅4⟩ . Following this practice, we would define i 𝜕 j̃r− = jr− − h(4) 4 𝜕p̄ r

(6.144)

So, that one can write 4

h(4) S|1⋅⋅⋅4⟩ = ∑ j̃r− V|1⋅⋅⋅4⟩ − SC r=1

(6.145)

6.5 Short orientation on recent developments

� 235

Then the authors of [273] argue that the equation can always be solved for S|1⋅⋅⋅4⟩ formally, no matter what contributions there are to the right-hand side, by dividing by h(4) . The operator (h(4) )−1 is however singular when all particles go on-shell. This singularity can only be avoided by requiring the right-hand side of (6.145) is proportional to h(4) . That is, we have to solve 󵄨󵄨 4 󵄨󵄨 =0 ( ∑ j̃r− V|1⋅⋅⋅4⟩ − SC)󵄨󵄨󵄨 󵄨󵄨 (4) r=1 󵄨h =0

(6.146)

for the vertex V|1⋅⋅⋅4⟩ . To the best of my knowledge, as of March 2023, it is still unknown whether there is any chance of finding a local solution in any way, or if the nonlocality is unavoidable. 6.5.3 The cubic chiral theory The cubic chiral theory that emerges out of the deliberations above has come to be known under the name chiral higher spin gravity. The theory has been researched along two directions: quantum properties and covariantization. Its cubic interaction is (see formulas (6.112), (6.119) and (6.135)) worth repeating 3ν

3ν

∑ ∫ d pδ (∑ pi )

λ1 ,λ2 ,λ3

λ +λ2 +λ3 −1

lp1

λ +λ +λ ℙ̄ 1 2 3

Γ(λ1 + λ2 + λ3 ) γλ1 γλ2 γλ3 1

2

3

Tr[Φλp11 Φλp22 Φλp33 ]

(6.147)

where the trace allows for nontrivial matrix-valued odd spin fields.15 Quantum properties As for quantum properties, there is a series of papers investigating the theory. The first one is a letter [283]. It is a bit short on details, but it reports the following three results: First, due to the particular form of the three-point couplings, all on-shell tree amplitudes vanish, and the S-matrix is 1. As the authors point out, this makes the theory consistent with the no-go theorems at least at the tree level. Remember that the Weinberg low energy theorem forbids massless higher spin fields to have long-range effects, while the Coleman–Mandula theorem forbids the S-matrix from having symmetry generators that transforms as tensors under the Lorentz group. Second, all vacuum diagrams (diagrams without external states) vanish. For the oneloop “bubble” diagram, this hinges on the need to regularize the total number ν of degrees of freedom. This sum is infinite, but may be regularized to zero [284], 15 The provenance of this simple higher spin theory is the initial discovery in the 1980s of the cubic couplings themselves [274, 160], the quartic analysis in the early 1990s that produced the particular numerical coupling constants [271, 272] and the mid-2010s renovation [273] that elicited the picture.

236 � 6 Light-front interactions ∞

ν = ∑ = 1 + 2 ∑ 1 = 1 + 2ζ (0) = 0 λ

s=1

(6.148)

where ζ (x) is the Riemann zeta-function. Using the value ζ (0) = −1/2, one may assign −x a value to the divergent sum 1 + 1 + 1 + ⋅ ⋅ ⋅ i. e., the naive value at 0 of ζ (x) = ∑∞ n=1 n . Such zeta-function regularization is not uncommon in theories with infinite number of states.16 According to the paper, all other vacuum diagrams vanish without the need to regularize, instead relying on the particular form of the coupling factors. Third, the paper argues that all loop diagrams with external legs vanish given that the total number of degrees of freedom is regularized to zero. The second paper [286] and third paper [287] provide much more detail and strengthens and extends the result of the first paper. It thus seems that the cubic chiral theory provides us with a UV-finite higher spin theory with a trivial S-matrix consistent with the Minkowski no-go theorems. All the same, there are actually interactions, gravitational as well, albeit cubic and nonunitary. Whether this theory will eventually turn out to be a subsector of a full, unitary and perhaps local, higher spin gravity theory, or if it will remain on its own—perhaps serving some yet unknown purpose in fundamental physics—is an open question. Covariantization There is ongoing work to covariantize the cubic chiral theory [288, 289, 290]. It should not be expected that this can be done using ordinary Lorentz tensors for the covariant higher spin gauge fields, as the no-go results are numerous and well established. Instead, the procedure is based on twistor theory, or rather, on two-component spinor language as introduced in the book [291] and developed by several authors (for references, see the papers cited above). It would take us too far to enter into this fascinating subject, suffice it to note a crucial basic idea without writing any formulas. As was discussed in great detail a long time by Weinberg [127] (see our Volume 1, Sections 2.6 and 3.5), there are several ways to describe massless arbitrary spin particle in field theory, corresponding to which representation of the Lorentz group one employs. A simple example is spin 1, which may be described by the gauge invariant field strength Fμν , or by the the gauge potential Aμ . In practice, both are used and they are related in the well-known way. However, one may imagine putting away this relationship (for a while) and see how far one gets by either picture. More to the point, one may imagine describing the two helicities in different ways: one helicity by the field strength, the other with the gauge potential. This is actually quite easy to do in the two-component spinor formulation, and has been employed in the twistor approach to field theory. Thus, one may say

16 It has been used in string theory as well [33] where the critical dimension was computed by using the value ζ (−1) = −1/12 for the infinite sum 1 + 2 + 3 + ⋅ ⋅ ⋅ . For the zeta-function, see for instance, Section 9.5 in [285].

6.6 Chapters 5 and 6 epilogue and a research question

� 237

that in covariantizing the light-front cubic chiral theory, a way forward would be to retain physical fields (field strengths) for half the helicities, while introducing gauge fields (potentials) for the other half of the helicities. Indeed, in this way the theory becomes chiral. The actual implementation of these ideas runs parallel to self-dual Yang–Mills and self-dual gravity theories. However, at the time of writing, it seems that one eventually has to resort to equations of motion and the formalism of free differential algebras, not unlike the formalism of the Vasiliev theory. Thus, again we encounter infinite sets of auxiliary fields, albeit with locality under control. Perhaps not unexpected, certainly interesting, but—at least in the opinion of the present author—running away from the initial simplicity of the light-front theory. This research is evolving rapidly.

6.5.4 Short guide to higher dimensions and supersymmetry For the reader who may be interested in higher dimensions and/or supersymmetry, there is a wealth of papers by Metsaev to study. For totally symmetric massless fields in arbitrary dimensions, the reader may consult the already mentioned [292] as well as [293]. Mixed symmetry fields were treated in [294]. Two comprehensive papers treating various aspects of cubic interactions in dimensions four and higher for massless as well as massive, bosonic as well as fermionic fields, are [295, 296]. These papers also contain classification results, often referred to in the literature. Furthermore, Metsaev have also studied light-cone cubic interactions in AdS space-time. Here, we only cite [297].

6.6 Chapters 5 and 6 epilogue and a research question It is now clear that we have three different approaches—the light-front vertex operator approach discussed in the present chapter, the covariant BRST approach of Section 5.2 and the covariant MKR generating function approach of Section 5.3, that all point to the same overall structure for the cubic higher spin gauge interactions in Minkowski spacetime. All these three approaches derive a Δ-operator of the generic form Δ ∼ Yrs αr ps + Yrstu αr αs αt pu

(6.149)

with appropriate adoption to the formalism at hand (with summations and contractions of indices and all). Δ itself, for sure, yields the correct Yang–Mills interaction. Higher powers produce higher spin interactions of some sort. For instance, Δ2 produce interactions that look like gravitation and Δ3 produce interactions, the terms of which are of the same form as in the BBvD spin 3 interaction. But, as far as I am aware of, the exact term-by-term comparison, has not been done, or at least not been published.

238 � 6 Light-front interactions It is certainly plausible that Δs produce the correct spin s cubic self-interaction, and one may argue that any other result would be very strange. Indeed, on the light-front, it is certainly the case that the higher spin cubic interactions factor in this way. It is inherent in the light-front approach, so within the Dirac program it is a proven fact. Still, within the Fronsdal program, it seems to me that the question needs further clarification. The problem can be made concrete as follows. It is clear that any spin-3 cubic selfinteraction whatsoever can be reproduced by operator expressions of the generic form ΣY(9,3) α9 p3 , but can it be factored into a cube of the Yang–Mills operator, i. e., can it be factored (ΣY(3,1) α3 p)3 ? The question is related to the exp(Δ) question and to the strong obstruction order g 2 result and to the existence of the cubic chiral theory. That is, can the covariant theory be cut short at the cubic level, too? Probably not, but the question deserves attention.

7 Higher spin in AdS and the Vasiliev theory The study of higher spin fields in anti-de Sitter space was started by C. Fronsdal in a long series of papers “Elementary Particles in a Curved Space” from 1965 to 1978 [298, 299, 300, 301, 302, 303]. The last paper in the series (with M. Flato) contained the very interesting group theoretic proof that the product of two singleton representations equals an infinite sum of higher spin representations. In retrospect, this result can be seen as the kinematic—free field theory—basis for the simplest higher spin AdS/CFT correspondence. There is a substantial follow-up literature to the last paper [303] in the series, mainly by Fronsdal and Flato, trying to exploit the result dynamically. It seems that the relevance of AdS space-time for higher spin gauge theory was suggested by E. S. Fradkin while working with M. A. Vasiliev on a supergravity paper [304]. They saw how the Aragone–Deser no-go result on minimal gravitational interactions of higher spin fields in Minkowski space-time (see our Section 2.3), could be circumvented by going to AdS space-time. Incidentally, the no-go result can also be circumvented in the light-front approach to higher spin interactions as reviewed above in Section 6.1.1.1 This fact was implicitly known—but not explicitly noted—from the 1987 light-front analysis in [160]. In any way, the perceived special relevance of AdS background was a strong impetus for the actual development of higher spin theory from the late 1980s up to quite recent times.

7.1 The geometry and algebra of anti-de Sitter space-time Anti-de Sitter space-time (AdS for short) is one of three types of constant curvature, maximally symmetric space-times. The other two types are de Sitter space-time (dS for short) and Minkowski space-time. Minkowski space-time has zero curvature, while AdS and dS are distinguished by their positive or negative constant curvatures. The sign of the curvature assignment is conventional. In all cases, the curvature is proportional to a cosmological constant in Einstein’s equations and corresponds to vacuum solutions. Maximally symmetric space-times Maximally symmetric spaces are characterized by a curvature constant K and the numbers of eigenvalues of the metric that are positive (or negative). Given two spaces, that are same in these respects, there is a coordinate transformation that carries the metric of one space to the other and vice versa. This gives the possibility 1 For further comments on this, see Chapter 8. Recently, the light-front interactions have been covariantized using twistor techniques; see further comments in Section 6.5.3 in the previous chapter. Thus, the problems with gravitational interactions in Minkowski space-time have more to do with the choice of fields, rather than with Minkowski space-time itself. Fronsdal fields are obviously not a good choice. https://doi.org/10.1515/9783110675528-007

240 � 7 Higher spin in AdS and the Vasiliev theory

to construct symmetric spaces with coordinates—and metric—that are convenient to use in particular circumstances, without losing generality. Maximally symmetric spaces are special cases of spaces with form invariant metrics: metrics whose functional dependence on the coordinates are the same under certain coordinate transformations, the isometries. As the name suggests, maximally symmetric spaces have the maximum number of isometries possible in D dimensions, namely D(D + 1)/2 of which D can be interpreted as the homogeneity of the space, and D(D − 1)/2 to the isotropy of the space. A space being maximally symmetric is therefore equivalent to it being homogeneous and isotropic [1]. In general, we can consider a flat space of p + q dimensions with the metric ημν dx μ dx ν = dx12 + dx22 + ⋅ ⋅ ⋅ + dxp2 − dt12 − dt22 − ⋅ ⋅ ⋅ − dtq2 = −dτ 2

(7.1)

A maximally symmetric space of dimension p + q − 1 can be obtained by the isometric embedding x12 + x22 + ⋅ ⋅ ⋅ + xp2 − t12 − t22 − ⋅ ⋅ ⋅ − tq2 = ±R2

(7.2)

with a choice of sign on the right-hand side. This sign can, however, be traded for an interchange of “space” and “time” coordinates. In any case, the isometry group is SO(p, q). Focusing on four-dimensional space-time, i. e., on three-dimensional space, we have two interesting cases, namely SO(3, 2) and SO(4, 1).

7.1.1 Construction of de Sitter and anti-de Sitter space-times With this background, let us turn to the concrete cases of de Sitter and anti-de Sitter space-times. Consider a D + 1 dimensional space with a flat metric given by − dτ 2 = ημν dx μ dx ν + ζdz2

(7.3)

where ημν is the Minkowski metric in D dimensions and ζ = ±1 parametrizes dS (adding a space coordinate: ζ = +1) and AdS (adding a time coordinate: ζ = −1) ambient spacetime. The embedding of dS or AdS is done by restricting the coordinates to either of the hyperbolical surfaces dS: AdS:

ημν x μ x ν + z2 = R2

ημν x μ x ν − z2 = −R2

(7.4) (7.5)

Both cases can be summarized in the generic equation ημν x μ x ν + ζz2 = ρR2

(7.6)

where ρ = ±1. The physically interesting cases (dS or AdS) are, however, ζ = ρ = ±1. Introduce also the constant K, where K −1 = ρR2 , to be interpreted as the curvature of the embedded space-time.

7.1 The geometry and algebra of anti-de Sitter space-time

� 241

Differentiating the embedding condition (7.6) to express dz, yields the metric for the embedded surface gμν = ημν +

ημρ x ρ ηνσ x σ

K −1 − ηρσ x ρ x σ

(7.7)

Note that the parameter ζ drops out of the computation of the metric. From an intrinsic D-dimensional perspective, this is as it should be. The embedding is nothing fundamental. Computing the inverse metric, the affine connection and the curvature tensor yield g μν = ημν − Kx μ x ν

(7.8)

Γλμν = Kx λ gμν

(7.9)

Rμνρσ = K(gμσ gνρ − gμρ gνσ )

(7.10)

Clearly, K = 0 gives D-dimensional Minkowski space-time back. Both the metric in D + 1 dimensions and the embedding equation are invariant under SO(D, 1) rotations (dSD ) or SO(D − 1, 2) rotations (AdSD ). The infinitesimal isometries of dSD are therefore generated by the rotation Lie algebra so(D, 1), while for AdSD the Lie algebra is so(D − 1, 2). From the Einstein field equations in vacuum, 1 Rμν − gμν R = Λgμν 2

(7.11)

with a cosmological constant Λ, follows Λ = 21 (D − 2)(D − 1)K. Which is which and what to choose? It is a matter of convention which of the two spaces dS or AdS should correspond to positive or negative curvature. In Riemann geometry, where the metric is positive definite, the convention is unanimous: a positive curvature means that geodesics converge, and vice versa. In dS, space-like geodesics converge while timelike diverge. In AdS, it is the other way around. Here, we focus on the space coordinates, and deem negative curvature to correspond to AdS space-time. This seems to be the most common choice of convention in theoretical particle physics contexts. What is not conventional, however, is the association of isometry groups. The AdS isometry group is isomorphic to SO(3, 2) and the dS isometry group is isomorphic to SO(4, 1). This leads to a curious naming conundrum. In the early days, when C. Fronsdal started to investigate these groups for particle physics application, he had the designations dS and AdS interchanged as compared to what is now common. In the Introduction to Essays on Supersymmetry [305], Fronsdal writes: It is crucial to choose the good de Sitter group and the good de Sitter space. The 4 + 1 de Sitter group was favored at first, but it can be excluded because it has no unitary representations with positive energy, and hence no particles. This leaves the 3 + 2 de Sitter group (henceforth: the de Sitter group).

242 � 7 Higher spin in AdS and the Vasiliev theory

This is also the choice made in the first paper [298] of the series of papers “Elementary Particles in a Curved Space” mentioned above. The name “anti-de Sitter” space for the (3, 2) case seems to have originated in the book [306] by Hawking and Ellis from 1973. The now dominating use of this designation came with the advent of gauged extended supergravity theories in the late 1970s. The Fronsdal convention did however live on into the late 1980s, with Vasiliev switching sides (from de Sitter to anti-de Sitter) in the middle of his development of the higher spin theory. For further comments, see Section 8.1. Leaving such trivia aside, a more important conundrum is the real physics. A positive energy spectrum is of course of greatest importance in quantum field theory, and the much researched AdS/CFT dualities are highly interesting, but as far as observations now tell us: the cosmological constant is positive rather than negative. The relation between the sign of the cosmological constant—the sign on the physical vacuum energy—and the type of symmetric space-time, is of course not a matter of convention. Observations seem to favor de Sitter space.

7.1.2 Symmetries of anti-de Sitter space-time The flat five-dimensional ambient space-time with metric ημν with signature (−, +, +, +, −) (where we now let the indices run over 0, 1, 2, 3, 5) is invariant under the Lie algebra so(3, 2) of infinitesimal rotations with generators Jμν obeying the commutation relations [Jμν , Jρσ ] = i(ημρ Jνσ − ηνρ Jμσ − ημσ Jνρ + ηνσ Jμρ )

(7.12)

A concrete representation as angular momentum operators is Jμν = −i(xμ 𝜕ν −xν 𝜕μ )

(7.13)

The embedding equation (7.5) is also rotationally invariant and the ten generators Jμν span the isometry algebra of four-dimensional AdS space-time. To make contact with Minkowski geometry and the Poincaré symmetry, we split the generators into four-dimensional rotations and translations. For that purpose, pick a point in AdS5 and let the 5th axis go through that point. The generators Jμν with the indices now running over 0, 1, 2, 3 then generate rotations around that point while Jμ5 generate translations [298].2 Define the translation generators Pμ as Pμ = √|K|J5μ

(7.14)

In this context, the Pμ are sometimes called transvections. The algebra then takes the form (with indices running over 0, 1, 2, 3) 2 For intuition, think of a 2-sphere embedded in 3-dimensional Euclidean space and pick the North pole. The three Euclidean rotations become one rotation (around the pole) and two (perpendicular) translations.

7.1 The geometry and algebra of anti-de Sitter space-time

[Jμν , Jρσ ] = i(ημρ Jνσ − ηνρ Jμσ − ημσ Jνρ + ηνσ Jμρ ) [Jμν , Pρ ] = i(ημρ Pν − ηνρ Pμ )

[Pμ , Pν ] = iη55 |K|Jμν = −i|K|Jμν

� 243

(7.15a) (7.15b) (7.15c)

The first two commutators are the same as for the Poincaré algebra. The last one says that performing two translations in different order corresponds to a rotation. The Poincaré algebra is recovered in the limit of vanishing curvature |K| → 0.3 For later use, define λ = √|K| to be interpreted as an energy scale. The semisimplicity of this algebra—in contrast to the Poincaré algebra—can be seen from Jμν appearing on the right-hand side of (7.15c) so that all elements of the algebra can be written as commutators of elements. Incidentally, it is the last equation (7.15c) that distinguishes between dS and AdS. The algebra so(3, 2) is noncompact and, therefore, its unitary representations must be infinite-dimensional. This is most conveniently brought out by first rewriting the algebra in terms of spin and energy raising and lowering operators and by subsequently rewriting these operators in an oscillator basis. In so doing, much will be learned about the foundations of AdS higher spin theory.

7.1.3 Jordan split of the AdS algebra The so(3, 2) Lie algebra can be split into a Jordan structure g(−) ⊕g(0) ⊕g(+) where g(0) will be a maximal compact subalgebra and the generators in g(+) and g(−) will act as raising and lowering operators [308].4 Let us work out the details. Start by interpreting the so(3, 2) generators according to the following scheme in close analogy to the Poincaré algebra (see Section 3.4.2 of our Volume 1): Space rotations: J = (J1 , J2 , J3 ) ≡ (J23 , J31 , J12 ) Boosts:

K = (K1 , K2 , K3 ) ≡ (J01 , J02 , J03 )

Space translations: P = (P1 , P2 , P2 ) ≡ (λJ51 , λJ52 , λJ53 ) 0

Time translation: H = P ≡ −λJ50

(7.16a) (7.16b) (7.16c) (7.16d)

The rotations and boosts span the Lorentz subalgebra so(3, 1) and the theory from Section 3.4.2 of Volume 1 carries over. Indeed, the algebra is the same except that we now have the nonzero commutators between translation generators stemming from (7.15c). Let us record them 3 This is actually a Wigner–Inönü contraction [307]. 4 This decomposition should not be confused with the “triangular decomposition” with a similar notation. See Section 3.11.2 in our Volume 1.

244 � 7 Higher spin in AdS and the Vasiliev theory [Pi , Pj ] = −iλ2 Jij

(7.17)

[Pi , H] = −iλ Ki

(7.18)

2

Comparing to the Poincaré algebra commutators (3.109) and (3.110)5 in Volume 1, [Ki , Pj ] = iHδij

and

[Ki , H] = iPi

suggest combining the three-momenta and the boosts into raising and lowering operators according to Ji(±) = J5i + (±1)iJ0i =

1 P + (±1)iKi λ i

(7.19)

where the notation (±) is used to disambiguate between upper and lower occurrences of ±. We see that the Ji(±) are energy raising and lowering operators with commutators [H, Ji(±) ] = (±1)λJi(±)

(7.20)

Note also the important commutators [Ji(+) , Jj(−) ] = 2(H/λδij − iJij )

while [Ji(+) , Jj(+) ] = [Ji(−) , Jj(−) ] = 0

(7.21)

that governs how the raising operators in g(+) and lowering operators in g(−) commute to produce the operators of g(0) . Furthermore, from the commutator [Ji , Jj(±) ] = iϵijk Jk(±)

(7.22)

one can expect to be able to form spin z-component raising and lowering operators by a further linear recombination J±(±) = J1(±) ± iJ2(±)

(7.23)

It then follows that [J3 , J±(±) ] = ±J±(±)

[H, J±(±) ]

=

(±1)λJ±(±)

(7.24) (7.25)

where again the notation (±) marks which plus/minus belongs to which. The four generators J++ , J+− , J−+ and J−− therefore raise and lower energy (in units of λ) and third component of spin (in units of 1) according to these formulas.

5 There is unfortunately a sign mistake in this formula in Volume 1.

7.1 The geometry and algebra of anti-de Sitter space-time

� 245

So far, we have defined generators H, J3 and the set of raisers and lowers {J++ , J+− , J−+ , J−− } as well as their third components J3(±) (through (7.19)). The final two so(3) generators J± are defined as usual (compare (7.23)) J± = J1 ± iJ2

(7.26)

It can now be checked that the generators fall into a Jordan structure g(−) = {J−− , J+− , J3− }

g(+) = {J++ , J−+ , J3+ } g(0) = {H/λ, J3 , J+ , J− }

(7.27)

with g(0) a maximal compact subalgebra so(2) ⊕ so(3).

7.1.4 Outline of representations Representations—of the lowest weight type—of the AdS algebra can be built by applying the raising operators in g(+) to a vacuum state |(E0 , s)⟩ of energy E0 and spin s, satisfying Ji(−) |E0 , s⟩ = 0. These will be denoted by D(E0 , s) with E0 the lowest energy and s the highest value taken by one of the so(3) generators. To characterize the excited states fully, we need to supply quantum numbers E, j and m (with the standard interpretation as energy, spin and spin z-component) and write them as |(E0 , s)Ejm⟩. We will not work out this in full detail, but rather confine ourselves to a few heuristic comments.6 Thus, starting from the ground state, we have the excited states n n n 󵄨 (J1(+) ) 1 (J2(+) ) 2 (J3(+) ) 3 󵄨󵄨󵄨(E0 , s)⟩

(7.28)

Each raising operator increases the energy by one unit +1. The states are organized according to energy level, thus at level E0 + n we have n1 + n2 + n3 = n. The states also have various values of the so(3) eigenvalues j and m. Indeed, on general grounds, they must organize themselves into multiplets of so(3), each of multiplicity 2j + 1. Spin 0 For the simplest case of spin zero, on the first excited energy level E0 + 1, there are three states corresponding the action of Ji(+) with i = 1, 2, 3. This is an so(3) j = 1 multiplet. On the second excited level with E0 + 2, there are in total six excited states corresponding the action of Ji(+) Jj(+) with ij = 11, 12, 13, 22, 23, 33. This corresponds to so(3) multiplets with j = 0 and j = 2.

6 “Heuristic” in the meaning of help for a study of the details. For these details, one may turn to the Fronsdal series of papers referred to in the Introduction to this chapter, or to the very useful review [309]. See also [310]. The review [311] provides concrete details. Our state diagrams (really weight diagrams) in this section are inspired by that latter reference.

246 � 7 Higher spin in AdS and the Vasiliev theory On the third excited level (E0 + 3), there are in total ten excited states corresponding the action of Ji(+) Jj(+) Jk(+) with ijk = 111, 112, 113, 122, 123, 222, 223, 133, 233, 333. This corresponds to a j = 1 and a j = 3 multiplet. The combinatorics should now be clear, and this pattern goes on indefinitely for higher and higher energy levels. The state diagram can be illustrated as in Figure 7.1. E Total # of states E0 + 5 7 3 11 21 E0 + 4 1 5 15 9 E0 + 3 7 3 10 E0 + 2 1 5 6 E0 + 1 3 3 E0 1 1 0 1 2 3 4 5 6 j Figure 7.1: States of the s = 0 representation. Multiplicities of the states are shown, as well as the total number of states at each energy level. This number equals the number of components in expression (7.28).

Spin 1 For the case of spin 1, the ground state |(E0 , 1)⟩ has m-degeneracy (m = −j, . . . , j) 3. All the spin-1 total energy state m-degeneracies turn out to be 3 times the corresponding spin-0 total energy state m-degeneracies. To account for this in the spin-1 state diagram, some states turn out to be doubly occupied. The representation is thus not irreducible. See Figure 7.2. Spin 1/2 For the case of spin 1/2, the ground state |(E0 , 1/2)⟩ has m-degeneracy 2. All the spin-1/2 total energy state m-degeneracies turn out to be 2 times the corresponding spin-0 total energy state m-degeneracies. This can be accounted for without any doubly occupied states. Unitarity and masslessness To get a handle on the question of unitarity of the representations, we will compute the quadratic Casimir operator Ĉ 2 for the algebra 1 Ĉ 2 = J μν Jμν = H(H − 3) + (Ji )2 − Ji(+) Ji(−) 2

(7.29)

Applying this operator to the ground state, we get 󵄨 󵄨 󵄨 Ĉ 2 󵄨󵄨󵄨(E0 , s)⟩ = (E0 (E0 − 3) + s(s + 1))󵄨󵄨󵄨(E0 , s)⟩ ≡ C2 󵄨󵄨󵄨(E0 , s)⟩

(7.30)

7.1 The geometry and algebra of anti-de Sitter space-time

E

� 247

Total # of states

E0 + 5 1

3

E0 + 4

3, 3 5

E0 + 3 1

3

E0 + 2

3, 3 5

E0 + 1 1

3

E0

3 0

1

5, 5 7

9, 9 11

7, 7 9

5, 5 7

13

63 45

11

9

30

7

18

5

9 3

2

3

4

5

6

j

Figure 7.2: States of the s = 1 representation. Multiplicities of the states are shown, as well as the total number of states at each energy level. Some states are doubly occupied since the representation is not irreducible. The states with unfilled circles corresponds to a spin zero representation as can be seen by comparing to Figure 7.1.

where C2 denotes the value of the Casimir C2 = E0 (E0 − 3) + s(s + 1). This value must be the same for all states in the representation. For a representation with s ≥ 1, there must be a state with E = E0 +1 and j = s−1 (see Figure 7.2). Computing the matrix element ⟨(E0 + 1, s − 1)|Ĉ 2 |(E0 + 1, s − 1)⟩ and equating with the value of C2 , we get 󵄨󵄨 (−) 󵄨󵄨 󵄨2 󵄨󵄨Ji 󵄨󵄨(E0 + 1, s − 1)⟩󵄨󵄨󵄨 = 2(E0 − s − 1)

(7.31)

Since this cannot be negative, we get a unitarity bound E0 ≥ s + 1. The choice made here may seem arbitrary, so let us be more general. Perform the corresponding computation for a general excited state |(E, j)⟩. This yields 󵄨󵄨 (−) 󵄨󵄨 󵄨2 󵄨󵄨Ji 󵄨󵄨(E, j)⟩󵄨󵄨󵄨 = E(E − 3) + j(j + 1) − E0 (E0 − 3) − s(s + 1)

(7.32)

Clearly, to find the lowest nonzero bound on E0 , it is enough to check the lowest laying excited states. The state chosen above serves that purpose. It is now interesting to study the lower bound at E0 = s + 1. From (7.31), we see that the state |(E0 + 1, s − 1)⟩ = |(s + 2, s − 1)⟩ is itself a ground state. It produces its own spectra of excited states. It can be illustrated in the spin-1 case. Comparing the spin-0 and spin-1 state diagrams, we see that the states built on the |(3, 0)⟩ ground state may account for precisely the j = 0 states and the extra j ≥ 1 states marked by circles in the spin-1 diagram. These states form an invariant subspace of the spin-1 state diagram. It is, however, not invariantly complemented. Interpreting these

248 � 7 Higher spin in AdS and the Vasiliev theory states as gauge modes, and subtracting them, we end up with a state diagram for massless spin 1 in Figure 7.3. E E0 + 5 E0 + 4 E0 + 3 E0 + 2 E0 + 1 E0 = 2

5 5 5 5 5

3 3 3 3 3 3 0

1

2

7 7 7 7

3

Total # of states 11 13 48 35 11 24 15 8 3

9 9 9

4

5

6

j

Figure 7.3: States of the s = 1 massless representation. Multiplicities of the states are shown, as well as the total number of states at each energy level. The spin-0 states have been subtracted.

In general, representations D(s + 1, s) with E0 = s + 1 can be interpreted as massless (in the AdS context) and correspond to gauge invariant fields of arbitrary spin s ≥ 1. Two original references that develop the field theory are [312, 313]. In the box below, we will state the complete results on unitary representations of so(3, 2). Unitary representations of so(3, 2) In [313], we find the following proposition that collects information on the unitary representations.7 D(E0 , s) is unitary if and only if one of the following conditions holds: (i) s = 0, E0 ≥ 21 (ii) s = 21 , E0 ≥ 1 (iii) s ≥ 1, E0 ≥ s + 1 The spectrum of the energy H is in all cases E = E0 , E0 + 1, . . . . For each E, there is a corresponding spectrum of j determined by the conditions: s + j is an integer, j ≥ 0 and |j − s| ≤ E − E0 . For each energy value, there appears irreducible representations D(j) of so(3). In certain cases, the multiplicity of these representations is one. Namely, for: (i’) D(E0 , 0), E0 > 21 , j = E − E0 , E − E0 − 2, . . . , 1 or 0 (ii’) D(E0 , 21 ), E0 > 1, j − 21 = E − E0 , E − E0 − 1, . . . , 0 (iii’) D(s + 1, s), s ≥ 1, j = s, s + 1, . . . , E − 1 Among the low spin representations, we find the limiting cases D( 21 , 0) and D(1; 21 ), called Rac and Di single-

7 Remember that in order to fully characterize a state belonging to any representations, one must supply the Ejm “quantum numbers”.

7.1 The geometry and algebra of anti-de Sitter space-time

� 249

tons, respectively, with a reduced spectrum of j = E − 21 . This spectrum of states is too small to support fields in three space-dimensions.8 The singleton representations were discovered by Dirac in 1963 in [314]. These representations came to be intensely studied by Fronsdal and collaborators from the 1970s and onward. Much of this interest stemmed from the quite remarkable relation between the singleton representations and the massless representations, discovered by Flato and Fronsdal in 1978 in [303]. They showed that in representation theory, direct products of singleton representations gave rise to directs sums of massless representations. More precisely, the showed the following equalities: Di ⊗ Di = ⨁ D(s + 1, s) ⊕ D(2, 0) s=1,2,...

(7.33a)

Di ⊗ Rac = ⨁ D(s + 1, s)

(7.33b)

Rac ⊗ Rac = ⨁ D(s + 1, s)

(7.33c)

2s=1,3,...

s=0,1,2,...

Much effort was expended in finding a field theory realization of these group theoretical equalities, starting in the papers [312, 313]. We do not have space to enter into this very interesting topic here. The singletonmassless correspondence has been discussed by J. Engquist, E. Sezgin and P. Sundell in the context of the Vasiliev theory; see, for instance, [315, 316]. Given that singleton fields may be viewed as propagating on the conformal boundary of AdS space-time, a proper setting for the correspondence is certainly the AdS/CFT conjecture and bilocal field theory (see [90] for a short overview). A modern review of singleton physics can be found in [317].

We may illustrate the bounds given in the box above, for spin 0 and spin 1/2 by applying the condition given by formula (7.32). For the spin-0 representation D(E0 , 0), choosing the excited state |E0 + 1, 1⟩ yields E0 ≥ 0. But we are looking for a positive, nonzero, bound. That is provided by the state |E0 + 2, 0⟩, which yields E0 ≥ 1/2. No other excited state can give a lower, nonzero bound. Applying the same reasoning to the spin 1/2 we find the lower energy bound for D(E0 , 1/2) to be E0 ≥ 1. We may also illustrate the phenomenon of multiplet shortening [310]. In our discussion of spin 1, we saw that when the unitarity bound was saturated, i. e., when E0 = s + 1 = 2, states corresponding to s = 0 decoupled. This is a general phenomenon for integer spin s ≥ 1 as stated in (iii’) in the box above. Consider now the lower bound E0 ≥ 21 for spin 0. The state |E0 + 2, 0⟩ used for deriving the unitarity bound is actually itself a ground state, as we see from the very computation yielding the bound. All the excited states built on |E0 + 2, 0⟩ form an invariant subspace of the spin-0 state diagram. This is similar to what happens for integer spin at the lower bound, and can be interpreted as a gauge phenomenon. This leads to the very sparse state spectrum j = E − 21 for the Rac singleton. The corresponding argument can be made for the Di singleton. Finally, let us make the direct product formula (7.33c) for singleton the Rac representation plausible.

8 Three-dimensional fields—solutions of wave equations—can be expanded over complete sets of eigenj functions, say FE (r)Ym (θ, ϕ) in spherical coordinates. With a fixed relation between E and j, only wave functions depending on two space variables can be realized. This was also noted by Dirac.

250 � 7 Higher spin in AdS and the Vasiliev theory Some Rac ⊗ Rac = ⨁s=0,1,2,... D(s + 1, s) numerology At first sight, it may seem implausible that the sparse number of Rac ⊗ Rac states are enough to fill out the full spectrum of ⨁s=0,1,2,... D(s + 1, s) states in formula (7.33c). But remember Cantor: two countably infinite sets can always be put into a one–one relationship by a clever arrangement. Here, we will offer a counting argument that makes the relationship very likely. In the diagram in Figure 7.4, we have marked the Rac spectrum of states and the spectra of massless integer spin representations. For every (j, E) point in the diagram, there is a 2j + 1 degeneracy for every representation D(s + 1, s). Now we can count the total number of states S(n) on each integer energy E = n level. A few low laying examples are S(1) = 1, S(2) = 6, S(3) = 19, S(4) = 44 and S(5) = 85. In general, S(n) can be found by recursion from the difference equations: S(n) − S(n − 2) = 4n2 − 8n + 6,

S(1) = 1 for n odd

(7.34)

S(n) − S(n − 2) = 4n − 8n + 6,

S(2) = 6 for n even

(7.35)

2

and the result is S(n) = (n + 2n3 )/3.

E 5 9/2 4 7/2 3 5/2 2 3/2 1 1/2

0

1 01

0

0 R 0

1 01 R 1

012 123 01234 R 12 0123 R 012 R

2

3

4

5

j

Figure 7.4: The diagram shows (j, E) locations of the Rac spectrum marked by “R” and locations of states of integer spin D(s + 1, s) representations marked by “0, 1, 2, . . .”. At each (j, E), point there is a (2j + 1)-fold m-multiplicity for all spin s states at the point. On the Rac ⊗ Rac side of the equality, we arrange the ⊗ multiplication 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 3 󵄨󵄨 5 󵄨󵄨 1 󵄨󵄨 3 󵄨󵄨 5 {󵄨󵄨󵄨 , 0⟩, 󵄨󵄨󵄨 , 1⟩, 󵄨󵄨󵄨 , 2⟩, . . .} ⊗ {󵄨󵄨󵄨 , 0⟩, 󵄨󵄨󵄨 , 1⟩, 󵄨󵄨󵄨 , 2⟩, . . .} 󵄨󵄨 2 󵄨󵄨 2 󵄨󵄨 2 󵄨󵄨 2 󵄨󵄨 2 󵄨󵄨 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨

(7.36)

according to total energy which is an integer n = 1, 2, 3, . . . . Summing up the products of the number of states for each n, one again finds the sum S(n) = (n + 2n3 )/3. That n + 2n3 = n(1 + 2n2 ) is divisible by 3 for all n follows from Fermat’s little theorem9 applied to the prime 3: If 3 does not divide n, then 1 + 2n2 is divisible by 3.

9 If p is a prime, not dividing n, then np−1 ≡ 1 mod p. See, for instance, Section 5.3 in [318].

7.1 The geometry and algebra of anti-de Sitter space-time

� 251

Flato and Fronsdal [303] prove the equality by employing group characters, but their proof also relies on the spectra of states on both sides of the equality.

7.1.5 Oscillator representations and singletons An intermediate step toward the higher spin algebra is to realize the so(3, 2) Lie algebra in terms of bilinear products of oscillators [308, 319].10 To that end, consider creation operators ai and annihilation operators ai where i = 1, 2 with commutation relations j

[ai , aj ] = δi

(7.37)

Precisely ten bilinear products can be formed. They naturally fall into a Jordan structure with 1 j (7.38) g(−) = {Sij = ai aj } g(+) = {S ij = ai aj } g(0) = {Si = (ai aj + aj ai )} 2 The singleton representations can be built by acting on the ground states with generators from g(+) . For the Rac ground state, we get spectrum of states 󵄨󵄨 1 󵄨󵄨 1 󵄨󵄨 1 󵄨 󵄨 󵄨 Rac = {󵄨󵄨󵄨 , 0⟩, ai1 ai2 󵄨󵄨󵄨 , 0⟩, ai1 ai2 ai3 ai4 󵄨󵄨󵄨 , 0⟩, . . .} 󵄨󵄨 2 󵄨󵄨 2 󵄨󵄨 2

(7.39)

The multiplet shortening—state reduction—comes about in a very nice way. Since any ai1 ⋅ ⋅ ⋅ ai2n is totally symmetric and the indices i run over 1 and 2, there is just 2n + 1 different states at energy level n, yielding the correct Rac spectrum. The Di singleton can be built in a similar way on the |1, 21 ⟩ = ai | 21 , 0⟩ ground state. 7.1.6 Gauging the AdS algebra In order to gauge the AdS algebra with gravitational fields, one now makes a linear recombination of the oscillators into sl(2, C) indexed oscillators ŷα and ŷ̄α̇ = (ŷα )† , where dotted and undotted indices run over 1, 2: ŷ1 = a1 + ia2

ŷ̄1̇ = a1 − ia2

ŷ2 = −a2 + ia1 ŷ̄2̇ = −a2 − ia1

(7.40)

For these oscillators, the commutation relations (7.37) translate into [ŷα , ŷβ ] = 2iϵαβ

[ŷ̄α̇ , ŷ̄β̇ ] = 2iϵα̇ β̇

[ŷα , ŷ̄β̇ ] = 0

(7.41)

10 A careful reading of Dirac’s 1963 paper “A remarkable Representation of the 3 + 2 de Sitter Group” [314] reveals that the oscillator representation is quite explicit already in this early paper (see the last section). Of course, the specific terminology is of a later date.

252 � 7 Higher spin in AdS and the Vasiliev theory The commutation relations of the AdS algebra are now realized by the bilinear operators [320] Lαβ =

1 λ 1 {ŷ , ŷ } L̄ α̇ β̇ = {ŷ̄α̇ , ŷ̄β̇ } Pαβ̇ = ŷα ŷ̄β̇ 4i α β 4i 2i

(7.42)

The explicit symmetric ordering of the oscillators in the algebra generators are called Weyl ordering. The space-time interpretation of these generators should be clear from the notation and their number.11 Indeed, there are four P corresponding to AdS translations—or transvections—and three plus three L corresponding to “rotations”. However, as compared to (7.16), we no longer have the explicit interpretation in terms of rotations and boosts, but we gain a more symmetric formalism. The algebra now reads [Lαβ , Lγδ ] = ϵαγ Lβδ + ϵαδ Lβγ + ϵβδ Lαγ + ϵβγ Lαδ

[Lαβ , Pγδ̇ ] = ϵαγ Pβδ̇ + ϵβγ Pαδ̇

(7.43) (7.44)

with analogous equations for the L̄ α̇ β̇ . Furthermore, [Pαβ̇ , Pγδ̇ ] = λ2 (ϵαγ L̄ β̇ δ̇ + ϵβ̇ δ̇ Lαγ )

(7.45)

AdS algebra dictionary For the benefit of the bathtub reader, we here record a dictionary between the different notation used here and in some of Vasiliev’s work. In our Chapter 8, where we excavate the original twenty papers setting up the theory, we will find that the N = 1 higher spin superalgebra can be built from generating elements [q̂α , q̂β ] = 2iℏϵαβ

[rα̂ ̇ , rβ̂ ̇ ] = 2iℏϵα̇β̇

[q̂α , rβ̂ ̇ ] = 0

(7.46)

the dictionary here is just q̂α = √ℏyα̂ and rβ̂ ̇ = √ℏyβ̄̂ ̇ . For the benefit of the reader fond of simple but convention dependent algebra, we leave the work of disentangling the exact translation between the three different realizations of the Jordan split of the AdS algebra in (7.27), (7.38) and (7.42).

A gauge field for the AdS algebra can now be introduced in a standard way12 2i αβ̇ h P ̇ λ μ αβ ̇ 1 1 ̇ ̇ = ωμαβ {ŷα , ŷβ } + ω̄ μαβ {ȳ̂α̇ , ŷ̄α̇ } + hμαβ ŷα ŷ̄β̇ 4 4

ωμ = iωμαβ Lαβ + iω̄ μαβ L̄ α̇ β̇ + ̇ ̇

(7.47)

11 The λ parameter is the same kind of energy parameter as in Section 7.1.3. 12 The i is for making the gauge field real and to conform to notation in the next section on the Vasiliev theory.

253

7.2 The Vasiliev theory: introductory remarks �

αβ α̇ β αβ where ωμ and ω̄ μ are Lorentz gauge fields, or spin connections, while the hμ are vierbeins. The representation in the second line of (7.47) is useful for an easy transition to the star product realization of the algebra, where operators are replaced by their symbols and quantum commutators and anticommutators are replaced by their star product counterparts. More on this in Section 7.3.2. Still another realization of the algebra is useful, and often used in the literature. Given the small algebra isomorphisms so(3, 2) ∼ sl(2, C) ∼ sp(4, R), the sl(2, C) oscillators ŷα and ŷ̄α̇ and corresponding generators Lαβ , L̄ α̇ β̇ and Pαβ̇ may be replaced by sp(4, R) oscillators Ŷ A with A = 1, 2, 3, 4 and ̇

̇

[ŶA , ŶB ] = 2iEAB

i and TAB = − {ŶA , ŶB } 4

(7.48)

The translation between the bases are quite direct with (ŷα , ŷ̄α̇ ) ↔ ŶA

and

ϵ EAB = ( αβ 0

0 ) ϵα̇ β̇

(7.49)

and will be used seamlessly below when discussing the nonlinear Vasiliev theory.

7.2 The Vasiliev theory: introductory remarks The construction of a new interacting field theory must be based on some principles, must employ some mathematical formalism and—if the theory is to explain some part of fundamental physics—be based on phenomenological input. When it comes to the theory of interacting higher spin gauge fields, the only phenomenological input is their absence in presently known physics. The endeavor must therefore be based entirely on theoretical principles and tools. In the absence of phenomenology, circumventing or accomodating the no-go theorems—which, by the way may be seen as going hand-inhand with the absence of phenomenology—may be taken as an input. This is certainly so for the Vasiliev theory, where the possibility to circumvent the “gravitational coupling no-go theorem” in Minkowski space-time by going to AdS space-time, offered such an input. The philosophers may debate whether one can derive a theory from principles or not. Perhaps not, and in theoretical physics what is important is to offer good arguments based on accepted principles and sound mathematics. For the Vasiliev theory, one may chisel out three major foundations: (1) It is based on a perceived phenomenological/theoretical need to work in an AdS space-time gravitational background and to reproduce the free field theory in the limit of no interaction. (2) It is based on the principle of diffeomorphism invariance and not treating the gravitational field in any more special way than necessary.

254 � 7 Higher spin in AdS and the Vasiliev theory (3) It is based on higher spin gauge invariance as manifested in an initially rigid global higher spin algebra determining the spectra of fields and spin. These foundations emerge from a close study of the development of the theory in the late 1980s and early 1990s. They are very explicitly stated in the first twenty papers on the theory. We will perform such a close study in the next chapter. For now, the reader will have to accept the foundations as reasonable as stated here, and they certainly remain valid. A few more comments can be made, however. There is a certain tension between the first and second foundation regarding the role of the spin-2 field. Item 2 is implemented by using the language of differential forms and not introducing any explicit metric. However, as soon a linearization is performed around a fixed background, there will be a special treatment of the spin-2 field. This is related to the fact that the spin 2 is the only gauge field that must have a nonzero background value. In the Introduction, we mentioned three major approaches to interacting field theory: gauging, deforming and differential geometry. The Vasiliev theory employs all three. It is built using differential geometric notions, it is based on gauging a global higher spin algebra and finally on deforming this algebra into an interacting theory. The differential geometric form language and the gauging aspects of the approach go together to almost unavoidably require us to us the frame formulation of free higher spin gauge fields rather than the metric formulation. The Vasiliev theory is an elaborate construction and in order to understand it in detail, one may find it useful to split up the work into coming to grips with the following theoretical concepts and methods: – AdS background geometry and free fields. – Higher spin algebras. – Star products and symbol calculus. – Free differential algebras. – Unfolding. Still, the question remains how to piece all parts together into a comprehensible picture of the theory? Should one work bottom-up, meticulously explaining all ingredients and put it together piece-by-piece? Or should one work top-down and just drop the equations on the reader, and try to disentangle them afterwards? Two long reviews that work bottom-up are references [321] and [322]. Of Vasiliev’s own review articles, one can point to [320, 323]. The first authors to take up the theory in the early 2000s was E. Sezgin and P. Sundell. In particular, [315, 324, 316] are helpful and clarifying. In the next chapter, the theory will be built bottom-up by following the historical route of the first twenty papers. Here, I will drop the theory on the reader in the next section, and then spend the rest of the chapter trying to make some sense of it. However, even working top-down, it is hard not to rely on bottom-up input. Since working out all the steps and formulas in full detail would require a book in itself, I will in some places

7.2 The Vasiliev theory: introductory remarks �

255

only indicate what is involved in coming to grips with the Vasiliev equations. In order not to proliferate notation, we will stay very close to the very well chosen notation of reference [322].

7.2.1 Dropping the Vasiliev equations on the reader The Vasiliev theory—or the Vasiliev equations as they are most often referred to—come in very many different types, depending on spectra of fields and dimension of spacetime. Here, we will only treat the simplest case in detail: the bosonic even integer spin model, sometimes designated the minimal bosonic theory. We will stay secure in four space-time dimensions and employ the two-component spinor formalism in order not to hide the essentials with irrelevant technical details.13 So, without further ado, here are the equations: d ∧ W + W ⋆ ∧W = 0 ̃ =0 dB + W ⋆ B − B ⋆ W dS + W ⋆ S − S ⋆ W = 0 S ⋆ B − B ⋆ S̃ = 0 α

α̇

S ⋆ ∧S = dz ∧ dzα (i + B ⋆ κ) + d z̄ ∧ d z̄α̇ (i + B ⋆ κ)̄

(7.50a) (7.50b) (7.50c) (7.50d) (7.50e)

What are we looking at here? Clearly, W , B and S are fields of some sort. The first equation has the looks of a Maurer–Cartan equation for an algebra, or rather an equation for a free differential algebra (see Sections 4.1 and 7.3.5). This is indeed the case, and the algebra is an AdS higher spin associative algebra. From the structure of the equation, we understand that the field W must be a space-time 1-form field. The wedge ∧ is an ordinary wedge product (and is not always written explicitly) but the star ⋆ stands for a particular way of multiplying the fields, or rather their components. We have already seen it in our Section 3.3 and we will explicate it in the context of the Vasiliev theory below in Section 7.3.2. As for the fields B and S, they must be space-time 0-forms, and the structure of the third equation may be interpreted as S being in the adjoint representation of the algebra. There is also a “twisted adjoint” representation of the algebra, denoted by the peculiar tilde ̃ decorating the B field in the second equation. The first three equations are therefore sometimes described as “trivial” in that they do not describe any interactions. They are at the level of gauging the higher spin algebra. The first equation can also be described as a “zero curvature” equation and the second and third as “covariant constancy” equations. Not very much can be said of the fourth

13 There are certainly enough of relevant technical details anyway.

256 � 7 Higher spin in AdS and the Vasiliev theory equation at this level of abstraction, except again noting the occurrence of the tilde ̃ on the S field. In order to introduce interactions, some kind of deformation procedure in necessary. One may suspect that the last two equations are responsible for that, although they also look conspicuously “nondeformed”. Clearly, the S field must play a role here. Before coming to that, let us first the introduce the variables that the fields depend on. The fifth equation stands out as being much more particular than the others. There are, what must be interpreted as differentials dzα and d z̄α̇ , of some variables z and z.̄ The S field is a 1-form in this space. The variables are also hinted at with the presence the objects κ and κ̄ in the last equations. These two objects are defined by Vasiliev as κ = exp(izα yα ) and

̇ κ̄ = exp(iz̄α̇ ȳα )

(7.51)

They are called Klein operators and play a special role in the theory. What we see here are two sets of spinorial variables (yα , ȳα̇ ) and (zα , z̄α̇ ). They are closely related to the operators introduced in Section 7.1.6. Their properties are similar to those of equations (7.41). Then there is the space-time dependence on x μ and d is the de Rahm differential d = dx μ 𝜕μ . We will see further down the road in Section 7.4.5, that the fourth and fifth equation taken together, actually have a quite simple interpretation as describing a deformed oscillator with S an oscillator and κB a deformation parameter. The two last equations therefore express their self-consistency. Before continuing with an exposition of the fields and the variables they depend on, let us temporarily remove the S field and the equations it appears in, writing ω for W and C for B. The dependence on the (zα , z̄α̇ ) is also dropped. Then one gets a form of the Vasiliev equations that one also often sees dω = −ω ⋆ ω

(7.52)

̃ dC = −ω ⋆ C + C ⋆ ω

(7.53)

These equations may be formally deformed by introducing powers of the 0-form C on the right-hand sides. One then gets dω = −ω ⋆ ω + V3 (ω, ω, C) + V4 (ω, ω, C, C) + ⋅ ⋅ ⋅ = F ω (ω, C)

(7.54)

̃ + V3 (ω, C, C) + ⋅ ⋅ ⋅ = F (ω, C) dC = ω ⋆ C − C ⋆ ω

(7.55)

C

This is the form in which the Vasiliev equations first appeared in the papers [325, 326, 327, 328],14 where the perturbative expansion was given up to 𝒪(C 2 ). The resulting expressions were very cumbersome, and it was not likely that the procedure could be extended

14 The papers of the FDA sequence as so designated in our Chapter 8.

7.3 Vasiliev free field equations

� 257

very far.15 Note that the fields ω and C do not depend on the variables z and z.̄ It is indeed the embedding of the equations in the larger space of these variables that allows the representation of interactions in the form of the equations (7.50). For the free field theory, the fields ω and C are sufficient, and we will start by studying them.

7.3 Vasiliev free field equations To arrive at the free field equations is an elaborate exercise in itself. But in the process much of the formalistic apparatus for the equations with interactions is constructed.

7.3.1 Higher spin fields in AdS We will first introduce the spectra of higher spin fields in the context of the ω and C fields, and then extend to the fields W , B and S, because this is closer to the physics. First, since ω = ωμ dx μ is a 1-form field, it will, for spin 2, contain as components the gravitational αβ αβ α̇ β vierbeins hμ and spin connections ωμ and ω̄ μ according to equation (7.47). We have already encountered the higher spin generalization of these fields in Section 5.7.5 in our Volume 1. They are multispinor fields ωμ,α(n),β(m) often abbreviated with ω(n, m). They ̇ can all be collected into one generating function ̇

̇

̇ ̇ 1 ωμα1 ⋅⋅⋅αn ,β1 ⋅⋅⋅βm (x)yα1 ⋅ ⋅ ⋅ yαn ȳβ̇ ⋅ ⋅ ⋅ ȳβ̇ 1 m n!m! n,m ∞

ωμ (y, y;̄ x) = ∑

(7.56)

As to the lower limit of the sum, spin 2 is given by m + m = 2 and spin 3/2 by m + n = 1. Spin 1 is represented by n = m = 0 in the Vasiliev theory. The spin spectrum is spin s : n + m = 2(s − 1)

physical fields: |n − m| ≤ 2

where {

extra fields:

|n − m| > 2

(7.57)

Of the physical fields, those with |n − m| ≤ 1 generalize the vierbeins (potentials) while those with |n − m| = 2 generalize spin connections. Thus, all fields with |n − m| > 1 will eventually be given in terms of derivatives of the physical fields with |n − m| ≤ 1. Note that for the vierbeins, integer spin is represented by |n − m| = 0 and half-integer spin by |n − m| = 1. The spinorial variables y and ȳ are commuting under the normal (concatenation) product, but noncommuting under the star product. We give the commutation relations explicitly here:

15 More details on this is reviewed in Section 8.5 in the next chapter.

258 � 7 Higher spin in AdS and the Vasiliev theory [yα , yβ ]⋆ = 2iϵαβ

[ȳα̇ , yβ̇ ]⋆ = 2iϵα̇ β̇

[yα , ȳβ̇ ]⋆ = 0

(7.58)

They will be explained in Section 7.3.2, but obviously there is a close relation to the quantum commutators of equations (7.41). Then we turn to the C fields. They are space-time 0-forms and they are generalizations of the gravitational Weyl tensor components. The idea is the following. For gravity, any solution to the Einstein equations determines all components of the curvature tensor. Some components are zero due to the field equations, while the nonzero ones are all contained in the Weyl tensor. One may then attempt to formulate the field equations directly in terms of the Weyl tensor.16 This is in fact possible based on the following equivalence: Rμν = 0

⇔

Rμν,ρσ = Wμν,ρσ

(7.59)

where the “,” separating index groups is related to the symmetry properties of the tensors; see the box below. Then assuming that higher spin fields behave in an analogous way, higher spin field equations are formulated in terms of higher spin generalizations of the Weyl tensor. From the Riemann tensor to higher spin Weyl tensors in two-component spinor form The Weyl tensor reorientation is an important part of the Vasiliev approach. It leads to the introduction of the C fields and “unfolding”. A brief sketch of the arguments may go as follows.17 To disambiguate notation, we write Rab for the 2-form Riemann tensor and Rab for the Ricci tensor. Remember the algebraic (first) Bianchi identity for the Riemann tensor R[μνρ]σ = 0 when the torsion is zero, and the fact that Rμνρσ is antisymmetric also in the second two indices when the metric is covariantly constant (the metric postulate). It follows that the Riemann tensor has the symmetry of the Young tableaux μ ρ . To indicate this symmetry in tensor notation, write R μν,ρσ . ν σ

Next, all considerations are translated to the frame-like approach by first going to Rμνab expressed in

terms of the spin connections ωμ ab in the standard way

Rab = d ∧ ωab + ωac ∧ ωc b

(7.60)

and then transferring also the differential form indices to the frame via Rab,cd = eμc ∧ eνd Rμνab . The index symmetry is still that of the Young tableaux a c . b d

The corresponding Weyl tensor C ab,cd may be introduced by splitting off the Ricci tensor Rab and curvature scalar R pieces in the same way as for Wμνρσ (see Formula (4.104) in our Volume 1).18 The Einstein equation Rab = 0 is then equivalent to

16 For further comments, see Section 4.5.7 in our Volume 1 where formulas can be found. 17 A much more thorough development can be found in [322], Section 4. 18 The slightly unfortunate notation C for the Weyl tensor has to do with Vasiliev using W for the gauge field generating function containing generalized vierbeins and spin connections ω. We follow Vasiliev notation in order not to introduce still more notational confusion.

7.3 Vasiliev free field equations

Rμνab = eμc ∧ eνd C ab,cd

� 259

(7.61)

This is interpreted as the nonzero components of the Riemann tensor being given by the Weyl tensor. One may ask: what determines the Weyl tensor then? Either it has to be specified—as for the AdS background—or more interestingly, being determined self-consistently from other field equations in the full theory. It is the higher spin generalizations of C ab,cd and the derivatives thereof—treated as space-time 0-forms—that are contained in the Vasiliev C generating functions. Thus, the theory is expressed in terms of 0-forms and 1-forms. The Weyl tensor is of mixed symmetry as indicated by its Young tableaux given above. For such tensors, it is possible to choose a different representation with different symmetry properties. For higher spin fields, it turns out to be convenient to have fields with groups of symmetric indices corresponding to the spin value and the number of derivatives, respectively. We therefore choose from now on to represent the Weyl tensor as C ac,bd with symmetry type a c , conveniently written as C a(2),b(2) . The comma is now separating groups of b d

symmetrized indices. Equation (7.61) is then replaced by Rμνab = eμc ∧ eνd C ac,bd

(7.62)

Next, we want to transfer the formalism to two-component spinor notation. Since we treat the Weyl tensor C ac,bd as a space-time 0-form, the indices are tangent space Lorentz indices. The relevant group theoretic—Lie algebraic—isomorphism is therefore so(3, 1) ∼ sl(2, C). While indices in the two Lie algebras can be traded through the matrices σ aαβ̇ one may instead trust the isomorphism and represent the ̄ ̇ right away. gravitational Weyl tensor by (Hermitian) conjugated multispinors Cα(4) and Cα(4) The higher spin generalization of the Weyl tensor can be traced back to the de Wit–Freedman generalized curvature tensors Rμ1 ⋅⋅⋅μs ,ν1 ⋅⋅⋅νs of Young type s . The corresponding Weyl tensors then translate s ̄̇ . into Cα(2s) and Cα(2s) Nota bene: the higher spin Weyl tensors are treated as independent degrees of freedom in the Vasiliev theory, although eventually they are to be expressed in terms of s derivatives of the basic higher spin potentials. Relevant sections in our Volume 1, pertaining to the discussion here, are 3.6.4, 5.2 and 5.7.4 to 5.7.6.

It is, however, not sufficient to work only with the higher spin Weyl fields Cα(2s) and C̄ α(2s) . We also need all orders of space-time derivatives of these fields. This may be mȯ tivated in several ways, the simplest one being “why not”? After all, higher spin theory is a higher-derivative theory. Such a frivolous motivation may, however, not be sufficient. A more technical motivation comes from looking back at the known gravity example. When we performed the Weyl tensor reorientation above, we did not take the second, differential, Bianchi identity into account. Subjecting the Riemann tensor in (7.62) to this identity, we get a differential constraint on the Weyl tensor that reads ec ∧ ed ∧ DC ac,bd = 0

(7.63)

What we want to do here is to express the covariant derivative D of the Weyl tensor in such a way that equation (7.63) does become an identity.19 19 This can be seen as mimicking the Weyl reorientation trick. It can also be seen as a resolution of the equation in a cohomological way, something that is in some places alluded to in the literature.

260 � 7 Higher spin in AdS and the Vasiliev theory To do that, first transfer the world index on the covariant derivative D to the fiber and study the index properties of DC ac|bd , which is tensor of type X a(2),b(2)|c . The | notation signifies that the symmetry properties of the added index with respect to the other indices is not yet analyzed. Sorting out and retaining those components C a(3),b(2) of X a(2),b(2)|c that make (7.63) an identity (i. e., discarding components that do not pass the (7.63) filter) allows us to write DC a(2),b(2) ∼ C a(3),b(2) . This equation is again subject to a Bianchi-type identity, leading to a new tensor C a(4),b(2) , and the process goes on iteratively, leading to an infinite tower of tensor C a(k),b(2) . To carry out this bottom-up procedure in practice, deriving the exact equations, is quite arduous,20 and since we decided on a top-down approach, let us turn back to that road. We will return to these equations in the much more transparent two-component formulation. For now, we just postulate as reasonable that the field C(y, y;̄ x) has an expansion in terms of components similar to that of the ωμ (y, y;̄ x) field in equation (7.56): 1 α1 ⋅⋅⋅αn ,β̇1 ⋅⋅⋅β̇m C (x)yα1 ⋅ ⋅ ⋅ yαn ȳβ̇ ⋅ ⋅ ⋅ ȳβ̇ 1 m n!m! n,m ∞

C(y, y;̄ x) = ∑

(7.64)

The components of C(y, y;̄ x) are related to derivatives of the basic higher spin Weyl tensors Cα(2s) and C̄ α(2s) . Since a space-time derivative 𝜕μ in two-component formalism ̇ corresponds to 𝜕αβ̇ , such higher derivatives, of order k say, will be given by components C and C̄ . We thus get a restriction |n − m| = 2s on the components. ̇ ̇ α(2s+k),β(k)

α(k),β(2s+k)

Nongauge fields are represented by n = m = 0 for the scalar and n + m = 1 for spin 1/2.

Anti-de Sitter space-time confusion Let us focus the role played by the anti-de Sitter space-time in the arguments behind the Vasiliev theory. ✓ Put aside, for the time being, any attempts to find an action for interacting higher spin gauge fields and focus instead on finding field equations. Take inspiration from Yang–Mills theory and its generalization to Poincaré and AdS gauge theories for gravity (see Section 4.6 in our Volume 1 and Section 2.5 in this volume). This forces the language of differential forms and first-order equations. This provides for a generally coordinate invariant formulation. ✓ Write the gravitational gauge fields in 1-form language as B = ea Pa + 21 ωab Mab with the generators Pa and Mab satisfying the AdS Lie algebra. Compute G = d ∧ B + B ∧ B to find 1 Λ G = T a Pa + (Rab + ea ∧ eb )Mab 2 3

(7.65)

with torsion T a and curvature Rab given by (7.60) (see also Section 4.6.3 in our Volume 1). Then G = 0 is equivalent to Einstein’s equations with a cosmological constant. The AdS space-time background is therefore a solution to the equation d ∧ B + B ∧ B = 0.

20 See Sections 4 and 5 of [322].

7.3 Vasiliev free field equations

� 261

✓ Implement the Weyl tensor reorientation. See the box [From the Riemann tensor...] for this. As an example, we then see that the nonzero Weyl components of the Riemann tensor is given by − Λ3 ea ∧ eb . There is though, a potentially confusing aspect of this procedure. The Vasiliev theory is supposed to be background independent, yet it seems here that we are actually building in AdS space-time from the very start by gauging the AdS algebra, and by extension, gauging the AdS higher spin algebras. One may compare to the Einstein action with a cosmological constant that can be discussed with no reference to AdS or dS background.

Vasilieviana, or some peculiar notions employed in the theory There are some notions that one comes across when reading up on the theory. It might be helpful for the first-time student to have some intuition on their meaning before studying them in detail. It is alright and completely natural to worry about them.21 Twisted adjoint representation. The operation ̃ changes the sign of all undotted spinors so that for a ̄ It is an automorphism of the algebra. One could just as well function f we have ̃f (z, z;̄ y, y)̄ = f (−z, z;̄ −y, y). have chosen to let the operation change sign on all dotted indices. It is often denoted by ̃f = π(f ). Bosonic theory. The original theory was developed with supersymmetry in focus. However, much of the detailed exploration of the properties of the theory, from the end of the 1990s and onward, was done for the much more manageable bosonic theory. The purely bosonic theory, i. e., with only integer spin fields, corresponds to the projection W (Y , Z; x) = W (−Y , −Z; x). In some early papers, the bosonic projection and the π twist seems not to have been explicitly distinguished. There is also a subtheory with only even spin. Deformed oscillators. Such were studied by Wigner a long time ago [329] in an entirely different context. Wigner asked if the quantum mechanical equations of motion determined the quantum commutation relations. Investigating the question for the harmonic oscillator, Wigner found the answer to be “no”. Rather, the solution for the commutator between the position x and the velocity v came out more general as ([v, x] + i)2 = −2(E0 − 1)2 , the standard case being E0 = 1/2. Chevalley–Eilenberg cocycle. This term comes from the advanced theory of Lie algebras.22 Here, it appears when deriving the unfolded form of the AdS background solution of the Einstein equations with a cosmological constant. It is a term quadratic in the gravitational fields, and linear in the higher spin fields, and thus cubic. It is therefore often said that the already the linearized theory “knows a little” of the interactions. Unfolding. The Vasiliev theory is based on an infinite set of variables, where for every physical higher spin field (the higher spin “multivierbeins”) there are also all their higher-order derivatives. Some of these derivatives arrange themselves as “spin connection” type fields—a finite number of them for each spin— and infinite chains of higher-order derivatives of the basic Weyl components of the higher spin curvatures.

21 Paraphrasing a well-known mathematician: you do not understand the Vasiliev theory, you just get used to it. 22 Original reference [330].

262 � 7 Higher spin in AdS and the Vasiliev theory 7.3.2 Star products (and symbol calculus) Here, we will explain in detail how the star product algebra and symbol calculus is actually used in the Vasiliev theory. In Section 7.1.6 above, we phrased the gauging of the AdS algebra in terms of quantum oscillators. It is, however, more convenient to work with star products of the corresponding symbols (of the oscillators), and this is for two reasons. First, star products produce expressions in terms of symbols and derivatives of symbols, making results of computations more explicit. Second, the Vasiliev theory actually relies on the associative algebra of the oscillators, rather than on the Lie commutator algebra.23 In Section 3.3, we discussed the Moyal product in some detail, in particular, the two ways—integral form and differential form—of representing it. There, the product was tailored to produce the standard basic quantum commutator between positions and momenta. Here, we need a product that produce the commutators (7.58). We will present it for the yα variables. The formulas for the ȳα̇ are analogous. The differential formula for the star product is ← 󳨀 → 󳨀 f (y) ⋆ g(y) = f (y) exp[i𝜕α ϵαβ 𝜕β ]g(y)

(7.66)

Our two-component spinor conventions from Section 3.6.4 in Volume 1 are in force here. ← 󳨀 → 󳨀 In particular, we define 𝜕α yβ = ϵαβ and 𝜕α = ϵαβ 𝜕β . Note that yβ 𝜕 α = 𝜕 α yβ .24 The corresponding integral formula can be written in two ways: 1 ∫ f (y + u)g(y + v) exp[iuα vα ]d 2 ud 2 v (2π)2 1 = ∫ f (u)g(v) exp[i(uα − yα )(vα − yα )]d 2 ud 2 v (2π)2

(f ⋆ g)(y) =

(7.67)

The differential form is convenient for polynomial functions. Clearly, the star product of two polynomials is a new polynomial. The integral form may be extended to more general classes of integrable functions. The basic formulation of the Vasiliev equations is in terms of formal power series, so the differential form is appropriate for practical calculations. Questions of convergence are not addressed. Two immediate consequences of the differential formula (7.66) are yα ⋆ yβ = yα yβ + iϵαβ

⇒

[yα , yβ ]⋆ = 2iϵαβ

{

{yα , yβ }⋆ = 2yα yβ

(7.68)

23 The fancy language of “symbols” is not essential. The symbols of operators are just ordinary classical variables. The symbol language was used in the early Vasiliev papers (see Section 8.3.2), but was subsequently dropped, and is seldom used nowadays. 24 For further clarification see that section in Volume 1 or Appendix F in [322].

7.3 Vasiliev free field equations

� 263

Indeed, all the commutators of (7.58) are reproduced (noting that the unbarred and barred variables commute under the star product). Some useful formulas may also be computed → 󳨀 yα ⋆ f (y) = (yα + i 𝜕 α )f (y) and

→ 󳨀 f (y) ⋆ yα = (yα − i 𝜕 α )f (y)

(7.69)

Therefore, the star commutator with y acts like a derivative, while the anticommutator with y acts like a concatenation product → 󳨀 [yα , f (y)]⋆ = 2i 𝜕 α f (y) and {yα , f (y)}⋆ = 2yα f (y)

(7.70)

Furthermore, one may compute → 󳨀 → 󳨀 → 󳨀 → 󳨀 yα yβ ⋆ f (y) = (yα yβ + i(yα 𝜕 β + yβ 𝜕 α ) − 𝜕 α 𝜕 β )f (y) → 󳨀 → 󳨀 → 󳨀 → 󳨀 f (y) ⋆ yα yβ = (yα yβ − i(yα 𝜕 β + yβ 𝜕 α ) − 𝜕 α 𝜕 β )f (y)

(7.71a) (7.71b)

from which formulas one finds → 󳨀 → 󳨀 [yα yβ , f (y)]⋆ = 2i(yα 𝜕 β + yβ 𝜕 α )f (y) → 󳨀 → 󳨀 {yα yβ , f (y)}⋆ = 2(yα yβ − 𝜕 α 𝜕 β )f (y)

(7.72a) (7.72b)

As already noted, formulas for the barred and dotted indexed variables ȳα̇ are analogous. But it is useful to explicitly note some examples of formulas where both types of variables occur. We therefore record (we now drop the right pointing arrow on derivatives acting to the right) yα ȳβ̇ ⋆ f (y, y)̄ = (yα ȳβ̇ + i(yα 𝜕̄β̇ + ȳβ̇ 𝜕α ) − 𝜕α 𝜕̄β̇ )f (y, y)̄

(7.73a)

f (y, y)̄ ⋆ yα ȳβ̇ = (yα ȳβ̇ − i(yα 𝜕̄β̇ + ȳβ̇ 𝜕α ) − 𝜕α 𝜕̄β̇ )f (y, y)̄

(7.73b)

and the commutators ̄ ⋆ = 2i(yα 𝜕̄β̇ + ȳβ̇ 𝜕α )f (y, y)̄ [yα ȳβ̇ , f (y, y)]

̄ ⋆ = 2(yα ȳβ̇ − 𝜕α 𝜕̄β̇ )f (y, y)̄ {yα ȳβ̇ , f (y, y)}

(7.74a) (7.74b)

All these formulas will come in handy when studying the Vasiliev free field equations in the next section. Referring back to Section 7.1.6, we will also see how these formulas come in when discussing the AdS higher spin algebra in Section 7.3.4 below. For now, we just note that the star product formulas derived here allow us to reproduce the AdS algebra of equations (7.43)–(7.45) with the generators Lαβ =

1 y y 2i α β

1 L̄ α̇ β̇ = ȳα̇ ȳβ̇ 2i

Pαβ̇ =

λ y ȳ ̇ 2i α β

(7.75)

264 � 7 Higher spin in AdS and the Vasiliev theory acting under star product commutators. Furthermore, we find the useful formulas: ̄ ⋆ = (yα 𝜕β + yβ 𝜕α )f (y, y)̄ [Lαβ , f (y, y)] ̄ ̄ ⋆ = (ȳα̇ 𝜕̄β̇ + ȳβ̇ 𝜕̄α̇ )f (y, y)̄ [Lα̇ β̇ , f (y, y)]

(7.76b)

̄ ⋆ = −iλ(yα ȳβ̇ − 𝜕α 𝜕̄β̇ )f (y, y)̄ {Pαβ̇ , f (y, y)}

(7.76d)

̄ ⋆ = λ(yα 𝜕̄β̇ + ȳβ̇ 𝜕α )f (y, y)̄ [Pαβ̇ , f (y, y)]

(7.76a) (7.76c)

Here, there is a natural branch point in the top-down approach. One could choose to continue with a study of the higher spin fields, or with the Vasiliev algebras. We chose the first option, after a technical note relevant to formulas to come. A note on combinatorial factors When acting with the operators of (7.76) one generating functions of the form (7.64), one is confronted with a perhaps trivial issue of combinatorial factors when considering equations for component fields. Remember first the standard higher spin convention25 of writing f a(n) for a completely symmetrized tensor of n indices. Symmetrizing a nonsymmetric tensor is done by adding the minimal number of needed terms with unit weight. Thus, as an example, symmetrizing gb f a(n) = gb f (a1 ⋅⋅⋅an ) needs n + 1 terms. This is always assumed to be done when the same letter is used for the indices, i. e., writing ga f a(n) means that the expression is understood to be symmetrized over all n + 1 indices. With this convention, we get the following two simple lines of schematic computation: gb xb (

1 a(n) 1 1 f (xa )n ) = gb f a(n) xb (xa )n = ga f a(n) (xa )n+1 n! n! (n + 1)!

(7.77a)

gb 𝜕b (

1 1 1 a(n) f (xa )n ) = ngb f ba(n−1) (xa )n−1 = (g ⋅ f )a(n−1) (xa )n−1 n! n! (n − 1)!

(7.77b)

It is thus clear that when acting with the generators of (7.76) no combinatorial factors occur for component formulas at any one level (xa )n . That is, terms at the level (xa )n comes with a combinatorial factor 1/n! and symmetrized coefficients g(b f a(n−1)) = ga f a(n−1)) and (g ⋅ f )a(n) , respectively. If, on the other hand, an expression ga f a(n−1) is not supposed to be symmetrized over all a-indices, then a factor of n must be supplied.

7.3.3 AdS background, free fields and linearized curvatures First of all, it is important to bear in mind that the equations (7.52) and (7.53) can only describe the dynamics of free fields. Therefore, it is necessary to understand how they actually describe free field equations for all spins. They are also the jump-off point for the interacting theory. But we must now address a potentially confusing point.

25 See Sections 5.1 and 5.7 in our Volume 1.

7.3 Vasiliev free field equations

� 265

The interacting theory is arrived at in a round-about way. First, the interacting spin-2 theory, which is known, is rewritten in the Weyl tensor reformulation of formula (7.61). This equation is then linearized around the AdS background. In this linearization, there will appear cubic terms of the form hhC coming from the right-hand side of (7.61). Next, the spin-2 equations are generalized to all spin. The hhC term remains, now with C containing higher spin Weyl curvatures. Although this is a reasonable argument, one may start to worry what happened to the equation dω + ω ∧ ω = 0 that does not, at least explicitly, contain any cubic term? As I understand it, here we have an inconsistency—not in the theory itself—but in the very way the theory was constructed historically, and often still is constructed, or explained. There is a leap of logic here. There was “never” any dω + ω ∧ ω = 0 equation because it contradicts the Weyl tensor reorientation. But there is an equation dΩ + Ω ∧ Ω = 0 for the background curvature Ω. We will now proceed to understand this. Weyl reorientated gravity in two-component formalism The spin-2 curvatures, written in two-component language are β̇

T αβ = deαβ + ωα γ ∧ eγβ + ω̄ ̇

αβ

̇

(7.78c)

γ

∧ω

2 α

(7.78a)

̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ R̄ αβ = d ω̄ αβ + ω̄ α γ̇ ∧ ω̄ γβ + λ2 eγ α ∧ eγβ

+ω

γβ

αγ̇

(7.78b)

= dω

α

e γ̇

βγ̇

R

αβ

̇

+λ e

γ̇

∧e

where the first equation defines the torsion. Einstein’s equations, in the Weyl reorientated form, become T αβ = 0

(7.79a)

̇

αβ

δ̇ αβγδ

= eγδ̇ ∧ eδ C

(7.79b)

̇ ̇ ̇ ̇ ̇ ̇ R̄ αβ = eδγ̇ ∧ eδδ̇ C̄ αβγδ

(7.79c)

R

These are general gravitational equations provided that the Weyl tensor satisfies the Bianchi identities. Background fields Since the approach to interactions is basically perturbative, we consider “weak” fields that are perturbations of some background, or “vacuum” values of the fields. For all fields, except the gravitational, the background values are zero, while for the AdS maximally symmetric background, the vierbein and spin connections take nonzero values ̇ ̇ that we denote by hαβ , ϖ αβ and ϖ̄ α̇ β , all written as 1-forms. Plugging these fields into the formula (7.47), we collect them into the AdS background 1-form Ω: ̇ 1 ̇ ̇ 1 Ω = ϖ αβ Lαβ + ϖ̄ αβ L̄ α̇ β̇ + hαβ Pαβ̇ 2 2

(7.80)

266 � 7 Higher spin in AdS and the Vasiliev theory Then computing the first Vasiliev equation (7.52) with ω = Ω, using the AdS algebra in Section 7.1.6, we get the explicit form of dΩ + Ω ⋆ ∧Ω = 0, namely ̇ β dhαβ + ϖ α γ ∧ hγβ + ϖ̄ γ̇ ∧ hαγ = 0 ̇

dϖ

αβ

d ϖ̄

+ϖ

α

γ

∧ϖ

γβ

+λ h

γ̇β̇

+ ϖ̄ α γ̇ ∧ ϖ̄

α̇ β̇

̇

̇

̇

2 α

(7.81a)

∧h

βγ̇

=0

(7.81b)

+ λ2 hγ α ∧ h

γβ̇

=0

(7.81c)

γ̇ ̇

This a good place to introduce the Lorentz background covariant derivative D Df α1 ⋅⋅⋅β1 ⋅⋅⋅ = df α1 ⋅⋅⋅β1 ⋅⋅⋅ + ϖ α1 α f αα2 ⋅⋅⋅α1 ⋅⋅⋅ + ⋅ ⋅ ⋅ + ϖ̄ ̇

̇

̇

β̇ 1

f β̇

α1 ⋅⋅⋅β̇ β̇ 2 ⋅⋅⋅

+ ⋅⋅⋅

(7.82)

That is, each dotted or undotted index is acted upon by the appropriate part of the spin connection. Note, in particular, that the “background torsion” equation (7.81a) says precisely that Dh = 0. For fields expanded over the (y, y)̄ internal coordinates, we have a simple representation for D, ̇ ̇ D = d + ϖ αβ yα 𝜕β + ϖ̄ αβ ȳα̇ 𝜕̄β̇

(7.83)

This definition is effective in any fixed background, Minkowski as well as AdS. There is, and must clearly be, a close relation to the Lie algebra generators of Section 7.1.6 and the star product realization of the preceding section. Indeed, the covariant derivative can be expressed as ̇ ̇ Df = df + ϖ αβ [Lαβ , f ]⋆ + ϖ̄ αβ [L̄ α̇ β̇ , f ]⋆

(7.84)

for a field f = f (x; y, y)̄ and appropriate representatives of Lαβ and L̄ α̇ β̇ . But more on this in the next section. For now, we turn to free fields and linear curvatures. Free fields and linearization: gauge potentials ̇ ̇ Let us denote general gravitational fields by ωαβ , ω̄ α̇ β and eαβ . In order not to introduce new notation for the fluctuations about the background, linearization may be thought of as a “shift”, ωαβ → ϖ αβ + ωαβ

̇ ̇ ̇ ω̄ αβ → ϖ̄ αβ + ω̄ αβ ̇

̇

̇

e α β → hα β + e α β ̇

̇

̇

(7.85)

where we retain the notation ωαβ , ω̄ α̇ β and eαβ for the “weak” fields. Upon performing this shift on the gravitational equations (7.79) using the curvatures of (7.78), keeping only linear terms (the zero order terms drop out due to (7.81)), we get ̇

̇

Deαβ + ωα γ ∧ hγβ + ω̄ ̇

̇

β̇

γ̇

∧ hα γ = 0

(7.86a)

̇

Dωαβ + λ2 (hα γ̇ ∧ eβγ + eα γ̇ ∧ hβγ ) = hγδ̇ ∧ hδ δ C αβγδ ̇

̇

̇

(7.86b)

7.3 Vasiliev free field equations

̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ Dω̄ αβ + λ2 (hγ α ∧ eγβ + eγ α ∧ hγβ ) = hδγ̇ ∧ hδ δ̇ C̄ αβγδ

� 267

(7.86c)

The left-hand sides of these equations are linearized spin-2 curvatures in AdS. Since ̇ ̇ the Weyl tensor for vacuum AdS is zero, the fields C αβγδ and C̄ α̇ βγ̇δ must be taken to first order. Here, we see that the right-hand sides of the second two equations, although linear in fluctuations, are actually cubic in fields. This is an important hint to the interacting theory. The equations (7.86) are now to be generalized to higher spin. In order to do that, we turn to the generating functions ω(y, y;̄ x) and C(y, y;̄ x) of formulas (7.56) and (7.64), respectively, of which we isolate the spin-2 parts26 ̇ 1 1 ̇ ̇ ω(y, y;̄ x)|s=2 = ωαβ yα yβ + ω̄ αβ ȳα̇ ȳβ̇ + λeαβ yα ȳβ̇ 2 2 1 1 ̇ ̇̇̇ C(y, 0; x)|s=2 = C αβγδ yα yβ yγ yδ C(0, y;̄ x)|s=2 = C̄ αβγδ ȳα̇ ȳβ̇ ȳγ̇ ȳδ̇ 4! 4!

(7.87a) (7.87b)

We will also need one more kind of covariant derivative, denoted by 𝒟 and defined by αβ̇

𝒟 = D + λh (yα 𝜕̄β̇ + ȳβ̇ 𝜕α )

(7.88)

which, in analogy with (7.84), can be implemented by (the wedge product is implicit) αβ̇

𝒟f = Df + h [Pαβ̇ , f ]⋆

(7.89)

All three equations of (7.86) are then subsumed under 𝒟ω(y, y;̄ x) = h

γα̇

∧ hγ β ̇

𝜕2

C(0, y;̄ x) + hαγ ∧ h ̇ ̇

𝜕ȳα̇ 𝜕ȳβ

β

𝜕2 C(y, 0; x) γ̇ 𝜕yα 𝜕yβ

(7.90)

evaluated under the restriction |s=2 . The generalization to all spin is done by lifting the restriction |s=2 and taking equation (7.90) as an equation for all fields contained in ω(y, y;̄ x) and C(y, y;̄ x). To see its contents in some detail, we compute the left-hand side to Dωα(n),β(m) + λhα γ̇ ∧ ωα(n−1),β(m)γ + λhγ β ∧ ωα(n)γ,β(m−1) ̇

̇

̇

̇

̇

(7.91)

where we note that the operator D does not “shift” indices on the component fields, while the h part of 𝒟 does so. We read off that all components except those for which n = 0 or m = 0 are set to zero. This gives (torsion-like) equations that allow for expressing fields ω(n − 1, m + 1) and ω(n + 1, m − 1) in terms of space-time derivatives of fields ω(n, m).27 26 Note that we write λeαβ for ωαβ . See further comments on the λ factors in the box below. ̇

̇

27 This includes the so-called “extra fields”; see Section 7.3.1 above.

268 � 7 Higher spin in AdS and the Vasiliev theory On the other hand, the pure ω(y, 0) and ω(0, y)̄ components are sourced by C(y, 0) and ̄ also in analogy with the spin-2 case (curvature-like equations). C(0, y), The expression for the left-hand side (7.90), taken at arbitrary spin s, i. e., the expression (7.91) defines the linearized higher spin curvatures. They generalize the linearized gravitational curvatures in equations (7.86). Note that there are contributions not just from the spin connections, but also from the vierbeins—a characteristic of the theory in AdS space-time. Phrased in a formula, we have Rlin (y, y;̄ x) = 𝒟ω(y, y;̄ x)

(7.92)

with the derivative operators given in formulas (7.84) and (7.89). Free fields and linearization: scalar Weyl curvatures The equations (7.90) for the gauge potentials must be supplemented by equations for the scalar Weyl fields in C. Such equations emanate from the second Vasiliev equation (7.53), which we will now proceed to interpret as an equation for scalar Weyl components Cα(n),β(m) in an AdS background. To get a grip, let us first approach the equation bottoṁ up. A first encounter with “unfolding” Consider the special case m = n of the expansion (7.64) with equal numbers of undotted and dotted indices. Suppose, for the sake of argument, that we are in Minkowski space-time and that such a C field expansion is subject to an equation ̇ dC(y, y;̄ x) = hαα

𝜕2 C(y, y;̄ x) 𝜕y α 𝜕y ̄α̇

(7.93)

with hαα the constant Minkowski metric. On the components, the equation spells out as dCα(n),α(n) = ̇ ̇

ββ Cα(n)β,α(n) for n ≥ 0, so that all the components, except the lowest one C, are derivatives according ̇ β̇ h to Cα(n),α(n) = 𝜕α1 α̇1 ⋅ ⋅ ⋅ 𝜕αn α̇n C. ̇ The components Cα(n),α(n) themselves, maintain (n + 1)2 “subcomponents” and this precisely corrė sponds to the number of components of a traceless symmetric tensor Cμ1 ⋅⋅⋅μn . This means that the equation (7.93), for the lowest components, implies Cμ = 𝜕μ C and Cμν = 𝜕ν Cμ = 𝜕μ 𝜕ν C. Tracelessness of Cμν then implies the Klein–Gordon equation ◻C = 0. Clearly, having a solution to the Klein–Gordon equation (with boundary conditions), all its derivatives can be computed. The unfolding approach is the reverse of this to consider all components of C at some space-time point as independent variables. Fixing them, or computing them, one may consider the Taylor expansion around that point, in this way reconstructing the field C. This is the basic intuition behind unfolding. Since d 2 C is zero, we must have d acting on the right-hand side of (7.93) also producing zero. That is, we must have ̇

hαα ∧ hββ ̇

̇

𝜕2 𝜕2 C=0 ̇ α α ̄ 𝜕y 𝜕y 𝜕y β 𝜕y ̄β̇

(7.94)

7.3 Vasiliev free field equations

� 269

This is indeed the case, as follows from the simple algebra of two-component formalism that yields hαα ∧ hββ = ̇

̇

1 αβ α̇β̇ 1 α̇β̇ αβ H ϵ + H ϵ 2 2

(7.95)

with H αβ = hα γ ̇ hβγ ̇

and

Hαβ = hα γ hβγ ̇ ̇

̇

̇

(7.96)

One conceptual conclusion that one can draw from the box above is that the internal coordinates (y, y)̄ in this example parametrize space-time derivatives. Indeed, the field C is a spinless scalar. On the other hand, as we have set up the theory so far, we also think of the (y, y)̄ coordinates as parameterizing spin, in particular, with regard to the expansion of the gauge potentials in (7.56). We now see that the internal coordinates do double duty to maintain both spin degrees of freedom and space-time derivatives. This double duty can be made explicit by separating the general expansion of C(y, y)̄ in equation (7.64) in terms of the spin s Weyl components Cs ∞

C(y, y)̄ = ∑ Cs (y, y)̄ 2s=0

(7.97)

where 1 ̇ ̇ (C α(n+2s),α(n) (yα )n+2s (ȳα̇ )n + C̄ α(n),α(n+2s) (yα )n (ȳα̇ )n+2s ) (n + 2s)!n! n=0 ∞

Cs (y, y)̄ = ∑

(7.98)

̇ ̇ For n = 0, the components C α(2s),α(0) and C̄ α(0),α(2s) correspond to the Maxwell tensor, the gravitational Weyl tensor, and so on. Clearly, the components with n ≥ 1 correspond to nth derivatives of the basic spin s tensors. We now want to understand how the second Vasiliev equation (7.53) with ω taken as the background Ω yields the AdS generalization of the Minkowski unfolded equation ̃ with Ω given by (7.80). of (7.93). For that purpose, consider computing dC + Ω ⋆ C − C ⋆ Ω ̃ this expression would just read Now, had it not been for the “twist” operation on Ω, dC + [Ω, C]⋆ . The twist—changing the sign on the y spinors—has no effect on the Lαβ and L̄ α̇ β̇ generators. However, it changes the sign on the transvections Pαβ̇ , effectively turning commutators [P, ⋅ ]⋆ into anticommutators {P, ⋅ }⋆ . Thus, using the formulas (7.76), we get the following computation:

̃ dC + Ω ⋆ C − C ⋆ Ω ̇ ̇ ̇ = dC + ϖ αβ [Lαβ , C]⋆ + ϖ̄ αβ [L̄ α̇ β̇ , C]⋆ + hαβ {Pαβ̇ , C}⋆

= DC + hαβ {Pαβ̇ , C}⋆ = DC + iλhαβ (𝜕α 𝜕̄β̇ − yα ȳβ̇ )C ̇

̇

(7.99)

̃ = 0 generalizes the unfolding equation (7.93) We can now see how dC + Ω ⋆ C − C ⋆ Ω to AdS space-time. First, d is replaced by the Lorentz covariant derivative D. Note that

270 � 7 Higher spin in AdS and the Vasiliev theory D acting on C does not shift among the components of C. Second, the 𝜕𝜕̄ derivative term in (7.93) is supplemented by the yȳ multiplicative term. As we saw in the box [A first encounter...] above, the 𝜕𝜕̄ derivative term serves to express the nonzero space-time derivative of C(n, n) in terms of an independent component C(n + 1, n + 1). The yȳ multiplicative term, however, refers back to a lower component C(n − 1, n − 1), introducing no new information. The “unfolding” interpretation remains. The presence of the yȳ multiplicative term can be seen as an effect of the noncommutative nature of the derivatives D. To see how it works in practice, we can write out the infinite chain of equations for the components C(n, m) that result from equating the expression (7.99) to zero DCα(n),β(m) + iλhγδ Cα(n)γ,β(m) =0 ̇ ̇ ̇ δ̇ − iλhαβ̇ Cα(n−1),β(m−1) ̇

(7.100)

̃.28 It is thus defined The “derivative-type” operator that occurs in (7.99) is denoted by 𝒟 by αβ̇

̃C = DC + h {P ̇ , C}⋆ 𝒟 αβ

(7.101)

to be compared to the derivative 𝒟 defined in (7.89). The occurrence of this operator, which is not really a derivative operator since it does not conform to the normal rules for linear derivative operators, is a strange feature of the Vasiliev theory.29 While the covariant derivative 𝒟 corresponds to an adjoint representation of the higher spin algẽ derivative furnish a twisted adjoint representation. We will return to it below. bra, the 𝒟 A note on the λ factors The factors of λ is a source of nuisance in the formalism. They are important if some comparison to reality is eventually to be made, or if the Minkowski limit is to be studied, but many authors—Vasiliev included—often simply put λ = 1 if the focus is on the construction of the theory itself. Note that in the above development, the factors of λ in the definition (7.88) of 𝒟 and in the spin-2 field (7.87a) ensures the λ2 factors in (7.86).

Summary: field equations and gauge invariance In the Vasiliev original papers on the theory, and in quite a few of his reviews, the infinite chain of equations (7.100) are derived from self-consistency of the linearized equations for the higher spin curvatures, i. e., from applying 𝒟 to the equation (7.90) and using 𝒟2 = 𝒟 ∧ 𝒟 = 0. For such an approach, see the review [322] (in particular Sections 4 28 Following [322]. See also this reference for further explication of the twist automorphism and the twisted adjoint representation discussed here. 29 “Strange”, because although there are, as we will see, formal justifications for it; nevertheless, it is hard to find any conceptual explanation for its occurrence in the literature.

7.3 Vasiliev free field equations

� 271

and 5) or the original paper [326] or our review of this paper in Section 8.5.2. The details are quite a bit intricate to follow but still informative. We relegate a discussion to a box below. For now, let us step back and take a birds eye view of what we have achieved so far. The free field theory of higher spin gauge fields in AdS space-time is captured by the equations: 𝒟ω(y, y;̄ x) = h

γα̇

∧ hγ β

𝜕2

̇

C(0, y;̄ x) + hαγ ∧ h ̇

𝜕ȳα̇ 𝜕ȳβ

̇

β

̃C(y, y;̄ x) = 0 𝒟

γ̇

𝜕2 C(y, 0; x) 𝜕yα 𝜕yβ

(7.102a) (7.102b)

subject to gauge transformations δω(y, y;̄ x) = 𝒟ϵ(y, y;̄ x)

(7.103a)

δC(y, y;̄ x) = 0

(7.103b)

These equations go under the name Central on-mass-shell theorem. As noted already, the second equation of (7.102) is not entirely independent, but rather a consistency consequence of the first equation for spin s ≥ 1. For the low spin nongauge fields, it is however independent. The equations are not really a “naive” linearization of the full Vasiliev equations since they also incorporate the Weyl fields in the cubic terms. Consistency of the field equations The occurrence of the twisted adjoint representation for the Weyl fields is a peculiar trait of the theory. It is “implicit” in the Vasiliev paper [326] (named FDA2 in the next chapter, Section 8.5.2) introducing the free differential algebra approach. It is certainly visible—in retrospect, if one knows what to look for—but hidden in the formulas and not at all pointed out as such in the early Vasiliev papers. Let us return to the two equations in (7.102). In many of Vasiliev’s own review papers, and in papers generalizing the theory to higher dimensions, it is pointed out that ̃2 = 0, the two equations are consistent with each other. Just as 𝒟2 = 0 one may compute 𝒟 so the self-consistency of equation (7.102b) is clear. For the first equation (7.102a), we apply 𝒟 on both sides, and the consistency requirement is that 𝒟 applied to the right-hand side should vanish. Here is an outline of the logic. The reader may fill in the details. Schematic consistency check The operator 𝒟 consists of two pieces, the Lorentz covariant derivative D and the frame dependent piece according to formula (7.88). Now upon applying 𝒟 to the right-hand side of the Lorentz covariant derivative D commutes through the h ∧ h factors due to (7.81a). It also commutes through the 𝜕𝜕 and 𝜕̄ 𝜕̄ derivatives as follows from direct computation using antisymmetry of the wedge product.

272 � 7 Higher spin in AdS and the Vasiliev theory

It remains to work through the frame-dependent piece. In a very schematic notation, focus first on the hy 𝜕̄ term. We get one set of terms ̄ ̄ ̄ = [hy 𝜕,̄ h ∧ h𝜕̄ 𝜕]̄ ∧ C(0, y)̄ + h ∧ h𝜕̄ 𝜕̄ ∧ (hy 𝜕C(0, ̄ hy 𝜕̄ ∧ (h ∧ h𝜕̄ 𝜕C(0, y)) y)) The commutator in the first term is clearly zero. The second term is also zero due to the three wedgeantisymmetrized 𝜕̄α̇ derivatives with indices taking just two values. Second, we get another set of terms ̄ 0)) hy 𝜕̄ ∧ (h ∧ h𝜕𝜕C(y, 0)) = [hy 𝜕,̄ h ∧ h𝜕𝜕]∧ C(y, 0) + h ∧ h𝜕𝜕 ∧ (hy 𝜕C(y, Here, the commutator in the first term is nonzero, but the remaining 𝜕̄ is zero on the field. The second term ̄ is zero for the same reason. The parallel argument goes through for the hy𝜕. So, what we now have is the requirement αγ ̇

hγ α ∧ hγ β 𝜕̄α̇ 𝜕̄β̇ DC(0, y)̄ + h ̇

̇

β

∧ h γ ̇ 𝜕α 𝜕β DC(y, 0) = 0

(7.104)

This does not imply DC(0, y)̄ = 0 and DC(y, 0) = 0, but rather that their second derivatives are zero. One ̃ may now see that this extra freedom is precisely captured by the twisted equation 𝒟C(y, y)̄ = 0. Computing ̃ ̃ 𝒟C(y, 0) = 0 and 𝒟C(0, y)̄ = 0 from formula (7.99), we get DC(0, y)̄ = iλhαβ yα yβ̄ ̇ C(0, y)̄ ̇

and

DC(y, 0) = iλhαβ yα yβ̄ ̇ C(y, 0) ̇

(7.105)

These equations solve (7.104) as can be ascertained by some meticulous two-component spinor and wedge product algebra. Note, looking forward to the next chapter, that in the paper [326] mentioned above, the consistency ̄ is claimed to lead requirement for the formulas of (7.102) (in terms of the components of ω(y, y)̄ and C(y, y)) to the chain of equations in (7.100).

7.3.4 Higher spin algebras Higher spin algebras—of which we will only discuss the simplest—are infinite-dimensional Lie algebras of the free field theory. They are not gauge algebras of the full interacting theory, which instead needs a deformation of the free theory algebra. For a general discussion of global higher spin algebras, there is the paper [331] to consult. The introduction in our Section 3.6.7 is a starting point for our discussion here. The simplest four-dimensional higher spin algebra—with only integer spin fields— is already implicit in what we have discussed in this chapter so far. The gauge field in (7.56), expanded over arbitrary powers of the oscillators yα and ȳα̇ , hints at a general algebra element of the form 1 α1 ⋅⋅⋅αn ,β̇1 ⋅⋅⋅β̇m g yα1 ⋅ ⋅ ⋅ yαn ȳβ̇ ⋅ ⋅ ⋅ ȳβ̇ 1 m n!m! n,m ∞

g(y, y)̄ = ∑

(7.106)

It is instructive to think through how to go from the Weyl algebra elements of Section 3.6.7 to the representation (7.106). Repeating formula (3.132), we had

7.3 Vasiliev free field equations

G(X, P) = ∑ S|A1 B1 |⋅⋅⋅|At Bt |At+1 ⋅⋅⋅Ar PA1 PA2 ⋅ ⋅ ⋅ PAr X B1 X B2 ⋅ ⋅ ⋅ X Bt 1≤t≤r

� 273

(7.107)

where we now think of the indices A, B running over 0, 1, 2, 3, 5. The four-dimensional AdS space-time is embedded in a five-dimensional ambient space as in Section 7.1.1. To begin with, no significance was attached to the grouping of the indices. Then considering the centralizer of the sp(2) subalgebra, we arrived at the subalgebra of elements with antisymmetry in each (Ai , Bi ) pair, and equal numbers of P’s and X’s. From there, it is almost immediate that the algebra elements can just as well be written in terms of factors of J AB = (X A PB −X B PA ). All traces can be removed by factoring out certain ideals. Then the ten so(3, 2) generators J AB are split up into six Lorentz generators J ab and four translations (transvections) Pa as in Section 7.1.2. Then one takes advantage of the existence of the isomorphic representation of the algebra in terms two-component spinors. A generic algebra element can then be written in terms of powers of Lαβ , L̄ ̇ α̇ β

and Pαβ̇ . But this is precisely what we have in formula (7.106) with the added requirement that the total powers of undotted and dotted oscillators differ by an even number, or |n − m| = 2s. We note that this is exactly what we have for C field. For the ωμ field, we have instead n + m = 2(s − 1) also an even number. The above argument is valid for integer spin. Should a half integer spin be included, one has to add generators yα and ȳα̇ to the AdS algebra leading to a superalgebra. It is almost inevitable to get a superalgebra when half-integral spin is introduced in AdS spacetime in a formalism of the type employed here.30 For some details on supersymmetry, see Section 7.4.3. The field expansions for C and ωμ remain valid when half-integer fields are present, as does the expansion of the general algebra element (then member of a superalgebra). Note, however, that in order to accommodate fermions and supersymmetry, further operators—the so-called Klein operators—must be included. Such operators are present also in the interacting bosonic theory, potentially leading to confusion, so we will defer this discussion to later. The obscure origin of the twisted adjoint representation The presence of the twisted adjoint representation may inspire some questions: What is it and what does it do? Why does it appear? When did it appear in the historical development of the theory? Let us start with the first technical question. Technicalities The twist operation, denoted by π(⋅) or a tilde ̃, depending on circumstance, changes the sign on the AdS transvections, i. e., π(P) = −P. It has no action on the Lorentz generators. It can be implemented through flipping the sign on either the undotted or the dotted oscillators. We will chose the undotted so that π(y) = −y

30 This is probably the reason behind scattered comments in the literature to the effect that supersymmetry was foreshadowed by Dirac in the 1963 paper [314].

274 � 7 Higher spin in AdS and the Vasiliev theory

and π(y)̄ = y.̄ The twist operation is an automorphism of the AdS algebra as can be seen from [π(P), π(P)] = [P, P] = L = π(L) = π([P, P]) and [π(L), π(P)] = −[L, P] = −P = π(P) = π([L, P]). Consider an associative algebra 𝒢, and its commutator Lie algebra [𝒢], acting on a vector space V by endomorphisms ρ : 𝒢 → End(V ), so that we have a representation31 v → ρ(v) of the algebra such that the structure is preserved, for instance commutators: ρ[v, u] = [ρ(v), ρ(u)]. The twisted action is defined as ρπ (v) = ρ(π(v)). The twisted action also defines a representation since ρπ ([v, u]) = [ρπ (v), ρπ (u)]. When the representation vector space is taken as the algebra itself, one may consider the adjoint representation and the corresponding twisted adjoint representation. While the adjoint action of the algebra on itself is [ρ, v], the twisted action is ρv − vπ(ρ). This discussion now applies directly to the higher spin case at hand. Both the C and ωμ fields are elements of the higher spin algebra. The background covariant derivative 𝒟 contains the adjoint action of the ̃ involves the twisted adjoint action of Ω. This we see in the linearized equations algebra element Ω while 𝒟 of motion (7.102). It remains to check that the twist automorphism extends to the full higher spin algebra, i. e., π(f ⋆ g) = π(f ) ⋆ π(g). We leave this to the reader.

Why? So much for the technicalities. Why does it appear? A conceptual reason for this is not, as far as I am aware of, available in the literature. We will have to do with pragmatic and matter of fact understanding. We must have the twisted covariant derivative in equation (7.102b) in order for it to be consistent with the equation (7.102a). This fact is propagated up to the fully nonlinear Vasiliev equations (7.50). But we can dig a little deeper. Although the expansions of the C and ωμ fields have the same outward appearance, their interpretations are quite different. For the ωμ field, for each and every spin, the expansion describes a finite number of vierbeins, spin connections and extra fields. For the C field, however, the expansion describes, for each and every spin, a basic generalized higher spin Weyl tensor together with an infinite “tail” of higher and higher space-time derivatives of the basic Weyl tensor. The Weyl tensors and their higher derivatives are considered as independent degrees of freedom, to be determined by the full set of equations. This is where the unfolding occurs as chains of equations, recursively connecting the components. The utilitarian reason for the twisted adjoint covariant derivative now appears. Since a space-time derivative, in this formalism, adds an index pair αβ̇ to the object it acts on, the covariant derivative better contain a differentiation 𝜕α 𝜕̇β̇ with respect to the oscillators in order to parametrize the derivative of a lower component with new component “one further step up the chain” so to speak. Such a 𝜕α 𝜕β̇ derivative operator sits in the twisted covariant derivative, but not in the “ordinary” covariant derivative. Technically, this is implemented as described in Section 7.3.3 above. There ̃ derivative also involves a term yα y ̄ ̇ making the chain of C equations more complicated, we saw that the 𝒟 β but the principle is captured by our verbal description here. I short: no twisted adjoint action, no unfolding. When? The infinite chains of C equations first appeared in the paper [326] as consistency equations. As the theory gradually began to settle down during the elaboration and review era in the 1990s, the twisted adjoint representation became explicitly noted as such, but still convoluted with the so-called Klein operators, introduced in order to accommodate supersymmetry and auxiliary fields.

7.3.5 Free differential algebras Before proceeding with setting up the free differential algebra apparatus appropriate for the Vasiliev theory, the reader may be helped by a few comments on terminology and context. 31 See Section 3.1.4.

7.3 Vasiliev free field equations

� 275

A brief historical reminder on FDA and unfolding Mathematically, free differential algebras can be seen as generalizations and developments of the Maurer– Cartan theory that we described in Sections 3.5.1 and 3.5.5. The concept was taken up in supergravity theory in the early 1980s and first went under the name of “Cartan integrable systems” (CIS) later morphing to “free differential algebras” (FDA). The approach was initiated in 1978 by Y. Ne’eman and T. Regge under the name “group manifold approach”, a name that can be understood in the sense of gauge theory of supergravity where one attempted to gauge a rigid space-time group such as the AdS group. This initial approach was based on a direct application of Maurer–Cartan theory to gravity and supergravity. FDA theory is a generalization of the group manifold approach to a theory containing higher p-forms than 1-forms. It was motivated by the problem of D = 11 supergravity. For more on this, see Section 4.1. Inspired by supergravity, and searching for a way to write field equations for higher spin in AdS spacetime, the technique was taken up by M. Vasiliev in 1988. In this context, it morphed into the “unfolding technique”. Unfolding occurs since the gauge algebra underlying the Vasiliev theory is infinite-dimensional and involves higher derivatives. Many of the papers in higher spin theory dealing with the Vasiliev theory refer to a mathematical paper by D. Sullivan [204] where the concept of free differential algebra is developed and applied to topological problems. This is indeed correct regarding priority, but Sullivan’s paper—having rather different mathematical objectives—may not be so helpful for the application of the theory to constructing field equations and actions for supergravity and higher spin theories.

Definitions and basic equations Now, let us jump in. We want to generalize the Maurer–Cartan equation (3.65). For that, let W A denote a set of differential forms on some d-dimensional manifold Md , with A denoting some unspecified, but relevant to the context, index set. The basic equation is RA = dW A + F A (W ) = 0

(7.108)

where the form degree of W A is denoted by |W A |. The equation can be thought of as a “zero curvature” condition. In the Maurer–Cartan case, one works with 1-forms on a Lie group manifold, and d is therefore the exterior differential on the group manifold. That is not really the case in the Vasiliev theory, where d is the exterior derivative on the space-time manifold. With W A a |W A |-form, F A (W ) must be a (|W A | + 1)-form. It is assumed that F A (W ) can be expressed in terms of exterior products of the fields W A F A (W ) = ∑ f A B1 ⋅⋅⋅Bk W B1 ∧ ⋅ ⋅ ⋅ ∧ W Bk k

(7.109)

where the sum is restricted by |W1B |+⋅ ⋅ ⋅ |WkB | = |W A |+1. The structure constants f A B1 ⋅⋅⋅Bk are subject to the symmetry condition that interchanging two nearby indices Bi and Bi+1 Bi

Bi+1

brings in a sign factor (−1)|Wi ||Wi+1 | . If equation (7.109) only involve 1-forms and higher, then the sum is necessarily finite. The application to the Vasiliev theory, however, also

276 � 7 Higher spin in AdS and the Vasiliev theory involve 0-forms so that the sum is potentially infinite. This is indicated in the deformed equations (7.54) and (7.55). The next crucial step is to impose integrability by demanding dR = 0. A one-line calculation using d ∧ d = 0 yields 0 = dRA = d 2 W A + dF A (W ) = dW B ∧

δF A (W ) δF A (W ) = −F B (W ) ∧ B δW δW B

(7.110)

One may be a little bit more sophisticated though. Not demanding the zero curvature condition yields dRA = (RA − F B (W )) ∧

δF A (W ) δW B

(7.111)

Then demanding that dRA be proportional to RA again yields the integrability condition FB ∧

δF A =0 δW B

(7.112)

and the Bianchi identity dRA = RB (W ) ∧

δF A (W ) δW B

(7.113)

Inserting the formula (7.109) into the integrability condition (7.112) yields an equation quadratic in structure coefficients that can be interpreted as a Jacobi-type identity. Note that in both these computations it is assumed that F A does not depend explicitly on space-time. Gauge structure Consider now gauge transformations. Forms W A of degree |W A | ≥ 1 enjoy gauge transformations with parameters ξ A of form degree |ξ A | = |W A | − 1. δξ W A = dξ A − ξ B

δF A δW B

󵄨 󵄨 for 󵄨󵄨󵄨W A 󵄨󵄨󵄨 ≥ 1

(7.114)

There are no independent gauge transformations of 0-forms. Instead, they transform homogeneously without the d term. To avoid confusion, we write C A for 0-forms. δξ C A = −ξ B

δF A δC B

󵄨 󵄨 󵄨 󵄨 for 󵄨󵄨󵄨C A 󵄨󵄨󵄨 = 0 and 󵄨󵄨󵄨ξ B 󵄨󵄨󵄨 = 1

(7.115)

As we have seen, the Vasiliev theory employs 1-forms W encoding gauge field potentials, and 0-forms C encoding low spin nongauge fields and spin s ≥ 1 Weyl-like curvatures. It can now be checked that the curvatures RA transform homogeneously

7.4 The nonlinear theory

δRA = −ξ B RC

δ2 F A δW C δW B

� 277

(7.116)

so that when δRA = 0 when RA = 0. One may also derive an equation relating gauge transformations to diffeomorphisms, generalizing formula (4.16) in Section 4.1.2. It reads Lξ W A = δiξ W W A + iξ RA . We leave it to the reader to define the needed operations and derive the formula; see also Section 3.5.2.

7.4 The nonlinear theory After having seen such an elaborate construction for the free theory, one may suspect a story ten times worse for the interacting theory. That is indeed the case, if one wants to work out the equations order-by-order on the lines of equations (7.54) and (7.55). However, on the high level of equations (7.50), the construction is quite easy to describe, although the conceptual underpinnings are weak, if not to say absent. When presenting the nonlinear Vasiliev equations, it is convenient to work in parallel with sl(2, C) and sp(4, R) oscillators, switching between writing equations in terms of (yα , ȳα̇ ) and YA as hinted at in Section 7.1.6 above. Index disambiguation Just to preclude unnecessary index confusion, the following conventions are used in this book for the Vasiliev theory: – Normal uppercase A, B, C, . . . for so(3, 2) indices. – Upright uppercase teletype A, B, C, . . . for sp(4) indices to be split into sp(2) ⊕ sp(2) undotted and dotted spinor indices α, β, γ, . . . and α,̇ β,̇ γ,̇ . . . . – Upright uppercase A, B, C, . . . for abstract-free differential algebra indices.

7.4.1 The fields and the doubling of the oscillators While the fields ω and C of the free theory are expanded over the algebra of the (yα , ȳα̇ ) oscillators, the fields W , B and S that we started with in Section 7.2.1 are expanded also over a new set of oscillators (zα , z̄α̇ ). Writing for short YA = (yα , ȳα̇ ) and ZA = (zα , z̄α̇ ), we will be using the notation W (Y , Z; x) = Wμ (Y , Z; x)dx μ

and

B(Y , Z; x)

(7.117)

for the extension of the ω and C fields where ω(Y ; x) = W (Y , 0; x) and C(Y , x) = B(Y , 0; x). For the new field S field, we write S(Y , Z; x) = SA (Y , Z; x)dZ A = Sα (Y , Z; x)dzα + S̄α̇ (Y , Z; x)d z̄α ̇

(7.118)

278 � 7 Higher spin in AdS and the Vasiliev theory The doubled oscillators enjoy the following commutators among themselves: [zα , zβ ]⋆ = −2iϵαβ

[z̄α̇ , zβ̇ ]⋆ = −2iϵα̇ β̇

[zα , z̄β̇ ]⋆ = 0

(7.119)

where we note the opposite sign in the nonzero commutators as compared to the Y oscillators. Furthermore, all Z oscillators commute with all Y oscillators. We have written the commutators here for the symbols of the oscillators, therefore, the commutators are to be calculated using the star product to be defined below. For the integer spin, bosonic theory, considered here, the fields are subject to the bosonic projection W (−Y , −Z; x) = W (Y , Z; x) and B(−Y , −Z; x) = B(Y , Z; x) while S(−Y , −Z; x) = −S(Y , Z; x) due to the extra spinorial index carried by the S field. 7.4.2 The Vasiliev star product The presence of the doubled oscillators forces a new definition of the star product. In integral form, it is given by (f ⋆ g)(Y , Z) =

AB 1 ∫ f (Y + U, Z + U)g(Y + V , Z − V )e[iUA VB E ] d 4 Ud 4 V (2π)4

(7.120)

For functions not depending on Z, it reduces to the star product of Section 7.3.2. A list of properties for this new star product is useful to collect. They all follow by direct computation form the definition. Indeed, computation with the integral formula may be algorithmitized in the following way. First, note 1 ⋆ f (Y , Z) = f (Y , Z) ⋆ 1 = f (Y , Z)

and

1⋆1=1

(7.121)

The first row of equations follows from observing that the star product formula in these cases are just Fourier transforms followed by inverse Fourier transforms when the U and V integrals are done consecutively. The second equation is just an instance of the first, the Fourier transform producing a delta function. Having noted this, four auxiliary formulas ePA YB E ⋆ f (Y , Z) = ePA YB E f (Y + iP, Z − iP) AB

AB

f (Y , Z) ⋆ ePA YB E = ePA YB E f (Y − iP, Z − iP) AB

AB

ePA ZB E ⋆ f (Y , Z) = ePA ZB E f (Y + iP, Z − iP) AB

AB

(7.122)

f (Y , Z) ⋆ ePA ZB E = ePA ZB E f (Y + iP, Z + iP) AB

AB

follow by a linear shift of variables in the integrals, taking into account sign changes due to the antisymmetry of E AB . These equations can be applied to computing the useful formulas

7.4 The nonlinear theory

𝜕 𝜕Y A 𝜕 f (Y , Z) ⋆ YA = (YA − i A 𝜕Y 𝜕 ZA ⋆ f (Y , Z) = (ZA + i A 𝜕Y 𝜕 f (Y , Z) ⋆ ZA = (ZA + i A 𝜕Y

YA ⋆ f (Y , Z) = (YA + i

𝜕 )f (Y , Z) 𝜕Z A 𝜕 − i A )f (Y , Z) 𝜕Z 𝜕 − i A )f (Y , Z) 𝜕Z 𝜕 + i A )f (Y , Z) 𝜕Z

� 279

−i

(7.123)

We just note here that [YA , YB ]⋆ = 2iEAB and [ZA , ZB ]⋆ = −2iEAB and that the AdS algebra generators can be written as TAB = − 2i YA YB . Note also the following equation: {ZA , f (Y , Z)}⋆ = 2i

𝜕 f (Y , Z) 𝜕Z A

(7.124)

that will be useful when solving the Vasiliev equations. The Klein operators and regularity of the star product This is a good place to discuss the Klein operators introduced in equations (7.51) above. This is because they are what is called inner Klein operators that can be realized as certain star product operations. In the original literature, the Klein operators were introduced in order to accommodate half-integer spin and supersymmetry in the theory. Such Klein operators are outer Klein operators, not part of the star product algebra. The inner Klein operators, however, lead to the twist operation π discussed at length in Section 7.3.3 above. We will discuss the outer Klein operators in the next section. For the inner Klein operators, let us start by noting what we want to achieve. We want to implement the π-twist operation as a certain star product. In formulas, F(π(Y , Z)) = κ ⋆ F(Y , Z) ⋆ κ

and

̄ , Z)) = κ ̄ ⋆ F(Y , Z) ⋆ κ ̄ F(π(Y

(7.125)

and

̄ y,̄ z, z)̄ = (y, −y,̄ z, −z)̄ π(y,

(7.126)

where π and π̄ flips the signs according to π(y, y,̄ z, z)̄ = (−y, y,̄ −z, z)̄

Further required properties for the Klein operators are κ⋆κ =1

κ ⋆ κ̄ = κ̄ ⋆ κ

κ̄ ⋆ κ̄ = 1

(7.127)

To study whether such Klein operators can be defined, let us start by first forgetting the Z oscillators and ask for an operator κy satisfying κy ⋆ yα = −yα ⋆ κy

and

κy ⋆ yᾱ ̇ = yᾱ ̇ ⋆ κy

(7.128)

From equations (7.70), and the corresponding equations for dotted oscillators, we have [yᾱ ̇ , κy ]⋆ = 2i 𝜕̄α̇ κy = 0 implying that κy does not depend on yᾱ ̇ . Furthermore, {yα , κy }⋆ = κy yα = 0. This last equation can be interpreted such that κy , acting under the star product integral formula, is concentrated at yα = 0, or that it may be represented by a delta distribution κy = 2πδ 2 (y).

280 � 7 Higher spin in AdS and the Vasiliev theory

The concept of regularity for the star product has to do with the function spaces over which the integrals can be defined. Polynomials are certainly included, as is clear from the equivalent differential definition of the product. The star product of two polynomials is again a polynomial. This is just as well, since the higher spin algebra elements are polynomials. With some stretch of imagination, power series may be included. As we see here, delta functions will appear. Regularity questions are discussed in a few places in the literature; see Section 6 of the original reference [332] and Section 13.2 of the review [321]. We will not enter this discussion further, and instead proceed in good faith. For instance, one checks that with the delta function representation for κy , it follows that κy ⋆ κy = 1. Returning to the case where the Z oscillators are present, we can repeat the argument, now neglecting Y , with the result κz = 2πδ 2 (z). With both Y and Z present, we define and compute κ = κy ⋆ κz = (2π)2 δ 2 (y)δ 2 (z) = exp(izα y α ) 2 2

2

α̇

̄ (z)̄ = exp(i zᾱ ̇ y ̄ ) κ ̄ = κȳ ⋆ κz̄ = (2π) δ (y)δ

(7.129a) (7.129b)

The following formulas will become important (and analogously for dotted variables): κ ⋆ f (y, z) = f (z, y) exp(izα y α )

and

f (y, z) ⋆ κ = f (−z, −y) exp(izα y α )

(7.130)

Note that there is no star product on the right-hand sides and that the arguments of the function are switched. This will become important since at a certain stage, the coordinate z will be set to zero, producing the C(0, y)̄ factors in the linear theory.

Let us continue with Klein operators. As already noted, there are two kinds of Klein operators in the theory: inner—like the ones studied above, and outer—to which we now turn.

7.4.3 Klein operators, supersymmetry and the twisted adjoint representation Although supersymmetry is not our main focus here, it was of central importance during the initial development of the theory in the late 1980s. It is also tied up with the Klein operators and the occurrence of the twisted adjoint representation in a way that may not at all be trivial for the first-time student to disentangle.32 For a supersymmetric AdS algebra, it is not good enough just to add yα and ȳβ̇ as generators, because these oscillators are commuting. To get a superalgebra, two so-called, Klein operators k and k̄ can be introduced with the following properties: k2 = 1

̄ k k̄ = kk

k̄ 2 = 1

(7.131)

and the following commutation relations with the symbols:

32 This section adds another layer to the story about the twisted adjoint representation from the box [The obscure origin ... ] above.

7.4 The nonlinear theory

{k, yα } = 0

[k,̄ yα ] = 0

{k,̄ ȳα̇ } = 0

[k, ȳα̇ ] = 0

� 281

(7.132)

This is a hybrid notation; the symbol oscillators are understood to be multiplied with the star product, and one may write kyα = k ⋆ yα without running into problems.33 Supersymmetry generators can now be defined by Qα1 = ikyα

̄ Qα2 = ky α

Q̄ α1 ̇ = ik̄ ȳα̇

Q̄ α2 ̇ = k ȳα̇

(7.133)

With this definition, the anticommutators between the supersymmetry charges can be computed to {Qα1 , Qβ1 } = 4ik 2 Lαβ

{Q̄ α1 ̇ , Q̄ β1 ̇ } = 4ik̄ 2 Lα̇ β̇

{Qα2 , Qβ2 } = 4ik̄ 2 Lαβ

{Q̄ α2 ̇ , Q̄ β2 ̇ } = 4ik 2 Lα̇ β̇

̄ {Qα1 , Qβ2 } = −2k kϵ αβ

̄ ̇ {Q̄ α1 ̇ , Q̄ β2 ̇ } = −2k kϵ α̇ β

2i 2i {Qα1 , Q̄ α1 ̇ } = −k k̄ Pαα̇ {Qα2 , Q̄ α2 ̇ } = −k k̄ Pαα̇ λ λ {Qα1 , Q̄ α2 ̇ } = 0 {Qα2 , Q̄ α1 ̇ } = 0

(7.134)

The Klein operator factors can be interpreted as building a two-dimensional so(2) matrix T ij with T 11 = k 2 = 1, T 22 = k̄ 2 = 1 and T 12 = T 21 = k k.̄ The rest of the osp(2|4) supersymmetry algebra can be computed likewise. Now, what does this have to do with the twisted adjoint representation? Consider the full higher spin algebra built from the Weyl algebra of all polynomials in the oscillators yα and yα̇ . To get a higher spin superalgebra, terms with factors of the Klein operators must also be included. Due to (7.131), only powers k 0 k̄ 0 , k 1 k̄ 0 , k 0 k̄ 1 and k 1 k̄ 1 must be included. This means that gauge fields and Weyl fields are indexed as ωAB μ and

CμAB with A, B = 0, 1. For deep technical reasons—that we cannot enter here, but see Sec-

11 01 10 tion 8.3—the gauge fields ω00 μ and ωμ are “physical” while ωμ and ωμ are “auxiliary”.

11 For the Weyl fields, the situation is the opposite. The ω00 μ and Cμ are “auxiliary” while

Cμ01 and Cμ10 are “physical”. The question is, how do the transvections Pαα̇ ∼ yα yα̇ act adjointly, i. e., when they act on algebra elements such as the fields? The normal action is through commutators, ̄ However, acting on so that on an element of the algebra f (y, y)̄ we have [Pαα̇ , f (y, y)]. algebra elements depending also on the Klein operators, something funny happens. First, note that the transvections anticommute with the Klein operators due to (7.132). Indeed, we get {Pαα̇ , k} = {Pαα̇ , k}̄ = 0

(7.135)

̄ It follows at once that Consider then an algebra element g(y, y,̄ k) = f (y, y)k.

33 See Section 5 of [333] and Section 4 of [326] for the original references (in a different notation).

282 � 7 Higher spin in AdS and the Vasiliev theory ̄ ̄ = {Pαα̇ , f (y, y)}k [Pαα̇ , f (y, y)k]

(7.136)

and similarly for algebra elements depending on k.̄ This shows that for “physical” fields, the transvections act by commutators for the gauge fields ωμ while they act by anticommutators for the Weyl fields C. This is the twisted adjoint action. This fact survives truncating down to a purely bosonic, non-supersymmetric theory. The reason we have focused this issue is that it is not at all explicit in the literature. Historically, the theory was developed with supersymmetry as a main objective, while the bosonic, non-supersymmetric theory was a special case. In the bosonic theory, there is no need for the Klein operators, but the twisted adjoint representation must be used for the Weyl fields for consistency as we have seen above. This was only gradually becoming explicit in the 1990s review literature. The issue is even more convoluted due to the existence of the inner Klein operators that may be used in the bosonic theory to implement the twist.34 7.4.4 The first three Vasiliev equations The nonlinear Vasiliev equations naturally splits into two sets, the first set being equations (7.50a)–(7.50c). These are not really nonlinear, rather they are zero curvature and covariant constancy equations for the fields W , B and S, respectively. As ̃ α , ȳα̇ , zα , z̄α̇ ) = we have explained by now, the twist operation is implemented by F(y ̄ ̄ ̄ ̄ π(F)(yα , yα̇ , zα , zα̇ ) = F(−yα , yα̇ , −zα , zα̇ ) for fields F for which it applies. For bosonic fields, one demands F(−yα , ȳα̇ , −zα , z̄α̇ ) = F(yα , −ȳα̇ , zα , −z̄α̇ ). Not very much more can be said about these equations except noting their gauge invariance under the gauge transformations δW = dξ + [W , ξ]⋆

δB = B ⋆ π(ξ) − ξ ⋆ B

δS = [S, ξ]⋆

(7.137)

The computations are direct, using associativity of the star product and observing that π(F ⋆ G) = π(F) ⋆ π(G). 7.4.5 The second two “nonlinear” equations The second two Vasiliev equations do not look very “nonlinear” either, though it may be argued that a square is not linear. Anyway, here they are again S ⋆ B − B ⋆ S̃ = 0 α

α̇

S ⋆ ∧S = dz ∧ dzα (i + B ⋆ κ) + d z̄ ∧ d z̄α̇ (i + B ⋆ κ)̄

(7.138a) (7.138b)

34 I would like to thank E. Skvortsov for a clarifying email pointing me in the right direction when trying to understand all this.

7.4 The nonlinear theory

� 283

On the face of it, the second of these equations looks positively odd and completely arbitrary. The two equations must, however, be looked upon as a whole, and by expanding them along the lines of the 1-form expansion of S in equation (7.118), we get [Sα , Sβ ]⋆ [SA , SB ]⋆ = ( ̄ [Sα̇ , Sβ ]⋆

{Sα , B ⋆ κ}⋆ = 0

{S̄α̇ , B̄ ⋆ κ}⋆ = 0

[Sα , S̄β̇ ]⋆ ϵ (1 + B ⋆ κ) ) = −2i ( αβ ̄ ̄ 0 [Sα̇ , S ̇ ]⋆ β

(7.139a) 0 ) ϵα̇ β̇ (1 + B ⋆ κ)̃

(7.139b)

The twisted commutator in equation (7.138a) is traded for the anticommutators in equations (7.139a) since for the components of S we have Sα ⋆ B = −B ⋆ S̃α and similarly for S̄α̇ . The factor of 2 in (7.139b) as compared to (7.139b) comes from the standard relation [F, G]∧ = 2F ∧ G. Abstracting a bit and focusing on the undotted spinor part of the equations, writing ̂ Q for B ⋆ κ and q̂ α for Sα and reverting to operator language, consider the equations [q̂ α , q̂ β ] = −2iϵαβ (1 + Q)̂

and

{q̂ α , Q}̂ = 0

(7.140)

These are the equations of a deformed oscillator, first studied by Wigner [329]. 7.4.6 Perturbative expansion of the equations Our goal is now to expand the Vasiliev equations perturbatively in an iterative way. We will be following the clear treatment of [166]. Let us first collect the equations in the form that we will work with them. The equations The first equation, we take as it is from (7.50a). For the second equation, we use the formula (7.125) for the twist operation and κ ⋆ κ = 1 to rewrite the twisted commutator as [W , B ⋆ κ]⋆ . The third equation, we take it as it is in (7.50c). The two last equations, we take from (7.139). All in all, we have, after rescaling W → −W to remove an awkward minus sign d ∧ W = W ⋆ ∧W

(7.141a)

d(B ⋆ κ) = [W , B ⋆ κ]⋆ = 0

(7.141b)

dS = [W , S]⋆ {Sα , B ⋆ κ}⋆ = 0 {S̄α̇ , B̄ ⋆ κ}⋆ = 0 [Sα , Sβ ]⋆ = −2iϵαβ (1 + B ⋆ κ) [S̄α̇ , S̄β̇ ]⋆ = −2iϵα̇ β̇ (1 + B ⋆ κ)̃

(7.141c) (7.141d)

[Sα , S̄β̇ ]⋆ = 0

(7.141f)

(7.141e)

Note that formally thinking of the Z as extending space-time and introducing an extended 1-form 𝒲 according to 𝒲 = Wμ dx μ + SA dZ A and computing (with appropriate

284 � 7 Higher spin in AdS and the Vasiliev theory definitions for wedge products) and requiring d 𝒲 = 𝒲 ⋆ ∧𝒲 and using (7.138b), one finds ̇ dW − W ⋆ ∧W = dzα ∧ dzα (i + B ⋆ κ) + d z̄α ∧ d z̄α̇ (i + B ⋆ κ)̄

(7.142)

This equation is reminiscent of the central on-mass-shell equation (7.102a). The fluctuations to first and second order Next, let us discuss what we want to achieve. To zeroth order, there are no propagating fields, and we just have the background ω(0) = Ω satisfying d ∧ Ω = Ω ⋆ ∧Ω, all other fields being zero. To first order, there are fluctuations and we write ω = ω(0) + ω(1) = Ω + ω(1)

and

C = C (0) + C (1) = 0 + C (1)

(7.143)

To first order, we want the Vasiliev equations to produce the equations (7.102) of the central on-mass-shell theorem. In particular, we need to reproduce the right-hand side of the first equation involving the terms bilinear in the background and linear in the Weyl fields. That is clearly impossible relying only on the two first Vasiliev equations. The equations for the S field must be used. For fluctuations up to the second order we write W = Ω + W (1) + W (2)

B = 2iB(1) + 2iB(2)

SA = ZA + 2iSA(1) + 2iSA(2)

(7.144)

The factors of 2i are inserted for convenience. The expansion for the S field needs some discussion. It looks as if we are thinking of a zeroth-order term ZA for SA . That is correct. At zeroth order, it is innocuous since it drops out of the equations completely (B = 0 at zeroth order). At first order, it plays a decisive role. The first-order fluctuation SA(1) appears in the computation, but drops out of the final result. It is helpful to have some advance intuition about the perturbative/iterative solution process to be undertaken. Only the first three Vasiliev equations involve the space-time derivative d. The second two (or three) equations are sometimes deemed as “algebraic” with regard to space-time. That is certainly so, but more to the point, they are differential equations in the auxiliary spinor space with coordinates ZA . Even more, they take the form—as we will soon see—of de Rahm-type cohomological equations. This is the way they are actually treated and solved with homotopy integrals. This also explains why the C fields are regarded as “initial conditions” in the sense C(Y ) = B(Y , 0). In this sense, the Vasiliev equations are spinor differential equations for space-time field equations. Preparing for the iteration To start the iteration, we need to process the equations some more. First, write the perturbations (7.144) as W →Ω+W

B → 2ib

SA → ZA + 2isA

(7.145)

7.4 The nonlinear theory

� 285

so that we are shifting the fields by their vacuum values. Inserting this into the last three of the equations (7.141), we get for (7.141d), 𝜕 b = sα ⋆ b + b ⋆ (κ ⋆ sα ⋆ κ) 𝜕zα 𝜕 b = sᾱ ̇ ⋆ b + b ⋆ (κ ⋆ sᾱ ̇ ⋆ κ) 𝜕z̄α̇

(7.146a) (7.146b)

using (7.124) after some judicious ⋆ and κ algebra. Likewise, for (7.141e) and (7.141f), we get 𝜕 s − 𝜕zα β 𝜕 s̄ ̇ − 𝜕z̄α̇ β 𝜕 s̄ ̇ − 𝜕zα β

𝜕 s = [sα , sβ ]⋆ + ϵαβ b ⋆ κ 𝜕zα β 𝜕 s̄ = [sᾱ ̇ , sβ̄ ̇ ]⋆ + ϵα̇ β̇ b ⋆ κ̄ ̇ α̇ 𝜕z̄β 𝜕 s =0 ̇ α 𝜕z̄β

(7.147a) (7.147b) (7.147c)

These equations can be solved through homotopy integrals. Before going into that, note that having shifted away the vacuum values, the equations are equations for perturbations. This means that when working at a certain order n for a field f on the lefthand side, bilinear terms g ⋆ h on the right-hand side contribute where the orders of g and h add up to n. Homotopy integrals de Rahm-type cohomological equations appear in the perturbative expansion of the Vasiliev equations already at the first, linearization, order. Generically, in n dimensions, they look like d ∧ fk = gk+1

and

d ∧ gk+1 = 0

(7.148)

with k < n and fk = fa1 ⋅⋅⋅ak dx a1 ∧⋅ ⋅ ⋅∧dx ak and likewise for g. The second equation is the integrability condition for the first. We will use the following notation for homotopy integrals: 1

Γn ⟨f ⟩ = ∫ t n f (zt)dt

(7.149)

0

In the Vasiliev theory, the system is 2 × 2-dimensional with variables zα and zᾱ ̇ . Focusing on the undotted variables, we have just the two cases for k = 0, 1, namely k = 0:

𝜕α f = gα

𝜕α gβ − 𝜕β gα = 0

k = 1:

𝜕α fβ − 𝜕β fα = gαβ

(7.150a) (7.150b)

It is quite interesting to work the cases explicitly. Consider first k = 0. We shall show that the solution is 1

f (z) = zα Γ0 ⟨gα ⟩ + c = zα ∫ gα (zt)dt + c 0

(7.151)

286 � 7 Higher spin in AdS and the Vasiliev theory

The first few steps in the computation is 1

1

1

1

0

0

0

0

𝜕β f = ∫ gβ (tz)dt + zα ∫ 𝜕β gα (tz)dt = ∫ gβ (tz)dt + ∫ zα 𝜕α gβ (tz)dt = ⋅ ⋅ ⋅

(7.152)

where the second equation in (7.150a) has been used. Then, noting that due to homogeneity zα 𝜕α = t𝜕/𝜕t, we may continue 1

1

0

0

⋅ ⋅ ⋅ = ∫ gβ (tz)dt + ∫ t

1 𝜕 g (tz)dt = [gβ (tz)]0 = gβ (z) 𝜕t β

(7.153)

where the last step follows from partial integration. It certainly helps intuition to see that the cohomological set of equations is solved by a surface term. Consider next the case k = 1. The solution is 1

fα (z) = zα Γ1 ⟨g⟩ + 𝜕α c(z) = zα ∫ tg(zt)dt + 𝜕α c(z)

(7.154)

0

where g = gαβ ϵ αβ . We leave the details to the reader.

The solutions to the cohomological equations in Z space can now be written b = C(Y ) + zα Γ0 ⟨sα ⋆ b + b ⋆ π(sα )⟩ + z̄α Γ0 ⟨sᾱ ̇ ⋆ b + b ⋆ π(̄ sᾱ ̇ )⟩ ̇

(7.155a)

β̇

sα = zα Γ1 ⟨sγ ⋆ sγ ⟩ + z̄ Γ1 ⟨[sβ̄ ̇ , sα ]⟩ + zα Γ1 ⟨b ⋆ κ⟩ + 𝜕α ξ

(7.155b)

α

sᾱ ̇ = z̄α̇ Γ1 ⟨sγ̄ ̇ ⋆ s̄ ⟩ + z Γ1 ⟨[sα , sᾱ ̇ ]⟩ + z̄α̇ Γ1 ⟨b ⋆ κ⟩̄ + 𝜕̄α̇ ξ

(7.155c)

γ̇

Next, turning to the first three equations, a technical issue concerning Lorentz covariance has to be resolved. It was observed in [323] that the component higher spin fields in the expansion over the oscillators yα and ȳα̇ did not transform properly as Lorentz tensors. It may not be so surprising since the theory contains also the oscillators zα and z̄α̇ and the oscillator valued fields sα and sᾱ ̇ , and not every object with a spinor index transforms as a spinor. The solution is quite simple, but the discussion leading to it is a bit involved, and we will not review it. Apart from [323] (Section 8.4), detailed discussions can be found in [324] (Section 4), [334] (Section 2.3) and a most clear one in [322] (Section 10.4). This is what has to be done. Apart from the Lorentz sp(2) ⊕ sp(2) generators Lαβ and y y L̄ α̇ β̇ , now to be denoted by Lαβ and L̄ ̇ ̇ , defined as in (7.75), there are two further sets of αβ Lorentz generators Lz , L̄ z ̇ and Ls , L̄ s ̇ . The exact definitions are αβ

α̇ β

αβ

α̇ β

i y Lαβ = − {yα , yβ } 4 i Lzαβ = {zα , zβ } 4

i y L̄ ̇ ̇ = − {ȳα̇ , ȳβ̇ } αβ 4 i L̄ zα̇ β̇ = {z̄α̇ , z̄β̇ } 4

(7.156a) (7.156b)

7.4 The nonlinear theory

Lzαβ =

i i {Sα , Sβ } L̄ sα̇ β̇ = {S̄α̇ , S̄β̇ } 4 4

� 287

(7.156c)

y

The LAB and LzAB may be considered kinematical, while the LsAB are dynamical in that they receive higher-order deformations. There is, however, just on the transvection generator, still defined by Pαβ̇ = − 4i {yα , ȳβ̇ }. The correct Lorentz generators are defined by y L̂ αβ = Lαβ + Lzαβ − Lsαβ

y L̂̄ α̇ β̇ = L̄ ̇ ̇ + L̄ zα̇ β̇ − L̄ sα̇ β̇

(7.157)

αβ

These definitions are effective beyond the AdS background given by W = Ω, B = 0 and y SA = ZA . In the background, we have L̂ αβ = Lαβ and Ω given by (7.80). Away from the background, one must use ̇ 1 1 ̇ ̇ Ω̂ = ϖ αβ L̂ αβ + hαβ Pαβ̇ + ϖ̄ αβ L̂̄ α̇ β̇ , 2 2

(7.158)

which reduces back to the ordinary expression in the background. Now, the first three equations of (7.141) involving the space-time derivative, can be written in the following way by shifting according to (7.145) with the background taken as in (7.158): yz

αβ

z

s

α̇ β̇

z

s

𝒟 W = W ⋆ ∧W − H (Lαβ − Lαβ ) − H̄ (L̄ α̇ β̇ − L̄ α̇ β̇ ) yz

(7.159a)

̃ B = W ⋆ B − B ⋆ π(W ) 𝒟

(7.159b)

𝜕A W = − adh sA − [W , sA ]⋆ + χA

(7.159c)

where, from now on, 𝜕A stands for 𝜕/𝜕Z A split up into undotted and dotted z derivatives. ̃yz are defined as The AdS covariant derivatives 𝒟yz and 𝒟 yz

yz

𝒟 F = dF − Ω

⋆ F + F ⋆ Ωyz = Dyz F − adh F

(7.160a)

̃ F ̃yz F = dF − Ωyz ⋆ F + F ⋆ π(Ωyz ) = Dyz F − ad 𝒟 h

(7.160b)

in close analogy to what we did in Section 7.3.3 above. The background connection Ωyz used here is ̇ 1 1 ̇ ̇ yz yz Ωyz = ϖ αβ Lαβ + hαβ Pαβ̇ + ϖ̄ αβ L̄ ̇ ̇ αβ 2 2 yz

(7.161)

y

with Lαβ = Lαβ + Lzαβ and analogously for the Lorentz generators with dotted indices. As usual, the twist operation only affects the transvections, and we have their adjoint and twisted adjoint action given by ̇ ̇ y adh F = [hαβ Pαβ̇ , F]⋆ = hαβ ((yα − i𝜕αz )𝜕̄ ̇ + (ȳβ̇ − 𝜕̄βż )𝜕αy )F

(7.162a)

̃ F = {hαβ̇ P ̇ , F} = −ihαβ̇ ((y − i𝜕z )(ȳ ̇ − 𝜕̄ ż ) + 𝜕y 𝜕̄ y )F ad h α α α ̇ ⋆ αβ β β

(7.162b)

β

β

288 � 7 Higher spin in AdS and the Vasiliev theory The explicit right-hand sides are effects of the extended star product in YZ space; see formulas in Section 7.4.2 above. In the Lorentz covariant Dyz computed with (7.161), the YZ star product conspire to produce the desired result ̇ ̇ y y Dyz = d + ϖ αβ (yα 𝜕β + zα 𝜕βz ) + ϖ̄ αβ (ȳα̇ 𝜕̄ ̇ + z̄α̇ 𝜕̄βż ) β

(7.163)

with cross-terms in Y and Z dropping out. All in all, this looks quite messy. One may suspect that we are working with too weak of tools. Given that the equations we are trying to solve are cohomological in their nature, the stronger tools of homological perturbation theory can be applied. That is, however, beyond our reach here. A reference to consult to get a foothold is [335]. Let us instead see how the equations (7.159) may be derived. Outline of the derivation of the space-time equations For the first equation (7.159a), making the shift W → Ω̂ + W one gets d Ω̂ + dW = Ω̂ ⋆ ∧Ω̂ + [Ω,̂ W ]⋆∧ + W ⋆ ∧W

(7.164)

The terms from dW and W ⋆ ∧W are good as they are and we just put them aside. The terms from d Ω̂ and Ω̂ ⋆ ∧Ω̂ almost cancel. Remember that dΩ = Ω ⋆ ∧Ω encode the torsion constraint and the zero-curvature constraints (7.81) in the AdS background. Scrutinizing d Ω̂ − Ω̂ ⋆ ∧Ω,̂ one finds that all terms cancel, except the curvature terms of the form h ∧ h in the “directions” of Lz and Ls . Precisely, the terms H αβ (Lzαβ − Lsαβ ) − ̇ ̇ H̄ αβ (Lzᾱ ̇β̇ − Lsᾱ ̇β̇ ) survive. Finally, contemplating the term [Ω,̂ W ]⋆∧ we may expect it to produce connection

terms in a Lorentz covariant derivative. That is certainly so in the y and z directions coming from Ly and Lz . The only trouble is what happens to the Ls generators. However, we are saved by the equation dS = [W , S]⋆ that implies [W , {SA , SB }]⋆ = 0. Putting together what we have, we get (7.159a). The second equation (7.159b) is comparatively straightforward. So, we turn to the third (7.159c) that needs more thinking. We make the shift W → Ω + W and SA → ZA + 2isA in equation (7.141c). This results in dZA + 2idsA = [Ω,̂ ZA ]⋆ + [W , ZA ]⋆ + 2i[Ω,̂ sA ]⋆ + 2i[W , sA ]⋆

(7.165)

Here, dZA = 0. The two terms [W , ZA ]⋆ = 2i𝜕A W and 2i[W , sA ]⋆ we keep as they are. The term [Ω,̂ ZA ]⋆ = 2i𝜕A Ω̂ has the form of a Z space gauge transformation. We will stow it away in a such. Finally, the two terms 2idsA and 2i[Ω,̂ sA ]⋆ combine into a Lorentz covariant derivative 2i𝒟L̂ sA plus an adjoint action adh sA of the transvections. The adjoint action we keep, while stowing the Lorentz covariant derivative away in a gauge term. All in all, the result is equation (7.159c) where χA denotes the stowed away gauge terms. To get the term χA out of the way, a gauge choice Z A χA = 0 is made. As the reader certainly perceives, the arguments become increasingly hand-waving. This has to do with the increasing complexity of the needed calculations, and the increasing difficulty of motivating the steps.

The equation (7.159c) can be integrated through homotopy integrals with the result W = ω(Y ) − zα Γ0 ⟨adh sα ⟩ − zα Γ0 ⟨[W , sα ]⋆ ⟩ − z̄α Γ0 ⟨adh sᾱ ̇ ⟩ − z̄α Γ0 ⟨[W , sᾱ ̇ ]⋆ ⟩ ̇

̇

(7.166)

7.4 The nonlinear theory

� 289

First-order computations After these preliminaries, we now proceed to first order, which will lead to the linearized free field equations. To first order, we have to plug b = B(1) , sA = SA(1) and W = W (1) into the Z space cohomology equations (7.155) and (7.166) and only retain terms of first order in perturbations. The result of such a maneuver is B(1) = C(Y )

(7.167a) 1

̄ ity⋅z dt Sα(1) = zα Γ1 ⟨C ⋆ κ⟩ = zα ∫ tC(−zt, y)e

(7.167b)

0

1

̄ ity⋅̄ z̄ dt S̄α(1)̇ = z̄α̇ Γ1 ⟨C ⋆ κ⟩̄ = z̄α̇ ∫ tC(y, −t z)e

(7.167c)

0

W (1) = ω(Y ) − zα Γ0 ⟨adh Sα(1) ⟩ − z̄α Γ0 ⟨S̄α(1)̇ ⟩ ≡ ω(Y ) + M ̇

(7.167d)

So far, we have only made substitutions in equations. Following reference [166], the term M in (7.167d) can be computed and written as y β

1

̄ ityz + c.c. M = −ihαβ zα 𝜕̄ ̇ ∫(1 − t)dtC(−zt, y)e ̇

(7.168)

0

using the formula Γn ∘ Γm = −(Γn − Γm )/(n − m) for nested homotopy integrals (provided n ≠ m). From there on it is possible to derive the central on-mass-shell equation (7.102a) 𝒟ω = V3 (Ω, Ω, C) with V3 (Ω, Ω, C) given by V3 = adh M|Z=0

(7.169)

reproducing the right hand side of (7.102a). The second central on-mass-shell equation (7.102b) is quite trivial. For details of the computation, see Section 10.3 of reference [322]. This is as far as we will take the Vasiliev theory in this book. Second-order computations and extracting interaction vertices is an intricate process. For this, we refer to the literature. One good inroad reference is [166].

7.4.7 Epilogue: some comments and where to go from here? We have only worked through, in some detail, the simplest AdS higher spin theory, but it is clear that the superstructure is quite general and can be adapted to various low level implementations. That has given rise to the so-called formal higher spin theory. Based on the recognition that the Vasiliev theory is really based on associative algebras quite generally, and generalizing the FDA and unfolding techniques to the general concepts of

290 � 7 Higher spin in AdS and the Vasiliev theory Lie∞ algebras and so-called Q-manifolds, one may abstract away from the more concrete Vasiliev AdS models. Inroad references to this research are [336, 337]. We may come back to the issue about locality that was briefly mentioned in Section 2.7 and was a central theme in our review of the antifield/antibracket approach to interactions in Section 4.3. Locality was never considered in the development of the Vasiliev theory (which will be clear from the next chapter), and we have not discussed it above in our review of the simplest theory. Indeed, there is no control of space-time locality in the theory. This raises the question of physical versus mathematical requirements on theory development. In the construction of the Vasiliev theory, consistency in the mathematical sense, was repeatedly stressed. In retrospect, it is clear that mathematical consistency of the deformation procedure and the correct linear, free theory, limit is not enough. The short list of underlying “physical principles” (apart from mathematical soundness) mentioned in the Introduction to Section 7.2 are not enough. In order to understand higher spin gauge theory, clearly more emphasis must be put on physical requirements and consistency rather than formal mathematical consistency. The latter must be there, but it does not work miracles. For models in other dimension than four and with supersymmetry, one may consult the original Vasiliev work—to be reviewed in the next chapter) or secondary review literature. Such references are not hard to find. The same goes for the highly technical— and extensive—literature on the AdS/CFT reconstruction of bulk vertices from the conformal boundary theory.

8 Archaeology of the Vasiliev theory In this chapter, we will try to reconstruct how the Vasiliev theory of interacting higher spin fields in four dimensions was invented in the late 1980s. The modern presentation of any theory—modern in the sense of contemporary to the present—is often very streamlined and perhaps simplified as compared to how the theory looked during its development. This is certainly true for the Vasiliev theory. The presentation chosen in the preceding chapter was an attempt at such a streamlined presentation. A further perspective on the Vasiliev theory can however be achieved from digging into the old papers from the first 7 years or so of the theory. This is the purpose of the present chapter. Hopefully, it can be of help for the reader who wants to turn to the original sources of the theory. Here, as in other fields of scholarship, returning to the original papers is often a good way to gain understanding. I hope to facilitate such an approach. Historically, as can be gathered from the initial papers, the theory was initially constructed—or discovered—around 1986, in a rather concrete way starting from the free field theory of higher spin gauge fields in anti-de Sitter space-time and by generalizing the MacDowell–Mansouri formulation of general relativity. The theory then developed, over the course of not that many years, into a quite abstract formulation where it can be thought of as a particular instance of a general mathematical structure.1 Already by the year 1992, the theory had acquired its mature form.

8.1 Archaeology? I will be quite explicit in my historical survey of the initial twenty papers. The reader may forgive me for pointing out and explaining aspects that is perhaps quite obvious for the expert, but may not be so for the newcomer. On the other hand, my survey of the first twenty papers does not cover all the contents of those very special papers. That would require a book on its own. The reader who actually puts in the effort to compare my text to the actual contents of the papers themselves, may therefore find my treatment uneven and skewed. Be that as it may. I have tried to focus on the essentials. The first twenty papers are certainly no easy read. For those who have had the privilege of seeing the original preprints—in the form of small booklets with orange covers, barely legible in crude type-write on paper, yellowed by time, with handwritten formulas, “archaeology” is perhaps not such a misplaced designation. It should be clear from the preceding chapters that “solving” the higher spin interaction problem at least requires two important tools: a formalism powerful enough to handle the inherent complexity of the problem and a good starting platform in the form

1 The Vasiliev theory is really a class of theories parameterized by space-time dimension and type of higher spin gauge algebra. https://doi.org/10.1515/9783110675528-008

292 � 8 Archaeology of the Vasiliev theory of physical principles or ideas from lower spin theory. The hope is to work around the no-go results. M. A. Vasiliev and E. S. Fradkin found the starting point in the MacDowell–Mansouri formulation [102] of general relativity. They had also noted that the problems of coupling higher spin fields to gravity could be overcome by working in anti-de Sitter rather than Minkowski space-time [304].2 The formalism was then developed during a period of 7 years, as recorded in the published papers. In perusing the Vasiliev published work from 1987 to 1992, I found it clarifying to classify them in the following sequences of papers, giving the titles and the month and year of submitting (within parenthesis). Some of the papers are written with collaborators (see the regular bibliography). Free fields sequence FF0 “Gauge” form of description of massless fields with arbitrary spin (Jan. 1980) [338]. FF1 Free massless fields of arbitrary spin in the de Sitter space and initial data for a higher spin superalgebra (1986) [339]. FF2b Free massless bosonic fields of arbitrary spin in the D-dimensional de Sitter space (Sept. 1987) [340]. FF2f Free massless fermionic fields of arbitrary spin in the D-dimensional de Sitter space (Sept. 1987) [341]. FF3 Linearized curvatures for auxiliary fields in the de Sitter space (Nov. 1987) [342]. Algebra sequence A1 Candidate for the role of higher-spin symmetry (Jan. 1987) [196]. A2 Extended higher spin superalgebras and their realizations in terms of quantum operators (1986, month unknown) [172]. A3 Superalgebra of higher spins and auxiliary fields (May 1988) [333]. A4 Massless representations and admissibility condition for higher spin superalgebras (June 1988) [343]. A5 Extended higher spin superalgebras and their massless representations (Feb. 1989) [344]. Cubic interaction sequence CI1 On the gravitational interaction of massless higher spin fields (Jan. 1987) [269]. CI2 Cubic interaction in extended theories of massless higher spin fields (Jan. 1987) [268]. 2 The cubic gravitational couplings of massless higher spin fields to gravity in the light-front formulation, found in [160] in 1987, seems to have gone unnoticed. These interactions are second order in space derivatives, and can therefore be classified as gravitational (see Section 6.1.1). It is a bit curious, since E. S. Fradkin was interested in the list of light-front interactions in [160] when meeting I. Bengtsson in London in 1987.

8.1 Archaeology?

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Free differential algebra sequence FDA1 Equations of motion of interacting massless fields of all spins as a free differential algebra (May 1988) [325]. FDA2 Consistent equations for interacting massless fields of all spins in the first order in curvatures (April 1988) [326]. FDA3 Triangle identity and free differential algebra of massless higher spins (Oct. 1988) [327]. FDA4 Dynamics of massless higher spins in the second order in curvatures (Dec. 1988) [328]. Vasiliev equations sequence VE1 Consistent equations for interacting gauge fields of all spins in 3 + 1 dimensions (March 1990) [345]. VE2 Properties of equations of motion of interacting gauge fields of all spins in 3 + 1 dimensions (Nov. 1990) [332]. VE3 Algebraic aspects of the higher-spin problem (Oct. 1990) [346]. VE4 More on equations of motion of interacting massless fields of all spins in 3 + 1 dimensions (Feb. 1992) [347]. The general contents of the papers are indicated by the names given to the sequences. Before entering into the details of them, it may be useful to have an overview of how they relate to each other. For the reader’s convenience, let us also write down the timeline of the papers in Table 8.1 below. Dates for reception at the journal, publishing date and preprint date are given when known. First, note the time gap between FF0 and FF1. The first paper has nothing explicit about AdS, and will be disregarded henceforth. Its relevance is the frame-like formalism it introduces, which is used subsequently in all of the Vasiliev theory. Vasiliev theory proper starts with FF1. Also, note the almost simultaneous submission dates of the papers A1, A2 and CI1, CI2 in January of 1987. Also, FF1 may have been submitted in the second half of 1986 judging from the preprint date. The papers were, however, written one after another, during the course of several years and only published when there was strong evidence that the construction would work at the cubic interaction level.3 It is no exaggeration to view these papers as marking the initial breakthrough in AdS higher spin theory. Furthermore, judging from the submission data, it can be surmised that the work contained in the papers A3, A4 and FDA1, FDA2 was done in parallel, but split into several papers according to the logic of the topic. Already in CI1—indeed already in the contemporary FF1 and A1—there was noted a difficulty, namely the “extra field problem” that did not go away, but rather forced 3 M. A. Vasiliev, private communication.

294 � 8 Archaeology of the Vasiliev theory Table 8.1: Time-line of the Vasiliev 1986–1992 papers. Papers that may be associated in time and are related in contents are grouped as indicated by vertical spaces. Reception dates are taken from the published papers. Publication dates are taken from various Internet sources. Legend: n. p. = dates not provided by the publisher. ? = information not found at the time of writing. M&M = action based on MacDowell– Mansouri action for gravity. Preprints are from Lebedev Institute (Moscow), except for those designated by IC, CERN or GOT.ITP. Note also that an Appendix in A1 appeared as preprint 257. Paper

Received

Publication

Preprint no.

Main contents

FF0

Jan. 3, ‘80

Sept. ‘80

14 (Jan. ‘80)

Frame-like free HS fields

FF1 A1 A2 CI1 CI2

n. p. Jan. 14, ‘87 n. p. Jan. 5, ‘87 Jan. 5, ‘87

Nov. 01, ‘87 July ‘87 Jan.01, ‘88 April 30, ‘87 Aug. 17, ‘87

233 (June ‘86) 258, 257 (July ‘86) 290 (Aug. ‘86) ? 309 (‘86)

Free fields in AdS Higher spin algebras Quantum operators for ext. HS algebras Cubic interactions (M&M) Cubic interactions (M&M), extended

FF2b FF2f

Sept. 17, ‘87 ?

n. p. April 25, ‘88

75 (‘87) 217 (‘87)

Free Bose fields in d-dim. AdS Free Fermi fields in d-dim. AdS

FF3 A3 FDA1 FDA2

Nov. 9, ‘87 May 18, ‘88 May 24, ‘88 April 19, ‘88

Sept. 19, ‘88 n. p. Aug. 11, ‘88 Feb. 15, ‘89

246 (‘87) 264 (‘87) ? 58 (‘88)

Lin. curvatures for aux. fields Algebra of HS and aux. fields Eq.’s of motion in FDA form (letter) Eq.’s of motion in FDA form (full paper)

A4 FDA3 A5 FDA4

June 22, ‘88 Oct. 26, 88 Feb. 17, ‘89 Dec. 19, ‘89

Jan. 16, ‘89 Sept. 25, ‘89 Feb. 12, ‘90 April 5, ‘90

94 (‘88) 153 (‘88) 58 (‘89) IC–89/201

Reps. of HS algebras Details of the FDA approach Ext. HS algebras and reps. Eq.’s of motion in FDA form

VE1 VE2 VE3 VE4

March 29, ‘90 Nov. 20, ‘90 Oct. 31, ‘90 Feb. 18, ‘92

July 5, ‘90 July ‘91 March 21, ‘91 July 9, ‘92

29 (‘90) 89 (‘90) CERN-TH-5871-90 GOT.ITP 92-6

Vasiliev equations (letter) Vasiliev equations (full paper) Vasiliev equations (algebraic aspects) Vasiliev equations (generalizations)

workarounds, and was partly responsible for the redirection of the development of the theory into the direction of free differential algebras (FDA) (including “unfolding”) and then to what became known as the Vasiliev equations. It thus seems that up to and including the CI papers, there was a hope that a generalization of the MacDowell–Mansouri approach to gravity would solve the higher spin problem and provide an action. Note that the extra field problem was addressed (provisionally) in CI1 and CI2 by employing constraints (proposed in FF1 and discussed in A1) expressing the extra fields in terms of the physical fields. The extra field problem was then returned to in the pair of papers FF3 and A3. The idea was to introduce further auxiliary fields such that the constraints on the extra fields could be derived from an action.4 This did however not work out, and it seems that 4 This is curiously not stated as an explicit goal, but is clear from the context. Perhaps it was too obvious to state in text, unless the authors were already cognizant of the problems with finding such an action.

8.1 Archaeology?

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this failure had the effect that the MacDowell–Mansouri inspired way was all but abandoned, instead leading to the FDA approach. Indeed, in the group of papers FF3–FDA2, the FDA papers take the place that in the earlier group FF1–CI2 were taken by the CI papers. The FDA sequence of papers can also be seen as intermediates between the CI papers and the VE papers: the free differential algebra method is introduced, but the consistency of the equations are only analyzed to low orders (although higher than in the CI papers). The VE papers eventually established the formal consistency to all orders. In the chronology of Table 8.1, the group of four papers A4–FDA4 also marks the transition to the Vasiliev equations and the form the theory then settled down into. Before entering the papers themselves, a few more general comments may be in order. First, a minor item is that what we now refer to as “anti-de Sitter” space was in the early papers referred to as “de Sitter” space. For instance, the titles of FF2b and FF2f were changed from preprint to the published version.5 Second, and more important, as compared to the light-cone approach, there is a slight difference in motivation. The focus of the Fradkin–Vasiliev papers is in the first place on higher spin fields interacting with gravity, higher spin self-interactions comes in second place. In the light-cone approach, the focus initially (1983) was on higher spin self-interactions, while all possible cubic interactions between arbitrary integer spin was derived in 1987, including higher spin, cubic gravitational interactions with two derivatives, i. e., of the minimal coupling type.6 Third, it is clear from the introductions to the early papers that much of the motivation and guidance was taken from supergravity and the wish to go beyond the N = 8 restriction related to the problems with gravitational interactions for higher spin. Also, techniques applied in supergravity were employed; apart from the MacDowell– Mansouri approach, most conspicuous perhaps the FDA technique that we reviewed in Section 4.1. A motivation for analyzing the Vasiliev original papers in this way, and in this detail, is the great importance—and dominance—of this work for higher spin theory. Furthermore, since the theory was worked out almost single-handedly, the kind of meta-data, historical notes and differing perspectives, that one often finds in the introductions to papers in subjects were several groups of authors have been involved, are not available here. On the other hand, the following review of the Vasiliev papers is focused on the overall logic of the approach, and not so much on the technical details. When it comes to crucial aspects of the theory, I have chosen to quote from the papers, rather than reformulating the text, in that way gaining faithfulness in the presentation. Although quoting text is not such a common practice in theoretical physics as it is in the humanities, it is a

5 This inverted terminology is also present in some of C. Fronsdal’s early papers. For a discussion on terminology, see Section 7.1 in the present volume. 6 This latter fact went unnoticed for 30 years, only surfacing around 2017 with the return of interest in Minkowski light-front higher spin theory. For a further discussion, see Section 8.8.1.

296 � 8 Archaeology of the Vasiliev theory way to gain some understanding of the thoughts behind the formulas.7 Perhaps, some researcher may be inspired to return to the early papers and explore directions not taken. It should be noted that I am not suggesting that it should be better to read the papers according to the sequences, as I have defined them here, rather than chronologically as they appeared. I have found myself jumping around in the papers a lot. The purpose of organizing the papers in sequences is rather to impose an alternative order, complementary to the time-line.8

8.2 The free fields sequence There is a paper [338] from 1980 where nonsymmetric free higher spin fields are introduced, modeled on the gravitational vierbeins written as eμ,a . This naturally suggests representing a spin s gauge field as eμ,a1 ...as−1 . It is symmetric in the last s − 1 indices ai valued in the tangent space.9 The formalism was called a “gauge” formalism. Nowadays it is called the frame-like formalism. These fields are accompanied by connection-like fields ωμ,b,a1 ...as−1 generalizing the gravitational Lorentz connections ωμ,ab . The paper— which treats half integer as well as integer spin fields—contains gauge transformations, trace and gamma-trace conditions on the fields and free actions. The paper is motivated partly by the need to generalize supergravity beyond spin 2. It is pointed out that since the vierbein formalism is useful for describing gravitational interactions for half-integer spin, the formalism may be convenient for introducing higher spin interactions from a geometrical point of view. Super-symmetry is also discussed.10 In the box below, we summarize some of the contents of the FF0 paper.11 Extract from the paper FF0 The action for a massless spin s field is given in terms of the fields eμ,a1 ...as−1 and ωμ,b,a1 ...as−1 as 1 Ss (e, ω) = (−1)s+1 ϵ μνρσ ϵ abcσ ∫ d 4 x[ωρ,b,ad(s−2) (𝜕μ eν,d(s−2)c − ωμ,ν,d(s−2)c )] 2

(8.1)

This action is invariant under local transformations with three independent kinds of parameters ξa(s−1) , αb,a(s−1) and βb(2),a(s−2) subject to symmetrization and trace conditions that we do not repeat here (see our Volume 1, Section 5.7.1). The transformations are

7 See footnote 13 in the preface to our Volume 1. 8 It may be added that there are Vasiliev papers predating the ones discussed here, that could perhaps be construed as “prehistory”, but I will not go further in that direction. 9 Vasiliev is employing a condensed notation for indices. See Section 5.7 in our Volume 1. 10 To consult the paper, one should try to get hold of the journal itself. The formulas are typeset very tiny, yielding photocopies that are almost impossible to read. 11 The notation is that of the FF1 paper, which employs condensed notation.

8.2 The free fields sequence

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δeμ,a(s−1) = 𝜕μ ξa(s−1) + αμ,a(s−1)

(8.2)

δωμ,b,a(s−1) = 𝜕μ αb,a(s−1) + βμb,a(s−1)

(8.3)

For spin 2, there is no parameter βμb,a , while ξa and αμ,a are linearized local coordinate transformations and Lorentz transformations, respectively. For future use, note that the parameter βμb,a(s−1) is the germ for introducing a further set of “connection-like” auxiliary fields—called “extra fields” in the paper FF1—for higher spin. The theory (without these extra fields) is the frame-like version of the metric-like Fronsdal free higher spin theory in Minkowski space-time. This is shown in the paper.

8.2.1 The paper FF1 Higher spin theory in AdS proper started in a pair of papers by Vasiliev (on free fields) [339], referred to as FF1 here, and by Fradkin and Vasiliev (on higher spin symmetry algebras) [196], referred to as A1. We will consider the free field paper here, the algebra paper in the next section. The paper FF1 starts by stressing the importance of the higher spin interaction problem; the self-interaction problem and in particular the interaction with gravity. Being written after a decade of supersymmetry and supergravity, the paper takes further motivation from these theories and the phenomenological hopes that were attached to them at the time. However, the most likely nonfiniteness of supergravity and the too restrictive spectrum for unification, point to the need for new theories, and the lesson is that “[. . .] a really new fundamental theory can only be built on the grounds of a new local symmetry.”. The argument is then that new local symmetries will require new gauge fields, and the natural candidates for such are massless higher spin fields. The difficulties of constructing interactions for higher spin gauge fields are mentioned. It is pointed out that the restriction of supergravity to N ≤ 8 is related to the inability to describe gravitational interactions for higher spin fields, thus closely relating the higher spin problem to the problems of supergravity. It is concluded that the solution of the higher spin problem “[. . .] rests mainly on the determination of an adequate symmetry, which should enable us to find a true set of fields of the theory and, at the second stage, a dynamics of these fields.”. Support for the belief in the existence of such a theory is gained from the existence of the covariant free field theory and the progress (at that time recent) with arbitrary spin cubic interactions in the light-cone approach and the BBvD self-interaction for spin-3 gauge fields. As to the contents of FF1, massless higher spin in AdS had been extensively studied by Fronsdal in the symmetric tensor-spinor formalism (metric-like) in the paper [312].12 The aim of FF1 is to reformulate the theory in the nonsymmetric “gauge” form of FF0 (nowadays called frame-like). This reformulation will then serve as “initial data” for the 12 As already noted, there is some confusion in early papers on which space to call “de Sitter” and which to call “anti-de Sitter”. So, what we now refer to as AdS may well be dS in early papers and vice versa.

298 � 8 Archaeology of the Vasiliev theory higher spin superalgebra of the paper A1.13 It may be surmised that some of the contents of these two papers were worked out in parallel. The reformulation is done in stages. First, vierbein-like and connection-like fields eν,a1 ,...,as−1 and ων,b,a1 ,...,as−1 , respectively, are introduced. These generalize the vierbein and Lorentz connection of gravity, which correspond to s = 2. Symmetry and trace properties are discussed as well as gauge transformation properties (see our Volume 1, Section 5.7.1). In the next stage, further “connection-like” fields ων,b(t),a(s−1) with 2 ≤ t ≤ s − 1 are introduced. One motivation can be gleaned from the following two transformation formulas: δων,a(s−1) = 𝜕ν ξa(s−1) + ξν,a(s−1)

(8.4)

δων,b,a(s−1) = 𝜕ν ξb,a(s−1) + ξνb,a(s−1)

(8.5)

generalizing spin 2. There ought to be a new gauge field ων,b(2),a(s−1) corresponding to the parameter ξνb,a(s−1) with transformation law δων,b(2),a(s−1) = 𝜕ν ξb(2),a(s−1) + ξνb(2),a(s−1)

(8.6)

This procedure is then iterated until there are as many new b indices as a indices. Note that the last field for each spin has no “Lorentz-like” transformation (just as for spin 2). Then, in a final step, the whole set of fields and parameters are replaced by twocomponent spinor fields and parameters. No actual rewriting has to be done, rather, the structure is argued from the representation theory of the Lorentz algebra. Twocomponent multispinors ων,α(n),β(m) are introduced with n + m = 2(s − 1). They are ̇ supposed to be real in the sense that ω†

̇ ν,α(n),β(m)

= ων,β(m),α(n) ̇ . The fields may be conve-

niently abbreviated by ω(n, m). The quite complicated symmetry and trace properties of the tensor-spinor fields are now replaced by simple symmetry in dotted and undotted indices, respectively. An extra bonus with this formalism is that integer and half-integer spin are described in a uniform way. The gauge transformation structure becomes quite symmetric14 δ δων,α(n),β(m) = 𝜕ν ξα(n),β(m) + nhναδ̇ ξα(n−1),β(m) ̇ ̇ ̇ ̇

δων,α(n),β(m) = ̇ δων,α(n),β(n) = ̇

γ 𝜕ν ξα(n),β(m) + mhνγβ̇ ξ ̇ ̇ α(n) ,β(m−1) δ̇ + 𝜕ν ξα(n),β(n) + nhναδ̇ ξα(n−1),β(n) ̇ ̇

for m > n for

(8.7a)

n>m

(8.7b)

γ ̇ α(n) ,β(n−1)

(8.7c)

mhνγβ̇ ξ

In these formulas, h denotes a background vierbein field. The price to pay for this uniformity—made possible by the extension of the number of fields—is that most of the fields are auxiliary. For integer spin, the physical components reside in just ω(s − 1, s − 1) and for half-integer, in ω(s − 3/2, s − 1/2) and ω(s − 1/2, s − 3/2). All the rest of the fields 13 The meaning of the term “initial data” in this circumstance will be explained below. 14 Compare formulas in Section 5.7.5 in our Volume 1.

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are auxiliary.15 It should be noted that the transformation formulas (8.7) are valid only in flat space even though in the sequel h will denote also the AdS background vierbein. There will be further modifications to the transformations. Having so secured the field content of the theory, the paper turns to the introduction of higher spin curvatures and dynamics. At this stage of the development of the theory, it was strongly influenced by the MacDowell–Mansouri approach to gravity [102]. Taking the importance of the de Sitter and anti-de Sitter background for given, this was natural since MacDowell–Mansouri gravity is a description of gravity in terms of gauge fields of the groups SO(4, 1) or SO(3, 2), respectively. The beginning of the third section of the paper offers a very clear outline of the strategy of the approach. Based on the analogy with MacDowell–Mansouri approach, the aim is to derive linearized higher spin curvatures, i. e., objects denoted by Rlνμ,α(n),β(m) ̇ that transform homogeneously under free theory gauge transformations. In the gravity case, the curvatures and the action are well known, and linearizing around an AdS background, one obtains the linearized curvatures and action for free spin-2 fields in AdS space-time. Note that this includes both the vierbeins and the connections. It is then noted that in the higher spin case “[. . .] neither complete curvatures nor a nontrivial dynamics are known presently [. . .]”, although the first problem is claimed to be solved in the paper A1 (to which we will turn below). The higher spin curvatures The higher spin curvatures are denoted by Rνμ,α(n),β(m) . They are related to the higher spin fields ων,α(n),β(m) ̇ ̇ where n + m = 2(s − 1). Although designated as curvatures, it is actually only the object with the highest number of dotted and undotted indices Rνμ,α(s−1),β(s−1) that is a curvature in the sense of gravity. The rest of these ̇ objects may more properly be thought of as generalized connections, in analogy of the generalized Christoffel symbols introduced by Freedman and de Wit in Minkowski space-time in the metric-like formulation [348]. See Sections 2.10.3 and 5.2 in our Volume 1.

The logic behind the method employed is gauge theoretic. For a generic gauge theory, curvatures and gauge transformations are related as in the formulas (quoted from the paper) Riνμ = 𝜕ν ωμi − 𝜕μ ων i + U ijk ων j ωμk i

i

δων = 𝜕ν ξ +

U ijk ων j ξ k

(8.8) (8.9)

The actual method employed is an ansatz-verification method. A general (the most general it is claimed) linearized form of the gauge transformations for the higher spin fields 15 Looking back at the papers from a vantage point in the future, warning bells could be ringing already at this stage. Auxiliary fields are to some extent arbitrary, and there is no guarantee that actions can be written in terms of them.

300 � 8 Archaeology of the Vasiliev theory is assumed modeled on the formulas (8.7) with partial derivatives replaced with AdS (or dS) Lorentz covariant derivatives and arbitrary coefficients for the hξ terms. From the generic gauge theory formulas, (8.8)–(8.9) are then taken the form of the linearized curvatures in terms of the arbitrary coefficients. Next, in a couple of technical pages of the paper, the form of the linearized curvatures are determined based on the argument that the gauge variation of them should be of zero order in the dynamical fields (deviations from the background).16 The AdS linearization and background The spin-2 gauge fields are the vierbeins ων,αβ̇ and the pair of connections ων,α(2) and ων,β(2) ̇ . The corresponding AdS background fields are denoted by hν,αβ̇ , wν,α(2) and w̄ ν,β(2) ̇ . Decorating a field expanded to linear order around the background by a superscript l, the linearization formulas are ων,αβ̇ = hν,αβ̇ + ωlν,αβ̇ ων,α(2) = wν,α(2) + ων,β(2) = w̄ ν,β(2) + ̇ ̇

(8.10)

ωlν,α(2)

(8.11)

ωlν,β(2) ̇

(8.12)

As explicit formulas, these equations occur first in the A1 paper with ων,α(2) = ων,α(2),β(0) and ων,β(2) = ̇ ̇

l ων,α(0),β(2) ̇ . The linearized curvatures themselves take the following form (the decoration ω is not used on the linearized fields in the paper) as derived in Section 3 of the paper γ ) ̇ μ,α(n) ,β(m−1)

l δ Rνμ,α(n), = DμL ων,α(n),β(m) + λ(nhναδ̇ ωμ,α(n−1),β(m) + mhνγ β̇ ω ̇ ̇ ̇ β(m) ̇

− (ν ↔ μ)

(8.13)

where the Lorentz covariant derivative is given by γ ̇ μ,α(n−1) ,β(m)

DνL ωμ,α(n),β(m) = 𝜕ν ωμ,α(n),β(m) + nwν,αγ ω ̇ ̇

δ + mw̄ ν,β̇δ̇ ωμ,α(n),β(m−1) ̇ ̇

(8.14)

As for the AdS background, formulas are given in FF1 for the background curvatures. We chose to write them here in a form that is more close to how they are used in the CI papers: rνμ,α(2) = 𝜕ν wμ,α(2) + wν,αγ wμ,αγ + λ2 hν,αδ̇ hμ,αδ − (ν ↔ μ) ̇

rνμ,β(2) = 𝜕ν w̄ μ,β(2)̇ + ̇

̇ w̄ ν,β̇δ̇ w̄ μ,βδ̇

rνμ,αβ̇ = DLν hμ,αβ̇ − DLμ hν,αβ̇

+λ

2

γ hν,γ β̇ h ̇ μ, β

− (ν ↔ μ)

(8.15a) (8.15b) (8.15c)

In CI1, primed fields ω′ are written for what we here have written as ωl , and W and W̄ are written for w and w.̄ The background curvatures are denoted by r instead of R in CI1. Apart from this, the notation is quite consistent throughout the papers (although perhaps not optimally chosen for easy reading). The parameter λ is related to the inverse of the AdS or dS radius λ ∼ r −2 .

16 The detailed argument can be found in the paper on page 751 below formula (3.12).

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The result of these deliberations can be interpreted as having determined the structure constants of a would-be higher spin gauge algebra. This is presumably the meaning of the phrase “initial data” used in the papers from this time. In the next section of the paper, the fourth, an action is proposed based on the linearized curvatures. The ansatz takes the form Sls =

̇ 1 Rlρσ α(n),β(m) ∑ As (n, m) ∫ d 4 xϵνμρσ Rlνμ,α(n),β(m) ̇ 2 n≥0

(8.16)

m≥0

It is pointed out that it can written in terms of differential forms, thus being general coordinate invariant without the explicit use of the metric. This allows for treating the gravitational field as one of the gauge fields. The action, however, contains the background fields of the chosen space-time. It is invariant under the linearized gauge transformations previously introduced (since the linearized curvatures are invariant). The paper then turns to the fixing the form of the arbitrary coefficients As (n, m). Hermiticity and space reflection invariance of the action is imposed. This restricts the coefficients somewhat. More important is however the following argument. The initial action for higher spin gauge fields written in the frame-like formulation does not contain the “extra fields” introduced in the present paper. It only contains the fields ω(n, m) with |n − m| ≤ 2. Remember that n + m = 2(s − 1). This means that for integer spin, say, only the initial vierbeins ω(s − 1, s − 1) and pair of connections ω(s, s − 2), ω(s − 2, s) occur in the action. It turns out that this can be arranged also for the action (8.16) by a proper choice of coefficients As (n, m). It is said in a footnote, that “more precisely, the fields ω(n, m) with |n − m| > 2 enter the action [our formula (8.16)] only through some topological invariants [. . .]”. The action finally takes the form of equation (8.16) with the coefficients given by As (n, m) =

γs in+m+1 δ(n + m − 2(s − 1))ε(n − m) 2 n!m!

(8.17)

with ε(n − m) = θ(n − m) − θ(m − n) where θ(n) = 1 for n ≥ 0 and zero otherwise. The factor γs is a normalization constant. The fifth section of the paper turns to the equations of motion and to the flat limit. It is again noted that the action does not depend (nontopologically) on fields with |n − m| > 2, and this fact is also utilized in the derivation of the field equations. It is shown that the action reproduces the known equations for Einstein gravity in the vierbein formulation. One gets a “zero torsion equation” that can be used to express the connections ω(s, s−2), ω(s−2, s) in terms of the vierbeins ω(s−1, s−1). Substituting for the connections yields the correct linearized equations for the vierbeins, gauge invariant to this order. For higher spin, the pattern is repeated, and the known free field equations are reproduced. There is one difference though, the connection fields are only expressed in terms of the physical fields up to gauge transformations corresponding to the gauge parameters not present for spin 2 but starting to appear for higher spin (see equations (8.4)–(8.6)

302 � 8 Archaeology of the Vasiliev theory and the surrounding discussion).17 The flat limit λ → 0 is taken, and then the equations of FF0 are reproduced. In order to take the flat limit, fields rescaled by a factor λ|n−m|/2 are used.18 The actual field equations are given in two forms, the second one—simplified involving the use of Bianchi-type identities—is reproduced here:19 ̃l =0 R ̇ νμ,α(s−1),β(s−1) ̇ l ̃ ϵνμρσ hρα δ R ̇ νμ,α(s−2),β(s−1) δ̇ νμρσ γ ̃ l ϵ h ̇ Rνμ,α(s−1)γ,β(s−2) ̇ ρ β

(8.18a)

=0

(8.18b)

=0

(8.18c)

̃ l are given in terms of the rescaled fields ω ̃l . Of these equaThe linearized curvatures R tions, (8.18a) resemble the zero torsion equation of gravity (to which they reduce in the case s = 2) and can be used to express the connection-like fields ω(s, s−2) and ω(s−2, s) in terms of the vierbein-like fields ω(s−1, s−1), and similarly for half-integer spin fields.20 In the higher spin case, this can only be done up to gauge transformations with the parameters ξ not present in the spin-2 case. This arbitrariness is removed by fixing a gauge. Then the two equations (8.18b) and (8.18c) reproduce the free field equations for the higher spin gauge fields.21 The last section of the paper (Section 6) turns to the “extra field problem”. Since the problem occurs for a specific—but well motivated—choice of coefficients As (n, m) in the general action of (8.16), there is a return to the general case. The action [our formula (8.16)] with arbitrary coefficients As (n, m) leads to some gauge invariant equations on the de Sitter [sic] background. The analysis is complicated [typo corrected] for the general case, and we have not carried it out. Nevertheless, we believe that all possibilities, related to the action [our (8.16)], possess some defects if they do not reduce to the case considered in the previous sections. These defects can be manifested in unboundedness [typo corrected] of the energy from below, indefiniteness of the Hilbert space, etc.22

An example is discussed, and the paper then states the problem which we quote below. In this quote, * refers to our formula (8.16) with coefficients given by (8.17). The emphasis in the last sentence of the quote is ours. 17 It is not explicitly said in what sense these transformations could designated as “gauge” since they do not remove any true degrees of freedom. We discussed this issue in Section 5.7.6 of our Volume 1. They do, however, remove Stueckelberg degrees of freedom. 18 For more on this rescaling, see the box [The λ parameter] in Section 8.4.2 below. 19 These are equations (5.5), (5.17) and (5.18) in the paper. 20 This should be quite clear from the formulas in the box [The AdS linearization...] above. In order not to be too repetitive, I generally only quote formulas for the integer spin case. 21 See equations (5.17) and (5.18) in the paper and the discussion preceding them. 22 There are minor misprints in the text here, some of which are corrected in pen by the author in the copy of the published paper I have available.

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In the rest part of this section, we discuss another possibility for the generalization of the equations of motion corresponding to the action [*]. The reason for this is that the proposed formulation for the description of massless higher spin fields possesses a shortcoming feature, which is irrelevant in the free case but can lead to some troubles for the case with interaction. This feature is that the action [*] does not depend on all “extra” fields ω(n, m) with |n−m| > 2. If higher spin interactions are introduced “naively” via transition to full (nonlinearized) curvatures of Ref. [our reference [196]], the variation of the corresponding nonlinear action over the “extra” fields becomes different from zero (for more details see Ref. [our reference [196]]). Hence, equations of motion for “extra” fields turn out to be essentially nonlinear. In some sense, this is another manifestation of the higher spin problem when higher spin interaction changes the number of degrees of freedom and turns out to be inconsistent.

The solution to the extra field problem proposed in FF1, and elaborated in later papers, is to introduce constraints that express the extra fields in terms of the physical fields. The following set of equations are introduced:23 ̃l =0 R ̇ νμ,α(n),β(m)

for n > 0, m > 0 n + m = 2(s − 1)

α ̃l ϵνμρσ R ̇ hρ β̇ = 0 νμ,α(2s−2),β(0)

(8.19a) (8.19b)

β̇

̃l hρα = 0 ϵνμρσ R ̇ νμ,α(0),β(2s−2)

(8.19c)

These equations not only constrain the extra fields, but also contain the equations for the physical fields. After a proof by induction, the paper states ̃ (n, m) It is thus proven that eqs. [**] indeed contain dynamical equations for “physical” fields ω ̃ (n, m) with n + m = 2(s − 1), with n + m = 2(s − 1), |n − m| ≤ 2 and express all “extra” fields ω 2 < |n − m| ≤ 2(s − 1) in terms of the “physical” ones (up to the pure gauge part) without imposing any further restrictions on the “physical” fields. Although eqs. [**] seem to be rather attractive providing solution of the “extra” field problem, unfortunately, we have not succeeded in the construction of an action leading to these equations. Perhaps a part of eqs. [**] should be considered as constraints.

In this quote, ** refers to our formulas (8.19). The paper then ends with a discussion of such constraints, that together with the equations for the physical fields (equivalent to our formulas (8.18b) and (8.18c)), reproduce the equations (8.19a)–(8.19c). The resulting equations (formulas (6.22) and (6.23) in the paper) are ̃l ϵνμρσ R hρα β = 0 ̇ νμ,α(n),β(m)

for n ≥ m ≥ 1

(8.20a)

=0

for m ≥ n ≥ 1

(8.20b)

̇

̃l ϵνμρσ R hραβ̇ ̇ νμ,α(n),β(m)

We will return, as the papers themselves do, to the discussion of the extra field problem, in particular when discussing the papers FF3, CI1 and FDA2.

23 These are equations (6.7), (6.8) and (6.9) in the paper.

304 � 8 Archaeology of the Vasiliev theory 8.2.2 Papers FF2b and FF2f The paper FF2b is written together with V. E. Lopatin. It concerns free fields in D-dimensional de Sitter space (AdS in modern terminology). It is motivated by the desire to see if the results in four dimensions can be extended to higher dimensions. In order to do that, there is a return to the tensor-spinor formalism. The paper seems a bit out of line with the general thrust of the theory at the time, and it contains quite a few misprints, which may be a sign that the focus was actually elsewhere.24 There is also a curious notation at the very beginning of the first section after the introduction, that seems not to be explained anywhere in the paper. An oscillator based generating function formalism is introduced in the paper, something that was to be done soon also in the four-dimensional theory. The paper FF2f written by Vasiliev alone, treats the corresponding theory for fermionic fields. Since the papers do not add much more to the understanding of the theory at this time, let us turn instead to the third paper in the sequence.

8.2.3 The paper FF3 The paper FF3 concerns the so called extra field problem. The paper stands in a similar relation to the A3 paper as the paper FF1 does to the paper A1, in that it provides “initial data” for the determination of a superalgebra of higher spin and auxiliary fields.25 The two cubic interaction papers CI1 and CI2 had been written and submitted in January 1988 (simultaneously with FF1 and A1 as we have noted) and as one consequence of that work, the extra field problem may have gone from a curiosity, even a good feature perhaps, to an acute problem. The paper FF1 is a bit ambivalent on this point. The proposed action in terms of linearized higher spin curvatures may be chosen so that it coincides with the action from FF0. The action then contains no nontrivial contributions from the extra fields. However, toward the end of FF1 this was noted as a problem as we saw above. Let us put the problem in sharper focus. The extra field problem The extra fields were introduced in FF1. We followed the argument in Section 8.2.1 above in connection to formulas (8.4)–(8.6) (see also our Volume 1, Section 5.7.3). Their introduction may be motivated in a few related ways.

24 One cannot escape the feeling that since “everyone” believed in higher dimensions at the time, the paper was meant to quickly plug that hole. 25 In FF3, it says that A3 is submitted to Int. J. Mod. Phys. However, FF3 is received at Nucl. Phys. B six months before A3 is received at Int. J. Mod. Phys. The discrepancy may be explained by the publication date for FF3 in September 1988. The comment was likely added in the proofs.

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Consider fields with the index structure ων,b(t),a(s−1) . The spin considered are given by the number of a indices. The case with t = 0 corresponds to the vierbein-like higher spin fields, and the case with t = 1 corresponds to the connection-like higher spin fields. They may be so-called since they generalize the case s = 2.26 The extra fields are precisely the fields ων,b(t),a(s−1) with 2 ≤ t ≤ s − 1 that one may introduce for higher spin. It is seen that such a step brings in a kind of “balance” between the a and b indices. As we saw above, the gauge transformation formulas may be generalized accordingly by introducing the corresponding extra gauge parameters. These extra parameters originates, in an iterative way, from the “Lorentz-like” transformations with parameter βμb,a(s−1) that are present for higher spin and which leave the action invariant, as noted in the box above on the FF0 paper. This “balance”27 of indices is even more tantalizing when the fields are rewritten in two-component spinor language. The paper FF1 motivates this step by first writing that “To construct consistent higher spin curvatures, it seems natural to introduce [fields with one more b index].”. Then “A more careful analysis shows that the true set of gauge fields being necessary for the construction of consistent curvatures should be extended even further [. . .]”, leading to the full set ων,b(t),a(s−1) with 0 ≤ t ≤ s − 1 for even spin (the paper treats both even and odd spin).28 One may, however, ask how well motivated the introduction of the extra fields is. The extra fields do not enter into the free higher spin action (8.1) of FF0. As we have already noted, in FF1, a general geometrical free higher spin action is given as (8.16). The paper argues that the coefficients in the action can be chosen so that it becomes independent of the extra fields. The logic is not very explicit here. Why is that desirable if one is then left with the problem of finding equations for the extra fields? To answer that, one must realize that just as the connection-like fields ων,b(1),a(s−1) may be expressed as first-order derivatives of the vierbein-like fields ων,a(s−1) , the extra fields ων,b(t),a(s−1) with t ≥ 2 correspond to t-order derivatives of the vierbein-like fields, thus bringing in higher derivatives in the free action. They also turn out to be necessary for a closed higher spin algebra.29

As to the actual contents of FF3, the introduction of the paper briefly reviews the, at that time, recent progress with higher spin interactions is AdS of the early 1987 papers FF1–CI2, in particular, the higher spin superalgebras and the interactions to cubic approximation. Then two “shortcomings” of the work so far is stated. First, there is the gauge algebra closure problem. In particular, the higher spin superalgebra is expected to be off-mass-shell nonclosed as is the case already for ordinary supergravity in the MacDowell–Mansouri formulation [reference to our ref. [102]]. The standard way to for closing [the] gauge algebra is to introduce appropriate auxiliary fields which are trivial on-mass-shell [. . .] but possess non-trivial off-mass-shell gauge transformation laws.

Second, there is the extra field problem. Although the extra field problem comes in second of the two shortcomings, it is clear from the context that it is really the driving force 26 Note that the indices are subject to a number of symmetry and trace conditions that we do not repeat here. 27 I refrain from using the word “symmetry” here without further discussion of what that could mean. 28 The quotes are from the last paragraph of FF1 on page 746. 29 Within the modern understanding of the theory, the spectra of fields and their jet-space structure, follows from the underlying FDA-Lie∞ theory. See [337] and subsequent work by the same authors.

306 � 8 Archaeology of the Vasiliev theory behind the introduction of auxiliary fields. In the paragraph of the introduction relating to the problem, the text notes that the constraints of FF1 were used in the CI papers, then continues with: Although such an approach is admissible when working with algebraic constraints, the situation seems to be more natural if some action exists leading to both the dynamical motion equations and constraints for extra fields. In particular, one can expect that the nontrivial problem of constructing an appropriate nonlinear generalization of the linearized constraints of ref. [FF1] will be simplified considerably in that case, the problem which is very important for the analysis of higher orders in the interaction.

This is then the motivation for finding auxiliary fields that can be used to construct an action that leads to the constraints for the extra fields. Such an action was not found, not then, and not later. There are actually no hints in FF3 of the form such an action may take.30 What is treated in the paper is what is called “new consistent curvatures” for auxiliary fields, linearized on AdS, and the related gauge transformations. There is also a long section (no. 3) on actions for the auxiliary fields. The auxiliary fields are denoted by aν,α(n)β(m) , where n ≥ 0, m ≥ 0, n − m = k ̇ and k = 0, ±1, ±2, . . . . In contrast to the fields ω(n, m), the “spectrum” of fields a(n, m) is parameterized by n − m = k rather than m + n = 2(s − 1), it is thus countable infinite. A further difference is the curvatures and their transformations, and subsequently the actions that are proposed for the auxiliary fields. The linearized curvatures take the following form: = DμL aν,α(n),β(m) − iλhνγδ̇ a Alνμ,α(n),β(m) ̇ ̇

γ δ̇ ̇ μ,α(n) ,β(m)

+ iλnmhναβ̇ aμ,α(n−1),β(m−1) − (ν ↔ μ) ̇

(8.21)

to be compared to the linearized curvatures for the physical fields given above in formula (8.13). We will comment further one the particular structure of the these two kinds of curvatures below in connection with the A3 paper. In Section 3 of the paper, actions for the auxiliary fields are discussed. An ansatz is made, quadratic in the curvatures (8.21), similar to the action (8.16) for the physical fields. It is then shown that the coefficients in the ansatz can be chosen so that the auxiliary fields have zero degrees of freedom. All in all, the construction is such that the auxiliary fields do not possess any nontrivial dynamical degrees of freedom. This is discussed in three technical sections (Sections 4–6) of the paper (which also discuss the flat limit). In this context, there is one 30 Actually, for massless fields with spin beyond 2, no manifestly covariant actions whatsoever are known, if minimal coupling to gravity is required. The light-front cubic chiral theory is the exception (see Section 6.5.3). One may wonder if there is something fundamental going on here. I was made aware of this curious, but quite obvious, observation by E. Skvortsov. At the time of writing, the covariantized cubic chiral theory is only known at the field equation level.

8.2 The free fields sequence

� 307

passage in the paper that relates to the role played by the auxiliaries. In the quote below, [***] refers to the linear equations of motion of the auxiliary fields that we do not reproduce here. They are equations (3.6) and (3.7) in the paper. Obviously, this does not mean a triviality of the proposed auxiliary fields from the point of view of their possible physical applications. Perhaps, in nontrivial higher-spin theories such fields may generate in the local fashion some nonlocal effective interactions of higher spins. Actually, the proposed auxiliary fields do not correspond to any particles. However, if one will succeed in constructing a consistent interaction of higher spin fields with auxiliary ones, eqs. [***] will be modified in some way and, in particular, they can obtain a nontrivial right-hand side depending on the higher spin fields. As a result, the solution of these equations, which must be unique (under appropriate boundary conditions) as the auxiliary fields do not possess their own degrees of freedom, will express these fields in terms of the physical ones. Generally, this relation should be nonlocal since eqs. [***] are differential equations. Thus, integration over the auxiliary fields in a functional integral can be nontrivial for the interacting case, and the presence of the auxiliary fields can be important for introducing a nontrivial higher spin interaction.

At this stage of theory development, the remarks made in the quote above, are of a general nature and may not yet have been substantiated by any explicit calculations, but they are reasonable as they stand. What is a bit puzzling though, is that no explicit mention is made of the extra field problem here. This is the one place in the paper that one would expect some discussion of the relation between the extra fields and the new auxiliary fields. There is one comment on the relation between the extra fields and auxiliary fields in the very last paragraph of A1 before the concluding section. Also, the conclusion of A3 contains some general comments, basically expressing the hope that the auxiliary fields should solve the extra field problem and the off-shell nonclosure of the gauge algebra. But there are no hints at how this could come about. There is one curious aspect of the theory developed in this paper. The action for the auxiliary fields, and the corresponding field equations, contain nontrivially only the fields a(u, v) and a(u + 1, v + 1). In order to have “[. . .] a closed system of equations for the whole set of fields a(n, m) with n ≥ 0, m ≥ 0, n − m = k for any fixed k [. . .]”, a new set of constraints are introduced that are analogous to the constraints introduced in FF1. It is not phrased so, but it seems that the extra field problem is reproduced once again, but now for the auxiliary fields themselves. The similarity between the equations for the auxiliary fields and the original higher spin fields is indeed noted in the last paragraph of Section 3 of the paper. To be critical, there is at this stage not very much that speaks for these auxiliary fields a(n, m), except what comes later, the “unification” with the higher spin fields ω(n, m) within the higher spin superalgebras of A3. Apart from this, it seems that much of the hope for the importance of the auxiliary fields was pinned to the corresponding role played by auxiliary fields in the Sohnius–West approach to supergravity [349], something that is referred to both in the introduction and conclusion of the paper. As to what came instead of an action leading to the constraints, we will return to this when reviewing the A3 paper in Section 8.3.3 below.

308 � 8 Archaeology of the Vasiliev theory

8.3 The algebra sequence of papers The higher spin superalgebras of Fradkin and Vasiliev played a central role for the AdS approach to higher spin interactions. They provided the kinematic gauge theoretic basis for the theory as we discussed in Section 8.2.1 on the paper FF1. Before reading the algebra sequence of papers, one should be aware of the fact that results of the FDA2 paper partly invalidates the relevance of the algebras of the A1 and A2 papers for an eventual application to interacting higher spins. The reasons for this will be reviewed below in connection to the papers A3 and A4. Essentially, the auxiliary fields from FF3 must be taken into the structure. However, the bosonic subalgebras of A1 and A2 are still relevant. Furthermore, reading the algebra sequence of papers in their order of appearance, may lead to some confusion for the reader as there are quite a few algebras in circulation. It helps to read papers from all sequences in their order of appearance (see the time-line of Table 8.1), but confusion may still be felt. The papers contain long technical stretches where one may loose track of what is actually discussed. Let us therefore take advantage of hindsight and imagine a time travel in 1987. Time travel to Section 5 of the paper A3 (from January 1987 to May 1988) At this stage, it was clear that one needed an algebra including elements corresponding to the auxiliary fields of FF3. Concrete oscillator type realizations of the algebras were also in place. Restricting to linear and quadratic polynomials in the oscillators, ordinary AdS symmetry algebras could be constructed à la [308, 350, 319]. Allowing polynomials of arbitrarily high order gave realizations of the higher spin superalgebras. Let us jump in and have a look at the overall structure. The complex superalgebra31 shsa(1, C) is constructed from two-component spinorial operators q̂α and rβ̂ ̇ obeying Heisenberg-type commutation relations [q̂α , q̂β ] = 2iℏϵαβ

[rα̂ ̇ , rβ̂ ̇ ] = 2iℏϵα̇β̇

[q̂α , rβ̂ ̇ ] = 0

(8.22)

{r,̂ R̂β̇ } = 0

(8.23)

and two operators Q̂ and R,̂ called Klein operators, obeying {Q,̂ q̂α } = 0

[Q,̂ rβ̂ ̇ ] = 0 [R,̂ q̂α ] = 0

and [Q,̂ R]̂ = 0 Q̂ 2 = 1

R̂ 2 = 1

(8.24)

As we will see, the Klein operators parametrizes the “doubling of fields” that was to be a consequence of the analysis of A3 when auxiliary fields were included. The actual construction of the Lie superalgebra starts by first defining an associative algebra with arbitrary elements of the form

31 Key to notation: super higher spin auxiliary.

8.3 The algebra sequence of papers

τ(q,̂ r,̂ Q,̂ R)̂ = ∑

n,m;A,B

̇ 1 AB,α(n),β(m) ξ (Q)̂ A (R)̂ B q̂α1 ⋅ ⋅ ⋅ q̂αn rβ̂ ̇ ⋅ ⋅ ⋅ rβ̂ ̇ 1 m n!m!

�

309

(8.25)

where A and B are summed over 0 to 1. The parameters ξ are assumed to be completely symmetric in all their indices, thus the operators in the expansion are Weyl ordered. The doubling of the physical sector being given by (Q)̂ A (R)̂ B = 1 and (Q)̂ A (R)̂ B = Q̂ R,̂ while the auxiliary sector is given by (Q)̂ A (R)̂ B = Q̂ and (Q)̂ A (R)̂ B = R.̂ The Lie superalgebra is defined, in terms of commutators and anticommutators, by first fixing a bosonfermion automorphism f , ̂ = τ(−q,̂ −r,̂ Q,̂ R)̂ f (τ(q,̂ r,̂ Q,̂ R))

(8.26)

so that even and odd elements of the algebra correspond to whether n + m is even or odd (as usual). The reality conditions read (q̂α )† = rα̂ ̇

(rβ̂ ̇ )† = q̂β

Q̂ † = R̂

R̂ † = Q̂

(8.27)

These reality conditions take us to the real algebra shsa(1). The extended algebras are constructed by introducing further Clifford generating elements ψ̂ i obeying anticommutators {ψ̂ i , ψ̂ j } = 2δij

(8.28)

and zero commutators with all other operators. The construction will be further discussed below in connection to the A3 paper. To connect with the A1 paper, the algebras considered there correspond to dropping the Klein operators.

Further orientation can be offered by the following observations. The paper A1 constructs N = 1 higher spin superalgebras ab initio from a set of basic hypotheses. The oscillator realization (to be used in all subsequent work) is introduced in A2 along with the so-called symbol calculus (also to be of continual use, although the actual designation “symbol calculus” was dropped). The main focus of the A2 paper is on extended algebras. The paper A3 sets out to generalize the construction of A1 to algebras containing auxiliary fields, but the focus is strongly shifted to oscillator realizations (the A1-type deliberations moved to an Appendix). The A3 paper only treats nonextended algebras. The two papers A4 (N = 1) and A5 (N > 1) address a problem of unitarity of representations of superalgebras raised in A3, as well a problem occurring in FDA2 having to do with the need for auxiliaries. 8.3.1 The A1 paper The algebra sequence of papers begins with the long paper [196]. It builds on the analysis of the free field theory of [339] where linearized curvatures was constructed. The algebras constructed are denoted by shsρ (1) (superhigher spins) where ρ takes the values 0 or 1, corresponding to two real forms of an underlying complex algebra shs(1, C), and the 1 denotes nonextended N = 1 supersymmetry. Extended algebras are

310 � 8 Archaeology of the Vasiliev theory considered in the subsequent papers in the series. The maximum finite-dimensional subalgebra is the N = 1 osp(1, 4) AdS superalgebra. The algebras are considered as rigid symmetry algebras G of the vacuum of the theory. In the introduction to the paper, it is noted that it remains to find the related local symmetry algebra L of the interacting theory. The authors point out the difference between Yang–Mills theories and gravity theories in this respect. It is interesting to quote at some length. [. . .] the question of the structure of local symmetries has two related aspects in theories incorporating gravity. The first concerns the rigid symmetry G which is a symmetry of the vacuum. This symmetry generates geometrical entities (connections and curvatures) that are used to describe the theory. The second aspect, which is more important, is the determination of the true local symmetry L leaving invariant an action of the theory. The relation between G and L is simple in nature for internal symmetries only, i. e., for Yang–Mills gauge fields when localization occurs in the standard fashion. In the more general case of theories of gravity and supergravity, the local transformations, corresponding to L, differ from the usual (Yang–Mills) gauge transformations generated by G by some terms that are proportional to the curvatures of G. Moreover, in such theories an action is determined simultaneously with the determination of L. So, if the symmetry G provides the main geometrical quantities of the theory necessary for describing the theory, the true local symmetry L has to be found in the procedure for constructing consistent dynamics.

This insight is referred to the development of supergravity, and a list of references are given (our references [102, 351, 352, 353, 354]). For spin-1 theories, the algebras G and L coincide. This means that the (global) gauge algebra exists independent of the fields. Gauging the global algebra motivates the introduction of the fields. This is a kinematical step. The dynamical step of finding the invariant action for the gauge fields is of course helped by the knowledge of the local algebra, but does not follow without further assumptions or input. However, this step is fairly simple for spin 1 where no amendments to the algebra G is needed. Gauging does not have to be followed by deformation. For spin 2, supposed to yield gravity, the issue is already quite convoluted, and we have discussed aspects of it in Chapter 4 of our Volume 1, and also in Section 2.5 of the present volume. It is important to examine the assumptions, explicit and perhaps hidden, contained in construction of the Fradkin–Vasiliev higher spin algebras, because it turns out that the program fails in the sense that no action is found for the theory.32 Instead, the “Vasiliev equations” are what results when the consequences are worked out. But we are running ahead of ourselves. What is studied in the first algebra paper are candidates for the higher spin algebras corresponding to G. One of the main results of the paper is the expression for the higher spin curvatures Rμν,α(n),β(m) expressed in terms of a tower higher spin fields ̇

32 At least no generally accepted action. The problem of finding an action is technically complicated as well as contentious.

8.3 The algebra sequence of papers

� 311

Rμν,α(n),β(m) = 𝜕μ ων,α(n),β(m) − 𝜕ν ωμ,α(n),β(m) ̇ ̇ ̇ + ∑ Cnm (p, q, s, k, l, t)α(p, q, s, k, l, t)ωμ,α(p)γ(s),β(k) ̇ ω ̇ δ(t) pqsklt

̇ γ(s) δ(t) ̇ ν,α(q) ,β(l)

(8.29)

In this formula, Cnm are combinatorial factors, while the α are structure constants of the higher spin algebra, the explicit form of which are determined in the paper: Cnm (p, q, s, k, l, t) = is+t−1

n!m! δ(n − p − q)δ(m − k − l) p!q!s!k!l!t!

󵄨 󵄨 α(p, q, s, k, l) = δ(󵄨󵄨󵄨(p + k)(q + l) + (p + k)(s + t) + (q + l)(s + t) + 1󵄨󵄨󵄨2 )

(8.30) (8.31)

The delta’s in Cnm connect the number of undotted and dotted indices on R with the summation over the corresponding indices on the right-hand side. The nontrivial structure of the algebra is captured by the α’s (the notation | ⋅ |2 means a modulo 2 addition). As we will see below, the Cmn are actually part of the ansatz for the algebra. Formulas with expressions like these, and more complicated ones in the extended case, and the case with auxiliary fields, abound in the papers. The result for α may look quite opaque, but a verbal explanation is given on page 75 of the A1 paper: α(p, q, s, k, l, t) = 1 if there are at least two odd numbers among the numbers (p + k), (q + l) and (s + t), and α(p, q, s, k, l, t) = 0 otherwise.

We will examine the α coefficients in Section 8.3.3 when discussing the A3 paper.33 The derivation of the structure constants is based on five hypotheses. The strategy of the approach can be discerned from the third section of the A1 paper. Let us start by focusing on this before looking at the hypotheses themselves. The main strategy A Lie superalgebra can be defined by generators TA obeying [TA , TB ] = U CAB TC

(8.32)

There is a one–one correspondence between the generators TA and the parameters ξ A so the algebra can also be expressed in terms of the parameters. This is done by writing a general element in the algebra (near

33 It is a bit curious that the authors do not introduce some more compact notation for the structure of the algebras. Instead, the very same expressions, sometimes reaching over several lines of formula space, are repeated over and over. Once understood, the higher spin algebras are not so complicated. Unfortunately, in my opinion, the basic simplicity was buried under complex descriptions of computations, and heavy presentation, in many of the original papers.

312 � 8 Archaeology of the Vasiliev theory

the identity) as34 T = ξ A TA and working out the commutator between two arbitrary elements T1 = ξ1A TA and T2 = ξ2A TA . The result is [ξ1 , ξ2 ]A = U ABC ξ1B ξ2C

(8.33)

The commutator is antisymmetric and obeys the Jacobi identity. One should understand that in the formula (8.33), the right-hand side is a definition of the meaning of the left-hand side. This is clear from its derivation. The approach pivots on the following gauging assumption: to each parameter ξ A there is a gauge field A A and gauge transformations ων and the corresponding curvatures Rνμ A Rνμ = 𝜕ν ωAμ − 𝜕μ ωAν − [ων , ωμ ]A

δωAν

A

(8.34)

A

= 𝜕ν ξ + [ων , ξ]

(8.35)

What the approach reaches for is to find the field equations, or better still, the action of the theory expressed through these curvatures, possibly or likely, in the process deforming the initial global gauge algebra. The A3 paper elaborates a bit by noting that it follows from the Jacobi identities that the curvatures transform homogeneously A δRνμ = [Rνμ , ξ]A

(8.36)

A ϵ νμρσ (𝜕ν Rμρ + [ων , Rμρ ]A ) = 0

(8.37)

and obey Bianchi identities

The reason for expressing the algebra in terms of the gauge parameters is not stated explicitly, but it is quite clear from the context. It is the gauge parameters ξ that are given by the free field theory (although their physical nature beyond spin 2 is not clear). The nature of the generators T is unknown, and they may perhaps best be thought of as abstract entities. Later in the development, they will be represented in terms of oscillators (as we hinted at above in the [Time travel...] box). Thus, having the gauge parameters ξα(n),β(m) of the free theory, the problem is how ̇ to combine them into a commutator algebra of the type (8.33) satisfying the Jacobi identities. In order to solve that problem, an ansatz is set forth: [ξ1 , ξ2 ]α(n),β(m) = ̇

∑

p,q,s,k,l,t

is+t−1

n!m! δ(n − p − q)δ(m − k − l) p!q!s!k!l!t! ̇ γ(s) δ(t) ̇ 2α(q) ,β(l)

× α(p, q, s, k, l, t)ξ1α(p)γ(s),β(k) ̇ ξ ̇ δ(t)

(8.38)

34 This is denoted the “Grassmann shell” by the authors of the paper and referred to a book in Russian by F. A. Berezin. See [12] of A1. Our reference [355] is an English translation. See also the box [Some special notions...] in our Section 8.3.2.

8.3 The algebra sequence of papers

313

�

This formula may look a bit daunting, but can be disentangled as follows. First, as the authors note, the factor is+t−1 (n!m!/p!q!s!k!l!t!) is for convenience (combinatorics). This is precisely the factor (8.30) in the formula for the curvatures (8.29). Second, one may think of taking two parameters ξ1 and ξ2 with arbitrary numbers of undotted and dotted indices and combining them into a contribution to the commutator with n undotted and m dotted indices. Focus on the undotted indices. Split the indices on ξ1 into p + s and on ξ2 into q + s, where the s indices are to be contracted over. Clearly, then p + q must be equal to n explaining the factor δ(n − p − q). Summing over p, q, s ensures that all such contributions are included. To be a little more specific, let us say that we want to combine two parameters with n1 = p + s and n2 = q + s undotted indices, respectively. Then n1 + n2 = n + 2s so that the combined number of undotted indices on the two parameters must exceed n by an even number or zero. The same reasoning applies to the dotted indices and the k, l, t summation variables. The coefficients α(p, q, s, k, l, t) are unknown functions of the nonnegative integers p, q, s, k, l, t. Those are the ones to be determined as they fix the structure of the algebra. In order to do that, Fradkin and Vasiliev require the Jacobi identities to hold. The problem cannot be solved without further assumptions about the algebra. These assumptions, called Hypothesis I–V, are both of a physical nature and a technical nature. Actually, the ansatz (8.38) written above is already guided by the Hypotheses I and II. The hypotheses are phrased in terms of generators T but the implementation is in terms of parameters ξ. The Fradkin–Vasiliev Hypotheses The hypotheses concern a tentative higher spin superalgebra g. Our formulation here stays close to the original, but is not verbatim.35 I It contains the Lorentz subalgebra oL (3, 1) and splits into a direct sum of finite-dimensional represeni tations of oL (3, 1). The generators are realized as multispinors Tα(n), with i enumerating generators ̇ β(m) of the same type. The commutators with the Lorentz generators Tα(2),β(0) and Tα(0),β(2) are ̇ ̇

II III

i i [Tγ(2),β(0) ] = nϵαγ Tγα(n−1), ̇ , Tα(n),β(m) ̇ ̇ β(m)

(8.39)

i i [Tα(0),δ(2) ] = mϵβ̇δ̇ Tα(n), ̇ , Tα(n),β(m) ̇ ̇ δ̇ β(m−1)

(8.40)

Any representation of oL (3, 1) occurs only once. That is i = 1. This corresponds to a N = 1 superalgebra. This hypotheses is for convenience and may be, and will be, relaxed. The algebra g contains the AdS Lie algebra spAdS (4) as a subalgebra which in turn contain oL (3, 1). The linearization of the curvatures corresponding to g with respect to spAdS (4) should give the linearized higher spin curvatures of FF1. (This is part of the “initial data” assumption.)

35 For instance, we keep the names given to the algebras, but change “de Sitter” to “anti-de Sitter”.

314 � 8 Archaeology of the Vasiliev theory

IV

The algebra contracts to the Lorentz algebra in the following way: [Tαγ(n),β(m) , Tα γ(n),β(m) ] = c(n, m)Tα(2),β(0) ̇ ̇ ̇

[Tα(n),δ̇β(m) ,T ̇

V

α(n)

,δ̇

̇ β(m)

] = d(n, m)Tα(0),δ(2) ̇

(8.41) (8.42)

with the coefficients c(n, m) and d(n, m) different from zero for all n ≥ 0 and m ≥ 0. This hypothesis is said to be technical. There exists an infinite sequence of algebras gk (k = 1, 2, . . . , ∞) for which the spinorial indices take values 1 . . . 2k. The structure constants do not depend explicitly on k and g = g1 satisfies Hypotheses I–IV. This means that the fact that the spinors are two-component is irrelevant. This hypothesis is clearly of a technical nature.

The authors discuss the nature of the hypotheses. The first three have physical content while the last two are technical. Hypotheses I and in particular III, relate to the “initial data” provided by the FF1 paper. Hypothesis I amounts to there being one algebra generator for each gauge field ωiα(n),β(m) . This is the standard kinematical gauging assumption. ̇ Hypothesis III is more specific. First, it says that the theory contains gravity described by the fields ων,α,β̇ , ων,α(2),β(0) and ων,α(0),β(2) ̇ ̇ . Second, it fixes the specific relation of the algebra g to the linearization of the paper FF1 through the linearization of equations (8.10)–(8.12). In this sense, the paper FF1 provides “initial data”, as the authors write, to the determination of g. As for Hypothesis II, there is a quite extensive discussion. Assuming just one gauge generator per spin is clearly for convenience. The authors consider it natural that there should exist higher spin superalgebras generalizing the extended AdS superalgebras osp(N, 4), so the hypothesis corresponds to taking N = 1. In this case, the higher spin fields do not carry any internal indices.36 The technical Hypotheses IV and V are discussed where they are used in the derivation of the algebra in the paper. Hypothesis IV is needed for the uniqueness of the solution, and Hypothesis V simplifies the solution of the equations following from the Jacobi identities. The introductory Section 2 of the paper ends with a discussion of why any theory of higher spin interacting with gravity most likely must contain an infinite number of higher spin fields with increasing spin.37 The philosophy behind the work is captured by the following quote:

36 It is curious that spin 1 sits a bit awkwardly in the Fradkin–Vasiliev scheme, something that we will come back to below. Here, the authors note that the solution for the superalgebra splits as shs′ (1) ⊕ o(2) with shs′ (1) simple and o(2) corresponding to the spin-1 generator Tα(0)β(0) ̇ , i. e., without indices.

37 The authors attribute the first realization of this circumstance to C. Fronsdal [148]. This is correct, Fronsdal understood this first. By the mid-1980s, it was a common point of view among the small number of researchers into massless higher spin interactions, an insight obtained from various attempts; see, for instance, [2, 151, 155].

8.3 The algebra sequence of papers

�

315

In the authors’ opinion, the situation is most satisfactory from an aesthetic point of view, when (elementary) particles of all spins exist in nature but for one reason or another only lower-spin fields can be detected at low energies. We do not know any arguments against an infinite number of higher-spin fields before their dynamics is not [sic] understood in detail.

The actual derivation of the structure of the higher spin algebra is in Section 3 of the paper with details relegated to two Appendices. It amounts to computing the Jacobi identities by applying the ansatz (8.38) twice. The description of the calculation runs over 20 pages in all. In very coarse outline, what is done is the following. The derivation of the gauge algebra in very coarse outline First, note that the ansatz (8.38) is the most general in agreement with Hypotheses I and II. Second, the object [[ξ1 , ξ2 ], ξ3 ]α(n),β(m) is computed in Appendix A of the paper. Then an expression for ̇ the Jacobi identity can be written (formula (3.13) of the paper). This involves a sum over a set of coefficients A depending on 12 nonnegative integers multiplied into a products of ξ1 , ξ2 and ξ3 with the dotted and undotted indices (their number depending on the arguments of A) contracted in various ways. The coefficients A in their turn are sums of three terms of products of the α structure coefficients of the algebra.38 Much of the solution to the equations are relegated to Appendix B of the paper. At this stage, a technical problem occurs. Due to the two-spinorial nature of the gauge parameters, there are Fierz-type identities among the products of ξ1 , ξ2 and ξ3 with the result that not all the equations following from the Jacobi identities are independent. Therefore, one cannot without further assumptions set all coefficients A to zero and solve the quadratic equations for the α structure coefficients. This further assumption is Hypothesis V. With k of Hypothesis V taken as k ≥ 2, there are no Fierz identities. Assuming that the structure of the algebra does not depend on k, the simpler equations A = 0 (without Fierz-type identities) should hold also for k = 1 (this is the force of the hypothesis). In the third step, the equations for the α’s are solved based on information provided by Hypotheses I, III and IV. This is the so-called “initial data”. The term can be understood in the sense that all three hypotheses concern commutators in the algebra involving the spin-2 part of the algebra. In effect, one gets equations for the α(p, q, s, k, l, t) equal to 1 or 0 with various combinations of the integers set to zero. This allows for the quadratic equations A = 0 to be solved linearly. Hypothesis I concerns the almost trivial fact that the higher spin generators are supposed to transform as Lorentz multispinors. Hypothesis III is directly related to the linearization around AdS space-time of the paper FF1. Hypothesis IV is deemed “technical” by the authors, and they write that they have not been able to solve the equations without it. It is used in the 11 pages long Appendix B where the actual application of the initial data to the solution is done.

The result is the higher spin superalgebra shs(1, C). It is complex since no reality assumptions are imposed. It is determined by the structure coefficients being given by formula (8.31), which we repeat here 󵄨 󵄨 α(p, q, s, k, l, t) = δ(󵄨󵄨󵄨(p + k)(q + l) + (p + k)(s + t) + (q + l)(s + t) + 1󵄨󵄨󵄨2 )

(8.43)

38 We do not copy these expressions here. The explicit formulas seem to have no more obvious structure than what is conveyed by the verbal description given here. They clearly have to be worked with at a detailed level to make sense.

316 � 8 Archaeology of the Vasiliev theory where the function |n|2 is defined by |n|2 = n − 2[n/2] and [n/2] stands for the integer part of n/2. Although the solution looks quite simple, one may perhaps venture that it is not very revealing either. As already noted, we will analyze the consequences for these structure relations further on in Section 8.3.3 when discussing the A3 paper. It is inherent in the approach—setting up a gauge algebra of all spins—that the algebra do indeed mix different spins. So, although it remains to find the dynamics, presumably in the process deforming the algebra, already at this kinematical gauging stage there is some “coupling” between fields of different spin. Having so understood the formula (8.38) and the ideas behind determining the coefficients (8.43), let us note that there is actually a slight shift of perspective—that may go unnoticed—in rewriting the algebra in terms of the parameters rather than the generators. When the generators of a Lie algebra are known, the commutator of any two generators can be computed, and the result must be a linear combination of generators, exactly as the algebra stipulates. However, in terms of parameters, there is no way of calculating the object [ξ1 , ξ2 ]A directly. As noted above, the formula (8.33) defines its meaning. This may actually raise a deeper question. A question of equivalence The step from the formula (8.32) to (8.33) is a one-line calculation. It may superficially be seen as showing that (8.32) implies (8.33). However, it is more appropriate to consider the calculation as providing a definition of the bracket [ξ1 , ξ2 ]A , as we have already noted. This raises the question of the equivalence of the two ways of writing the Lie-algebra. Now one may object and say: Wait, if the generators are viewed abstractly, then there is no way of directly computing their commutators either. Then the right-hand side of (8.32) may only be viewed as a definition of the commutator bracket. Furthermore, examining the “one-line” calculation in more detail, it is seen that it hinges on the parameters commuting (or anticommuting when relevant) with the generators and with themselves. What comes into play in both ways of writing the algebra is the requirement that the Jacobi identity must be satisfied. It is the Jacobi identity that fixes properties for the structure constants U CAB . For the classical Lie algebras, all this is well understood, and causes no problems. It may however be that in the higher spin case, these questions may have to be reconsidered; see Section 4.2.1.

Section 3 of the paper ends by writing down the resulting formulas for the algebra, the gauge transformations and the curvatures. Using a compact notation Umn (p, q, s, k, l, t) = Cnm (p, q, s, k, l, t)α(p, q, s, k, l) for Cmn with α from (8.30) and (8.31), the formulas can be collected as [ξ1 , ξ2 ]α(n),β(m) = ∑ Unm (p, q, s, k, l, t)ξ1α(p)γ(s),β(k) ̇ ̇ ξ ̇ δ(t) δωνα(n),β(m) = Dν ξα(n),β(m) = 𝜕ν ξα(n),β(m) ̇ ̇ ̇

̇ γ(s) δ(t) ̇ 2α(q) ,β(l)

̇ γ(s) δ(t) ̇ α(q) ,β(l)

+ ∑ Unm (p, q, s, k, l, t)ωνα(p)γ(s),β(k) ̇ ξ ̇ δ(t)

(8.44)

(8.45)

8.3 The algebra sequence of papers

̇ γ(s) δ(t) ̇ α(q) ,β(l)

= ∑ Unm (p, q, s, k, l, t)Rνμ,α(p)γ(s),β(k) δRνμ,α(n),β(m) ̇ ̇ ξ ̇ δ(t)

� 317

(8.46)

where the summation over p, q, s, k, l, t is to be understood. Sections 4, 5 and 6 of the paper A1 are technical in nature, treating Hermiticity conditions and parity transformations, subalgebras and contractions of the algebra. From Section 4, we record the result that there are two inequivalent real forms of the algebra shs(1; C) denoted by shsρ (1) parameterized by ρ = 0, 1. We move on to the last Section 7 before the conclusions. Section 7 is about higher spin interactions. It is written as a preliminary discussion of the interaction problem, and some questions are referred to as “currently under investigation”. It is not clear whether this refers to issues that was considered as settled in the cubic interaction papers, or to issues that goes beyond that stage of investigation.39 Anyway, reading Section 7 of A1 is of considerable help in understanding the CI papers. The section starts by discussing the need to deform the higher spin algebra. An action, Hermitian and parity invariant with respect to the analysis of the preceding sections is proposed S=

̇ 1 in+m+1 β(n, m)ϵ(n − m) ∫ d 4 xϵνμρσ Rνμ,α(n),β(m) Rρσ α(n),β(m) ∑ ̇ 2 n≥0,m≥0 n!m!

(8.47)

n+m>0

It is a generalization of the free action of FF1 (see also discussion below in Section 8.4.1). The gauge variation under the transformations (8.46) is computed and it is noted that the variation is nonzero. Thus, the gauge transformations must be deformed. The reader may ask why other kinds of actions are not tried, but the authors write: In our attempts to solve the higher spin problem, we avoid those variants for higher spin actions, which contain the metric tensor explicitly. The reason for this is that the presence of the gravitational metric tensor, related to the massless spin-2 field, destroys uniformity in the description of different higher spins and leads to too broad a spectrum of possible actions, being however, nongeometric in their form and much more complicated than those without explicit use of metrics, as in geometric approaches to gravity and supergravity [references removed].

The paper discusses the gauge variation of the action. It is argued that the terms that are proportional to the gravitational background fields and diagonal in gauge parameters ξ(n, n) can be removed (at least in the lowest order in the dynamical fields). More problematic are the terms with nondiagonal gauge parameters ξ(n, m) with n ≠ m. Here, the extra field problem resurfacea. The last page of the section contains a discussion of this. 39 Note again the chronology of papers in Table 8.1. Reception dates at the journals for the cubic interaction papers precedes the first two algebra papers. As already noted, the reason for this has to do with these papers not being sent for publication before the cubic interaction results were secured.

318 � 8 Archaeology of the Vasiliev theory 8.3.2 The A2 paper In the second paper of the algebra sequence, [172], the higher spin algebra is realized in terms of bosonic oscillators. This both simplifies and clarifies the structure, and offers a way to introduce extended algebras with N > 1. The idea of realizing the algebra in terms of polynomials in oscillators—and consequently in terms of coordinates and momenta (derivatives)—is, in retrospect, very natural. The idea had been considered before in the higher spin literature going back to Fronsdal’s symplectic basis [2]. However, it was first in the Vasiliev AdS setting that the idea came to be realized. One key point was to expand in both creators and annihilators (both coordinates and derivatives) and not just in derivatives that had been done previously. The general introduction to the paper is very brief, amounting to outlining the previous work leading up to and including the A1 paper. It is noted, as we saw above, that the superalgebras of A1 were obtained by solving the Jacobi identities using information from the FF1 paper as initial data. The rationale for this is explained, writing: A priori, the very fact of existence of non-Abelian higher-spin superalgebras satisfying the physical “initial data” seemed to be problematic enough. Therefore, the most direct method was adopted in ref. [A1], and no concrete realization of the higher spin superalgebras was initially assumed in their derivation that provided the most general problem setting.

This motivation for the method of A1 is indeed rational, but it may be suspected that the concrete—and simpler—realizations of A2 were already known to the authors, at least in their general outline. This assumption is supported by the preprint dates—just a month apart—of the two manuscripts as can be seen from Table 8.1. See also a comment in the box below.40 The paper A2 then sets itself two tasks: first the determination of a simple realization of the superalgebra shs(1, 4), and second, the construction of extended higher spin superalgebras shs(N, 4). As the reader may notice, the notation for the algebras is changed. Catalogue of higher spin superalgebras The algebras studied in A1 are denoted by shsρ (1) where the parameter ρ can take two values 0 and 1. These are real forms of the complex algebra shs(1, C). The maximum finite-dimensional subalgebra of shsρ (1) with the N = 1 is the AdS superalgebra osp(1, 4). The 1 in the notation stands for N = 1, preparing for a generalization to higher N, alluded to in a footnote on the first page of the paper that points to A2 (then in press).

40 It is curious that a close reading of the Dirac 1963 paper [314] and the Flato–Fronsdal 1978 paper [191], could have revealed the oscillator representation of the higher spin algebras, without the detour over the quite arduous explicit tensor computations in the A1 paper.

8.3 The algebra sequence of papers

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319

In A2, there is a change in notation. First, it is said that several arguments indicate that it is the real case with ρ = 0 that is of physical interest. Then shs0 (1) is renamed as shs(1, 4), presumably in order for the notation to conform to the pattern of osp(1, 4) of the finite subalgebra. The complex form is denoted shs(1, 4; C). This paves the notational way for the extended algebras shs(N, 4) generalizing the N extended AdS superalgebras osp(N, 4), and their complex forms. The extended algebras discussed in A2 are based on oscillator realizations of the algebras osp(N, M; C). Two alternatives are given: shsE (N, M; C) and shs(N + 1, M; C) to be defined below. The following isomorphisms are proven in A2. For odd N: shs(N, M; C) ∼ shsE (N, M; C) and for even N: shs(N, M; C) ∼ shsE (N −1, M; C)⊕shsE (N −1, M; C). Furthermore, there is an o(2) factor in shsE (N, M; C) ∼ shs′E (N, M; C)⊕ o(2) (see footnote above in the section on the A1 paper). Real forms of the algebras as well as subalgebras are discussed at length in the paper. We will comment on the real forms below. Looking ahead, it turned out the algebras discussed so far were inadequate for higher spin interactions, something that was discovered in the FDA2 paper. The auxiliary fields introduced in FF3 lead to the higher spin superalgebras shsa(1) introduced in A3 (the real form of shsa(1, C)). Further subalgebras of shsa(1) are discussed at length in the algebra paper A3 and in FDA2. The important “admissibility condition” (unitary particle representations matching field theory components) is discussed in the papers A4 (simple algebras) and A5 (extended algebras).

The rest of the introduction is an exposition of the results of the paper. First, there is the realization of shs(1, 4; C) in terms of arbitrary polynomials41 P(̂ q,̂ r)̂ of operators q̂ α and rβ̂ ̇ satisfying Heisenberg-type commutation relations [q̂ α , q̂ β ] = 2iℏϵαβ

[rα̂ ̇ , rβ̂ ̇ ] = 2iℏϵα̇ β̇

[q̂ α , rβ̂ ̇ ] = 0

(8.48)

The product of the operators is assumed to be associative. A parity π, taking values 0 or ̂ q,̂ −r)̂ corresponding to the 1, for the polynomials are defined through P(̂ q,̂ r)̂ = (−1)π P(− oddness of its overall power in q̂ and r.̂ The commutators and anticommutators (called “products” in the paper) are defined through ̂ P̂ 2 (q,̂ r)} ̂ = P̂ 1 (q,̂ r)̂ P̂ 2 (q,̂ r)̂ − (−1)π1 ⋅π2 P̂ 2 (q,̂ r)̂ P̂ 1 (q,̂ r)̂ [P̂ 1 (q,̂ r),

(8.49)

Second, there is a discussion of extended higher spin superalgebras, which is the main object of the paper. These are introduced through oscillator realizations42 of the algebras osp(N, M; C). Consider N “anticommuting operators” ψ̂ i (i = 1, . . . , N) and M/2 canonical pairs of “commuting” operators q̂ aα (a = 1, . . . , M/2).43 The anti-commutators and commutators are 41 In our Section 7.1.5, we saw that the Weyl ordered (symmetric) polynomials of order 1 and 2 form the complex AdS superalgebra osp(1, 4; C). 42 The paper refers to a conference talk by M. Günaydin. An original reference is [308]. The review [319] from 1989 contains material on higher spin algebras as well as further references. 43 Here, “anticommuting” means fermionic and “commuting” means bosonic. The reason for this slightly confusing terminology (since the brackets are nonzero) is explained in the paper at the bottom of page 43. It seems that the chosen terminology is just as confusing as the one avoided.

320 � 8 Archaeology of the Vasiliev theory {ψ̂ i , ψ̂ j } = 2ℏδij

[q̂ aα , q̂ bβ ] = 2iℏϵαβ δab

[ψ̂ i , q̂ aα ] = 0

(8.50)

Here, there is no split over dotted and undotted Greek indices. That comes later. The space of quadratic Weyl ordered polynomials in the operators ψ̂ i and q̂ aα provides a realization of the superalgebra osp(N, M; C). The parity π for the polynomials are now defined by P(̂ ψ,̂ q)̂ = (−1)π P(̂ ψ,̂ −q)̂ corresponding to the oddness of its overall power in q.̂ The commutators and anticommutators are defined through [P,̂ Q}̂ = P̂ ⋅ Q̂ − (−1)πP ⋅πQ Q̂ ⋅ P̂

(8.51)

As is noted in the paper, and providing a link between the N = 1 and N > 1 cases, taking N = 1 and M = 4 here yields an isomorphic realization of osp(1, 4; C). This, in fact, extends to shs(1, 4; C) so that polynomials of overall even power yield an isomorphic realization to the one given above. This comes about since the factor ψ can be factored out and the operators q̂ aα can be linearly recombined into q̂ α and rβ̂ ̇ .44

This argument leads to the higher spin superalgebras shsE (N, M; C) where E stands ̂ ̂ Another possibility is to generfor all polynomials even in the sense P(̂ ψ,̂ q)̂ = P(−ψ, −q). alize the first realization of shs(1, 4; C) by considering all polynomials constructed from the operators in (8.50). These algebras are denoted by shs(N + 1, M; C) where N is the number of ψ operators. Then a number of isomorphisms are discussed (see box above). As regards the parameter M, it is noted that in practice M = 4 because higher M would entail more than one spin-2 field. The restriction comes for the inability to describe interactions for more than one spin-2 field. It is hoped that this restriction will be overcome in a consistent higher spin theory.45 Section 2 of the paper contains general material on various operations, such as automorphisms, conjugations (abstraction of complex conjugation) and involutions (abstraction of Hermitian conjugation) over algebras. Real and complex algebras are discussed. The use of automorphisms to define gradings and subalgebras is discussed.46 The section is concluded with a discussion (Comment 10) of the relation between gauge fields and the Lie superalgebras (see formulas (8.34)–(8.35)). The operator realizations of the higher spin superalgebras are all based on underlying associative algebras. The concepts of first class and second class Grassmann shells of Berezin are discussed.47 Since these notions figure quite conspicuously in the theory, let us highlight some special parts of it here. Most of the more common contents can be found in our Section 3.1. 44 Factoring out ψ effectively means that polynomials of even order become of arbitrary order. 45 It turned out that the case with M > 4 was relevant for the theory in d dimensions, although with a different interpretation; see [197]. 46 This section is actually a very neat review of these topics. 47 In the second class case, the gauge fields and the basis vectors of the algebra may anti-commute as well as commute.

8.3 The algebra sequence of papers

� 321

Some special notions from Section 2 in the A2 paper The concept of Grassmann shells is used in the paper. In the box [The main strategy] above in connection with the paper A1 we introduced what is called the first class Grassmann shell. In this case, the objects of the theory such as gauge fields, gauge parameters and curvatures are expanded over a basis of the Lie superalgebra as, for instance, ων = ωkν ek = ek ωkν , i. e., the fields ωiν commute with the algebra generators ei . For the second class Grassmann shell, one allows for the objects of the theory to anticommute with the basis elements, as for instance, ωiν ej = (−1)π(ei )π(ej ) ej ωiν . The shell is then defined as ων = i −π(ek ) ωkν ek . This definition is said to be more convenient in “many cases”. When in Section 4 of the paper, Lie superalgebras are defined in terms of commutators and anticommutators, this come into play. The interplay between complex and real superalgebras is important in the paper. The following definition is used: If for a real algebra AR and a complex algebra AC , bases {eRi } and {eCi } exist for which the structure constants coincide uijRk = uijCk , then the algebra AC is called a complexification of AR while the algebra AR is called a real form of AC . Real forms can be extracted from complex forms using conjugations σ(a) = a. It is noted that for Lie superalgebras, conjugations σ and involutions μ are in a one-to-one correspondence through μ(a) = −i π(a) σ(a) (called canonically related). The reality condition can then be expressed as μ(a) = −i π(a) a.

In Section 3, the technique of symbols of operators, that is to be used throughout the rest of the development of the theory, is introduced.48 The general formalism is defined using generic operators ẐΩ indexed from 1 to N + M where N of them are “anticommuting” (in the sense intended above) and M (an even number) are “commuting”. They span a complex associative algebra denoted by aq(N, M; C). No explicit formulas are given for the products of the algebra, rather it must be surmised that such derive from the commutators and anticommutators of the generating elements ẐΩ .49 These are given by conditions ẐΩ ẐΦ − (−1)ηΩ ηΦ ẐΦ ẐΩ = 2ℏΛΩΦ

(8.52)

where ηΩ = 1 for 1 ≤ Ω ≤ N and ηΩ = 0 for N < Ω ≤ N + M. The numerical matrix ΛΩΦ is assumed to be nondegenerate and to obey the relation ΛΩΦ = −(−1)ηΩ ηΦ ΛΦΩ . A more concrete form of the operators are also introduced that are subsequently used in the following papers up until FDA3 when a new notation is introduced. The set of operators ẐΩ is split into ψ̂ i , q̂ α , rβ̂ ̇ with indices running as i = 1, . . . , N, α = 1, . . . , M/2 and β̇ = 1, . . . , M/2. Compared to the commutators and anti-commutators of (8.50), there 48 The formalism was later to be condensed into the star product formalism, in the process much of the surrounding concepts and formalism being shed. The star product formalism is discussed in quite a few review articles; see our Section 3.3. 49 The paper does not state it explicitly, but this comes about when assuming a certain ordering (symmetric Weyl ordering) of the generating elements for the basis of the algebra. Reordering a product of two basis elements then produces the structure relations of the algebra. The commutators of the corresponding Lie superalgebra are defined in the standard way.

322 � 8 Archaeology of the Vasiliev theory is now a split into operators with dotted and undotted indices, and the brackets (8.52) read [rα̂ ̇ , rβ̂ ̇ ] = 2iℏϵα̇ β̇

[q̂ α , q̂ β ] = 2iℏϵαβ

[q̂ α , rβ̂ ̇ ] = 0

{ψ̂ i , ψ̂ j } = 2ℏδij [q̂ α , ψ̂ i ] = 0

[rβ̂ ̇ , ψ̂ i ] = 0

(8.53a) (8.53b) (8.53c)

At this stage, one does not have to assume that the number of dotted and undotted indices are the same. That assumption comes in later (Section 5 of the paper) when real higher spin superalgebras are considered. This is related to the Hypothesis V of the paper A1. In practice, the algebra aq(N, M; C) is the algebra of all polynomials constructed out of the operators ψ̂ i , q̂ α and rβ̂ ̇ . The paper next turns to the symbols of operators. Let

us focus it in some detail as it is introduced and used in the Vasiliev theory.50 It is an essential tool in order to work efficiently with polynomial associative algebras. The symbols of operators formalism Vasiliev considers a basis for aq(N, M; C) in terms of homogeneous polynomials of the form ̂ ei(k),α(n), = ̇ β(m)

1 ψ̂ ⋅ ⋅ ⋅ ψ̂ ik ] q̂{α1 ⋅ ⋅ ⋅ q̂αn } r{̂ β̇ ⋅ ⋅ ⋅ rβ̂ ̇ } 1 m k!n!m! [i1

(8.54)

which are symmetrized or antisymmetrized according to the Grassmann nature of the operators. This is called Weyl ordering. A general element â in the algebra is written a(̂ Z)̂ =

̇ 1 ai(k),α(n),β(m) ψ̂ i1 ⋅ ⋅ ⋅ ψ̂ ik q̂α1 ⋅ ⋅ ⋅ q̂αn rβ̂ ̇ ⋅ ⋅ ⋅ rβ̂ ̇ 1 m k!n!m!

∞

∑

k,n,m=0

(8.55)

with c-number coefficients ai(k),α(n),β(m) antisymmetric in the i indices and symmetric in the undotted and dotted spinor indices. Only a finite number of a coefficients may be nonzero in practice in any algebra element. In the general formulas, Z ̂ is a collective notation for the operators ψ,̂ q̂ and r,̂ as is Z for the corresponding classical variables ψ, q and r. Next the “symbols of operators” are introduced. Given an operator a(̂ Z)̂ its symbol is ̇

a(Z) =

∞

∑

k,n,m=0

̇ 1 ai(k),α(n),β(m) ψi1 ⋅ ⋅ ⋅ ψik qα1 ⋅ ⋅ ⋅ qαn rβ̇ ⋅ ⋅ ⋅ rβ̇ 1 m k!n!m!

(8.56)

50 If one only reads the Vasiliev paper, one may come away with the impression that the formalism is wholly due to F. A. Berezin. It however goes back to H. Weyl and E. Wigner with contributions from other authors, among them in particular H. J. Groenewald and J. E. Moyal (see short historical notes in our Section 3.3.3 where further references can be found). This is in fact clear if one follows Vasiliev’s reference to Berezin. The paper A2 refers to the book [355] by Berezin and the paper [356] by Berezin and M. S. Marinov and “references therein”, which indeed do go back to the original work. The symbol calculus can be found in Appendix B of the paper [356].

8.3 The algebra sequence of papers

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323

where ψ, q and r are classical anticommuting and commuting variables (i. e., take ℏ = 0 in formulas (8.53)) The step from a(̂ Z)̂ to a(Z) is just a replacement of basic operators with their corresponding symbols.51 As we have discussed in our Section 3.3, when two algebra elements a(̂ Z)̂ and b(̂ Z)̂ are multiplied, the operators in the result are no longer Weyl ordered. To restore the Weyl ordering and thereby write the result as a sum of elements in the algebra, the basic commutators and anticommutators (8.53) must be used. The symbols of operators formalism offers an alternative way of doing this. Precisely how this works was explained in Section 7.3.2. The symbol for the operator product a(̂ Z)̂ ⋅ b(̂ Z)̂ is denoted by (a ⋆ b)(Z) and an integral formula can be written as (a ⋆ b)(Z) = (−1)N(N−1)/2

ℏN−M ∫ a(Z1 )b(Z2 ) exp(ℏ−1 [(Z, Z1 ) + (Z1 , Z2 ) + (Z2 , Z)])dZ1 dZ2 (2π)M

(8.57)

where dZ = dψ1 ⋅ ⋅ ⋅ ψN d M/2 qd M/2 r and ∫ ψi dψj = − ∫ dψj ψi = δij and the integrals over the q and r are treated as for real variables. The object (Z1 , Z2 ) is a skew symmetric, bilinear form and is given by (Z1 , Z2 ) = Z1Ω Z2Ψ (Λ−1 )ΩΨ . In terms of the concrete variables ψ, q and r it reads β̇

(Z1 , Z2 ) = ψ1i ψ2i + iq1α q2α + ir1β̇ r2

(8.58)

Vasiliev also writes down two differential formulas for the star product but we refer the reader to the paper itself for those (see also our Section 3.3.2).52 To clarify a bit, along the general theory of our Section 3.3, Vasiliev works with Weyl ordered operators. Then, if we denote the Weyl map by W and the symbol map by S, we have the inverse pair of mappings: S: W:

âb̂ → a ⋆ b

(8.59)

a ⋆ b → W (a ⋆ b) = W (a)W (b)

(8.60)

The mappings are inverse to each other, since obviously W (a) = â Weyl ordered, and S(a)̂ = a, where a is obtained from â by replacing Z ̂ → Z, provided â is Weyl ordered (which it is in the Vasiliev theory). The last caveat is needed in order not to run into utter confusion. It is all too easy to think of the symbol map as just replacing basic operators with their symbols. It works, but only for Weyl ordered operators. For products of Weyl ordered operators, formula (8.59) must be used. See our Section 3.3 for further clarification.

The rest of Section 3 of the paper continues with an extensive discussion of automorphisms of the associative algebras. The application of the formalism comes in Section 4 where complex higher spin Lie superalgebras are defined. A certain involutive automorphism f (a(ψ, qα , rβ̇ )) = a(ψ, −qα , −rβ̇ ) is fixed defining parities π in accordance with the usual fermion-boson statistics. For the algebra shs(N + 1, M; C), the first Grassmann shell is used when defining the gauge fields of the theory. For the algebra shsE (N, M; C), the second Grassmann shell is employed. A complication arises in the first case in that the fields and the symbols must obey independent Grassmann algebras leading to complicated commutation relations for general gauge fields ων (Z). In the second case, fields and symbols obey 51 The inverse step, replacing symbols with operators in a Weyl ordered way, is given by the Weyl map. See our Section 3.3.1. It is not used in the paper. 52 The designation “star product” is not used in the paper A2.

324 � 8 Archaeology of the Vasiliev theory the same Grassmann algebra and the general gauge fields become even elements of the Grassmann algebra, i. e., ωE1ν (Z1 )ωE2μ (Z2 ) = ωE2μ (Z2 )ωE1ν (Z1 ). The section also contains computations showing that the curvatures built using the symbol formalism coincides with the ones arrived at in A1 and with the linearized curvatures of FF1. Section 5 treats real higher spin superalgebras as these are the ones that are relevant for physics. The section also discusses parity reversal automorphisms and invariant forms. Real forms are extracted through conjugations σ or involutions μ as mentioned in the box [Some special notions...] above. As is usual, there exist various inequivalent real forms. Only certain of those are of interest: those that are “in accordance with the Lorentz structure” and admit “nontrivial (exact) unitary representations in which Hermitian conjugation is identified with the involution μ extracting the real form under consideration”. The first requirement means that the conjugation or involution (extracting the real form) should interchange dotted and undotted indices by interchanging q̂ α into rα̂ ̇ and vice versa. The second amounts to the involution μ being possible to implement as Hermitian conjugation in a positive definite Hilbert space. These requirements are shown to delimit the possible involutions to the simplest choice of μ(ψ̂ i ) = ψ̂ i , μ(q̂ α ) = rα̂ ̇ and μ(rα̂ ̇ ) = q̂ α . The relevant real forms are denoted by shs(N + 1, M) and shsE (N, M). It is noted that the algebra denoted by shs0 (1) in the paper A1 “[. . .] coincides with shs(1, 4) (equivalently shsE (1, 4)) [. . .]”. The algebra shs1 (1) of A1 corresponds to a real form inadmissible by the requirements discussed here.53 Section 6 of the paper (the last before the conclusion), that we will not review here, concerns subalgebras and contractions of algebras. We will instead continue with the paper A3, which returns to the constructive method of A1 but now includes the auxiliary fields from FF3. 8.3.3 The A3 paper In the A3 paper, the authors construct a new higher spin superalgebra denoted by shsa(1), which at the linearized level generates both the higher spin curvatures of FF1 and the auxiliary field curvatures of FF3. The algebra shs0 (1) is contained as a subalgebra. Only the case N = 1 is treated in the paper. Again, the hope is expressed that this construction will provide a solution to the extra field problem as well a the problem of the nonclosure of the gauge algebra. As the time-line shows, A3 was received at the journal a month after the FDA2 paper that entailed a radical reorientation of the project, but no explicit mention of this appears in A3.54 However, a new potential problem appeared; there was a doubling of the higher spin fields. 53 See discussion starting with the next to last paragraph on page 53 of the paper. 54 There are references to “questions under investigations” in the conclusion of the paper. FDA2 strongly indicated that the algebras allowing for auxiliaries was the right way to go in this approach to higher spin interaction.

8.3 The algebra sequence of papers

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An important and somewhat unexpected feature of shsa(1) consists of the fact that any field of spin s ≥ 1 occurs twice among the gauge fields originating from this superalgebra. Specifically, two massless spin-2 fields are present (this property takes place not only for shsa(1) but also for those its subalgebras, which pretend themselves for the role of superalgebras for higher spins and auxiliary fields).

This is indeed a positively strange feature of the theory. Authors with a “no-go inclination” may very well have scrapped the project at this stage, since two gravitons normally spell disaster. Instead, they continue by writing: This can lead to important cosmological implications if the spin-2 massless fields will serve both as gravitational fields in two worlds whose mutual interaction will be suppressed by powers of the gravitational constant.

The authors return to this speculation in Sections 6 and 7 and in the conclusion to the paper.55 The technical reason for the doubling is outlined in the Appendix, which describes the derivation to the algebra using the constructive method of A1. Essentially, solutions to certain structure coefficients of the algebra turn out to involve two inequivalent sets of parameters. Both has to be included in order to get a real algebra, and this leads to the doubling. Historically, the root of the doubling phenomenon is the extra fields problem, the connection being the auxiliary fields.56 This is perhaps the place to comment on a puzzling aspect of the overall development of the theory. One reason for introducing the auxiliary fields was the hope that they would solve the extra fields problem by providing the constraints expressing them in terms of physical fields. This would, however, require an action. With no such on the horizon, the project swerved into the FDA road. In that approach, the auxiliary fields and the physical fields, together with their extra fields, just occur alongside each other. There seems to be no explicit comment about this in the papers from the time, nor in the latter literature. Going back to a section by section perusal of the paper, Section 2 states the results of the paper. Here, we find a reiteration of the basic gauge-algebraic framework from A1 (see the box [The main strategy] above). Commutators of the algebra and curvatures are given explicitly for the complex superalgebra shsa(1, C). These, as well as the gauge fields themselves, are now decorated by a new index taking discrete values 0 and 1, for ̇ α(n)β(m)

instance, as in TFG for the generators of the algebra. The physical space is given by F = G while the auxiliary space is given by F +G = 1, thus we have the doubling. The algebra shs(1, C) of A1 coincides to the subalgebra of shsa(1, C) corresponding to F = G = 0. 55 One may guess that it was a feature that bothered them.

56 This is written from the perspective of the understanding at the time. In retrospect, the doubling of the fields can be understood as a necessary consequence of the Flato–Fronsdal theorem (see the formulas (7.33) in Section 7.1.4). It is clear that tensoring a supermultiplet of Rac and Di with itself will produce a doubled set of higher spin fields. The doubling can also be seen from the expression (8.25) for a general algebra element.

326 � 8 Archaeology of the Vasiliev theory The algebra—demystified One may certainly find the appearance of the higher spin superalgebras a bit intimidating with their structure coefficients running over several lines of formula space. Here, it is in the form given in Section 2 of the paper A3: [ξ1 , ξ2 ]FGα(n)β(m) = ̇

1 2

∑

pqskltABCD

i s+t−1

n!m! p!q!s!k!l!t!

× δ(|F + A + C|2 )δ(|G + B + D|2 )δ(n − p − q)δ(m − k − l) × [(−1)C(p+s)+D(k+t) − (−1)A(q+s)+B(t+l)+s+t+(p+k+s+t)(q+l+s+t) ] × ξ1AB α(p)γ(s),β(k) ̇ δ(t) ̇ ξ

̇ CD γ(s) δ(t) ̇ 2 α(q) ,β(l)

(8.61)

The structure can be disentangled in the following way. The first line is, of course, just summation over all variables p, q, s, k, t, l from 0 to ∞ and A, B, C, D over 0 and 1 with, not surprisingly, a certain combinatorial factor. The second line connects the left-hand side free variables to the summation variables: the sum of p and q must equal n, and the sum of k and l must equal m. The notation57 | ⋅ |2 means a modulo 2 addition. Thus, for F = 0 the allowed combinations of A, C are 0, 0 and 1, 1, while for F = 1 we may have A, C equal to 0, 1 or 1, 0. Likewise for the variables G and B, D. The third line contains the rest (and most) of the nontrivial structure of the algebra together with the last line which tells how the parameters ξ1 and ξ2 “fuse”, so to speak, into the commutator [ξ1 , ξ2 ] on the left-hand side. Considering the special case F = G = 0 and A = B = C = D = 0, we see that the formula coincides with the structure from the paper A1 quoted in formulas (8.38) and (8.43) above. To see this, use the identity (−1)n − (1−)m = 2(−1)n δ(|n + m + 1|2 )

(8.62)

2

and that (s + t) + (s + t) is always an even number. Now, let us dig deeper. Having two objects with undotted and dotted indices, there are only two ways the respective types of indices can be combined; either symmetrizing them58 or contracting them. Thus, we see that p + q = n undotted and k+l = m dotted indices are symmetrized, while s undotted and t dotted indices are contracted in ξ1AB ξ2CD . The left-hand side of the formula is supposed to be a gauge parameter for a spin s = (n+m)/2+1 ≥ 3/2 gauge field, and the formula tells us how such a gauge parameter for spin s0 say, can be built from gauge parameters corresponding to gauge fields of various spins s1 and s2 . The other way around, having gauge parameters of certain spins s1 and s2 , it tells us what set of spins {s0 } result from the commutator. The actual combinations that occur is given precisely by the third line of the formula. Although obvious, in a way, it is not superfluous to spell it out explicitly. A few examples may also serve to gain intuition. We will do that in the next box below.

It may seem a bit curious that in none of the twenty papers analyzed, are there any examples of consequences of the basic higher spin superalgebra formula (8.61).59 Not that it is difficult to derive such examples, but given the novelty of the subject at the time, 57 Defined as |n|r = n − [n/r]r where [n/r] denotes the integer part of n/r. 58 Antisymmetrizing is trivial in two dimensions.

59 Indeed, such examples, or even general statements relating possible values of the spins on the righthand side with possible values of the spins on the left-hand side of formulas for the higher spin algebras, seem only to have been made in the light-front approach papers.

8.3 The algebra sequence of papers

� 327

such consequences could have been helpful to see for a prospective reader. Indeed, one may conclude that the only thing the algebra does, is to connect sets of spin values for the parameter [ξ1 , ξ2 ] with sets of spin values for the parameters ξ1 and ξ2 . In a sense it should be obvious, but still worth pointing out. The algebra—explicated To gain some further intuition on the algebra, we imagine fixing two spin values s1 and s2 for the parameters ξ1 and ξ2 on the right-hand side of the formula (8.61). We then have spectra of values {(n1 , m1 )} and {(n2 , m2 )} with n1 + m1 = 2(s1 − 1) and n2 + m2 = 2(s2 − 1) for ξ1 (n1 , m1 ) and ξ2 (n2 , m2 ), respectively. For the resulting spin on the left-hand side, i. e., the actual commutator, we use the notation sL with spectra {(nL , mL )}. It will be useful to write sH = s1 + s2 occasionally. We further simplify by choosing the pure FG = 00 sector. For easy reference, the commutator formula then reads [ξ1 , ξ2 ]α(n)β(m) = ̇

1 n!m! ∑ i s+t−1 δ(n − p − q)δ(m − k − l) 2 pqsklt p!q!s!k!l!t!

× [1 − (−1)s+t+(p+k+s+t)(q+l+s+t) ]ξ1α(p)γ(s),β(k) ̇ δ(t) ̇ ξ

̇ γ(s) δ(t) ̇ 2α(q) ,β(l)

(8.63)

As already noted in the box above, the factor in the second line of the formula contains crucial properties of the algebra. Clearly, it only takes values 0 or 2, so it is a modulo(2) function, which is also clear from equation (8.62). Using this equation, the second line can be written in various useful ways, for instance, as60 󵄨 󵄨 2δ(󵄨󵄨󵄨󵄨(p + k)(q + l) + (p + k)(s + t) + (q + l)(s + t) + 1󵄨󵄨󵄨󵄨2 )

(8.64)

The factor 2 multiplies the factor 1/2 in front of the sum, so what the second line of the commutator formula tells us is (p + k)(q + l) + (p + k)(s + t) + (q + l)(s + t) + 1 ≡ 0

mod (2)

(8.65)

Together with the crucial relation (8.65), let us now collect all basic formulas relating the summation variables and spin values. From the structure of ξ1 ξ2 on the right-hand side, we have p + s = n1

(8.66a)

k + t = m1

(8.66b)

q + s = n2

(8.66c)

l + t = m2

(8.66d)

From the formulas relating (n, m) values to spin, we have n1 + m1 = 2(s1 − 1)

(8.67a)

n2 + m2 = 2(s2 − 1)

(8.67b)

nL + mL = 2(sL − 1)

(8.67c)

60 See also formulas (A.3) and (A.4) in the Appendix of the paper A3.

328 � 8 Archaeology of the Vasiliev theory

And finally from the balance of indices on the right- and left-hand sides, p + q = nL

(8.68a)

k + l = mL

(8.68b)

One consequence is now almost immediate. Adding the formulas (8.66) and using the formulas (8.67) and (8.68), one derives sL = s1 + s2 − 1 − (s + t),

(8.69)

sL ≤ s1 + s2 − 1

(8.70)

which implies, since s + t ≥ 0, that61

This formula allows sL = 3 for the pure spin 2 case of s1 = s2 = 2, something we do not expect. To see what spin values sL may actually occur on the left-hand side of the formula for the commutator, we analyze the δ(| ⋅ ⋅ ⋅ |2 ) factor. By adding appropriate combinations of the four formulas (8.66) and using the first two formulas of (8.67), it is possible to eliminate the summation variables p, q, k and l while keeping s and t. The result is the following relation: 4(s1 − 1)(s2 − 1) − (s + t)2 ≡ 1

mod (2)

(8.71)

Thus, unless both s1 and s2 are half-integer, s+t must be an odd integer. This excludes the unwanted spin-3 lefthand side in the pure spin-2 case.62 Speaking about spin 3, let us consider the pure spin-3 case of s1 = s2 = 3. Then equation (8.69) together with the oddness of s+t implies the possible spin values for sL to be 4 and 2. This may at first appear a bit “odd” since one certainly ought to expect spin 3 itself to turn up in the commutator of two spin-3 parameters. However, from the light-front cubic analysis, we know that odd integer spin fields carry internal indices. Let us then take a brief look at the extended higher spin algebras of A2. An explicit formula for the commutator of two higher spin parameters cannot be found in the A2 paper, but the structure coefficients can be read of from the corresponding formula (4.13) on page 50 for the curvatures. The parameters now carry an internal set of antisymmetrized indices denoted by i. If the lefti(f ) hand side [ξ1 , ξ2 ] carries f of them, then the right-hand side, apart from some further numerical and ̇ α(n)β(m)

combinatorial factors and summation over three new integer variables u, v and r, contains the crucial factors δ(f − u − v)δ(n − p − q)δ(m − k − l) 󵄨 󵄨 × δ(󵄨󵄨󵄨󵄨(p + k)(q + l) + (p + k + q + l)(s + t) + u ⋅ r + v ⋅ r + u ⋅ v 󵄨󵄨󵄨󵄨2 ) × ξ i(u) 1

̇ j(r)i(v) γ(s) δ(t) ̇ δ(t) ̇ ξ2 ̇ j(r),α(p)γ(s),β(k) α(q) ,β(l)

(8.72)

where r is a new summation index for the internal indices. The new delta function on the first line connects the number of antisymmetrized indices u and v on right-hand side to the number of indices f on the left-hand side. Crucially, there is now an extra contribution from the number of internal indices to the δ(| ⋅ ⋅ ⋅ |2 ) factor. Let us be concrete and consider the first nontrivial case of N = 2. Then we have, focusing on just the new internal indices, parameters ξ, ξ 1 , ξ 2 and ξ 12 = −ξ 21 . From ξ11 and ξ22 , by antisymmetrizing, we get a new parameter η12 . From ξ11 and ξ212 , by contracting, we get a new parameter η2 . Likewise, from ξ12 and ξ212 , by contracting, we get a new parameter η1 . Working out the details,

61 Note that s + t is also bounded from below so that sL is never negative.

62 Incidentally, this shows that the AdS sp(4) is a subalgebra. Similar arguments including spin 3/2 singles out the N = 1 AdS superalgebra osp(4).

8.3 The algebra sequence of papers

�

329

this yields a realization of su(2). For these three cases, we find that the term u ⋅ r + v ⋅ r + u ⋅ v is always equal to 1. Then redoing the analysis of the δ(| ⋅ ⋅ ⋅ |2 ) factor adding this information, we now find that for both s1 and s2 integer, s + t must be an even integer. Thus, in the pure spin-3 case of s1 = s2 = 3, we now get possible spin values on the left-hand side to be 5, 3 and 1. Including also the “singlet” parameters without internal indices, allows for the spin values 4 and 2 from above. Other combinations of spins s1 and s2 may be studied analogously, and by reintroducing the summation variables p, q, k and l, one may work out the exact combinations of ξ1 and ξ2 (i. e., with specific sets of undotted and dotted indices) producing new parameters η = [ξ1 , ξ2 ].

Having so understood the algebra a little better, it remains to gain some intuition on how the inclusion of the auxiliary fields affects the picture. That can now be done by including the A, B, C, D indices in the analysis. Overall structure of the algebra including auxiliaries The N = 1 algebra with auxiliary fields were quoted in formula (8.61) above. Working along the lines of the two above boxes, we may now start by focusing on the algebra of the AB indices. It is clear that we have a modular addition for this kind of index F ≡A+C

mod (2) and

G ≡B+D

mod (2)

(8.73)

The “physical” fields are indexed by 00 and their “doppelgangers” by 11. The auxiliary fields are indexed by 01 and 10.63 Clearly, as remarked above, the algebra of 00 parameters, is a subalgebra. Also, the 00 and 11 parameters, form a subalgebra in that two 11 parameters yields a 00 parameter while a 00 parameter and a 11 parameter yields a 11 parameter. The auxiliaries, however, do not form a subalgebra. Combining two 01 parameters or two 10 parameters yield a 00 parameter, and one may continue investigating particular cases in this way, using the modular arithmetic of (8.73). Now, as before, the second line of the formula (8.61) for the algebra, governs what combinations of parameters are actually permitted. The crucial delta function can now be written 󵄨 󵄨 δ(󵄨󵄨󵄨󵄨C(p + s) + D(k + t) + A(q + s) + B(l + t) + (p + k)(q + l) + (p + k)(s + t) + (q + l)(s + t) + 1󵄨󵄨󵄨󵄨2 )

(8.74)

This expression certainly looks a bit opaque, but the structure is actually quite easy to disentangle, given the work already done. We want to see what spins sFG may occur on the left-hand side given certain spins s1AB and s2CD on the right-hand side. The terms not involving ABCD again reduces to the relation (8.71) with the spin values indexed accordingly. Then looking back at the formulas (8.66), we find that also the ABCD terms have a simple form AB CD CD in terms of spin parameters n and m. Indeed, they reduce to CnAB 1 + Dm1 + An2 + Bm2 . The m parameters are a bit redundant, so we may want to replace them using formulas n + m = 2(s − 1). Doing this, we arrive at the following relation: CD AB CD AB CD 2 (C − D)nAB 1 + (A − B)n2 + 2D(s1 − 1) + 2B(s2 − 1) + 4(s1 − 1)(s2 − 1) − (s + t) ≡ 1

mod (2) (8.75)

Formula (8.69) still holds, and we may now write it as sFG = s1AB + s2CD − 1 − (s + t) 63 Who is a doppelganger of whom, becomes clear from the subsequent analysis.

(8.76)

330 � 8 Archaeology of the Vasiliev theory

CD As an example, consider only integer spin. Then the modulo relation reduces to (C − D)nAB 1 + (A − B)n2 − 2 (s + t) ≡ 1 mod (2), and in all sectors of the algebra where A = B and C = D, s + t is an odd integer. We leave the reader to experiment with the formulas.

Let us now continue with the contents of the paper. The real form shsa(1) of the algebra is extracted through a Hermiticity condition on the fields, which reads (ωAB ν,α(n)β(m) ) = (−1)An+Bm ωBA ν,β(m)α(n) ̇ ̇ †

(8.77)

Section 2 ends with a short discussion of parity reversal and of bilinear forms on the algebras. It is also noted that the algebras are not simple, containing an o(2) piece (related to spin 1 as it transpires later on). Section 3 of the paper treats AdS subalgebras and linearization. In this case, because of the doubling of fields, there is an arbitrariness in the choice of gravitational sector. It is stated that “without loss of generality”—indicating a choice—one can choose the objects labeled by AB = 00 to correspond to the Lorentz connection part of the gravitational sector. However, from our analysis in the box above, choosing the AB = 11 sector of the algebra would not work, since commuting two 11-labeled parameters yield 00 parameters.64 It thus seems that the AdS subalgebra must be identified with the generators ̇ α(2)β(0)

̇ α(0)β(2)

̇ α(1)β(1)

, T00 and T00 , which is in fact also proved.65 Having fixed the gravitational sector, linearized curvatures can be written based on a background split of the gravitational fields. Apart from notational details, this background split is the one quoted in formulas (8.10)–(8.12) above.

T00

Understanding the linearized curvatures The linearized curvatures are given in the paper by RlFG νμα(n)β(m) = DLν ωFG μα(n)β(m) + ̇ ̇ ×i

s+t−1

(−1)

F+G

∞

∑ δ(n − p − q)δ(m − k − l)δ(p + s − 1)δ(k + t − 1)

pqsklt=0

n!m! δ(|F + G + s + t + 1|2 ) q!l! ̇ FG γ(s) δ(t) ̇ μα(q) ,β(l)

× hνα(p)γ(s),β(k) ̇ δ(t) ̇ ω

+ (ν ↔ μ)

(8.78)

Let us focus on the significant 𝒪(hω) term. The last two delta functions in the first line of the formula, severely restrict the values for the p, s and k, t summation variables. This together with the first two delta functions (that we recognize from the algebra) allow us to set up Table 8.2 explicating the form of the 𝒪(hω) terms.

64 Therefore, vierbeins may come from the 11 sector, but not connections. 65 There is some arbitrariness in representing the translation part of the AdS algebra sp(4), discussed in the paper, related to a possible automorphism of the superalgebra. Similar statements are proved for the AdS superalgebra osp(4).

8.3 The algebra sequence of papers

�

331

Table 8.2: Structure of the 𝒪(hω) terms in the linearized curvatures. p

s

k

t

hω-term

q

l

index structure

n!m!/k!l!

FG γ δ̇ hνγ,δ̇ ω ̇ μ α(q) ,β(l) FG γ hνγ,β̇ ω ̇ μ α(q) ,β(l) δ̇ hνα,δ̇ ωFG ̇ μ α(q),β(l)

n

m

contr.undot. contr.dot.

�

n

m−�

contr.undot. sym.dot.

m

n−�

m

sym.undot. contr.dot.

m

n−�

m−�

sym.undot. sym.dot.

nm

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

hνα,β̇ ωFG ̇ μ α(q),β(l)

The meaning of the “index structure” entry in the table should be clear from studying the actual 𝒪(hω) terms. We will comment further on this below in context of the FDA2 paper. In order to ascertain which curvatures should be considered as physical, respectively auxiliary, we look to the modulo-2 function |F + G + s + t + 1|2 that should be zero. We then see that for F = G, i. e., the 00 and 11 sectors, s + t must be an odd number. That singles out lines 2 and 3 in Table 8.2, which compared to the linearized curvatures of formula (8.13), correspond to the physical sector. Likewise, for F + G = 1, i. e., the 01 and 10 sectors, s + t must be an even number. That, singles out lines 1 and 4 in Table 8.2, which compared to the linearized curvatures of formula (8.21), correspond to the auxiliary sector.

As for the doubling of fields, the paper states that the two copies of any field (00 and 11 for the physical, and 01 and 10 for the auxiliary) are completely equivalent on the linearized level but different at the nonlinear level. At this stage in the reading, the meaning of this statement is a bit difficult to understand, since the gravitational sector 00 is in fact special as regards the structure of the algebra, as we have seen. The explanation comes in the Section 4 of the paper that treats Hermitian conjugation and P-transformations (parity). Considering Hermitian conjugation, it is clear from the formula (8.77), that the sectors 00 and 11 behave differently.66 In Section 4, the most general Hermitian conjugation in accordance with the Lorentz structure is presented as (ωAB ν,α(n)β(m) ) = ∑ βABCD (n, m)ωCD ν,β(m)α(n) ̇ ̇ †

CD

(8.79)

with nondegenerate matrices βABCD (n, m) for all n and m. The choices for these matrices are limited by two sets of requirements: (i) that the higher spin subalgebra shs(1, C) corresponding to the 00 sector is invariant so that the β00 CD (n, m) elements are proportional to δ0C δ0D and (ii) that the linearized curvatures satisfy Hermiticity conditions analogous to the ones for the fields in (8.79).67 A restriction on the coefficients βABCD , spanning five lines of formula space follows. Solving this restriction, together with (i), leads to a solution for the coefficients that involve five arbitrary parameters taking values 0 or 1.

66 They do so also under P transformation, the formula of which we have not quoted.

67 Just replace ω with Rl in the formula.

332 � 8 Archaeology of the Vasiliev theory Further arguments, based on certain involutive automorphisms of the algebra, lead to a remaining arbitrariness parametrized by ρ = 0 or 1. This is precisely the ρ arbitrariness from the A1 paper, which could not be resolved there. Here, as shown in Section 5 of the paper, only the ρ = 1 choice remains, leading to the real shsa(1) algebra. Section 4 ends with a discussion about P-transformations, similar to the one for Hermitian conjugation. It ends with there being a three-fold Z2 grading of the shsa(1) algebra corresponding to three superselection rules related to the numbers |n + m|2 (fermion/boson), |A + B|2 (physical/auxiliary) and 21 |n + m + |n + m|2 − 2|4 for which no physical interpretation is given. Let us now turn to Section 5 where an operator realization is introduced. The operators needed for the operator realization was mentioned above in the box [Time travel...] at the beginning of our Section 8.3. An arbitrary element of the algebra is written in the form ̇ 1 AB,α(n),β(m) ξ (Q)̂ A (R)̂ B q̂ α1 ⋅ ⋅ ⋅ q̂ αn rβ̂ ̇ ⋅ ⋅ ⋅ rβ̂ ̇ 1 m n!m! n,m;A,B

τ(q,̂ r,̂ Q,̂ R)̂ = ∑

(8.80)

and the Lie superalgebra is introduced as described in the same box. Based on the operator realization, the main topic of Section 5 is a discussion of the question of unitary representations of the superalgebra. This is the start of an exploration that is continued in the last two papers A4 and A5 of the algebra sequence of papers. Let us focus on this. In Section 8.3, we quoted the oscillator realization of the superalgebra shsa(1) that is given in Section 5 of the A3 paper. It is argued that the operator realization does not lead to unitary representations. The argument is based on introducing creation and annihilation operators constructed from q̂ and r̂ and studying the Fock space built on the vacuum structure corresponding to the Klein operators. The representation space turns out not to be positive definite. The question of unitary representations of the superalgebra Since the algebra is supposed to organize the states of the theory, its representations must be unitary. Having the operator realization, this question can be investigated via a Fock space F formulation. Creation αi† and annihilation αi operators are introduced (with i = 1, 2) from q̂α and rβ̂ ̇ . The details of such a construction can be found in our Section 7.1.5 in a slightly different notation. The Fock space vacuum is denoted by |0⟩ and as usual all states have positive definite norm. The question concerns the states generated by the Klein operators Q̂ and R.̂ The full representation space V is ̂ ̂ ⊗ F) ⊕ (Q̂ R|0⟩ ̂ ⊗ F) F ⊕ (Q|0⟩ ⊗ F) ⊕ (R|0⟩

(8.81)

̂ ̂ and This representation space is nonpositive definite no matter what values are chosen for ⟨0|Q|0⟩ = ⟨0|R|0⟩ ̂ ⟨0|Q̂ R|0⟩.

8.3 The algebra sequence of papers

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I find the discussion in Sections 5 to 7 of the paper quite hard to follow, and get the impression that the issues discussed have not yet settled down completely. One way to get an overview of the text and its contents, is the following. As we reviewed in the box [Time travel...] at the beginning of Section 8.3, the oscillator formalism yields a realization of the complex algebra shsa(1, C) while the reality conditions take us to the real algebra shsa(1). This algebra, however, admits no unitary representations. Therefore, one gets a mismatch between the needed group theoretical spectrum of physical states and the field theoretical component fields ωABν,α(n),β(m) . This is unacceptable. Here, it is ̇ best to let the paper speak for itself. Perhaps, this [the nonpositive definiteness of the representation space] implies that shsa(1) admits no unitary representations at all and one may wonder whether any consistent field theory exist in this case based on shsa(1). For the first sight, it cannot because it seems that shsa(1) should serve as a global symmetry of a symmetrical physical vacuum state, and physical states must form some unitary representation of shsa(1) if physical ghosts are assumed to be absent. However, a more careful analysis is needed here because there are auxiliary fields among the gauge fields related to shsa(1), the fields which were shown in Ref. [FF3] to possess no physical degrees of freedom after complete gauge fixing. In fact, it is likely that shsa(1) will be deformed in such a way after appropriate localization in a related physical theory that only its higher spin subalgebra, shsf (1), will lead to nontrivial global symmetry acting on physical states, the subalgebra which gives rise to massless fields ωAB with A = B (evidently, shsf (1) coincides with the subalgebra of shsa(1) ̇ ν,α(n),β(m) ̂ formed by polynomials constructed from q,̂ r̂ and the Klein operator K̂ = Q̂ R).

Here, there is a “glitch” in the paper. The new subalgebra shsf (1) mentioned in the quote is not explicitly defined anywhere in the paper. Its properties can be surmised, but only by adding up information in several places of the text. Close reading of the quote uncovers that the “f” in shsf (1) refers to the “full” higher spin subalgebra of the real superalgebra shsa(1), i. e., without the auxiliary fields. Let us continue with the quote. Indeed, after complete gauge fixing auxiliary fields, ωAB with A + B = 1, cannot form nontriv̇ ν,α(n),β(m) ial representations at all while nonunitarity of pure gauge degrees of freedom is both well known and nondangerous. Thus, it is only necessary to require the full higher spin subalgebra of shsa(1) to possess unitary representations. Remarkably, shsf (1) does indeed possess unitary representations. For example, this can easily be seen by identifying a representation space [removed footnote] V of shsf (1) with ̂ ̂ F ⊕ (K|0⟩ ⊗ F) (here K̂ = Q̂ R)̂ and by setting ⟨0|K|0⟩ = 0.

The paper continues with a couple of more pages devoted to this issue. The conclusion being that the higher spin subalgebra shsf (1) of shsa(1) can admit unitary representations. Section 6 of the paper concerns subalgebras of shsa(1) that may themselves serve as bases for higher spin theories. We will not go deeper into this, but just note—for further clarification—that the just mentioned shsf (1) is one among those. In terms of the T generators, it is spanned by TAA

̇ α(n),β(m)

with both A = 0 or A = 1 and extracted by

334 � 8 Archaeology of the Vasiliev theory the conditions ωABν,α(n),β(m) = (−1)n+m ωABν,α(n),β(m) on the fields. Furthermore, there is a ̇ ̇

pure bosonic subalgebra of even spins (even spins since the algebra is nonextended).68 It is stressed that shsa(1) and all its subalgebras (considered in the paper) contain “[. . .] twice boson fields of any even spin.”. This is the “doubling of fields” problem that we have already met. The discussion is continued in the last Section 7 of the paper, before the conclusions.

8.3.4 The A4 and A5 papers The paper A4 paper, together with its companion paper A5, was concerned with the unitarity questions raised in A3. The A4 paper, arrived after FDA2 and addresses questions raised in that paper. Note also that the A4 paper is after the CI papers, to which it refers. An admissibility criterion is defined. As already pointed out, it has to do with the match between the group theory representations of the algebra and field theory content. A more detailed definition can be found in the Introduction to the A4 paper, which we quote almost verbatim here. The admissibility condition for higher spin algebras according to the A4 paper Massless gauge fields with spin s ≥ 1 are assumed to be described by connection 1-forms in the adjoint representation of the chosen higher spin superalgebra h. The spectrum of physical field with spin s ≥ 1 is then fixed by the results of the FF1 paper. If there exists some fully consistent theory based on h, then there should exist an appropriate unitary representation of h describing just the same spectrum of massless particles with spins s ≥ 1 and, possibly, some set of lower spin fields with s < 1, which cannot be fixed be the pure gauge methods of FF1, CI1 and CI2. Whenever such a unitary representation exist, we say that h obey the admissibility condition. If h does not obey the admissibility condition, it is impossible to construct a fully consistent theory starting from h.

Let us continue to quote a passage, in extenso, also from the Introduction to A4. It captures well a reorientation of the project—that we will come to below—taking place between the CI papers and the A3 and FDA papers. [. . .] the results obtained in ref. [FDA2] exceed considerably the cubic approximation of ref. [CI1, CI2]. Taking this into account, it was argued in ref. [FDA2] that the sets of fields in this reference is complete. (One of the important advantages of ref. [FDA2] is that it enables one to handle gauge (s ≥ 1) and nongauge (s < 1) massless fields in a quite uniform way.) It is the superalgebra of higher spins and auxiliary fields of ref. [A3], which was used in ref. [FDA2] for the description of higher spin field dynamics. Surprisingly enough, it turned out that, in the presence of fermions, there exist

68 This is the “simplest” Vasiliev theory, and it has subsequently often been used as a laboratory to understand the theory.

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no consistent truncations of equations of motion of ref. [FDA2] that correspond to “pure” higher spin algebras of refs. [A1, A2]. It was concluded in ref. [FDA2] that, most likely, the superalgebras proposed in refs. [A1, A2] cannot be used for construction fully consistent theories (when fermions are present), and one is forced to use the broader superalgebras of ref. [A3]. [. . .] It will be shown that, in agreement with the results of ref. [FDA2], the superalgebras of higher spins of ref. [A3] obey the admissibility condition while those of refs. [A1, A2] do not.

The only subalgebra of the algebras of [A1, A2] that survive the admissibility criterion is the bosonic higher spin algebra. We will have more to say about this below in connection to the FDA2 paper. The companion paper A5 concerns the extended algebras. We will not review any of that.

8.4 The cubic interaction papers Perhaps not surprisingly, the cubic interaction papers radiate high hopes for the imminent full solution of the higher spin problem. They came very early in the timeline.69 The expectations could well be regarded as realistic at the time, since the frame-like formulation of the free theory made provisions for a generally covariant theory in terms of higher spin curvatures, naturally related to higher spin algebras. The MacDowell– Mansouri formulation of gravity and supergravity offered the blueprint for the form of the action. It may have seemed that only technical details remained to sort out. The conspicuous, and a bit strange, aspect of the theory was the need for AdS space-time, which allowed for working around the problems with coupling higher spin fields to gravity (see our Section 2.3). And it was a grand scheme. It may be regarded as the third large scale construction work on the higher spin problem based on a set of explicitly stated assumptions. The first proposal, the generalized Gupta program of Fang and Fronsdal of 1979, gave some positive results with the work of Berends, Burgers and van Dam in 1984. The second attempt, the light-front approach of 1983, which together with the 1987 follow up paper, solved the higher spin problem at the cubic level in Minkowski space-time, can be regarded as an instance of the Dirac program. It was not so large scale in its execution, consisting of just three papers, but it was the first positive results to be reported. If we stay in the time frame of the present archaeology, then one should also add the two Metsaev papers of 1990, analyzing the quartic order of the light-front theory, which as a by-product, lead to a purely cubic light-front higher spin theory.70 For a more detailed history, see our Section 2.12 in Volume 1 and Sections 2.6 and 6.5 in the present volume. 69 For the reader who read these sections consecutively, we are now returning back in time, as can be seen from Table 8.1. 70 This theory now goes under the name chiral higher spin theory.

336 � 8 Archaeology of the Vasiliev theory Judging from the submission and publication data for the two CI papers, it seems safe to assume that they were written largely unaware of the problems that would a little later appear with the extra fields, the doubling of fields and need to generalize the higher spin algebras including auxiliary fields.71 The two CI papers are the ones most often referred to when the first interactions for higher spin fields in AdS are mentioned. Although natural from a bibliographic perspective, it is a bit strange logically, since the approach ended in a cul-de-sac. It is not entirely clear what is the standing of the results in relation to the later Vasiliev equations, although the cubic interactions coincide according to the VE sequence of papers. Questions regarding action principles for AdS higher spin theory are still open to research. As already noted, the first two papers on cubic interactions are modeled on the MacDowell–Mansouri and Stelle–West theory of gravity and supergravity.72 The CI papers are focused on the gravitational interaction of higher spin fields, rather than selfinteractions between higher spin fields. In the introductions to both papers, it is pointed out that it is the nonanalyticity in the cosmological constant that allows the theory to circumvent the no-go theorems for higher spin coupled to gravity. This was the so-called hypergravity problem that mainly concerned coupling spin 5/2 fields to gravity to which Vasiliev refers (see our Section 2.3 and Section 2.10.4 in our Volume 1).

8.4.1 The CI1 paper As explicitly stated, the paper builds on FF0, FF1 and A1. After a short two-paragraph introduction, the paper goes through the reasoning leading up to the main result in a systematic fashion. It can be divided into seven steps. In the first step, the higher spin gauge fields ω(n, m) with n + m = 2(s − 1) of FF1 are introduced, each spin s occurring once corresponding to the N = 1 higher spin superalgebra of A1. In the second step, the curvatures and infinitesimal gauge transformations corresponding to the algebra are introduced.73 In the third step, a specific expansion procedure around the AdS background is defined, again following FF1, in order to study the cubic order. A slightly different expansion procedure is used in CI2 and we will describe that below. The difference is said to be inessential. In the fourth step, an action is proposed. It is essentially the action from FF1 but now with the curvatures interpreted as the nonlinear curvatures corresponding to the higher spin algebra of A1. The actual formula reads

71 The extra field problem was known already in FF1, but the other problems appeared later in connection with A3. 72 Essentially a gauge theory of gravity with gauge group the AdS isometry group SO(3, 2). See references [102, 103]. 73 We will quote formulas in connection with the CI2 paper.

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S=

̇ 1 in+m+1 Rρσ α(n),β(m) β(n, m)ϵ(n − m)λ−|n−m| ∫ d 4 xϵνμρσ Rνμ,α(n),β(m) ∑ ̇ 2 n+m>0 n!m!

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(8.82)

This action is already written in the paper A1 but without the factor λ−|n−m| that comes from rescaling the fields. It does not say so at this place, but β(n, m) are coefficients to be determined in order to assure invariance of the action to cubic order. The action is motivated by the following facts. If the linearized curvatures are inserted into the action instead of the nonlinear curvatures, then the action becomes a sum of free actions for all spins s > 1. It generalizes the MacDowell–Mansouri action for gravity and N = 1 supergravity with a cosmological term. It is also generally coordinate invariant since it can be written entirely in terms of exterior differentiations, wedge products and differential forms (although it is not done so in the papers). In this context, a large part of the text (the fifth step) is devoted to the extra field problem. Although no new insight into the problem is offered in this paper, it is anyway interesting to quote how the problem is viewed here. The very important property of the action [****] consists in the fact [preprint N 233] that the variation of its quadratic part over all “extra” fields, ω(n, m) with |n − m| > 2, vanishes identically. However, when the interaction is switched on, the variation of the action [****] over “extra” fields turns out to be different from zero [reference to A1]. The fact that the extra field motion equations are essentially nonlinear prevents from a sensible interpretation of the action [****] when these fields are regarded as independent dynamical variables. In order to solve this problem one is enforced to express from the very beginning the extra fields, ω(n, m) with |n − m| > 2 in terms of the dynamical ones, ω(n, m) with |n − m| ≤ 2. This can be achieved with the aid of some constraints.

In the quote, [****] refers to our formula (8.82). The constraints of FF1 are then given in the form h δ=0 ϵνμρσ Rlνμ,α(n),β(m−1) ̇ δ̇ ρα

at n ≥ m

(8.83a)

=0

at m ≥ n

(8.83b)

̇

γ ρ β̇

h ϵνμρσ Rlνμ,α(n−1)γ,β(m) ̇

This corresponds to the true constraint part of equations (8.19a)–(8.19c) referred to in the section on FF1 above, neither containing the field equations for the vierbein-like fields, nor the equations for the connection-like fields.74 Note that these constraints are written in terms of the linearized curvatures. The text notes that “[. . .] an appropriate nonlinear generalization [. . .] is not yet known to us that leads to consistent full dynamics.”. It is then argued that, nevertheless, the linear constraints are sufficient to study the dynamics at the desired cubic order. Next, it is noted that the constraints express all the fields ω(n, m) with |n − m| > 1 in terms of the physical fields ω(n, m) with |n − m| ≤ 1 and their derivatives up to a gauge 74 See further comments in Section 8.5.2 on the FDA2 paper.

338 � 8 Archaeology of the Vasiliev theory part corresponding to the linearized gauge transformations.75 A formula is given for the order of highest derivatives, namely 21 (|n − m| − |n − m|2 ).76 It is also noted that this fact is responsible for the higher derivatives in higher spin interactions. In the sixth step, the variation of the action under the gauge transformations is studied. The overall structure of the variation is quite simple due to the general transformations law of equation (8.36), which for the theory at hand is given by (8.46), but the coefficients—governed by the algebra—appears unwieldy at first sight. The formula appeared already in the A1 paper, in Section 7 that contains a tentative discussion of interactions. Anyway, the aim is now [. . .] to show that for some special choice of the coefficients β(n) the variation [of the action] vanishes in the approximation under consideration when the constraints [our formulas (8.83a), (8.83b)] and the free motion equations of the physical fields hold. In this case, an appropriate deformation of the gauge transformations [reference to formula for δω(n, m)] can be found, which will guarantee the off mass shell invariance of the action [our formula (8.82)] in the cubic order.

We recognize this as an instance of the standard Noether procedure to lowest order: find an action that is invariant to lowest nontrivial order under the field equations and from there one find the lowest nontrivial order of the deformed gauge transformations. The proof is roughly outlined in the paper. Technically, it hinges on a representation of the linearized curvatures represented in terms of higher spin generalized Weyl tensors, already introduced at the very end of FF1. Another crucial ingredient is that the constraints (8.83a) and (8.83b) must hold. Then for a constant parameter β(n), the variation of the action is proportional to the free higher spin action over the physical fields. Then there exists a modified gauge transformation leaving the action invariant, in this lowest-order approximation. More details are provided in CI2. 8.4.2 The CI2 paper The first CI paper can be regarded as the letter version of the second, but CI2 goes beyond CI1 not just in the technical detail it provides, but also in that it concerns interactions corresponding to the extended higher spin algebras. It also completes the analysis of the deformed gauge transformations. This paper marks the end of a line of work before the theory development took a turn from actions to field equations. As many of the early papers, the introduction reiterates the importance of the higher spin interaction problem, especially the interaction with gravity, which is universal. The paper do so toward a backdrop of the problem of unification of all particles and forces in 75 This means that the equations for the fields with n − m = 1, that actually follow from the action, are also included in the constraints. 76 The actual formulas for the extra fields in terms of the physical are given in FF1 as equations (6.24) and (6.25).

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the face of the possibility that supergravity theories or superstring theories may fail to provide such a unification. Reading today, more than 30 years downstream, it is best to disregard such motivations, as they are most likely too naive, and anyway not essential for the actual content of the theory. Having noted this, one should however be aware of the fact that supergravity theory was an inspiration and source of techniques for the development of the Vasiliev theory and, therefore, did have an effect on the form the theory took. We will return to this question in Section 8.8 when we try to evaluate what the initial 1986–1992 period of papers achieved. There we will also appraise the motivations as they were formulated in the early papers. The steps followed in the CI2 paper parallel those of CI1. Since the paper treats extended higher spin algebras, the fields ωi(k),α(n),β(m) are now indexed by a number of ̇ internal indices i(k), where k denotes the number of indices, taking values 1 . . . N. In the second section, after a discussion of various higher spin algebras, the paper settles on shs(N + 1, 4), and curvatures and gauge transformations are given. These formulas are taken from the second algebra paper A2 that treats extended algebras. Again, the curvatures are expressed according to the general scheme of equations (8.8) and (8.9) with structure coefficients given by the algebra. Then, as in CI1, the introduction of the λ parameter is discussed. The λ parameter It may perhaps be easy to skip over lightly the precise introduction of the parameter λ into the theory, but it is pointed out in several of the early papers, and very much so in CI2. The curvatures used in the CI papers are related to the curvatures of FF1 (where they were first introduced) through the following field redefinition: ωold (k, n, m) →

1 1−|n−m|/2 new λ ω (k, n, m) 2iℏ

(8.84)

The real parameter λ is the inverse radius of the background AdS space-time. The cosmological constant Λ is proportional to −λ2 with a positive numerical factor or order 1. It is stressed that: [. . .] in fact, eq. [our (8.84)] provides the only admissible way for introducing the parameter λ, which leads to the meaningful flat limit, λ → 0, in the noninteracting case [footnote removed] [reference to more details later in paper]. Nevertheless, λ cannot serve as a contraction parameter for the full (nonlinearized) superalgebra shs(N+1, 4) since the curvatures [reference to formula for the curvatures] contain both positive and negative powers of it. As for ℏ, it is related to the “quantum” realization of the algebra, and does not concern the issue at stake here. It is in practice set equal to 1. In order to evade unnecessary confusion, the reader should be aware of the fact that the λ-parameter is introduced with a slight difference between the N = 1 case and the extended case.

Next, the expansion procedure is discussed. It is said that it is different from the one in CI1 but that the difference is inessential. In CI1, an explicit representation of the gravi-

340 � 8 Archaeology of the Vasiliev theory tational sector is used. It is recorded above in formulas (8.10)–(8.12) and (8.15a)–(8.15c), although in a slightly different notation. The background AdS vierbein h and connections w and w̄ are assumed to be of zero order. The deviations of the gravitational fields (denoted by primes) from the background, as well as all higher spin fields, are of first order. The background fields are assumed to be such that the zero-order curvatures (formulas (8.15a)–(8.15c)) corresponding to them vanish. In CI2, the gravitational fields are identified with hν,αβ̇ = ων,i(0),α(1),β(1) and wν,α(2) = ̇ ̄ ων,i(0),α(2),β(0) = ων,i(0),α(0),β(2) and are supposed to be of zero order. All other fields ̇ , wν,β(2) ̇ ̇ are again of first order. Thus, there is no explicit split of the gravitational sector into a background and a deviation, in this way preserving general coordinate invariance. Next, linearized higher spin curvatures are given as well as the gravitational curvatures, which appear as our formulas (8.15a)–(8.15c). The crucial point in the expansion scheme seems to be the following, quoting the authors’ own wording: It should be stressed that although the gravitational fields w, w̄ and h are supposed to be of the zero order, the curvatures [(8.15a)–(8.15c)] will be regarded as having at least the first order. This implies that the gravitational fields w, w̄ and h are supposed to deviate weakly from the background fields describing the anti-de Sitter space, which corresponds to the vanishing curvatures [(8.15a)–(8.15c)]. In other words, instead of expanding over the powers of the ordinary Riemann tensor, which is described by eqs. [(8.15a), (8.15a)] at λ = 0 (supplemented by the zero-torsion condition rνμ,αβ̇ = 0), we shall expand over its deviation from the curvature tensor of the background

anti-de Sitter space (the terms proportional to λ2 on the right-hand sides of eqs. [(8.15a), (8.15a)]). It is this expansion procedure, which turns out to be adequate for the description of the gravitational interaction of massless higher-spin fields while the standard expansion over the Riemann tensor, used in refs. [[99], [98]] turns out to be meaningless due to the nonanalyticity in the cosmological constant.

The discussion of the expansion procedure continues in the paper, but let us not pursue it further. After the discussion of the curvatures and the expansion procedure, the paper turns to the action. It is a generalization of the action of CI1 from N = 1 to extended algebras, but it retains its basic form, being an integral over a sum of terms ̇ i(k),α(n),β(m)

ϵνμρσ Ri(k),νμ,α(n),β(m) Rρσ with generalized coefficients to be determined. It is ̇ again (as in CI1) noted that the quadratic part of the action reduces to a sum of free actions for massless higher spin fields, and that it coincides with the MacDowell–Mansouri action for gravity for the gravitational fields. However, at this stage there appears a peculiarity of the approach related to the spin-1 gauge fields. The N = 1 higher spin action of CI1 does not contain any spin-1 gauge field.77 But for N > 1, the action of CI2 contains spin-1 nonlinearly, but no quadratic kinetic term. The authors are then forced to introduce a Yang–Mills-type action (with a coefficient to be determined) containing a spin 1 kinetic term. This action involves the

77 Indeed, spin 1 has been conspicuously absent from the theory up to this stage.

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metric tensor and, therefore, the spin-2 field explicitly, and thus “[. . .] spoils an equalfooting for the fields of different spins.”. After this, the paper turns to the extra field problem and the discussion largely parallels that of CI1. However, some opaque details regarding the constraints (8.83a) and (8.83b) that were relegated to footnotes in CI1 are here raised to the level of text. It is stated that the constraints for n = m “[. . .] coincide with the linearized equations of motion for the auxiliary higher spin fields ω(k, n, m) with |n−m| = 2, which turn out to be algebraic and are equivalent to the equations Rl (k, n, m) = 0, the higher spin counterparts to the zero-torsion equation in gravity.”. This is quite clear. Then, however, the authors write “The extra fields are excluded by [our formulas (8.83a) and (8.83b)] at n ≠ m. Supplementing the latter by the linearized equations of motion for auxiliary fields, we are going to use the 1.5 order formalism known from supergravity [references removed].”. The emphases in these quotes are ours. It is not clear what is meant by excluded. In both quotes, it is not clear what auxiliary fields are intended. It may be some subset of the extra fields. It is likely not the auxiliary fields of FF3, which were introduced much later in 1987. Assuming that we have a conflation of “extra” and “auxiliary”, the sentences can be read as meaning that the intended fields can be expressed in terms of derivatives of physical fields (as the Lorentz connection in Einstein–Cartan gravity). It seems that the constraints do not account for all the extra fields. At this point, we will discontinue our exposition of the contents of the paper. The rest is concerned with the Noether-type deformation, in the chosen expansion scheme, and the construction of cubic interactions in analogy of CI1 but in much more detail. This takes up 15 pages of the paper, and although the authors quite often resort to phrases such as “it is easy to see” this is clearly no simple calculation. Even without trying to reproduce the calculations of these two papers, it should be clear—to anyone with some familiarity of computations in higher spin theory—that a continuation to higher orders must be very hard.78 Let us instead turn to the reorientation of the project that followed.

8.5 The free differential algebra sequence of papers As already noted, the FDA sequence of papers entails a reorientation of the project from the search for actions to the search for field equations in the formalism of free differential algebras. The FDA papers can be seen a transitory in this shift from the MacDowell– Mansouri inspired action approach to what became the Vasiliev equations. It is not so easy to read out from the preceding papers what brought about this shift, but some elements that seem to be involved are the following items: – The vogue (at that time quite recent) in using free differential algebras in supergravity theory as an attempt to approach the problems those theories (extended

78 As indeed admitted in the next step of the development of the theory.

342 � 8 Archaeology of the Vasiliev theory

– –

–

and gauged versions) had found themselves in. It offered a systematic procedure of setting up such theories. Problems with the MacDowell–Mansouri route, which seemed too difficult an approach to higher than cubic orders. The extra fields/auxiliary field problem also seemed to be intractable in the MacDowell–Mansouri approach. In particular, the quite simple action (our (8.82)) was insufficient, and no alternative was at hand.79 The “doubling of the fields phenomena” discovered in the algebra sequence of papers.

The first FDA paper mentions technical difficulties in pursuing the route of the CI papers to “highest order”. Let us turn to this paper.

8.5.1 The FDA1 paper The first FDA paper was submitted a little more than a month after the second paper, but since it is a letter, it was published 6 months earlier than the second one. It is written as an announcement of the elaborate results of FDA2. Let us briefly review its contents as an introduction to the considerably more demanding second paper. In the second paragraph of FDA1 (after a short—now very standard—introductory paragraph), we read the following. A straightforward generalization of the approach of ref. [our refs. CI1 and CI2] to highest orders in interactions is technically very cumbersome forcing us to look for more efficient methods. Here, we announce our first results in this direction (a more detailed discussion will be given elsewhere [FDA2]). Namely, it turns out that the equations of motion of massless fields of all spins admit a quite natural formulation in terms of an appropriate free differential algebra. This formulation enables one to investigate consistent equations of motion for massless fields of all spins by expanding nonlinearities in powers of higher-spin curvature tensors (Weyl zero-forms) generalizing the ordinary gravitational Weyl tensor. As for the contribution of gauge potentials (one-forms), it is accounted for completely in every order in powers of the Weyl zero-forms. In this paper consisting [sic] higherspin equations are found in first order in the Weyl zero-forms. It is worth mentioning that these results exceed considerably those of ref. [CI1 and CI2].

What follows then is a brief collection of results. The fields are gauge one-forms ωAB (Z) and Weyl zero-forms C AB (Z), where the indices A, B and the variables Z are the ones introduced in the A3 paper. The fields ω and C both belong to the adjoint representation of the superalgebra shsa(1), also from A3.80 The symbols of operators formal-

79 According to M. A. Vasiliev, the main motivation was that the FDA approach was the only way to resolve the extra field problem (private communication). 80 It had not yet become clear that the C should belong to the twisted adjoint representation.

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ism81 of A2 is then employed to write down the curvatures RAB (Z) and covariant derivatives 𝒟(C AB (Z)). It is not said so explicitly (in this part of the paper), but the formulas given must be supposed to cover both physical fields and auxiliary fields (and of course the doubling of fields).82 The expansion of the physical fields are given as in our formula (8.112) below. Then the main result of the paper is stated, explicit formulas for the equations of motion to order ωωC in the fields. The form is generically R+ωωC+ 𝒪(C 2 ) = 0 with corresponding contributions to the covariant derivative of C of order ωCC. The equations are not very illuminating to the casual reader as they contain quite complex coefficients in terms of the variables and indices involved.83 It is best to try to disentangle them when we come to the second paper in the FDA sequence. The paper ends with discussions of three properties of the given equations: [. . .] (i) they obey the Frobenius consistency condition and are therefore gauge invariant; (ii) in the linearized approximation eqs. [ref. to the equations in the paper] reduce to the free equations of motion for massless fields of all spins 0 ≤ s < ∞ (any spin appears twice) supplemented by the equations for auxiliary fields of ref. [to FF2f but should probably be to FF3]; (iii) [the given equations] are explicitly general coordinate invariant and contain Einstein equations in the sector of spin-2 gravitational field described by one-forms ω00,α1 ...αn ,β1 ...βm with n + m = 2. ̇

̇

The properties (i)–(iii) are then discussed in more detail in the last three pages of the paper. 8.5.2 The FDA2 paper The second FDA paper—48 pages long—is pivotal in the reorientation of the theory from searching for an action and instead settling with equations of motion. The introduction to the paper is quite extensive, which is appropriate for a paper that is clearly central to the whole enterprise. We will not reiterate it here, but rather return to it in Section 8.8 where we try to evaluate—with the benefit of hindsight—the achievements and shortcomings of the full set of twenty papers. Section 2 of the paper offers, upon close reading, a clear exposition of the road from the higher spin linearized curvatures of the first papers, over the introduction of auxiliary fields as an answer to the extra fields problem, to the unification within the algebra shsa(1), which however leads to the doubling of fields phenomena. The structure is suggestive of a FDA formulation of the theory, and it contains the germs of the 81 It is not explicitly given that designation in the paper. 82 Note also that the formulas are not linearized, but are the ones corresponding to the global higher spin algebra. The object of the sequence of papers is to deform out of this global algebra. 83 The two formulas run over 3/4 of a Physics Letters page with the coefficients taking up six lines of formula space.

344 � 8 Archaeology of the Vasiliev theory unfolding technology. It is in this sense the first two FDA papers entails a quite dramatic reorientation of the project, that it has retained to the present day. But let us not run ahead of ourselves. The road described in Section 2 of FDA2 hinges on a shift of perspective as regards the field equations. In pure gravity, Einstein’s field equations are normally written as Rμν = 0. Equivalently, at least on the level of field equations, one may write the equations as Rμν,ρσ = Wμν,ρσ , specifying the components of the curvature that do not vanish in terms of new fields, in this case the Weyl tensor (see our Volume 1, Section 4.5.7). In the box below, we will explicate how Vasiliev employs this perspective for higher spin. At the very end of the FF1, the field equations had been expressed in terms of the nonvanishing generalized Weyl tensors.84 The Weyl tensor reorientation Remember first the argument itself for gravity: The curvature tensor can be split up in terms of the traceless Weyl tensor, denoted by C instead of W in accordance with the Vasiliev notation, as 1 1 Rμν,ρσ = Cμν,ρσ + (gμρ Rνσ − gνρ Rμσ − gμσ Rνρ + gνσ Rμρ ) − (gμρ gνσ − gνρ gμσ )R 2 6

(8.85)

In then follows that the tracelessness of the Weyl tensor implies that the Ricci tensor vanishes. Conversely, the Ricci tensor being zero implies that the curvature tensor equals the Weyl tensor. In formulas, this means Rμν,ρσ = Cμν,ρσ ⇔ Rμρ = 0

(8.86)

Now imagine that we solve the Einstein field equations Rμρ = 0 explicitly in some context. Then we get an expression for the metric. Computing the curvature tensor, the nonzero components will give the Weyl tensor. This is not what we want to do in the Vasiliev theory. Instead, we want to formulate the field equations directly in terms of the curvature tensor. It is not the solutions—at least not for now—that we are interested in, but setting up the equations themselves. Thus, since we cannot require the too strong equation Rμν,ρσ = 0, we must parametrize the nonzero components of the curvature in terms of the unknown Weyl tensor components. With this background understanding, we can proceed to the Vasiliev theory.

The higher spin free field equations in AdS can be written as (as shown in FF1) ̇ γ h γβ Cα(n)γ(2) ν β̇ μ

Rlνμ,α(n),β(m) = δ(m)h ̇

+ δ(n)hνγδ hμγδ C̄ β(m) ̇ ̇ δ(2) ̇

̇

(8.87)

in terms of the linearized curvatures from FF1 given above in formulas (8.13) and (8.14). Note that the parameter λ is set to 1 in FDA2. The background fields are given by equations (8.15a)–(8.15c) above. So, what is new here are the formulas (8.87) for the free field equations. For the interpretation of the formula, the author writes as follows:

84 See formulas (6.26) and (6.27) on page 768 of the FF1 paper.

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The quantities Cα(n+2) and C̄ β(m+2) on the r.h.s of Eq. [our (8.87)] generalize the gravitational Weyl ̇ tensor to massless fields of arbitrary spins s ≥ 3/2. In fact, it was shown in Ref. [FF1] that, for any fixed n+m = 2(s−1) > 0, the part of Eq. [(8.87)], which do not contain C and C̄ is equivalent to the free equations of motion for spin-s physical massless fields, ω(n, m) with |n − m| ≤ 1, supplemented by some constraints expressing all other fields, ω(n, m) with |n − m| > 1, in terms of the physical fields and their derivatives [notational parenthesis]. In practice, it is convenient to treat the “higher-spin Weyl tensors”, C and C,̄ as independent dynamic variables, which however, can also be expressed in terms of (derivatives of) physical fields by means of Eq. [(8.87)].

For the proof of these statements, the author refers back to the paper FF1 where the relevant text can be found in the second half of Section 6 of that paper, pages 766–768, that treats the extra field problem. Perhaps this is a good point to remind ourselves of the logic of the relation between the extra field constraints and the physical field equations. Sorting out the logic of the equations for physical fields and extra fields In Section 8.2.1 on the FF1 paper, we quoted the linearized field equations as (8.18). Then we quoted a set of equations (8.19), that contain both the field equations for the physical fields and the constraint equations for the extra fields. How can we move from here to the equations (8.87), claimed in the quote above to contain field equations for the physical fields as well as constraint equations for the extra fields? Furthermore, going back to Section 6 in FF1, we find the “pure” constraint equations in (8.20). The paper states that these equations, together with the “dynamical equations” (8.18b) and (8.18c) are equivalent to the set of equations (8.19). Coming then to the equation (8.87), we can note the statement in FF1, at the very end of the paper, that in the presence of the “pure constraints” (8.20), all components of the curvatures except the following ones: l γ ϵ νμρσ Rνμ,α(n), ̇ hραγ ̇ hσα β(0) ̇

and

γ σ β̇

l ϵ νμρσ Rνμ,α(0), hργ β̇ h ̇ β(m)

(8.88)

vanish on the mass-shell. One should convince oneself that the formula (8.87) for the nonzero Weyl curvatures parameterize precisely these (8.88) components of the curvatures.

Next comes a crucial passage that we also choose to quote in verbatim: Although one can easily express C and C̄ in terms of physical fields on the linearized level by solving Eq. [(8.87)] (for explicit expressions see [FF1]), a nonlinear generalization of these expressions would be complicated, and it is convenient to deal with C and C̄ (as well as the 1-forms ω(n, m) with |n − m| > 1) without referring to an explicit form for their relation to the physical fields assuming, however, that this relation exists and is governed by some nonlinear generalization of Eq. [(8.87)].

The emphasis in the quote above is ours. One should keep track of whether this assumption turns out to be true or not.

346 � 8 Archaeology of the Vasiliev theory Warning bells? Even if the assumption in the box [The Weyl tensor...] above is true, one should be aware of the fact that treating the higher spin Weyl-like tensors as independent, brings in another infinite set of fields into theory. Perhaps warning bells should be chiming here. We already have the “extra fields”, then the “auxiliary fields” and the “doubling of fields” and now the generalized Weyl tensors. Below, the “auxiliary fields” will be accompanied by their own “generalized Weyl fields”. Clearly, one may risk losing control of the field structure of the theory. In particular since, as emphasized in several of the papers, the only space-time derivative that explicitly appears in the theory is the exterior derivative, and this only linearly. All higher derivatives that are known to have to appear in higher spin interactions, are implicitly given by the equations for all the different kinds of nonphysical fields in terms of the physical fields. This must be considered a serious weakness of the theory, although at the kinematical level, all the fields are brought into order by the higher spin algebra shsa(1). Looking a bit ahead, hiding away all the higher-derivative structure of the theory in “nonphysical fields” allowed for turning the interaction problem into a partly algebraic problem of setting up and solving nonlinear, nonderivative constraints. It is ingenious, but still, there may be cause for worry.85

Then it is pointed out that the Frobenius consistency conditions for the curvatures (8.87) lead to restrictions on the C and C̄ 86 ̇ γ h γβ DLρ Cα(n)γ(2) ν β̇ μ ̇ ̇ ϵνμρσ hνα δ hμαδ DLρ C̄ β(m) ̇ ̇ δ(2)

ϵνμρσ h

=0

(8.89a)

=0

(8.89b)

These equations, in their turn (and for the same reason), are equivalent to DLρ Cα(n+2) − ihργδ Cα(n+2)γ,δ̇ = 0 ̇

DLρ C̄ β(m+2) − ̇

̇ ihργδ C̄ γ,β(m+2) ̇ δ̇

=0

(8.90a) (8.90b)

where Cα(n+3),δ̇ and C̄ γ,β(m+3) are new arbitrary fields. This process can be iterated indeḟ initely, leading to infinite chains of differential relations for Weyl-like quantities C(n, m) ̄ m) with n undotted and m dotted indices: and C(n, DLρ Cα(n),β(m) − ihργδ Cα(n)γ,β(m) =0 ̇ ̇ ̇ δ̇ + inmhραβ̇ Cα(n−1),δ(m−1) ̇

(8.91a)

85 One may argue, that the method of treating higher derivatives in this way is standard in the jet bundle approach to the theory PDEs. It is clear that any PDE can be reformulated in the jet bundle machinery, but it is another question whether every set of equations formulated in jet bundle language corresponds to a well-posed PDE. For this point of view, see [357]. The question is contentious in Vasiliev theory. However, it can be argued that just writing unfolded FDA equations of the kind of the Vasiliev equations, without imposing locality, will not guarantee mathematical consistent field equations as standard PDEs; see [166]. 86 Essentially, since they are underdetermined first-order differential equations. Equivalently, the conditions follow from the differential Bianchi identities for the linearized curvatures (8.13), also a reflection of the underdetermination of the equations.

8.5 The free differential algebra sequence of papers

̇ ̄ DLρ C̄ α(n),β(m) − ihργδ C̄ α(n)γ,β(m) =0 ̇ ̇ ̇ δ̇ + inmhραβ̇ Cα(n−1),δ(m−1)

� 347

(8.91b)

̄ m) occur only for n − m = 2s > 2 and m − n = 2s > 2, It is pointed out that C(n, m) and C(n, respectively. Although they appear in a similar way in the equations, and are complex conjugates of each other, they will turn out to be distinguished at the nonlinear level. For the further interpretation of this system, let us quote the paper again. The full linearized system of equations [(8.87), (8.91a), (8.91b)] splits into independent subsystems corresponding to all spins s ≥ 3/2 (any spin emerges once and only once). Each spin s ≥ 3/2 is ̄ m) described here by the fields ω(n, m) with n + m = 2(s − 1), C(n, m) with n − m = 2s and C(n, with m − n = 2s. Physical spin s fields are identified with the 1-forms ω(n, m) at |n − m| ≤ 1 and n + m = 2(s − 1). Equations [(8.87)] contain dynamic equations for physical fields and express all ̄ 2s) in terms of derivatives of the physother 1-forms ω(n, m) as well as 0-forms C(2s, 0) and C(0, ical fields [ref. to FF1]. It is also clear from the construction above that Eqs. [(8.91a), (8.91b)] ex̄ m) with n ⋅ m > 0 in terms of C(2s, 0) and press consistently all Weyl 0-forms C(n, m) and C(n, ̄ C(0, 2s), respectively, thus expressing all Weyl 0-forms in terms of derivatives of the physical fields, too.

The paper concludes that the new form of the equations of motion are equivalent to the equations of motion for the physical fields supplemented by equations for all the auxiliary87 quantities expressing them in terms of derivatives of physical fields. It should be noted that an infinite chain of fields C and C̄ are introduced for each spin. It is then shown that the new equations describe also the dynamics for fields of lower spin s ≤ 1. That was not feasible with the 1-forms ω but can be done with 0-forms C and C.̄ Thus, the results in the quote above is extended into the following. Any spin is described [. . .] by means of a chain of fields formed by 1-forms ω(n, m) with n+m = 2(s−1) ̄ m) with m − n = 2s. Physical fields (when s ≥ 1), 0-forms C(n, m) with n − m = 2s, and 0-forms C(n, ̄ 2s) when s < 1. coincide with ω(n, m) at |n − m| ≤ 1 when s ≥ 1 or with C(2s, 0) and C(0,

Again, all other fields in the chains are expressed algebraically through the new equations in terms of the physical fields and their derivatives. The paper then turns to the equations for the auxiliary fields of FF3. From Section 8.2.3 above, we know that the auxiliary fields are described by 1-forms aν,α(n)β(m) ̇ where n − m = k and k = 0, ±1, ±2, . . . . Their nondynamical equations can also be recast in a similar way to the fields ω.

87 Not to be confused with the auxiliary fields introduced in FF3 to be discussed below. Too disambiguate, one could use the term “supplementary” for the extra fields and the Weyl fields. To be precise: physical fields are ω(n, m) with |n − m| ≤ 1, supplementary fields are ω(n, m) with |n − m| > 1 and C(n, m) ̄ m). and C(n,

348 � 8 Archaeology of the Vasiliev theory Weyl-like reformulation for the auxiliary fields Let us compare the linearized curvatures for the auxiliary fields given in formula (8.21) with those of the physical and extra fields given in formula (8.13). We will repeat both formulas here for easy reference: γ ̇ μ,α(n) ,β(m−1)

l δ Rνμ,α(n), = DμL ων,α(n),β(m) + λnhναδ̇ ωμ,α(n−1),β(m) + λmhνγ β̇ ω ̇ ̇ ̇ β(m) ̇

Alνμ,α(n),β(m) = DμL aν,α(n),β(m) − ̇ ̇

γ δ̇ iλhνγ δ̇ a ̇ μ,α(n) ,β(m)

− (ν ↔ μ)

(8.92)

+ iλnmhναβ̇ aμ,α(n−1),β(m−1) − (ν ↔ μ) ̇

(8.93)

The covariant derivative acts in the same way and is given by formula (8.14). Note that λ = 1 in FDA2. So, what is the difference? The terms in the Al curvatures that are of order ha are constructed in two ways, either contracting spinor indices of the vierbein h into an a field with n + 1 undotted indices and m + 1 dotted indices (bringing down the number of indices to n and m, respectively), or symmetrizing the spinor indices on the vierbein with the corresponding indices on a field a with n − 1 and m − 1 undotted and dotted indices (thus bringing up the number of indices to the correct values). The factors in front of these terms are −i and inm, respectively. Now, looking at the Rl curvatures we see that we have one term with coefficient n where dotted indices are contracted while undotted are symmetrized, and one term with coefficient m the other way around. Thus, up to a multiplicative factor of i we can reproduce all the terms if we associate coefficients with contractions and symmetrizations as recorded in Table 8.3 below. Table 8.3: Structure behind coefficients of order hω and ha in linearized curvatures for physical and auxiliary fields, respectively. Two of the numbers shall be multiplied and multiplied with a factor i to get the appropriate coefficient corresponding to its construction. symmetrizing undotted: in symmetrizing dotted: −im

contracting undotted: � contracting dotted: −�

The equations of motion for the auxiliary fields can be written in a way similar to equation (8.87), namely as follows: γ Ė μ β̇ β(m−2)

Alνμ,α(n),β(m) = δ(m)θ(n − 2)hναδ̇ hναδ 𝒟α(n−2) + δ(n)θ(m − 2)hνγ β̇ h ̇ ̇

(8.94)

We here have symmetrizations in both terms, while for the curvatures Rl we have contractions in both terms. There are no factors of n or m in either types of curvatures, but there are extra factors of θ’s in the Al . The fields 𝒟(n − 2, 0) and E(0, m − 2) are, citing the paper, “[. . .] ‘auxiliary Weyl 0-forms’, which can be viewed as some new variables analogous to the higher-spin Weyl 0-forms introduced previously.”.88 Similar to the chain of consistency conditions (8.91a) and (8.91b), there are chains of conditions for fields 𝒟(n, m) and E(n, m), γ ̇ α(n) ,β(m−1)

=0

(8.95a)

γ ̇ α(n) ,β(m−1)

=0

(8.95b)

δ DLρ 𝒟α(n),β(m) + nhραδ̇ 𝒟α(n−1),β(m) + mhργ β̇ 𝒟 ̇ ̇ ̇

DLρ Eα(n),β(m) + ̇

δ̇ nhραδ̇ Eα(n−1),β(m) ̇

+ mhργ β̇ E

88 There does not seem to any other reason for the curious choice of notation 𝒟 (as compared to E) than to disambiguate with respect to the covariant derivative. Note also that 𝒟 is used for the index D in the formula (2.29) in FDA2 for the curvatures.

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� 349

For the respective chains of consistency conditions, we see a pattern (again up to a factor of i) as recorded in Table 8.3. The equations express 𝒟(n, m) and E(n, m) in terms of 𝒟(n, 0) and E(0, m) occurring in (8.94). It is pointed out in the paper that no nontrivial dynamics are contained in the equations quoted here. The auxiliary fields a are considered to be complex, still certain Hermiticity conditions are imposed on the 𝒟(n, m) and E(n, m), in their turn imposing “some additional restrictions” on the fields a.

In the box above, we made some notes on the structure of the equations for physical fields and auxiliary fields. A further note, made in the paper, is the close resemblance between the structure of the chain of consistency conditions for the C, C̄ Weyl forms, and the auxiliary curvatures Al . Such a close resemblance is also seen between the chains of consistency conditions for the auxiliary 𝒟, E and the higher spin curvatures Rl . Based on this observation, the paper states the following: In fact, this important observation enables us to suppose that both the 1-forms (ω, a) and the Weyl 0-forms (C, C,̄ 𝒟, E) belong to the adjoint representation of some Lie superalgebra incorporating both the massless fields of Ref. [FF1] and the auxiliary fields of Ref. [FF3].

This is precisely the superalgebra shsa(1) of the paper A3 with the concomitant doubling of fields. So far into the paper, one can say that the groundwork is done for introducing the approach of free differential algebra (FDA). The notation of A3 is returned to and the gauge fields are given as ωAB and the corresponding formula for the nonlinear ̇ ν,α(n)β(m) curvatures are repeated (Formula (2.12) in A3). The “Weyl tensor inspired” form of the linearized field equations (that we have recounted above) is collected using the A, B indexing (remember A, B = 0, 1 give doubled physical fields and A + B = 1 give auxiliary fields). Then the linearized curvatures Rl and Al (formulas (8.92) and (8.93)) and covariant derivatives for physical fields (formulas (8.91a) and (8.91b)) and auxiliary fields (formulas (8.95a) and (8.95b)) are collected into comprehensive formulas. We will not reproduce them here (they are formulas (2.33) and (2.34) in the paper),89 but just note a circumstance that may seem odd to the casual reader. Along with the linearized curvatures Rl,AB taking the doubling of fields into account, there are generalized Weyl curvatures C AB . The association is such that C 01 and C 10 appear in the formulas with Rl,00 and Rl,11 . Vice versa, C 00 and C 11 appear in the formulas with Rl,01 and Rl,10 . That is, C and C̄ are represented by C 01 and C 10 , while 𝒟 and E are represented by C 00 and C 11 . The explanation for this resides in the observation (noted in the paper on page 68) that the structure of the covariant derivatives in formulas (8.95a) and (8.95b) resemble that of the curvatures (8.92). Vice versa, the structure of the covariant derivatives in formulas (8.91a) and (8.91b) resemble that of the curvatures (8.93). See our box [Weyl-like

89 The notational clash having to do with D and 𝒟 continues here in denoting the covariant derivatives with 𝒟.

350 � 8 Archaeology of the Vasiliev theory reformulation...] above.90 What we see here is a hint of the twisted adjoint representation for the 0-form fields, not yet recognized as such. All this leads up to the introduction of FDAs. It is stated that the structure of the linearized equations are suggestive of writing the full equations for interacting fields in the form R + f (ω, C) = 0 and 𝒟C + g(ω, C) = 0 (8.96) Here, R are supposed to the full, nonlinear, curvatures of shsa(1) while 𝒟 is a covariant derivative and f and g are constructed from 1-forms ω and 0-forms C, all in the adjoint representation of shsa(1). The formulation in terms of exterior algebra is understood. It is assumed that f and g are of at least first and second orders in C, respectively (as suggested by the linearized germs for the equations). The theory of FDAs as employed in the FDA papers is developed in Section 3 of the paper. In the box below, we will be a little more explicit than the paper itself. The free differential algebra of FDA2 It starts with assuming an arbitrary set of differential forms p-forms W A with p ≥ 1. Curvatures RA are defined through RA = dW A + F A (W )

(8.97)

where F A (W ) are functions of W A . At this stage, nothing is said about the nature of the exterior derivative d or the nature of the index A. Instead, continuing and formally computing dRA one gets dRA = (RB − F B )

δF A δW B

(8.98)

Then equation (8.97) is said to define “some free differential algebra” if dR is proportional to R, i. e., if FB

δF A =0 δW B

(8.99)

δF A δW B

(8.100)

holds. In that case, we have “Bianchi identities” dRA = RB

“which ensure that Frobenius consistency conditions are satisfied for the equations” RA = 0

(8.101)

Now, what is going on here?91 Setting the curvatures to zero yield first-order differential equations for the W A . Then clearly one must also have dRA = 0, which are the Frobenius consistency conditions in this context. They can be ensured if equations (8.99) and (8.100) hold. 90 There are also some comments on Hermitian conjugation in the paper that we do not enter into here. 91 The quotes “ ” in the text above are from the paper.

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First, looking back to our Section 3.5.1, we see that equations (8.99) correspond to Jacobi identities for the algebra. Second, equations (8.100) can be thought of as (differential) Bianchi identities in this context. If they hold, R = 0 implies dR = 0. Indeed, in the special case of p = 1, the formalism describes an ordinary Lie algebra if d is interpreted as the exterior derivative on the group manifold. In order to see how this abstract formalism may be used for higher spin gauge theory, the first step is to interpret the W A as the higher spin fields. Then the rest of the formalism needs to conform to the general gauge theory scheme outlined in the box [The main strategy] above. The structure of the algebra must reside in the functions F A (W ).

The paper defines infinitesimal gauge transformations in this context by δW A = d ℰ A − ℰ B

δF A δW B

(8.102)

The curvature then transforms as δRA = −RC

A δ B δF ) ( ℰ δW B δW C

(8.103)

For the commutator of two gauge transformations, the paper writes [δ2 , δ1 ]W A = δ1,2 W A + Δ1,2 W A

(8.104)

Here, δ1,2 W A is a transformation of the same type as (8.102) with parameter A

A B

deg(W B )

ℰ1,2 = ℰ1 ℰ2 (−1)

δ2 F A δW C δW B

(8.105)

while Δ1,2 W A contains the curvatures RA and third functional derivatives of F with respect to the forms W . With Δ1,2 , the commutator resembles that of an “open algebra”. In the higher spin case at hand, the physical fields, along with the extra fields and the corresponding auxiliary fields are meant to be contained in 1-forms ωAB (n, m), also taking the doubling of fields into account. The Weyl components of all the curvatures are likewise contained in 0-forms C AB (n, m) and C̄ AB (n, m). Then it is inherent in the formalism that the ωAB (n, m) can only occur in (low) polynomial order, while there may be arbitrarily high powers of the C AB (n, m) and C̄ AB (n, m). This paves the way for a perturbative approach to the interactions in which the dependence on ωAB (n, m) is exact in every order. The paper then considers a “subclass” of FDAs such that92 ω a

R = dωa + F a (ω, C)

C a

R = dC a − C b

a

δF (ω, C) δ ≡ −C b b ω Ra δωb ω

92 No reason is given for using new indexing apart from them signifying a linear space.

(8.106a) (8.106b)

352 � 8 Archaeology of the Vasiliev theory No explicit motivation is given, at this point, for studying such FDAs, but the reason must be the noted close resemblance between the coefficients in the curvatures and the chains of consistency relations that can be read of from the formulas, (8.92), (8.93) and (8.91), (8.95). The consistency condition (8.99) for the FDA of (8.97) is replaced by Fb

δF a (ω, C) δF b (ω, C) δF a (ω, C) − Cd =0 b δω δωd δC b

(8.107)

which is claimed to ensure the consistency of both equations of (8.106). A perturbative deformation of the equations (8.106) is then outlined. It is stated that any FDA with equations (8.106) can be treated as a deformation of the same FDA with F̃ a (ω, C) = F a (ω, 0). Some general properties of the deformation scheme A deformation parameter η is introduced through C = ηC ′ and the equations (8.106) are rewritten (and normalized) into (with primes dropped) ω a

R = dωa + F a (ω, ηC) and

C a

R = dC a − C b

δF a (ω, ηC) δωb

(8.108)

Here, the zero-order FDA is identified with the Lie superalgebra shsa(1) for the 1-forms supplemented by 0-forms in the adjoint representation. This is so because at zeroth order, the algebra is closed according to the discussion above. When it comes to the order by order deformation, we may just as well cite what the paper states since it is crucial for the following development: Let us emphasize that if one starts from a zero-order FDA governed by some Lie superalgebra (and 0-forms in its adjoint representation) in accordance with Eqs. [(8.108)] at η = 0, it is natural to expect that full consistent “curvatures” ω R and C R can also be reduced to the form [(8.108)] in all orders (if they exist). Indeed, this is supported by the observation that if Eqs. [(8.108)] are shown to be consistent up to the nth order, and if one also finds ω R in the (n + 1)st order such that the consistency condition of the type [(8.107)] holds in this order, then the (n + 1)st order deformation of C R, defined via Eq. [second eq. of (8.108)] turns out automatically to be consistent due to Eq. [(8.107)]. It is also important here that one need only know ω R and C R up to the nth order when determining ω R in the (n + 1)st order; i. e., ω R in the (n + 1)st order can be found independently of C R in the (n + 1)st order. After this, field redefinitions are discussed. As always, trivial nonlinear terms due to field re-definitions of the η = FDA must be factored out.

The third section ends with a discussion of an action. We will not review that. In the fourth section of FDA2, the basic formalism is set out—largely based on A3—that is then retained up to and including the VE papers with minor changes in notation. For the subsequent theory development, we will however leave the FDA2 paper—after a few more comments—and continue with FDA3 where the formalism is starting to settle down. In particular, FDA3 employs the integral form of the star prod-

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uct93 that is a little bit more easy on the formalism, than the differential form used in FDA2. As for the actual achievement of the FDA2 paper, apart from the FDA reorientation itself, is the first-order deformation in the zero-form Weyl-type curvatures C. This takes up the almost 8 pages long Section 5 of the paper. It is highly technical. The equally long Section 6 of the paper examines which of the automorphisms of the algebra shsa(1) from the paper A3 admits generalizations to the FDA of FDA2. Again, we will not review any details, but just note that analysis is based on three automorphisms, denoted by f , k and t already used in A3 to construct subalgebras of shsa(1), and now used to extract nontrivial deformed FDAs. Here, f is the fermion-boson automorphism, k is related to the simultaneous change of sign of the Klein operators Q̂ and R̂ (hence ̂ while t has a more complicated definition. Two the denotation k related to K̂ = Q̂ R), interesting special cases are given in the paper. First, f (ω) = ω and f (C) = C extract the even bosonic sector. Second, k(ω) = ω and k(C) = C extract all massless fields ωAA and C AB with A + B = 1. Thus, the paper writes “[. . .] in the framework of the deformation constructed, all auxiliary fields can be consistently set equal to zero.”. The statement is a bit cryptic, but is presumably meant to be interpreted in view of the equations of motion approach inherent in the FDA formalism. Next comes a passage that is both interesting and a bit confusing. It starts by saying that “[. . .] one cannot now avoid the doubling of massless fields [. . .]”. Then a few lines later, adding the caveat “It seems that doubling of bosonic massless fields is inevitable, at least in the presence of fermions.”. The passage is ended (after an argument) with “[. . .] therefore, one can consider bosonic theories containing each spin once and only once (or each spin belonging to some simple associative algebra).”.94 The parenthesis about associative algebras has to do with the discussion in the section about the possibility to let the fields be valued in associate matrix algebras. Section 6 also stresses that the higher spin superalgebras of A1 and A2 are insufficient as bases for interacting theories in FDA approach. The extension with auxiliary fields is necessary. This fact could, however, not be seen in the cubic order of the CI papers. Section 7 of the paper discusses the relation between the equations of motion interaction results obtained, and the results on the action level interactions of the CI papers.

8.5.3 The FDA3 paper The first paragraph of the introduction to the paper outlines the new strategy well.

93 Not yet so named. 94 The full passage can be found in the middle of page 90 of the paper.

354 � 8 Archaeology of the Vasiliev theory

It was argued recently [FDA2] that consistent equations of motion for interacting massless fields of all spins can naturally be described in terms of an appropriate free differential algebra (FDA), which is denoted from now on as Fη . This FDA can be thought of as the deformation of the FDA F0 originating from the infinite-dimensional superalgebra of higher spins and auxiliary fields, which was proposed in ref. [A3]. Fη was constructed in ref. [FDA2] to the first order in the deformation parameter η.

This reminds us where we are in the development of the theory. The paragraph continues with the following illuminating sentences: In practice, the expansion in powers of η is equivalent to the expansion in powers of higher spin curvatures (0-forms) generalizing the spin-2 gravitational Weyl tensor to massless fields of all spins. This implies that consistent equations of motion for interacting massless fields of all spins (including the gravitational field) were found to the first order in the higher spin curvatures in ref. [FDA2]. Since the contribution of gauge potentials (1-forms) is accounted completely in every order in η, the class of interactions constructed in ref. [FDA2] is broader than the class of purely cubic interactions analyzed in ref. [CI1, CI2].

There is thus no deformation in powers of ω to be expected, only in C and C.̄ Note also, by the way, that [FDA3] does not go beyond [FDA2] in the expansion, rather it offers further and alternative details of the construction. The paper continues with a discussion of Einstein’s spin 2 equations as a free differential algebra. Then the basic formalism is set up in Section 3. It is based on a minor change of notation as compared to the variables introduced in the paper A3. The basic Vasiliev formalism (with changes of notation and disambiguation) The basic variables employed in order to parameterize the higher spin fields are the oscillator-like twocomponent spinor operators q̂α and rβ̂ ̇ obeying commutators given above in (8.22) that we repeat here for easy reference with ℏ = 1 as the FDA and subsequent papers have it [q̂α , q̂β ] = 2iϵαβ

[rα̂ ̇ , rβ̂ ̇ ] = 2iϵα̇β̇

[q̂α , rβ̂ ̇ ] = 0

(8.109)

There are also the Klein operators Q̂ and R,̂ that parametrize the doubling of fields, obeying {Q,̂ q̂α } = 0

[Q,̂ rβ̂ ̇ ] = 0

[R,̂ q̂α ] = 0

{R,̂ rβ̂ ̇ } = 0

(8.110)

and [Q,̂ R]̂ = 0 Q̂ 2 = 1

R̂ 2 = 1

(8.111)

The gauge field 1-forms are ω(̂ q,̂ r,̂ Q,̂ R)̂ = ∑ n,m A,B

̇ 1 ω̂ AB,α(n),β(m) (Q)̂ A (R)̂ B q̂α1 ⋅ ⋅ ⋅ q̂αn rβ̂ ̇ ⋅ ⋅ ⋅ rβ̂ ̇ 1 m 2in!m!

(8.112)

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and the corresponding expansion for the Weyl-like 0-forms C. The corresponding symbols of operators are in all cases denoted by the same letter without the hat ̂. In FDA3, there is a change of notation q̂α → yα̂ and rβ̂ ̇ → yβ̂̄ ̇ that is adhered to from there on. A collective variable Y is introduced according to Y ̂ = (y ̂ , y ̂̄ ̇ ). Note, however, that in the papers FDA1 and FDA2 a collective α

α

variable Z was used for (q̂α , rα̂ ̇ ) that should not be confused for the Z introduced later in the VE sequence of papers for the “doubling of variables”. The Klein operators also undergo a few minor changes of notation. In FDA3: Q̂ → K ̂ and R̂ → K ̂̄ . In FDA4: K → κ and K ̄ → κ ̄ for the symbols. And finally, in VE1: κ → k and κ ̄ → k ̄ collectively denoted by K . Taking these changes of notation into account, all commutators and anticommutators, between operators and/or symbols can be read off from the formulas (8.110)–(8.111).

For the following development of the theory, the paper turns to the integral formulation for the free differential algebra (i. e., the star-product). Fields—expanded as in formula (8.112) in the box above—are described by 1-forms ωAB (Y ) with A = B for physical components and A + B = 1 for auxiliary components, and by 0-forms C AB (Y ) with A + B = 1 for physical components and A = B for auxiliary components (i. e., the other way around). For the free field theory curvatures, with generic form R ∼ dω + ω ∧ ω, one needs a product to multiply gauge fields ωAB (Y ) so that all the components gets multiplied together correctly. The paper first quotes a quite complicated (and a bit unsymmetrical) formula in terms of the fields ωAB , then goes on to a simpler expression. It is based on retaining the Klein operators both for “operator formulas” and “symbol formulas”. Thus, one writes 1

̂̄ = ∑ K̂ A K̂̄ B ω̂ AB (Ŷ ) ̂ Ŷ , K,̂ K) ω( A,B=0

(8.113)

and similarly for the symbol fields ω but keeping the Klein operators as operators. The operator curvatures are given by ̂̄ = d ω( ̂̄ + ω( ̂̄ ∧ ω( ̂̄ ̂ Ŷ , K,̂ K) ̂ Ŷ , K,̂ K) ̂ Ŷ , K,̂ K) R(̂ Ŷ , K,̂ K)

(8.114)

To compute the product using the symbol calculus, the paper introduces new collective variables ̂̄ ̂̄ Φ = (Y , K,̂ K) Φn = (Yn , K,̂ K)

(8.115)

where n is a label on different variables Yn . It is noted that the Klein operators are not enumerated by n. The symbol version of (8.113) is then 1

̂ n ) = ∑ K̂ A K̂̄ B ωAB (Yn ) ω(Φ A,B=0

The symbol of the free curvatures can then be reduced to the formula

(8.116)

356 � 8 Archaeology of the Vasiliev theory 2

R0 (Φ0 ) = dω(Φ0 ) + ∫ d 4 Y1 d 4 Y2 exp[−i ∑ (−1)n+m (Ym , Yn )]ω(Φ1 ) ∧ ω(Φ2 ) n,m=0 n>m

(8.117)

where (Yn , Ym ) = −(Ym , Yn ) = ynα yαm + ȳnα̇ ȳαm ̇

(8.118)

The indices n, m = 0, 1, 2 just enumerate different variables entering the product formula.95 It is noted that all Klein operators are supposed to be moved to the left before performing the integrals over the symbols Yn . In a similar way, the covariant derivative of the 0-form symbol C is computed according to 2

D0 (C0 (Y0 )) = dC(Φ0 ) + ∫ d 4 Y1 d 4 Y2 exp[−i ∑ (−1)n+m (Ym , Yn )] n,m=0 n>m

× (ω(Φ1 )C(Φ2 ) − C(Φ1 )ω(Φ2 ))

(8.119)

The rest of the paper is essentially concerned with a derivation of the first-order deformation of the curvature, a property called “regularity” and the connection to the differential formulation of the product. We will not go into the details of this, but rather quote from the paper, outlining the general strategy. The curvatures R0 [(8.117)] serve as a generalization of the gravitational curvatures [equations of our type (8.15a)–(8.15c)] to the case of higher spins and auxiliary fields. The equations R0 = 0 are dynamically trivial describing pure gauge fields on the anti-de Sitter background of unit radius [. . .]. Equations R0 = 0, D0 (C) = 0 obey Frobenius consistency conditions as consequence of the Jacobi identities for shsa(1) [. . .]. Therefore, these equations define some FDA. It is this FDA, which has been denoted by F0 [in the first quote above]. We are looking for a deformation Fη of F0 , which is described by full curvatures R and covariant derivative D(C) of the form ∞

R = ∑ ηn Rn (ω, C) n=0

∞

D(C) = ∑ ηn Dn (ω, C) n=0

(8.120)

such that the equations [R = 0, D(C) = 0] obey the Frobenius consistency conditions and reduce to equations of motion of massless and auxiliary fields similar to [the case of pure gravity]. [. . .] The requirement that eq. [our (8.120)] obey Frobenius conditions implies that they correspond to some FDA.

This quote summarized quite well the strategy as it stood at this time. Next the paper gives the formula for the first-order 𝒪(ωωC) deformation R1 of R, the derivation of which is one of the main results of the paper. We will just quote it (formula (3.18) in the paper) but make no attempt to explain its detailed structure: R1 (Φ0 ) = ∫ d 4 Y1 d 4 Y2 d 4 Y3 d 4 Ud 4 Sω(Φ1 ) ∧ ω(Φ2 ) ∧ C(Φ3 )

95 The 0 on R0 means “free theory” however.

8.5 The free differential algebra sequence of papers

3

3

n,m=0 n>m

n=0

� 357

× exp(−i[ ∑ (−1)n+m (Ym , Yn ) + ∑ (−i)n (Yn , S) + 2(U, S)]) ̂ u)Δ(y ̄ × (μKδ( 0 + u, s + u + y2 − y3 , y3 − u) ̂̄ ̄ + μ̄ Kδ(u)Δ( ȳ0 + u,̄ s̄ + ū + ȳ2 − ȳ3 , ȳ3 − u))

(8.121)

Perhaps it would be naive to expect anything more simple, or intuitive, in higher spin theory. There is, however, a structure to the formula. We have a Gaussian integral (from the star product) over the variables Yn parameterizing the three fields entering the interaction as well as over two more spinorial variables S = (sα , sᾱ ̇ ) and U = (uα , ū α̇ ). Then there is a “measure function” of three variables, denoted by Δ, responsible for “intertwining” the various components of the fields into the form demanded by the deformation. One ingredient in the derivation of the formula is the so-called “triangle identity” for the function Δ that is subsequently used in the following papers.96 The following section of the paper treats the consistency (Frobenius) of the equations, the connection to the differential formulation of the previous two papers of the sequence, and a property called “regularity”. Since the interaction terms are defined through integrations involving exponentials over the integration variables and products of fields ω and C, the question arises as to the functional class of the result. It is a argued in Section 4, that with the fields ω and C being polynomials in the basic variables, so is the case also for the resulting interaction terms.97

8.5.4 The FDA4 paper This papers treats the next, second, order of deformation. Again, we will not quote the actual formulas. These are quite complicated, as any perturbative expansion is likely to produce (see Section 3 of the paper). Instead, we will focus on the general strategy. The basic formalism is that of FDA3. We will quote part of the paper at (almost) full length. [. . .] the equations of motion of interacting massless fields are assumed to be of the structure R(ω, C) = dω + ω ∧ ω + ω ∧ ωC + ω ∧ ωCC + 𝒪(C 3 ) = 0 2

(8.122) 4

D(ω, C) = dC + (ωC − Cω) + (ωC − Cω)C + (ωC − Cω)C + 𝒪(C ) = 0

(8.123)

(d = dx ν 𝜕/𝜕x ν ). The most important condition imposed on equations [(8.122), (8.123)] is their formal consistency, i. e., the equations d 2 ω = 0 and d 2 C = 0 should automatically hold as consequences of eqs. [(8.122)] and [(8.123)]. In accordance with the terminology of ref. [our references [204, 358, 353]],

96 The form and properties of the function Δ (a kind of characteristic function for triangles) is discussed at length in the paper FDA3, but we will not review that. 97 See pages 516–517 of the paper.

358 � 8 Archaeology of the Vasiliev theory

we say in this case that eqs. [(8.122)] and [(8.123)] define some FDA. (Footnote: Strictly speaking, it is often assumed that FDA’s are based on p-forms with p > 0 while in our scheme 0-forms C are also allowed. Obviously, the 0-forms C serve as a source of the non-polynomiality of eqs. [(8.122)] and [(8.123)].) Because of using the language of exterior algebra, the equations of motion [(8.122)] and [(8.123)] are explicitly general coordinate invariant. Simultaneously, the consistency of eqs. [(8.122)] and [(8.123)], ensures [FDA1, FDA2] that these equations are invariant under higher spin gauge symmetries. As a result, the consistency of eqs. [(8.122)] and [(8.123)] is in fact the only essential requirement that fixes their explicit form in the highest orders. Let us rewrite the left-hand sides of eqs. [(8.122)] and [(8.123)] in the form ∞

R(ω, C) = ∑ Rn (ω, C), n=0

where

Rn (ω, λC) = λn Rn (ω, C),

∞

𝒟(ω, C) = ∑ 𝒟n (ω, C)

(8.124)

𝒟n (ω, λC) = λn+1 𝒟(ω, C)

(8.125)

n=0

(the lowest term of the expansion of 𝒟(ω, C) is linear in C). For example, R0 = dω + ω ∧ ω, R1 = ω ∧ ωC. Generally, the zero-order parts R0 (ω) and 𝒟0 (ω, C) describe, respectively, a curvature 2-form, which corresponds to some Lie superalgebra with gauge potentials ω, and the covariant derivative in a representation of this superalgebra realized by the 0-forms C. In ref. [FDA1, FDA2], it was suggested that this zero-order superalgebra coincides with the superalgebra of higher spin and auxiliary fields, shsa(1), proposed in ref. [A3]. As for the 0-forms, these were argued in ref. [FDA1, FDA2] to belong to the adjoint representation of shsa(1). The main result of refs. [FDA1, FDA2, FDA3] consists of the derivation of the explicit forms for R1 (ω, C) and 𝒟n (ω, C) which describe some nontrivial deformation of the original FDA based on R0 and 𝒟0 and, what is in fact most important, lead to the correct equations of motion of free massless fields of all spins at the linearized level. As observed in ref. [FDA1, FDA2], in this case it is reasonable to assume that the following relation holds in all orders: δ (8.126) 𝒟n (ω, C) = −C i i R(ω, C) δC

(it is essential here that C belongs to the adjoint representation of shsa(1) as is manifested by the zero-order relation 𝒟0 (ω, C) = dC + ωC − cω).

This is as clear as any, a summary of the project, as it must have stood at this time. The reason for our emphasis of a sentence in the quote is the question of whether “consistency” is enough of a constraint on the theory to yield correct higher spin interactions, or not. We will return to that question in the summary of the present chapter; see Section 8.8. In the third section of the paper, the expansion is carried to the second order in the 0-forms C. The fourth section discusses the question of regularity.

8.6 The Vasiliev equations sequence of papers In this series of papers, the theory acquires its mature form. In the FDA sequence, consistency to second order in the Weyl-type field C was established. Now the step is taken to all orders consistency. But there is a price to pay for that. Yet another layer of fields must be introduced through the introduction of a new pair of spinorial variables zα and z̄β̇ . This is the so-called “doubling of variables”.

8.6 The Vasiliev equations sequence of papers

� 359

The motivation for the introduction of the new layer of variables is not very clear from the papers. It can however be discerned from close reading of the text and examination of the formulas. It must have been clear to the author—and it is indeed commented upon—that an order by order continuation along the lines of equations (8.122) and (8.123) is next to impossible, given the complicated results for the two lowest nontrivial orders. The idea is to move all complications into one place where it can hopefully be controlled (algebraically) and subjected to further systematic study.98 There are naturally a lot of technical details in this series of papers. It would not be appropriate here to reproduce them. Many are of a transitory nature and clearly stepping stones to the final form of the theory. So, what follows is a sketch of the main contents of the papers—more so than in our overview of the earlier papers—hopefully helping the reader to navigate should he or she want to consult the original papers themselves. Two more comments are in order. First, I have drawn the boundary between the FDA sequence of papers and the VE sequence at the introduction of the new Z variables. However, the now familiar compact form of the Vasiliev equations, does not appear until the VE3 and VE4 papers. The VE1 and VE2 papers look very much like FDA papers in their contents. Second, as we saw in Section 8.5.4 on the FDA4 paper, the C fields were still considered as belonging to the adjoint representation. This persist throughout the VE sequence of papers, as far as I can tell.99 8.6.1 The VE1 paper The first paper in the sequence was probably meant as a letter, briefly announcing what had been achieved since last. It describes the results, but there is not much in the way of motivations. Furthermore, there is a 7 months lapse between the submit dates of the first and the second paper, and it is quite clear that in the full length paper VE2, the approach had been refined further, both conceptually and technically. We will therefore be quite brief in our study of the first paper. But let us anyway see what we can learn from it about the new approach. [. . .] the present paper consists of the explicit formulation of totally consistent equations of motion for interacting massless fields of all spins in 3 + 1 dimensions (in all orders). These equations are explicitly general coordinate invariant, contain the Einstein equations with the cosmological term, posses all necessary gauge symmetries and reduce to the usual free equations of massless fields of all spins in the linearized approximation. 98 From the modern point of view, where solving the Vasiliev equations in order to get deformations of the free equations is treated as a cohomological problem, the introduction of the extra variables can be understood in terms of homological perturbation theory; see Section 7.4 of the previous chapter. 99 I haven’t been able to pinpoint where in the published record the twisted adjoint representation first appeared.

360 � 8 Archaeology of the Vasiliev theory New generating functions W (Z, Y ; K|x) for higher spin gauge fields (1-forms) and B(Z, Y ; K|x) for on mass-shell nontrivial higher spin curvatures as well as lower spin nongauge fields (0-forms) are introduced without further explanation of the reason for introducing the new spinorial variable Z. The doubling of variables The doubling of variables in VE1 consisted of the introduction of Z = (zα , zᾱ ̇ ). The corresponding operators are subject to the following commutators as given in VE2: [zα̂ , zβ̂ ] = 0

[zα̂ , yβ̂ ] = −iϵαβ

[yα̂ , yβ̂ ] = 2iϵαβ

(8.127)

and {k,̂ zα̂ } = 0

{k,̂ yα̂ } = 0

k ̂2 = 0

(8.128)

The condition that the sets of variables (z,̂ y,̂ k)̂ and (z,̂̄ y,̂̄ k)̂̄ are mutually commuting (and mutually conjugates) supply the rest of the needed commutation relations. The generating functions for the fields are now supposed to be expanded in powers of the full set of variables Z, Y and K and their conjugates.

Next, a set of formulas are given. Two of those—numbered (2) and (3) in the paper— are quite simple in their structure. They express the exterior derivatives of W and B as Gaussian star product integrals of W ∧ W and WB − BW , respectively. If one reads the VE2 paper in parallel, one will suspect that no higher orders are supposed to be added to these formulas. Rather, the old C fields will be related to the new B fields in the lowest order of expansion, while moving all the complications into formulas that will maintain this relationship. Indeed, four more equations are given—numbered (4) and (5) in the paper (only one of them explicitly)—that express 𝜕z𝜕α W (Z, Y ; K) as a star product integral—with a complicated measure factor—over products of the new W and B, and similar equations for 𝜕𝜕z̄α̇ W , 𝜕z𝜕α B and 𝜕𝜕z̄α̇ B. These equations together, are claimed to be the main results of the paper. As already remarked, the equations (2) and (3) in the paper are apparently supposed to be exact since they do not contain any expansion parameter. This is confirmed by the following two quotes from the paper: Eq. (2) can be thought of as the flatness (zero strength) condition for the 1-form W interpreted as a connection of an appropriate infinite-dimensional Lie algebra g. Eq. (3) implies in that case that the 0-form B, which takes its values in the adjoint representation of g, is covariantly constant.

And further down we read Although equations (2), (3) are dynamically trivial in the sense that their general solution is pure gauge, the total system of equations (2)–(5) describes nontrivial degrees of freedom. This happens because eqs. (4), (5), which are algebraic with respect to the space-time coordinates x ν , reduce both the number of independent components of the fields W and B and the number of independent

8.6 The Vasiliev equations sequence of papers

� 361

gauge transformations which preserve the forms of eqs. (2)–(5). As a result, some degrees of freedom survive which are not pure gauge with respect to the restricted gauge transformations.

The paper then argues that the new formulation reproduces the lowest nontrivial deformation of the FDA papers. To understand this, and the overall logic, one is helped by comparing the contents and formulas of the two papers VE1 and VE2. 8.6.2 The VE2 paper After the Introduction, reviewing what has been achieved, Section 2 reiterates the FDA approach to pure Einstein gravity and to scalar fields, thus illustrating the “unfolding method” (not yet designated so) inherent in the infinite chains of equations that we saw in equations (8.91) above. In Section 3 of the paper, the FDA equations of motion are written as ∞

dω = ∑ Fn (ω, C) n=0

∞

dC = − ∑ C k n=0

δ Fn (ω, C) δωk

(8.129)

where ω and C are the generating functions ω(Y ; K|x) and C(Y ; K|x) expanded over the spinorial variables Y and Klein variables K as in the box [The basic Vasiliev formalism...] (with minor notational changes as noted in the box) in Section 8.5.3 above. In (8.129), Fn (ω, λC) = λn Fn (ω, C). The lowest-order expressions for F0 and F1 from FDA1 and FDA2 are then given: 2

F0 (Y0 , K) = ∫ d 4 Y1 d 4 Y2 exp(−i ∑ (−1)n+m (Ym , Yn ))ω(Y1 ; K) ∧ ω(Y2 ; K) n,m=0 n>m

(8.130)

F1 (Y0 , K) = ∫ d 4 Y1 d 4 Y2 d 4 Y3 d 4 Ud 4 Sω(Y1 ; K) ∧ ω(Y2 ; K) ∧ C(Y3 ; K) 3

3

n,m=0 n>m

n=0

× exp(−i[ ∑ (−1)n+m (Yn , Ym ) + ∑ (−i)n (Yn , S) + 2(U, S)]) 2

̄ × (μkδ (u)Δ(y 0 + u, s + u + y2 − y3 , y3 − u)

̄ 2 (u)Δ(ȳ + u,̄ s̄ + ū + ȳ − ȳ , ȳ − u)) ̄ + μ̄ kδ 0 2 3 3

(8.131)

where we recognize (8.121) up to notation in the second formula. Note that in (8.131), the expansion parameter is denoted by μ (not by λ). Properties of these equations are discussed in the rest of the third section. It is noted that in the lowest order, the equation (8.129) for dω, reduces to the zero-strength equation dω = F0 for the higher spin superalgebra of A3.100 Linearization of the equations are also discussed. 100 Here, the paper writes F0 = 0 for “zero-strength equation”, which makes no sense.

362 � 8 Archaeology of the Vasiliev theory In the fourth section of the paper, the doubling of variables (see also the box [The doubling of variables] in the previous section) is motivated as follows. The crucial point that makes it possible to solve the problem consists of the observation that the deformed higher-spin FDA [(8.129)] can be embedded into zero-strength equations for some broader Lie superalgebra.

In this context “zero-strength equation” must be interpreted as an equation of the generic form dω = ω ∧ ω. The new 1-form W (Z, Y0 ; K) is introduced and defined as W (Z, Y0 ; K) = ω(Y0 ; K) + ∫ d 4 Y1 Y2 d 4 Uω(Y1 ; K)C(Y2 ; K) 2

2

n,m=0 n>m

n=0

× exp{−i[ ∑ (−1)n+m (Ym , Yn ) + 2(U, ∑ (−i)n Yn )]} ̄ × (μδ2 (u)Δ(z + y0 + u, y0 + u, y2 − u)

̄ 2 (u)Δ(z̄ + ȳ0 + u,̄ ȳ0 + u,̄ ȳ2 − u)) ̄ + 𝒪(C 2 ) + μδ

(8.132)

This can be thought of as the “embedding” equation. It then follows from (8.129), (8.130) and (8.131), the paper says, that 3

dW (Z, Y0 ; K) = ∫ d 4 Y1 d 4 Y2 exp(−i ∑ (−1)n+m (Ym , Yn )) n,m=0 n>m

× W (Z, Y1 ; K) ∧ W (Z + Y0 − Y2 , Y2 ; K)

(8.133)

In an analogous way to (8.132), one can define a new 0-form B(Z, Y ; K) by replacing W → B and ω → C. It then follows from (8.129), (8.130) and (8.131) that 3

dB(Z, Y0 ; K) = ∫ d 4 Y1 d 4 Y2 exp(−i ∑ (−1)n+m (Ym , Yn )) n,m=0 n>m

× [W (Z, Y1 ; K)B(Z + Y0 − Y2 , Y2 ; K)

− B(Z, Y1 ; K)W (Z + Y0 − Y2 , Y2 ; K)]

(8.134)

From the drift of the argument, one may surmise that the two last formulas are supposed to be exact. This is confirmed by reading on. The paper says: Strictly speaking, it follows from [(8.132)] that [(8.133), (8.134)] hold up to terms which are of first order in powers of the Weyl 0-forms. However, it turns out using the results from [FDA3], one can modify [(8.132)] in such a way that [(8.133), (8.134)] will be valid up to the second order in Weyl 0-forms. This suggests the idea that there exists some generalization of [(8.132)] which ensures [(8.133), (8.134)] to be valid to all orders. Then the problem of formulating higher-spin equations reduces to searching an appropriate embedding, which generalizes [(8.132)] and is consistent with [(8.133), (8.134)].

8.6 The Vasiliev equations sequence of papers

� 363

Much of the rest of the paper is concerned with this. We are now approaching the Vasiliev equations in their final form. The 1-forms ω and 0-forms C are embedded in a larger spinorial space by the doubling of the variables as W and B, respectively. The W and B obey simple—essentially trivial—equations, while all the complications, and presumably interesting interactions, are moved into the embedding equations. These embedding equations are algebraic in the sense that they do not contain any space-time derivatives at all. They are the same equations for 𝜕 W , 𝜕𝜕z̄α̇ W , 𝜕z𝜕α B and 𝜕𝜕z̄α̇ B as given in VE1. Their form is not very illuminating, and we 𝜕zα will just reproduce one of them here: 𝜕 W (Z, Y0 ; K) = μ ∫ d 4 Y1 d 4 Y2 d 4 Uδ2 (u)̄ 𝜕zα 2

2

n,m=0 n>m

n=0

× exp(−i[ ∑ (−1)n+m (Ym , Yn ) + 2 ∑ (−i)n (U, Yn )]) × [∇α (z + y0 + u, y0 + u)W (P, Y1 ; K)

− ∇α (z + y0 + u, y2 − u)W (Z, Y1 ; K)]B(Q, Y2 ; K)k

(8.135)

In this formula, and the similar formulas for 𝜕𝜕z̄α̇ W , 𝜕z𝜕α B and 𝜕𝜕z̄α̇ B, the variables P and ̄ are defined as Q, that should be thought of as Z-type variables of the form Z = (z, z), P = (−z − 2y0 − 2u, z)̄ and Q = (−z − y0 − y2 , z̄ − ȳ0 − ȳ2 ). The spinoral function ∇ is given in terms of the triangle function Δ as ∇α (z, y) = 𝜕/𝜕zα Δ(z, x, y)|x=y . The rest of the equations can be found in Section 5 of the paper, containing a discussion of the “embedding equations” by which it is meant equations of the type (8.132). According to the paper: To fix the form of the embedding [(8.132)], we restrict W (Z, Y ; K) and B(Z, Y ; K) by certain differential-type equations with respect to the variables Z. These equations are required to be totally consistent among themselves and with [(8.133), (8.134)]. The original higher-spin fields are identified with the initial data ω(Y ; K) = W (0, Y ; K), C(Y ; K) = B(0, Y ; K). Equation [(8.132)] then arises as a perturbative solution of these equations in the lowest order in C.

Let us now try to summarize and compare the logic as it is expressed in the VE1 and VE2 papers. The logic in VE1 and VE2 To the first nontrivial order, the equations arrived at in FDA3 correspond to dW (Y0 ; K ) = F0 (Y0 ; k) + F1 (Y0 ; K ) with left-hand side given by (8.130) and (8.131). In VE2, the left-hand side consists of (3.5) and (3.6). In VE1, the full equation is numbered as (15) and is the result of that paper. Further correspondences between formulas are collected in Table 8.4 below. In VE1, the logic is as follows. In zero order in μ and μ,̄ the differential embedding implies W0 (Z, Y ; K ) = ω(Y ; K ) and B0 (Z, Y ; K ) = C(Y ; K ) (obviously a “boundary value”). Next, solving the differential embedding

364 � 8 Archaeology of the Vasiliev theory Table 8.4: Formula correspondences between the papers VE1, VE2 and the present chapter. VE1

VE2

This chapter

Mnemonic

Interpretation

(2) (3) (4), (5) (11)

(4.2) (4.3) (5.1a), (51.b) (4.1)

(8.133) (8.134) (8.135) (8.132)

dW = ⋅ ⋅ ⋅ dB = ⋅ ⋅ ⋅ 𝜕W /𝜕z = ⋅ ⋅ ⋅ W (Z, Y ; K ) = ⋅ ⋅ ⋅

zero-strength equation zero-strength equation differential embedding embedding equation

equations in first order in μ and μ̄ yield the embedding of formula (11). The paper then considers equation (2) at Z = 0. It says that from equation (11) it follows that W (0, Y ; K ) = ω(Y ; K ) (consistent with the “boundary value”). Then a formula for dω(Y0 ; K ) is quoted that can be read of from equation (2) at Z = 0: 3

dω(Y0 ; K ) = ∫ d 4 Y1 d 4 Y2 exp(−i ∑ (−1)n+m (Ym , Yn ))ω(Y1 ; K )W (Y0 − Y2 , Y2 ; K ) n,m=0 n>m

(8.136)

Then, inserting the embedding equation (11) into the right-hand side of this equation, yields the first-order equation for dω(Y0 ; K ) that was the main result of the first three FDA papers. Now, as the paper VE2 stresses (see quote above), one wants to keep the zero-strength equations exact by generalizing the embedding equation. Note that the embedding equation of the two papers, as reproduced here according to Table 8.4, is first order in C.

Section 6 discusses regularity. This is a property that has to do with the functional spaces over which the fields W and B of the theory are defined. Essentially (for details, consult the paper on p. 1401), in order for such fields to make sense, they must be polynomial expansion over the basic variables with coefficients that can be interpreted as higher spin fields. A proposition is proved to the effect that if the functions W (Z, Y ; K) and B(Z, Y ; K) are regular, so is the right-hand sides of formulas like (8.133) and (8.134), too. Note, however, that regularity is not identical to polynomicity, as there might be certain exponential integral coefficients multiplying the polynomials.101 The problem, having to do with ordering questions for the basic variables, is discussed in the last paragraph of the section. We leave it to the reader to explore further. Section 7 discusses other forms—by field redefinitions—of the equations. This section is the stepping stone toward the more “streamlined” formulation of the theory in the two coming papers VE3 and VE4. Without at least a casual reading of this section, the form of the equation in these two latter papers may seem disconnected from the previous papers. A formal 1-form V is introduced V (Z, Y ; K) = dx ν Wν (Z, Y ; K) + dx 5 B(Z, Y ; K) + dzα sα (Z, Y ; K) + d z̄α sα̇ (Z, Y ; K) ̇

(8.137)

101 To be clear, regularity means that inserting polynomial expressions into the vertices must give polynomials out. It does not mean that the vertices themselves must be polynomial.

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All fields depend on the space-time coordinates x ν but not on the purely formal fifth component x 5 (not to survive beyond this paper). The crucial new item here is the field S = (sα , sα̇ ). It appears in the course of studying nonlinear field redefinitions of the form W → W ′ = W + WB + WB2 + ⋅ ⋅ ⋅ and B → B′ = B + B2 + ⋅ ⋅ ⋅ , preserving the zero-strength equations. The embedding equations are, however, not preserved. Section 7 is clearly of central importance for the transition to the “final” form of the Vasiliev equations. One may discern the emergence of the embedding equations in a form that they will subsequently keep. The contents of the section are however highly technical, and we will content ourselves (in our next section) with the result of that analysis as it is presented in the VE3 paper. Section 8 of VE2 discusses gauge symmetries, and finally Section 9 discusses automorphisms and reality conditions. 8.6.3 The VE3 paper The third paper in the VE sequence was actually received at the journal a month before the second paper, so we may conclude that their contents were largely worked out in parallel. It is also clear that while the second paper supplies the low level formula grinding that is necessary to show that the equations are indeed self-consistent, the third paper offers instead the large scale overview of the new theory. In particular, the embedding equations are given in a quite simple form by the introduction of still one more generating function S(Z, Y ; K|x) that appeared in the analysis of field redefinitions in the VE2 paper.102 It is however clear from this paper that these equations are still in a transitory state. Let us step back and try to get an overview. The first two Vasiliev equations have now stabilized to the following now familiar form: dW = W ⋆ W

dB = W ⋆ B − B ⋆ W

(8.138) (8.139)

The first equation can be interpreted as a zero-strength equation for an infinitedimensional Lie superalgebra based on the associative algebra built from the basic spinorial and Klein-type variables of the theory (as we have seen in several places in our exposition of the theory). The second equation can be interpreted as the 0-form B being covariantly constant. These equations are meant to be exact to all orders. Dynamically, they are trivial. Then there are three embedding equations expressed in terms of the new Z-space 1-form field S(Z, Y ; K|x) = dzα sα (Z, Y ; K|x) + d z̄α sᾱ (Z, Y ; K|x) and two new “fixed operators” Q and P. We will not quote these formulas here, as they will be considerably 102 The paper uses a lower case s that was changed to an upper case S in the next paper.

366 � 8 Archaeology of the Vasiliev theory simplified by a further field redefinition involving the operators Q and P. The result of this is the following three further Vasiliev equations: dS = W ⋆ S − S ⋆ W

S ⋆ S = c1 + c2 B

S⋆B−B⋆S =0

(8.140) (8.141) (8.142)

where c1 and c2 are given by 1 ̇ ̄ z̄α̇ d z̄α̇ ) c1 = i(dzα dzα + d z̄α̇ d z̄α ) c2 = (μκdzα dzα + μ̄ κd 2

(8.143)

with new operators ̄ κ = k exp(2izα yα ) κ̄ = k exp(2iz̄α̇ ȳα )

(8.144)

Since the exponential factors of these operators themselves act as Klein-type operators, κ and κ̄ anticommute with all spinors except for the differentials dz and d z.̄ The Klein operators, automorphisms and all that Ever since the doubling of fields was introduced in the algebra sequence of papers, the Klein operators have played an important role in the theory. The reader should be aware of the fact that we have not been careful in our review of the detailed workings of these operators. The same goes for the detailed analysis of automorphisms and other operations of the underlying algebra.

The first of the embedding equations (8.140) is clearly of a type similar to the second equation (8.139), while the third embedding equation (8.142) simply expresses starcommutativity of the S and B fields. Thus, we may surmise that the second embedding equation (8.141) is responsible for the essential embedding. The paper stresses that the equations become dynamically nontrivial precisely because the operators c1 and c2 are different. On the other hand, the last two equations (8.141) and (8.142) stands out in being algebraic, involving no explicit space-time derivatives. It thus seems that the higher spin interaction problem have been reduced to a purely algebraic problem. Further clarification may be read off from the conclusion of the paper. 8.6.4 The VE4 paper We will not have much to comment on the fourth paper in the sequence. It treats generalizations of the basic system of equations, an improved and simplified method of linearization and ambiguity in the interactions due to field redefinitions. For a historical overview of the development of the basic Vasiliev theory, it seems that the equations

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as they were given in the VE3 paper is a good end station. But for the record, and for connecting with our treatment in Chapter 7, let us record the equations as presented in VE4: dW = W ⋆ W

(8.145a)

dB = W ⋆ B − B ⋆ W dS = W ⋆ S − S ⋆ W

(8.145b) (8.145c)

̄ ̄ S ⋆ S = −i[dzα dzα (1 + F(B) ⋆ κ) + d z̄α̇ d z̄ (1 + F(B) ⋆ κ)] α̇

S⋆B=B⋆S

(8.145d) (8.145e)

We recognize the Vasiliev equations—up to minor notational differences—as we have discussed them in Chapter 7. There we had F(B) = B. As far as I can judge, the twisted adjoint representation is not explicitly ackowledged in the VE4.

8.7 The review and generalization era After 1992, there is a hiatus in the published record before a sequence of review papers started to appear. This also marks the start a period of generalization of the theory, and in 1998 new researchers entered the subject, leading to further elaborations. There are ten review papers between 1991 and 2014 written by M. Vasiliev.103 Of these, three can be pointed to in particular for the student eager to get quickly at the equations. The first one is [359] from 1991, which contains few details but offers a good overview of the theory as it stood at that time. The second one is [320] from 1996. This is, in my opinion, one of the most accessible basic reviews of the subject. Both these two reviews treat the simplest purely bosonic theory, and quote the Vasiliev equations in the form dW = W ⋆ W

(8.146a)

dB = W ⋆ B − B ⋆ W̃

dS = W ⋆ S − S ⋆ W S ⋆ B − B ⋆ S̃ = 0 α

(8.146b) (8.146c) (8.146d) α̇

S ⋆ S = dz dzα (i + B ⋆ κ) + d z̄ d z̄α̇ (i + B ⋆ κ)̄

(8.146e)

where κ = exp(izα yα ) and κ̄ = exp(iz̄α ȳα ). The operation ̃ changes the sign of all undotted spinors so that for a function f we ̄ This we recognize as the twisted adjoint have f ̃(dz, d z;̄ z, z;̄ y, y)̄ = f (−dz, d z;̄ −z, z;̄ −y, y). 103 At least ten that I found in my folders. There may well be more.

368 � 8 Archaeology of the Vasiliev theory operation, but it is not designated as such. It should not be confused with automorphism of the algebra that is responsible for the truncation of theory to bosonic fields. That operation entails invariance of all fields under the interchange of the sign of all spinors, not just the undotted ones. The second review is quite clear on this, but the first one seems to conflate the two automorphisms.104 Equations (8.146) may be taken as a baseline form of the theory. As any student of the subject will soon learn, there are many versions of the equations. The third review to mention is the long paper [323] that gives many details, and seems to have been the blueprint for the later review [321] cowritten with X. Bekaert, S. Cnockaert and C. Iazeolla. This latter review is often referred to and may be considered the standard review. A very thorough and quite recent review is [322] by V. E. Didenko and E. D. Skvortsov. It is written by “students” of the theory, thus offering a slightly different perspective and approach.

8.8 What was achieved during 1986–1992? In rereading the papers from a 30-years perspective, I am struck by grandness of the approach, and the vigor with which it was pursued. But there is also another side to it. I get the impression of a big construction machine, of great generality, being built in the very first papers. It is set in motion. The terrain is however unknown and difficult, and the machine soon runs into problems, and it is running the risk of bogging down. Workarounds are forced, more structures are added, new techniques are developed and honed. The machine comes out with a solution, but is it what was hoped for? Here, it is up to the student of the theory to judge for oneself. My own judgment—biased by my personal view of the subject as a whole—is that the end result finds itself quite far from the initial intuition of the problem. As I wrote in Section 1.2 of the Introduction: What indeed is higher spin? 8.8.1 Motivations for Vasiliev theory It is interesting to look a little closer into the explicit motivations given in the early papers for the approach followed in trying to solve the higher spin problem. These motivations were (as is common) given in the introductions to the papers. The substantial part of the motivation is the problem with the gravitational interaction of higher spin fields. Both the CI papers retells the discovery of the need for AdS 104 There are no comments in the text surrounding the equations in these two review papers explaining the reason for introducing this ̃ operation. This is a somewhat annoying feature of the papers, that the formulas change slightly from paper to paper without much of an explanation for the reason for the change. Both review papers mentioned here, refer back to VE3, where as far as I can tell, there is no explicit twist operation.

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in unpublished work by Fradkin and Vasiliev. It was realized that the Aragone–Deser problem with gravitational interactions for higher spin gauge fields could be circumvented by working in an AdS background rather that in a Minkowski background. This led to the strong belief that higher spin gauge fields could not have gravitational interactions of the standard two-derivative minimal coupling type in Minkowski space-time. It is a strange twist of history that just such gravitational interactions were found in the Minkowski light-front formalism already in 1987. But no one noticed this fact until 2013, and it was not stressed by the authors of the discovery paper [160], because their focus was on other questions.105 It is in many places in the papers stressed that the theory has features similar to, or generalizing, features of supergravity theories. It was quite clear, already in the late 1980s, that supergravity was not likely to fulfill the early hopes for unification and quantum finiteness. Therefore, a generally covariant supersymmetric theory, that went (far) beyond supergravity was interesting to investigate. The fact that the need for AdS had been found also in gauged supergravity theories, added to the interest. On the other hand, one has to remember that in the late 1980s, superstring theory was in full swing. Ten-dimensional superstring theory seemed to be the natural generalization of, and step beyond, the 4- to 11-dimensional supergravity theories. Where then, could massless higher spin theory fit into the conjectured picture of grand unification? Well, it could fit in as an alternative to string theory, should string theory eventually fail to fulfill the hopes. This idea is mentioned in the introduction to CI2. More often in the papers, though, is the idea of higher spin field theory as an unbroken high-energy phase of string theory, put forth. This is done in the long introduction to FDA2.

8.8.2 The Vasiliev theory and physics A particular point raised in the introductions to many of the papers is the need for a cosmological constant of the sign corresponding to AdS space-time. It is stressed that this is a feature also of gauged N-extended supergravity theories (with N = 2, 3 at that time of writing). However, from a phenomenological perspective, that argument is not stronger than the argument for the supergravity theories themselves. As far as we know today, these theories have not lived up to the initial high hopes. If the Vasiliev theory is considered only to be an interesting, and very nontrivial field theory, then the presence of a cosmological constant—of the phenomenologically wrong sign, as far as we know today—may not be so disturbing. If the theory however pretends to play a role in the structure of reality, it becomes more worrying. Presumably, and this is the standard view, a Vasiliev-type theory should then come into play at extremely high

105 I noted it while recomputing the light-front interactions [252]. It became more generally known in the higher spin community around 2017.

370 � 8 Archaeology of the Vasiliev theory energies. It is not at all clear why a cosmological constant of that sign, and of what size, should be present; that the question worried the author/authors is clear from several of the early papers. Let us quote representative of passage from FDA1. The quote is in the context of property (iii) mentioned above in the quote from the FDA1 paper (see last paragraph of Section 8.5.1). [. . .] the proposed higher-spin equations contain inverse powers of the cosmological constant (which is set equal to unity throughout the paper) in higher-spin interactions that makes it impossible to take the flat limit in these equations. Let us however emphasize that this property is only relevant to the phase of unbroken higher-spin gauge symmetry, and there are no reasons preventing one from taking the flat limit in an appropriate “physical” phase in which higher-spin gauge symmetries should be broken spontaneously.

The emphasis in the quote is ours. Another speculation, mentioned in some of the papers, is that superstring theory could correspond to a broken phase of a higher spin theory. Such a symmetry breakdown should then be responsible for both generating large masses for all but a few lowspin gauge fields, while also push the cosmological constant to a physical value. During the long dominance of superstring theory in high energy theory, this seems to have been a quite common point of view among workers in the higher spin community. I find this scenario unconvincing. The string spectrum, based on an infinite set of oscillators, is infinitely larger than the higher spin spectrum, based on a finite number of oscillators. True, given the freedom in choosing higher spin algebras, a better match may be arranged, but that would then need a motivation beyond string theory itself. The question is discussed at some length in the conclusion to the paper FDA2. A serious problem for symmetry breakdown of massless higher spin to massive higher spin is the scarcity of states in the massless theory. In four dimensions, there are just two helicity states for any spin s, while a massive field of the same spin require 2s + 1 states. The paper FDA2 offers three scenarios to overcome this mismatch: (i) A generalization of the kinematical higher spin algebras to infinite dimension, presumably by having a countable infinite number of basic oscillators. (ii) Explicitly introducing extra massive multiplets (as in supergravity theories). (iii) Compactification from higher dimensions. This later option incidentally providing a motivation for exploring AdS higher spin theories in higher than four dimension. For any of these three tentative directions, a connection to superstring theory may be hoped for. Perhaps the strangest, and most worrying, aspect of the AdS theory—as it emerges from the early papers—is the “doubling of fields”.106 As we have already remarked, this feature clearly worried the author(s). The issue is commented upon at some length in Sections 7 and 8 in the A3 paper. We will say no more about it here.107 106 With higher supersymmetries, there will be the corresponding multiplicities of fields. 107 In my humble opinion, if higher spin gauge theory turns out to play any role in fundamental physics, it will not be along any “high energy unification of all forces” scenario. Rather, the weight of all the

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8.8.3 Higher derivatives One item that may escape the reader, struggling with coming to grips with the theory, is the absence of any deeper discussion of the number and structure of derivatives in the interactions. True, it is stressed that all the “supplementary fields” of various kinds can be expressed as derivatives of physical fields. There are infinite chains of nonphysical fields that can, in principle, be expressed in terms of indefinitely high derivatives of the basic physical fields. This is, of course, an ingenious idea, more or less explicit in mathematical work that we referred to in Section 4.4.3. Rather than working with indefinitely high space-time derivatives of the physical fields, one opts for treating all higher derivatives of all the physical fields as independent degrees of freedom, eventually to be solved for by constraints. It is remarked in many places in the papers that the only space-time derivative in the theory is the exterior derivative d = dx μ 𝜕μ . This may seem strange since we know that higher spin gauge theory is a theory of higher derivatives to infinite order. However, these higher derivatives are hidden in the infinite levels of extra fields, auxiliary fields and Weyl-like curvatures. This is indeed the essence of the unfolding technique. Smart as this technique is, it is also dangerous, because one may lose control of locality in the field theory. This has in fact become a serious issue for the theory [166]. As we have remarked elsewhere, the locality issue is contentious and still an object of research. Contrast this with the light-front approach, where the structure of derivatives is very explicit and up on the stage all the time. Indeed, when the Vasiliev and light-front theories are compared, it is often pointed out that they are each other’s opposites in that while the light-front theory deals only with physical components of the gauge fields (i. e., no gauge d. o. f.’s at all), the AdS theory works with several infinite layers of auxiliary fields. Perhaps more to the point, in the light-front approach, the space-time derivatives acting on the fields are entirely explicit, there is complete control. In the AdS theory, one only sees the differential d, all other space-time derivative structure is implicit in the powers of the Weyl tensor-like components of the 0-form generating field C.

negative interaction results is that either “they are not there at all” (the mainstream view) or “they play a very special role” (which we do not know what it may be) in the scheme of things. With the benefit of hindsight 35 years downstream, string theory (and supersymmetry) has not lived up to its promises, and connecting massless higher spin theory to string theory in any phenomenological way, is in my opinion, now totally unrealistic. The need for “doubling of fields”, discovered in the algebra sequence of papers, is a worrying aspect of the general theory, leading to two spin-2 fields, although only one of them is interpreted as a graviton. There is, however, a purely bosonic theory, with a simple spectrum of higher spin in four dimensions. I find that attractive.

372 � 8 Archaeology of the Vasiliev theory 8.8.4 Epilogue and a personal reflection It is impossible to know the size of the readership of the papers at the time they appeared. There was no higher spin community at that time. Secondary literature from other authors did not appear until the paper [360] by E. Sezgin and P. Sundell in 1998. Myself, I made efforts to read the papers as they appeared, both in their preprint format and the printed papers. I found them difficult to read and the formulas did not speak to me. I could see no inroad for myself into the theory. The absence of secondary literature certainly contributed to my difficulties, as such literature often provide hints and clues to what is going on. Being a top-down thinker, I have great difficulties spending time with formulas unless I see what they are good for. Perhaps more importantly, I did not believe in the necessity for going to AdS spacetime. I could, of course not, refute the arguments for AdS, but the possibility of doing higher spin interactions in AdS, in my opinion, did not rule out the possibility of doing it in Minkowski space-time in a different way. After all, we had the light-front theory, at that time, largely unexplored. Although I made efforts to understand the AdS theory, it seemed to me completely impossible to contribute myself. New papers appeared quite regularly, and it was clear that anyone trying to enter the subject had a steep learning curve. Trying to foresee the next step, and arrive there first, was clearly both useless and impossible. If one wanted to contribute, one had to think up, or see, a new direction within the AdS approach.

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Index action-at-a-distance interactions 11 acyclic – homology 150 adjoint action – Lie algebra 86 adjoint representation – Vasiliev theory 255 admissibility – Vasiliev theory 334 admissibility condition – Vasiliev theory 319 AdS/CFT 2, 239 algebra homomorphism 82 algebraic Poincaré lemma 157 Anti-de Sitter space-time 239 antibracket 90, 143 antifield 90, 143 antifield method 90, 130 antighost number 146 associative algebra 81 automorphism – algebra 83 Baker–Campbell–Haussdorff formula 102 BBvD hypothesis 169 Bell–Robinson tensor 190 Bianchi identity – free differential algebra 276 bilocal theory – singletons 249 bosonic projection – Vasiliev theory 278 bosonic theory – Vasiliev theory 261 boundary operator 88 bra–ket notation – light-front field theory 205 bracket 92 bracket identity – strongly homotopy algebra 161 BRST cohomology 87 BRST symmetry operator 144 BRST transformations 144 Buchdahl 49 Cartan Integrable System 131 Casimir operator – AdS 246 https://doi.org/10.1515/9783110675528-010

central on-mass-shell theorem 271, 284 centraliser – algebra 127 centralizer 273 chiral higher spin gravity 201, 235 chiral higher spin theory 335 Clifford generating element 309 closed elements – cohomology 88 cluster decomposition 41 co-chain complex 88 co-frame 115 coboundary operator 88 cocycles 89 cohomology 87 – BRST 87 – differential 87 Coleman–Mandula theorem 54 commutator 92, 103 commutator Lie algebra 81 conjugation 83 constraint hypersurface 37 contracting homotopies 151 contracting homotopy 89 coordinate basis 115 coordinate transformation – group manifold approach 135 covariant derivative – Lorentz background in AdS 266 cubic chiral theory 2, 233 current algebra 9 de Rahm cohomology 87 de Sitter space-time 239 de Wit–Freedman curvature 196 deformed oscillator 256, 283 delta vertices 209 demarcation problem – in philosophy of science 5 derivation 86, 107 – inner 86 descent equations 157 differential 87 differential commutator 209 differential complex 88 differential operator – higher order 94, 121 Dirac forms of dynamics 13

388 � Index

Dirac generator formalism 45 Dirac program 7, 70 directional derivative 107 discretized string 17 doubling of fields 308, 324, 331, 342 doubling of oscillators see also doubling of variables doubling of variables – disambiguation 355 – Vasiliev theory 358, 360, 362 dual models 9 dynamical group 9, 10 Einstein relation 12 embedding equation – Vasiliev theory 362, 363, 365 exact 89 exterior derivative 87 extra field problem 293, 302–304 extra fields – Vasiliev theory 294, 297, 301 factorization 133 fake interaction 74 field ghost number 145, 146 field redefinition 140 – BRST approach 182 Fierz–Pauli program 6 filtered algebra 85 first class Grassmann shell 321 flow – vector field 110 Fock space – internal higher helicity 213 form invariant metric 240 formal higher spin theory 289 forms of relativistic dynamics 13 frame 115 frame-like – higher spin fields 297 frame-like formalism 296 free algebra 79 Free Differential Algebra 131 free differential algebra approach 130 Frobenius consistency condition – Vasiliev theory 350 Frobenius integrability condition see integrability condition Fronsdal program 6, 70, 171 Fronsdal tensor 196

front form – Dirac dynamics 13 functional 90 gauge transformation – group manifold approach 135 general coordinate transformation – Lie algebra 109 generalized dynamical trajectory – Komar sense 37 generating function – for higher spin fields 257 Gerstenhaber algebra 145 ghost field – antifield method 143 ghost number – anti 146 – field 146 – pure 146 goodness 229 graded algebra 80, 85 graded differential algebra 88 graded Leibniz rule 86 Grassmann parity 80 Grassmann shell 312, 321 – first class 321 – second class 321 gravitational interaction – higher spin 204 Groenewald theorem 101 group manifold approach 130, 134 Gupta program – generalized 6, 70 Hamiltonian – light-front 229 Hamiltonian cotangent approach – to higher spin gauge algebra 120 Heisenberg algebra 92 Heisenberg-type commutation relation – Vasiliev theory 308, 319 higher order differential operator 94, 121 higher order differential operator approach – to higher spin gauge algebra 120 higher spin algebra 116, 123 – AdS 126, 127 – Minkowski 116, 197 – off-shell 128 higher spin Lie derivative 124

Index

holonomic basis see coordinate basis holonomy – Utiyama paper 63 homology 87 homotopy integral 285 horizontal – curvature 136 horizontal curvature 137 horizontality 133 Howe dual pair 126 hypergravity 48, 336 image – map 88 infinite component fields 9 infinite jet bundle 162 infinitesimal generator 110 inner derivation 86 inner Klein operator 279 inner product 108, 136 instant form – Dirac dynamics 13 integrability condition – free differential algebra 276 – Maurer–Cartan equation 106 integral curve 110 interior product 109 involution 84 involutive – algebra automorphism 83 isometric embedding 240 isometry 240 isometry group 240 isotropic 240 isotropy 240 Jacobi identity – BBvD theory 139 Jacobiator – strongly homotopy algebra 167 jet bundle 162 jet space 162 Jordan structure 243, 245, 251 kernel – map 88 Klein operator 256, 273, 280, 308, 354 – inner 279 – outer 279 Koszul–Tate differential 148, 150

� 389

Lakatos 5 left and right functional derivatives 143 left invariant 1-form 113 left invariant vector field 112 Leibniz rule 93 Lie algebra 81, 109 – general coordinate transformations 109 Lie bracket 81, 92, 107 Lie derivative 108 Lie group 110 Lie∞ algebra 159 linearized higher spin curvatures – Vasiliev theory 268 little group scaling 228 local function 153 – jet space 162 locality – in space-time 142 – of antifield formalism 153 longitudinal differential 148 M function 52 MacDowell–Mansouri gravity 291, 299 magma 79 manifest relativistic covariance 45 master action 145, 154 master equation 145 Maurer–Cartan equation 106, 255 Maurer–Cartan theory 105 maximally symmetric spaces 239 minimal bosonic theory – Vasiliev theory 255 module – algebra representation 82 Moyal Bracket 101 Moyal product 95, 262 multiplet shortening 249 Nambu action – of bosonic string 15 nilpotent – differential 87 no-go theorem 49 No-Interaction Theorem 10, 13 Noether coupling 130 Noether procedure 184 non-coordinate basis 115 non-holonomic basis see non-coordinate basis normal ordering 94

390 � Index

observable – antifield theory 150 – Komar sense 37 obstruction – to deformation 153 off-shell 117 off-shell higher spin algebra 128 on-shell 117 on-shell higher spin algebra – off-shell 128 operator constraint – BBvD spin 3 theory 172 O’Raifeartaigh theorem 54 oscillator realization – of superalgebra 308 outer Klein operator 279 parameter redefinition 141 parity – Grassman 80 Poincaré lemma 156 – algebraic 157 point form – Dirac dynamics 13 Poisson algebra 92, 93, 104 Poisson bracket 92 predictive mechanics 32 product – Moyal 95 – star 95 product identity – strongly homotopy algebra 167 proper solution – to the master equation 152 proper time 11 pull back map 111 pure ghost number 146 push forward map 111 quantum mechanics 92 Rac and Di singletons 249 reducible – gauge transformations 145 Regge slope 15 Regge theory 9 Regge trajectory 15 regularity 151 – Vasiliev theory 357, 358, 364

relativistic harmonic oscillator 27 relativistic invariance 45 reparametrization invariance 11 representation – algebra 82 research program – Lakatos theory of science 5 research programs in higher spin theory – Dirac 7 – Fierz–Pauli 6 – Fronsdal 6 – generalized Gupta 6 – tensor field Lagrangian 6 – Vasiliev 6 – wave-equation 6 resolution 151 rest frame 11 right and left functional derivative 143 rigid string 18 S-matrix theory 9 Schouten bracket 120 second class Grassmann shell 321 Siegel algorithm 21 singletons 249 skew symmetry 81 source commutator 209 source constraint 73 – BBvD spin 3 theory 172 spectrum generating algebra 10 spinor helicity formalism 228 stability group 230 Standard Model 9 star product 95, 321, 323 – non-linear Vasiliev theory 278 stationary surface 150 string theory 9, 15 strong obstruction – BBvD theory 198 strongly homotopy Lie algebra 159, 161 structure coefficients – algebra 79 structure preserving map – algebra 82, 83 substantial variation 56 super-commutative algebra 87 symbol – of higher order differential operator 121 – of operator 322, 355

Index � 391

symbol calculus 309, 322 symbol of an operator 96 symbol of operator – Vasiliev theory 321 tachyon 15 tangent space 107 tangent vector 106 tensor field Lagrangian program 6 the world-line condition 14 transport term – total variations 66 transvection 242, 252, 273 transversality constraint 12 triangle function – Vasiliev theory 357, 363 triangle identity – Vasiliev theory 357 triangular decomposition – light-front 229 twisted action 274 twisted adjoint representation 270, 274, 350, 368 – Vasiliev theory 255 unfolding 130, 268, 274 – Vasiliev theory 161, 258, 361 unital – algebra 79 unitarity bound – AdS representations 247 universal enveloping algebra 82, 91, 103

variation – local and total 56 variational bicomplex 160 Vasiliev equations 255, 310, 341 – first two 365 – second three 366 Vasiliev program 6 Vasiliev theory 2 vector field 107 Velo–Zwanziger problem 50 wave-equation program 6 Weinberg low-energy theorem – no-go 52 Weyl algebra 94, 122 Weyl map 96, 323 Weyl ordering 94, 252, 309, 322 Wigner map 96 world line 11 world line condition 13, see No-Interaction Theorem, 43 world-sheet – of the bosonic string 15 Yang–Mills 8 Z2 graded algebra 80 zero-slope limit 19 zeta-function regularization 236

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