Introduction To Symmetry And Supersymmetry In Quantum Field Theory 9971501600, 9789971501600

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An Introduction to SYMMETRY and SUPERSYMMETRY in QUANTUM FIELD THEORY Jan topuszanski

World Scientific

An Introduction to SYMMETRY and SUPERSYMMETRY in QUANTUM FIELD THEORY

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An Introduction to SYMMETRY and SUPERSYMMETRY in QUANTUM FIELD THEORY

Jan topuszanski Institute of Theoretical Physics University of Wroclaw

World Scientific • Hong Kong Singapore • New Jersey • London Lon

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

AN INTRODUCTION TO SYMMETRY AND SUPERSYMMETRY IN QUANTUM FIELD THEORY Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo­ copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 9971-50-160-0 9971-50-161-9 (pbk)

Printed in Singapore by JBW Printers and Binders Pte. Ltd.

ACKNOWLEDGEMENTS

These notes originated in a course of lectures given by me in the Spring Semester of 1984 at the University of Goettingen on symmetry and supersymmetry in quantum field theory. The very warm hospitality extended to me there by Dr. Helmut Reeh and his constant friendly interest in my work are cordially acknowledged. I am grateful to Dr. Jerzy Lukierski for critical read­ ing of my manuscript and to Dr. Detlev Buchholz for assisting me in putting Section 6.2.9 into a proper shape, and to both of them for valuable advice as well as many remarks and comments. I wish to thank Drs. Wlodzimierz Garczynski, Andrzej Hulanicki, Witold Karwowski, Jan Mozrzymas, Zbigniew Oziewwicz, Maciej Przanowski, Helmut Reeh and Erhard Seiler as well as my students in Wroclaw, Wojciech Hann and Robert Olkiewicz for discussions and advice I made use of while writing this book. And last but not least I wish to record my gratitude to Dr. K. K. Phua, Editor-in-Chief of World Scientific, who encouraged me to write this book. The research reported here was supported in part through funds provided by the Polish Department Program C.P.B.P. 01.03.

J. Lopuszanski

V

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TABLE OF CONTENTS

Acknowledgements

v

1.

1 1

Introduction 1.1. Introductory remarks about symmetries in physics 1.2. Introductory remarks about quantum free theory, in particular axiomatic quantum field theory References and Comments to Chapter 1

8 13

2.

Example of a Classical and Q u a n t u m Scalar Free Field Theory 2.1. Classical scalar free field 2.2. Q u a n t u m scalar free field References and Comments to Chapter 2

15 15 18 27

3.

Scene and Subject of the Drama. Axioms 1 and 2 3.1. Scene of the d r a m a 3.1.1. Axiom 1 3.1.2. Complete normed linear space and its application in q u a n t u m field theory 3.1.3. Operators in the Hilbert space and their application in q u a n t u m field theory 3.2. Subject of the d r a m a 3.2.1. Axiom 2 3.2.2. The space of test functions Sn 3.2.3. The space of tempered distributions S„

28 28 28

vii

28 33 38 38 39 40

Vlll

4.

5.

Table of Contents

3.2.4. Some explanatory comments References and Comments to Chapter 3

43 44

Principle of Relativity. Causality. Axioms 3, 4 and 5 4.1. Basic geometrical transformations of the field 4.1.1. Axiom 3 4.1.2. The group SL(2,C) 4.1.3. The symplectic tensor and the contravariant components of the spinor 4.1.4. The Lorentz group 4.1.5. The representation spaces of the SL(2,C) group 4.1.6. The proper spinorial group ^ o 4.1.7. Infinite dimensional representations of t h e ^ o group. Stone's Theorem. Common domain of definition of the generators of ^ o 4.1.8. Projective representations of P\ 4.1.9. The transformation properties of the fields 4.2. Spectral condition in Minkowski momentum space 4.2.1. Axiom 4 4.2.2. Some explanatory remarks 4.2.3. Properties of the vacuum. The set Dn 4.3. Causality 4.3.1. Axiom 5 4.3.2. Some explanatory comments 4.3.3. The Theorem on Spin and Statistics 4.3.4. Example of a free neutral spinor field References and Comments to Chapter 4

45 45 45 46

Irreducibility of the Field Algebra and the Scattering Theory. Axiom 6. Axiom 0 5.1. Irreducibility of the field algebra 5.1.1. Weaker version of Axiom 6 5.1.2. Associative, involutory, normed algebra 5.1.3. Irreducibility and cyclicity of a vector with respect to a field algebra 5.2. Scattering theory 5.2.1. General remarks concerning the scattering theory 5.2.2. Outline of the mathematical formalism of the scattering theory

54 56 57 64

66 71 73 75 75 78 79 81 81 82 83 85 86

89 89 89 89 90 91 91 93

Table of Content*

5.2.3. 5.2.4. 5.2.5.

Stronger version of Axiom 6 Scattering states The weaker version as a consequence of the stronger version of Axiom 6 5.3. The 5-matrix 5.4. Superselection rule. Axiom 0 References and Comments to Chapter 5 6.

Preliminaries about Physical Symmetries 6.1. General theory of physical symmetries 6.1.1. Wightman functional 6.1.2. Definition of a symmetry group 6.1.3. Antilinear operations and PCT symmetry 6.1.4. Relativistic geometrical symmetries 6.1.5. Borchers' classes. The global symmetries of the 5-matrix 6.1.6. Poincare symmetry is a global symmetry of the 5-matrix. Conserved quantities 6.1.7. Internal symmetry is a global symmetry of the 5-matrix 6.2. Currents and charges 6.2.1. Classical theory of charges and currents. Noether's Theorem. Some remarks concerning the quantal case 6.2.2. Quasilocal and (strictly) local states 6.2.3. Some theorems concerning the sesquilinear forms of the field operators 6.2.4. Definition of the translationally covariant, locally conserved quantum current 6.2.5. Inverse of the quantal Noether Theorem 6.2.6. Outline of the proofs of Theorems 6.2.3-6.2.13 6.2.7. Some characteristic features of the charges 6.2.8. Spontaneously broken symmetry. Goldstone's particles 6.2.9. Identically conserved currents, Gauss' Law and gauge charges 6.2.10. Translationally noncovariant currents. Currents associated with the Lorentz group

IX

101 101 102 102 104 106 108 108 108 110 113 115 115 124 125 126

126 129 131 133 135 142 154 157 161 167

X

Table of Contents

6.2.11. Charges of massless particles. Dilatations and special conformal symmetry. The Weinberg-Witten Theorem 6.2.12. Vanishing of the current in the one-particle Hilbert space implies vanishing of the charge 6.2.13. Examples of internal symmetry. Carruthers' Theorem 6.2.14. Fermionic charges 6.3. A comment on anomalies References and Comments to Chapter 6 7.

Global Symmetries and Supersymmetries of the 5 - m a t r i x 7.1. The structure of the generators of the symmetries of the 5-matrix and the "No-Go" Coleman-Mandula Theorem 7.1.1. Bosonic charges in the framework of the free asymptotic fields 7.1.2. Special interpolating fields 7.1.3. The "No-Go" Coleman-Mandula and O'Raifeartaigh Theorems 7.1.4. Outline of the proofs of the O'Raifeartaigh and "No-Go" Coleman-Mandula Theorems 7.1.5. Affluence of symmetries in the free field theory 7.2. The Lie algebra of the bosonic charges 7.3. The structure of generators of supersymmetry 7.3.1. Fermionic charges in the framework of the free asymptotic fields 7.3.2. The extension of the Lie algebra of bosonic charges to the super-Lie algebra 7.3.3. Translationally invariant spinorial charges in a massive field theory. The degree of a charge 7.3.4. Commutation relations among the spinor and scalar charges. Central charges. Intertwiners 7.3.5. Summary concerning the super-Lie algebra for m ^ 0 7.3.6. Historical note about supersymmetry in physics References and Comments to Chapter 7

169 175 176 179 179 183 189 189 189 195 197 202 218 221 223 225 225 231 236 242 244 244

Table of Contents

8.

9.

Representations of the Super-Lie Algebra 8.1. The relation between the highest spin value and the number of spinor charges 8.1.1. Derivation of the relation 8.1.2. Simple examples 8.2. Representations of the super-Lie algebra 8.2.1. The case without central charges. Fock construction and the fundamental irreducible multiplet 8.2.2. The N = 1 case without central charges. Field theoretical approach 8.2.3. A model for the JV > 1 case (extended supersymmetry) without central charges 8.2.4. The case with central charges 8.2.5. A comment on groups we are concerned with 8.3. The bounds for central charges References and Comments to Chapter 8 The Case of Massless Particles 9.1. Bosonic charges 9.1.1. Bosonic charges of degree 1 9.1.2. Bosonic charges of degree 2 9.1.3. Bosonic charges of degree higher than two do not exist 9.1.4. Some facts about the dilatational and conformal symmetries in quantum field theory 9.1.5. Summary concerning the algebra of the would-be S-matrix symmetries for m = 0 9.2. Fermionic charges 9.2.1. Fermionic charges of degree 1 9.2.2. Supplementary algebraic relations involving fermionic charges of the would-be S-matrix supersymmetries for m = 0 9.3. Representation of the super-Lie algebra for m = 0 References and Comments to Chapter 9

XI

246 246 246 251 252

252 255 259 264 274 276 277 278 279 279 281 283 284 290 291 291

300 301 305

xii

Table of Contents

10. Concluding Remarks References and Comments to Chapter 10

307 307

Appendix Appendix Appendix Appendix Appendix

309 313 329 339

1. 2. 3. 4. 5.

Bispinors Free quantum fields Scattering theory of Haag, Ruelle and Hepp The Reeh-Schlieder and Jost-Schroer Theorems The Wightman formalism and the Reconstruction Theorem Appendix 6. Noether Theorem Appendix 7. Some useful formulae References and Comments to the Appendices

344 351 354 357

Index

359

An Introduction to SYMMETRY and SUPERSYMMETRY in QUANTUM FIELD THEORY

Chapter 1 INTRODUCTION

1.1. I n t r o d u c t o r y R e m a r k s a b o u t S y m m e t r i e s i n P h y s i c s The main subject of my lectures will be symmetry and supersymmetry occurring in q u a n t u m field theory. The word ''supersymmetry'' makes one think of something akin to the notion of symmetry, a more general kind of symmetry. This is correct although one could argue whether the name is chosen properly. Supersymmetry is indeed an extension of the standard concept of symmetry used in classical textbooks. To make this clear we first have to learn something about the symmetries. Thus let us say a few words about them. Since childhood we encounter in daily life more or less clear-cut symmetries. Most people believe t h a t the world reflected in a mirror is exactly as good or as bad as t h a t real one in which we live, and are definitely wrong as shown by Lee and Yang in 1957. 1 We are fascinated by the perfection of proportions of sculptures and edifices created by great masters. The ornaments discovered in excavations in Egypt involve all 17 known crystallographic 2-dimensional groups; the same is true with the ornaments of Alhambra. 2 Not all of us, however, are aware of the fact t h a t the concept of symmetry entered into physics in a mathematically consistent way relatively late. 3 It was of course, obvious to every physicist and almost everybody, t h a t crystals (as an example) display symmetry; but what exactly was meant by the notion — symmetry — and how to formulate it in mathematical terms was not clear. In the 1870s and 1880s Curie was among the first to introduce a modern concept of symmetry which discussed the structure of crystals. It took a long time to create theoretical foundations of the mathematical concept of space l

2

introduction to Symmetry and Svperiymmetry

groups which describe the symmetry of the crystals (Fedorov in Russia, Schoenflies in Germany around 1890). The next great development in the concept of symmetry in physics was special relativity (Poincar£, Einstein, Minkowski). Then came general rela­ tivity. Einstein started from a very general symmetry principle and derived the gravitational equations on this basis. In this way symmetry dictated the interaction among fields, as in this particular case of the gravitional field and the m a t t e r as its singularities. Finally came the striking concept of gauge fields which developed from the geometrical symmetry consideration. The original idea goes back to Weyl (1918) and climaxed in the beautiful and profound theory of nonabelian fields of Yang and Mills. 4 Not all of us are aware of a fact well known to artists that exact symmetry is not an attribute of perfection. 5 A piece of art can be truly beautiful only when its regular symmetry is accentuated by fine, almost imperceptible breaking of this symmetry. 6 These aesthetic canons have its counterpart in nature. Ideal exact symmetries do not occur in nature; all of them are approximate. The defect of human experience method forces the scientist to separate certain class of events he is investigating from the integral whole of occurring phenomena, to adopt an abstract model of the world as well as to introduce and use idealized notions like t h a t of exact symmetry of a physical system. 7 This procedure is, in my opinion, the only one left to us by Providence in exploring nature in a systematic way; it cannot be improved although we are aware of the deficiency of this method. Let us illustrate these considerations with the following example. In the nonrelativitistic theory one considers the Galilean symmetry group as t h a t one which in the best possible manner simulates the symmetries encountered in nature. But it turns out that this exact symmetry does not mimic nature too well as soon as we examine events in which the observable velocities are comparable with the speed of light, in which the principle of relativity of Einstein plays a role. The special theory of relativity which in turn rests on the theory of Lorentz transformations, and which rules in the experiments in the laboratories created by man, loses its unshaken position as soon as we turn to problems of gravity and cosmology. Suppose we agreed on a certain investigation method based upon an ab­ stract mathematical model. Then there arises a paradoxical situation: it seems rather self-evident that a notion of approximate symmetry should be intuitively easier for us to grasp than that of an exact symmetry; this should be the con-

Introduction

3

sequence of our everyday experience in which we have dealt exclusively with approximate symmetries. On the contrary, however, it is by far easier for us to define an exact symmetry and to make use of it than to give a clear-cut and nonambiguous definition of an approximate symmetry. This lies in the logi­ cal structure of our experience; the case of exact symmetry can be distinctly separated in mathematical terms from the case of approximate symmetry; the approximate symmetry cannot be sharply defined and admits a variety of in­ terpretations. This does not create a problem for an artist who is guided in his work by his artistic taste. It is not even a problem for an experimental physicist who is accustomed to using his intuition in his work; in his opinion the problem is solved as soon as he comes to the conclusion that the deviation from the exact symmetry is minor. If, however, the theoretical physicist tries to formulate in mathematically rigorous terms what the experimental physicist meant by small deviation he gets, in general, into serious trouble which often cannot be overcome. This is the reason why physicists as well as mathematicians first concen­ trated their efforts on solving the problem of exact symmetry; they either subject the events occurring in nature to a process of abstract idealization or they restrict the domain of applicability of the theory, or both. We shall here also deal mainly with exact symmetries. However, before we do it we first have to decide the model of nature we are going to use. We intend to restrict ourselves to problems of symmetry which do not exceed the size comparable with human proportions, in other words, first and foremost to problems of symmetry of quantum systems which are encountered in laboratories here on Earth. We will refrain from problems going beyond this scale such as that of the theory of general relativity or supergravity. Our model of nature will then have a limited range of applicability. We demand that this model be logically coherent and based on properly chosen axioms which reflect in a possibly simple, idealized way some basic facts observed in nature; we require also that the theory founded on these axioms is mathematically rigorous so that it should be possible to check by confronting the theory with experimental data whether the axioms reflect the properties of nature properly. Such a theory is called axiomatic. Our main topic will be the quantum field theory, thus we shall deal here with the axiomatic quantum field theory. The reader will find more about this model in Sec. 1.2 and Chaps. 3, 4 and 5. We could readily choose a different model of the world to start with, e.g. to use a model based upon the functional integral "over all paths" of Feynman. This method, although not yet mathematically completed, seems to be a very

4

Introduction to Symmetry and Supersymmetry

handy tool for performing computations in which mathematical rigour is not strongly emphasized. It has the advantage that it fits very well into the modern scheme of gauge field theories and ensures good, though not always rigorous, results. This method is very often used in recent work on supersymmetry as well as on supergravity; the bibliography is enormous. The reason for deciding to abide by the somewhat outdated method, the axiomatic field theory, is that these lectures are meant as an introduction to the problem; the foundation of symmetry and supersymmetry can then be rigorously expressed. This helps to get a more consistent and reliable picture of standard field theory. The moment the choice of the model has been decided we may again turn towards the symmetry. To start, we shall try to make the notion of symmetry in physics clearer. The meaning of symmetry of a physical system is frequently influenced, if not shaped, by the guidelines of our investigation. It is obvious that the symmetry of a physical system is closely related to the transformations of the parameters describing this system. Notice, however, t h a t not every transformation of parameters is linked to a symmetry of the system; such symmetric transformations have to satisfy certain conditions. The necessary condition is t h a t the physical system remains the same object of our perception before as well as after the transformation. Next, one has to distinguish between invariance of laws of physics under certain symmetries and the covariance of its realization in nature; if the symmetry is subtantiated the laws of nature must be invariant with respect to it. The realization, however, can but does not need to be symmetrical. In case the laws of nature as well as the realization are invariant under the transformation, the tranformed object looks exactly the same before as well as after the action. We say that a 3-dimensional sphere has a rotational symmetry because the picture of it does not change while we rotate it through an angle around an abitrary axis going through the center of this sphere. A more complicated situation arises when the realization is not symmetric. We discard here the case when the symmetry is explicitly broken by introducing interaction with external sources. Even in the absence of such sources the realization does not need to be symmetric. Then the sufficient condition to maintain symmetry of the theory is that the transformed realization behaves covariantly, not invariantly. Both realizations, the original one as well as t h a t obtained as a result of transformation, are equivalent in the sense t h a t both can be equally well carried out and neither of them displays any special properties not shared by the other. To adapt George Orwell's adage from "Animal Farm": no one realization is more equal than the others.

Introduction

5

Let us take as an example the relativistic field theory. This theory as a whole is symmetric with respect to the Lorentz transformations. This means that independently of the choice of the frame of reference, the same field theory is the object of our investigation; changing from one frame to another the fields transform covariantly according to the rule imposed by the principle of relativity. The question arises on how to choose the parameters whose transformation leaves the physical system invariant. This choice is to some extent ambiguous since it depends on our views and goals (as mentioned before) and on whether we regard a parameter as a dynamical variable or material constant. In quan­ tum field theory, for example, all physically meaningful continuous symmetries leave the mass, spin or charge of a particle invariant; admittedly the renormalization group changes the mass and charge of the particle but it is not viewed as a physical symmetry. To make this consideration clearer let us examine the following example. The massless free fields are dilatationally covariant. This comes about as there is no length standard available8 and one cannot record any change of the system when the space and time variables as well as the field itself are accordingly enlarged or diminished (the latter is required for a quantum field to preserve the proper canonical commutation relations). A similar procedure is applied for constructing aerodynamical models; one makes use of the dimen­ sional analysis; one simulates using a miniature model, say, in a aerodynamical tunnel, with conditions closely approximating what a real aeroplane is likely to encounter during its flight; extensive use of so-called Reynolds number is made here. The similarity between both mentioned cases is, in my opinion, superficial as the dimensional analysis is not what we would call symmetry in quantum field theory (notice that the dimensional analysis is related to the renormalization formalism). Should we apply transformations of this kind in a theory of massive and charge fields we would be forced to regard the mass as well as the charge as dynamical variables; these new dynamical parameters would undergo a transformation. One has, however, to admit that it is essen­ tially a matter of convention whether one considers the dimensional analysis to be a symmetry or not. The canonical dilatation of the massless free field for example, can be identified without a doubt with the special case of transforma­ tion of dimensional analysis. This canonical dilatation symmetry holds only for free fields and every free field displays canonical dilatation symmetry; for those fields which display dilatational covariance but are no longer free (anomalous dimension) the close link between dilatational and dimensional transformation

6

Introduction to Symmetry and Supertymmetry

is destroyed. It is easy to see that the symmetries of a physical system can be described mathematically in the most elegant way in terms of group theory. Two trans­ formations performed one after another can be replaced by one transformation of the same kind. There exists an identity transformation consisting in that no transformation of the system is performed; to each transformation corresponds an inverse one as if this would not be the case should there exist some transfor­ mations which violate the principle of equivalence of all the transformations. We shall be, of course, interested in a narrower class of symmetries which not only may exist in our model of the world but to which we can assign a physical meaning. We shall encounter for example symmetries which are trivial and to which no physical sense whatsoever can be attributed. We shall try to reject the latter by applying the notion of Borcher's classes as well as the scattering matrix formalism (see Sec. 6.1., in particular 6.1.5). Some symmetries are organically built in into the model. Since we restrict ourselves in these lectures to a model linked intrinsically to the principle of relativity, the symmetries of the space and time expressed by the Lorentz transformations will be of this kind. It will turn out that what we actually need is the covering group of the Lorentz group — the SL(2, C) group (see Sec. 4.1). In addition there is a symmetry which is also closely linked to the principle of relativity and follows from the homogeneous structure of the space and time continuum — the space-time translations (see Sees. 4.1 and 4.2); if the whole universe happened to be displaced we would not be able to notice and ascertain this fact. As far as the time shift is concerned the time variable in the theory of Einstein is linked to the space variable by a linear relation. This invariance or covariance of the model with respect to the pseudo rotations and translations in Minkowski space is taken mathematically into account by incorporating both groups, Lorentz and translation groups, into our model. These two groups are welded into one — the Poincare' group. It turns out that particular representations of the latter group, appearing in our model, are characterized by two parameters, by the eigenvalues of the so-called Casimir operators of this group. These two quantities are closely re­ lated to the mass and spin of the investigated physical system. It will turn out in the course of our lectures that we have to differentiate carefully between the massless and massive cases. The massive case was comprehensively and thor­ oughly investigated as far as the theory of the scattering operator is concerned — and it should be emphasized that this theory will play an important role in our considerations (see Sees. 5.2 and 5.3). The situation is somewhat different

Introduction

7

as far as the massless case is concerned although in the last ten years consider­ able progress has been achieved. This progress is due to the pioneering work of Buchholz (see Chap. 9). One is often tempted to extend the results obtained for massive systems to the massless ones; in most cases such an extrapolation can be justified, but not always. The opposite is definitely not true: there are symmetries of massless systems, like dilatational and conformal mappings, which can never be implemented in the massive case. Moreover, in the mass­ less case a phenomenon of an approximate symmetry which can be described in mathematically rigorous terms can arise — so-called spontaneously broken symmetry (see Sec. 6.2.8). This and the related Higgs phenomenon cannot appear in the absence of massless systems. It is worthwhile to notice that the case of spontaneous breaking of symmetry is remarkable in that it mixes the internal and geometrical symmetries in a nontrivial way; such a mixing is ex­ cluded in the massive case by the famous "No-Go" Theorem of Coleman and Mandula (see Sec. 7.1). It was discovered fifteen or so years ago that in axiomatic field theory there is still some room left for extending the notion of symmetry to a more general notion called supersymmetry. No further extension is possible. The main nov­ elty is that the multiplication rule for elements in the symmetry algebra which was until now effected by commutators has been supplemented also by anticommutators in the new extended supersymmetry algebra. The infinitesimal transformations generated by this algebra can change bosons into fermions and vice versa; so the spin is no longer preserved under these generalized symmetry transformations. Supersymmetry is one of the main topics of Chaps. 7, 8 and 9. One can separate the symmetries and supersymmetries encountered in field theory into two classes: i) global ones which do not depend on the position of the physical system in space and time, ii) local ones depending on the position in space and time. To the first class belong symmetries and supersymmetries which in princi­ ple are experimentally observable, specifically in the scattering experiments of elementary particles. One could also rate into this class the so-called dynamical symmetries (e.g. in the nonrelativistic quantum mechanics the equation for the hydrogen atom displays the 0(4) symmetry). To the second class belong symmetries which are hidden from our eyes but not from our intellect, for example, symmetries involved in the structure of

8

Introduction to Symmetry and Supertymmetry

the equations of the theory. They are as a rule linked to the appearance of massless fields. To give some examples: the gauging of the first kind, the SU(2) or SU(Z) mapping, the Poincare group, global supersymmetry as well as space or time reflection, charge conjugation and the P T C mapping number to the first class. Gauge transformations of the second kind, whether abelian (as in electro­ dynamics) or non-abelian (as in the Yang-Mills theory), and supersymmetry depending on space-time, belong to local symmetries. Among the examples given above are discrete transformations like t h a t of space or time reflection, charge conjugation and the P T C mapping. These transformations do not fit into the mathematical theory of groups depending on continuous parameters; in other words, they do not belong to the realm of Lie groups. In these lectures we shall occupy ourselves first with the symmetries which depend on finite number of continuous parameters. Then we shall try to ex­ tend and generalize this formalism to the case of supersymmetry. Not much attention, however, will be paid to discrete symmetries. The next section will be devoted to the ideological background of the quan­ tum field theory, in particular of the axiomatic quantum field theory. 1.2. I n t r o d u c t o r y R e m a r k s a b o u t Q u a n t u m F i e l d T h e o r y , in particular Axiomatic Q u a n t u m Field Theory The conventional approach to quantum field theory arose as a n a t u r a l but formal extension of classical field theory as well as quantum mechanics. Its vehement development falls into the thirties and forties of this century. The classical analytic mechanics of systems of finite number of degrees of freedom is well known. 9 One is then able to extend formally the m a t h e m a t ­ ical workshop of classical mechanics to systems of infinitely many degrees of freedom, for example, hydrodynamics or electromagnetism. The mathematical formalism of general q u a n t u m mechanics of systems of finite number of degrees of freedom is also well established. There are some fundamental, philosophical aspects as that of the Einstein-PodolskiRosen paradox 1 0 which are not yet solved, but do not have any decisive impact upon the computational technique. One may use the same idea as in the case of classical mechanics and extend formally the methods of quantum mechanics of finite number of degrees of freedom to the case of infinite degrees of freedom. In this way one gets the conventional quantum field theory.

Introduction

9

The fundamental notion of conventional field theory is the Lagrangian and the main tool is the Lagrange formalism, based upon the variational princi­ ple. Prom this principle we get in a formal way the equations of motion for the fields; we may also establish the standard canonical formalism of Hamil­ ton. So far everything is classical. To get a quantized theory one exploits the celebrated rule consisting in replacing the classical Poisson brackets by com­ mutators of operators acting in the Hilbert space; all commutation relations are taken at equal time. Equipped with this model one tries to evaluate some experimentally measurable quantities; the computations are performed first and foremost by using the formal perturbation procedure paying little or even no attention to convergence problems. In this context the problem of renormalization arises; to make the theory consistent to some extent one is forced to remodel the original theory in such a way that former quantities appearing in the theory are replaced by new quantities which differ from the former ones by a numerical factor. These newly introduced quantities are known as renormalized ones; this procedure also affects the commutation rules of the field variables. To apply this formalism one has to postulate the functional form of the Lagrangian. This is relatively easy in the case of free fields which do not inter­ act with each other; the invariance or covariance of these fields with respect to symmetries supplies sufficient information to derive the form of the Lagrangian in a unique way. The situation becomes more involved as soon as one tries to do the same in a theory with interaction; there is no unique choice. One may, however, reduce the ambiguity by accepting the principle of simplicity. This situation is not satisfactory from the philosophical point of view as there can be various forms of microscopic (quantum) interaction, not particularly simple in structure, which correspond to the same macroscopic (classical) interaction. One can also raise serious objections against the renormalization procedure. This procedure has a deep physical meaning. The problem of renormalization appears already in classical field theory and is well justified; what makes it meaningless, however, is t h a t the numerical factors mentioned before become either infinitely large or infinitely small. It is not advisable to use such vague expressions in a theory which one would like to see selfconsistent and mathe­ matically precise. The physicists were well aware of the mathematical shortcomings of this model of nature, nevertheless they had to live with it and to use it for lack of anything better. They did not pay too much attention to mathematical prob­ lems. On the one hand they were dazzled by the spectacular success of this

10

Introduction to Symmetry and Superiymmetry

formalism in quantum electrodynamics, and on the other hand, they relied on intuition. Some of them argued that although the rules seem to be mathemat­ ically nonsensical they nevertheless rest on intuition which is a substitute for a rigorous mechanism not invented by mathematicians yet. In support of this point of view one could as an example quote the anecdote concerning the delta function of Dirac: the latter was considered by some famous mathematicians as nonsense until Schwartz and others founded the theory of distributions (see Sec. 3.2). That there is some truth to that reasoning defending these piratic developments can be inferred from the later trend called constructive field theory developed in the sixties and seventies which is a synthesis of formal perturbative and axiomatic approaches. After the brilliant success of quantum electrodynamics came setbacks suf­ fered in mesodynamics. 11 The development came to a stop and the theory seemed to be in a dead end street. This strengthened the opposition against the light-hearted use of methods banned by mathematicians, the arbitrariness in choosing the assumptions and against the strange recipes on how to handle the infinities popping up in the formulae. This rebellious atmosphere inspired some critically minded but creative physicists in the early fifties to propose an essentially new formulation of quantum field theory; to name only a few: Glaser, Haag, Kallen, Lehmann, Symanzik, Wightman, Zimmermann. 12 This approach is called axiomatic. What is new in this approach and how does it differ from the conventional one? In the axiomatic quantum field theory the Lagrangian is not even men­ tioned; consequently one does not have the Euler equations of motion, in this case, field equations; the canonical formalism as well as the equal time com­ mutation relations for the fields are also not used. Instead one introduces a set of axioms of possibly general character, which most physicists would be inclined to accept, and rejects all assumptions which may raise some doubts or the presence of which is not sufficiently justified by physical arguments, which appear artificial or of very special character; these axioms are, of course, not contradictory and independent from each other. 13 Equipped with the axioms one constructs a model of nature using to this aim the full mathematical rigor. As mentioned in Sec. 1.1 this is a very important issue; should such a theory fail, that is, should there appear a discrepancy between theory and experiment, we may immediately conclude that the choice of axioms was not proper and that some of them have to be altered. The axiomatic approach can, of course, be realized in many different ways.

Introduction

11

The model we are going to use is constructed in a very ingenious manner by scientists named before as well as their followers. The task to construct such a theory was not easy. One had to overcome, for example, the difficulties concerning the definition of a product of fields at the same space-time point, linked to the problem of equal time commutation relations (for renormalized fields). It will soon become clear why a quantum field cannot be viewed as an ordinary function in the Minkowski space like a classical field; the difficulty in defining such a product is clearly exhibited in the so-called Thirring Model. 1 4 One has to give up a careless exploitation of perturbative methods; it has to be emphasized that there is nothing wrong in applying perturbation calculus but the procedure must be mathematically well defined. In conventional approach one does not even care for interchanging of limits; the following example of a Poisson distribution tn e *— n!

t >0,

n = 0,1.2,... .

where

°° lim Ye-* -*" =1 n!

t—oo *—' n=0

and 0 0

Y lim ^—' t-*oo

j.n

e-'Uo n!

n=0 shows that even in pedestrian calculations one has to be careful. There is no contradiction between the conventional Lagrangian and ax­ iomatic approaches; this can be seen distinctly in the example of free fields (see Chap. 2), where the conventional and axiomatic theories coincide. The results achieved so far by using the axiomatic model of nature are of very general character but are, unfortunately, not numerous; they are not all spectacular and not always immediately relevant for applications. There are few exceptions as the celebrated PTC Theorem, Spin and Statistics Theorem or Dispersion Relations. Until now nobody succeeded in constructing a solution for this model in (3+1) space-time dimensions which would differ essentially from the trivial free field solution; the latter proves that the axioms are not contradictory; but the danger still exists that the impressive mathematical formalism can be applied with success only to the case of free fields. This definitely would not be what we are looking for. We shall in the course of these lectures take for granted that the theory is not trivial; then the results obtained so far by axiomatic methods are almost sufficient for our purposes.

12

Introduction to Symmetry and Superiymmetry

As it was already pointed out the axiomatic approach to q u a n t u m field theory we are going to use is not the only one which was invented in the fifties and sixties. One can, roughly speaking, distinguish between the LSZ formal­ ism (Lehmann, Symanzik, Zimmermann) in which unbounded field operators, particle representation of the ^ - m a t r i x mechanism in the m o m e n t u m space are used and the Wightman formalism in which unbounded operators are also used but the emphasis lies with (anti-) local fields in Minkowski configuration space. Approximately at the same time the Araki approach and its continua­ tion and extension by Haag and Kastler came into being; the latter formalism uses as the basis bounded operators, specifically the so-called C* algebra of local observables. It is difficult to tell how much these different approaches have in common. It was shown in the middle of the sixties that the Wightman approach (see Sec. 6.1.1) we are going to use encompasses the LSZ formal­ ism (see Sec. 5.2). The most promising development which also started in the middle sixties is the so-called constructive quantum field theory which is a kind of renaissance of the older ideas and is a fine synthesis of the naive con­ ventional theory and the mathematically rigorous axiomatic field theory. The results of this approach show distinctly that the primitive and naive methods of conventional theory resting partly on physical intuition were pointing in the right direction. Nevertheless, without axiomatic theory — as an antithesis to conventional approach — this synthesis could not be effected. In this introductory note I tried to justify and emphasize the importance of the mathematical rigour for the investigations of the foundations of the quan­ tum field theory and theory of symmetries based on it. It is appropriate to warn the reader not to draw wrong conclusions from these remarks and not to expect me to adhere to high mathematical standards in the following lectures. To present the subject with full mathematical rigour without exceeding by far the planned size of this book would be impossible for me and — even more important — beyond my professional abilities; my mathematical culture is too poor to take up such a task. So my attitude towards the reader is somewhat like that of some preachers who tell you: do as I preach, don't do as I do. Of course, some mathematical notions and theorems are indispensable when presenting the theory in a most coherent and consistent way. In most cases, however, I shall skip the proofs, except of statements of particular relevance for which I shall present the outline of the proofs.

Introduction

13

References and Comments to Chapter 1 1 T. D. Lee and C.N. Yang, Phys. Rev. 105 (1957) 1671, C. S. Wu et al., Phys. Rev. 105 (1957) 1413. 2 H. Weyl, Symmetry, Princeton University Press, 1952. E. Miiller, Gruppentheoretische Ornamente aus der Alhambra in Granada, Zurich, ETH Doktor Disserta­ tion, 1944. Weyl wrote about the Egyptian edifices: "The art of ornamentation contains in hidden form the oldest fragment of higher mathematics known to us. No doubt that until the 19th century no proper notational tools existed to for­ mulate in an abstract way these problems, as the mathematical notion of a group of transformations did not yet exist; not before this basis was established one was able to prove that there are no other symmetries besides the 17 already known, in their essence of the matter, to the Egyptian artisans. It is amazing that the proof was given first in 1924 by G. Poly a!". I am grateful to Dr. J. Mozrzymas for information concerning these topics. 3 C. N. Yang, Fields and Symmetries — Fundamental Concepts in 20th Century Physics, 1984, Talk given at the Gakushuin University, Japan, August 1983. See also Ref. 2. 4 C. N. Yang and R. L. Mills, Phys. Rev. 96 (1954) 191; there are some unpublished notes of W. Pauli concerning non-abelian gauge field theory written in 1953, see C. N. Yang, Selected Papers 1945-1980 With Commentary, W. H. Freeman and Co., San Francisco, 1983. 5 C. N. Yang, in 7th Winter School of Theoretical Physics, Karpacz, 1970. See H. Weyl (Ref. 2) as well as some pictures of M. C. Escher in The World of M. C. Escher, H.N. Abrams, Inc. Publ., New York, 1971, which illustrate well the unconventional aspects of symmetry. 6 Notice that the two towers of the Frauenkirche (Cathedral and Parish Church of Our Lady) in Munich look about the same but are not identical, one is slightly higher than the other one; the same is true of the St. John Cathedral in my hometown, Wroclaw. 7 Notice that in classical statistical physics one very often uses a picture consisting of particles interacting with each other by means of a potential, even in cases when the interaction is so strong that the separation of a system of free particles from the whole system looks somewhat artificial. 8 We are going to use such units that

2ir 9 Notice, however, that recent investigations on stochastic dynamics show distinctly that the standard classical mechanics does not cover the majority of mechanical problems. 10 See e.g. Symposium on the Foundations of Modern Physics, ed. P. J. Lahti and P. Mittelstaedt, Turku, World Scientific Publ., Singapore, 1985.

14

Introduction to Symmetry and Supersymmetry

11 Speaking here about mesodynamics, we refer to the period of late forties and early fifties. In contemporary approach, mesons are viewed as composite particles in the framework of chromodynamics. 12 See the textbooks: R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, W. A. Benjamin, Inc., New York, Amsterdam, 1964, N. Bogolubov, A. Logunov and I. Todorov, Introduction to Axiomatic Quantum Field Theory, original in Russian, Nauka, Moscow, 1969, English edition 1975, R. Jost, The General Theory of Quantized Fields, Amer. Math. Soc, Providence, 1965, H. Araki, Einfuhrung in die axiomatische Quantenfeldtheorie, lectures given at the E.T.H. in Zurich 1961-62 and the literature quoted therein. 13 See Ref. 12 and e.g. R. Haag and B. Schroer, Journal Math. Phys. 8 (1962) 248. 14 Review article on quantum Thirring model, see B. Klaiber, in Boulder Lectures in Theoretical Physics, 10A (1967) 141, Gordon and Breach, New York, 1968; the original paper of W. Thirring was published in Ann. Phys. S (1958) 91.

Chapter 2

E X A M P L E OF A CLASSICAL A N D Q U A N T U M SCALAR FREE FIELD THEORY

The goal of this chapter is to get the reader acquainted with the theory of free fields using the most simple example of scalar massive field. The free fields will prove to be a handy tool in our investigations. As mentioned before, they can be treated by axiomatic as well as conventional (Lagrangian) methods. 2 . 1 . C l a s s i c a l S c a l a r Free F i e l d The scalar free field theory is controlled by the so-called Klein-Gordon Equation (KG Equation); in the 4-dimensional Minkowski space this equation reads (D* + m 2 ) ~ l j - 1 ) - 1 ) , x„ = r / ^ x " . 15

(2.1.2b)

16

Introduction to Symmetry and Supertymmetry

Here summation over repeated indices is understood, i.e. we skip the summa­ tion sign as soon as the same index occurs twice as a super- and subscript. With this notation

q, = d^

= (3 0 ) 2 - (ao 2 - (3 2 ) 2 - (d3f ,

(2.1.3)

where 3» = ~ -

(2-1-4)

OXft

In (2.1.1) the parameter m ^ 0 is a real number. The quantity -'lkxellr^/!llc2

—

*

/

-

(2*?]

A ( T > -

£±(x) —

(2-\3 J

m2)

e(Ko)0(K

" */

-iunn

2Wfc{e

C

m )

iut*(,\ikx ,C

(2.2.16)

where e[k0) is a distribution defined by . £l

. _ f

1

°' ~ \ - 1

for fc0 > 0 for fc0 < 0 .

The distribution (2.2.16) is called the delta of Jordan and Pauli. It follows immediately t h a t A(x) = — A(—x) A(x) = A(x)

(skewsymmetric) (real) .

(2.2.17a) (2.2.17b)

W h a t are the transformation properties of the fields and other quantities related to it under the Poincare' transformation? As suggested by (2.1.12) we should have on the one hand ip'(A.x + a) = u = 0 .

(3.1.2c)

Notice that (3.1.2) implies p(u) > 0. For two elements u, v e ft we define the distance between them by p{u - v) . We say that {un € ft, n = 0 , 1 , 2 , . . . } converges strongly towards u S ft, viz. u n —► u

(3.1.3a)

n—*oo

if lim p(un - u) = 0 .

(3.1.3b)

n—*oo

Then the normed linear space ft is a linear space with the convergence given by the norm. A sequence { u „ } is fundamental (or a Cauchy sequence) if for any e > 0 we can find an integer N(s) such that p(u,n — u m ) < e

for n, m > N(e)

or lim

p(un - u m ) = 0 .

min(n,m)—»oo

If u n S ft converges strongly to u e ft then { u „ } is a Cauchy sequence; notice, however, that not every fundamental sequence converges to a limit u G ft. If the limit of every fundamental sequence { u „ } e ft belongs to ft then

30

Introduction to Symmetry

and

Supereymmetry

the normed linear space ft is called complete or a Banach space. It can be shown that every normed linear space can be completed to a Banach space. A Hilbert space )l is a Banach space equipped with a scalar product (u, v), u, v € X, satisfying the following requirements (u, Aiwi + A 2 v 2 ) = Ai(u, t>i) + A 2 (u,u 2 )

(linearity) , (3.1.4a)

(u,v) = (v,u)

(antilinearity) ,

(3.1.4b)

(u,u)>0,

(3.1.4c)

(u,u)=0=>u = 0,

(3.1.4d)

where Ai, A2 are complex numbers and u, v, Vi, t>2 € )t. We may introduce in )i the norm p(u) = +\/(u,u) = ||u|| . (3.1.5) According to (3.1.3) the sequence {un,n towards u, where un and u 6 M, if

= 0 , 1 , 2 , . . . } converges strongly

lim ||u„ — u|| = 0 .

(3.1.6)

n—t-oo

We shall need in the future another topology of H namely the notion of weak convergence as this notion plays an important role in physics. We say t h a t the sequence ( u „ , n = 0 , 1 , 2 , . . . } converges weakly towards u if lim («,u„) = («,«)

V«€^.

(3.1.7)

n—^oo

Of course, the strong convergence implies the weak convergence but not the other way around. A functional / over the space ft relates Vu 6 ft a complex number / ( u ) . If Vu, v S ft and V complex numbers a, /? we have f(au + Pv)=af{v.) + 0f{v)

,

we call the functional a linear one. Assume that ft is a normed space and u„ € ft as well as u S ft. If lim p(un — u) = 0 n—*oo

implies

/{«„) — f[u) n—*oo

Scene and Subject of the Drama. Axtomt 1 and 2.

31

in the sense of convergence of complex numbers, then we say that / is contin­ uous in the topology of ft at u. We say that / is bounded if Vu € ft we have |/(«)| 0 is dependent of u. One can show that every functional bounded on ft is also continuous on ft. If a functional is linear and is continuous at one single point it has to be bounded and consequently has to be continuous on ft. For fixed v e % the scalar product (v, u), where u & )(, is a continuous (bounded) linear functional over M in u. Every continuous linear functional over )(, /(u), can be represented in the form of a scalar product. /(u) = [v,u) where u is arbitrary and v a fixed element of M (Theorem of Riesz). Given a normed linear space ft with the norm p and a continuous func­ tional /o(u) defined over an arbitrary subspace fto C ft, there exists a linear functional /(u) defined on the whole space ft such that /(u) = / 0 (u)

Vuefto

and p*, = p*. where pi is the smallest number Cj in (3.1.8) (Theorem of Hahn-Banach). The Hilbert space )i is separable if there exists a denumerable complete orthonormal system of vectors e i . e a , . . . € M with {ei,ei) = 6ij.

(3.1.9)

Here a physically motivated comment is in order. The separable Hilbert space is for our means perfectly sufficient. As it will turn out soon we may omit the adjective "separable" in the formulation of Axiom 1. In quantum mechanics of finite number of degrees of freedom one has always to do with a separable Hilbert space. The L^ space of square-integrable func­ tions forms such a space as e.g. the Hermite functions form an orthonormal, complete system. In case of field theory, however, the first impression is that we have to use a nonseparable Hilbert space as the number of degrees of freedom becomes

32

Introduction to Symmetry and Supcriymmetry

infinite. It looks like as in every point of the 3-dimensional space an (in gen­ eral anharmonic) oscillator had to be attached. This impression is, however, misleading. This point was discussed in detail for a scalar free field in Sec. 2. In case of an interacting field it turns out that it is also sufficient to use a separable Hilbert space. The construction of the Hilbert space for a field the­ ory which admits particle interpretation and which we are going to use (see stronger form of Axiom 4), consists then in composing a direct sum of Hilbert spaces each of which is separable and can be numbered: the space Mo consists of states representing absence of particles in the physical system (in our case Mo will consist of one state only — the vacuum; see Sec. 4.2.1), Mi represents the space of one particle states only, M2 two particle and so on. Then M = M0eMi®M2®...

(3.1.10)

(so-called Fock construction) is separable. This follows from the observation that a countable set of countable sets is again countable. The direct sum (3.1.10) consists of vectors u = (u0,ui,u2,...),

u3e)ij,

with the scalar product defined by CO

>=0

with the norm 00

H 2 = X>II2j=0

All vectors differing by a multiplicative numerical factor belong to the same ray. In quantum theory we deal mostly with non-zero vectors whose length — in conformity with the probabilistic interpretation — does not matter; thus we may normalize them as follows 1 :

Ml = 1 • Taking this into account the unit ray is a set u = {{e* a u} : u fixed S M,a arbitrary real number} .

(3.1.11)

Scene and Subject of the Drama. Aaomt 1 and S.

33

Even if u is not a unit vector we shall, in the following, use the notion of a ray in the sense of (3.1.11). Physical justification for use of rays is that in physical calculations based on experiments the important quantity is |(u, v)\2 and not (u, v) itself; the first one remains the same for each choice of u £ u and v & v_ while (u, v) goes into e -«{«x-«»)( U | 1 > )

where u = { e x p { t a i } , « = {exp {%a2}v}. There is still one issue concerning the Hilbert space which needs a com­ ment. In axiomatic field theory, like in quantum mechanics, one assumes that the underlying Hilbert space is a space of positive definite metric. This as­ sumption can be questioned as the probabilistic interpretation concerns only physically realizable states as e.g. those measured in scattering experiments; the states of the system appearing in the period of time between the initial and final scattering experiments, where any interference of the observer into the quantum process is excluded, could be as well of a different nature e.g. they could have a negative norm. Moreover, in the theory of gauge fields (elec­ trodynamics as well as Yang-Mills theory) the requirement of positive definite metric in Hilbert space causes violation of locality as well as of Lorentz covariance of the theory; in this case the standard procedure is to use a space with indefinite metric. In spite of these objections we shall in these lectures consistently adhere to the positive definite metric approach. To save locality and Lorentz covariance of massless fields we shall refrain from models where it is unavoidable to introduce non-local, Lorentz non-covariant potentials like in Yang-Mills gauge theory. 3.1.3. Operators in the Hilbert space and their application in quantum field theory A set D is dense in M if Vu G M and e > 0 3 a vector v € D such that ||u — u|| < e . By adding all accumulation points to D we get the closure of D and denote it by D. Let us make a digression. What is the difference between closure of a set and the completion of a linear space by means of Cauchy series? We cannot close a linear space as long as we do not know the topology. As soon as we have a complete normed linear space we know what its topology is and then

34

Introduction to Symmetry and Supenymmetry

it makes sense to speak about closure in its topology of a subset belonging to this complete space. One can show that D = M. A necessary and sufficient condition for a set D to be dense is (u, v) = 0

Vv e D => u = 0 .

Consider a linear operator A in M defined as follows: A(u + v) = Au + Av , A{Xu) = XA{u) . u, v, A(u + v), Au, Av 6 )t and A is a complex number. A linear operator is, in general, not defined in the whole M; let us call the linear subset DA on which the operator A is defined as the domain of definition of A. The linear subset AD A is called the range of the linear operator A. We may introduce the norm of an operator A by |J4| = sup ||v4u|| . ||u|| = 1

ueDA We say that an operator is bounded in DA C H if \A\ is finite. If |.A| is infinite the operator is said to be unbounded. It turns out that A is bounded in DA if \\Au\\2 < CA\\u\\2

VueD,,

where 0 < CA < oo and independent of u. If A is bounded in DA it can be extended to the whole M with the same norm (the same CA) and it remains linear; if DA = M the extension is unique. An operator is continuous at v when u - t t = > Au —► Av . It can be shown that every bounded linear operator is continuous; conversely, a linear operator continuous at v = 0 is bounded and consequently continuous everywhere in )(. Thus the notions of boundedness and continuity of a linear operator coin­ cide in a similar way as in the case of linear functionals.

Scene and Subject of Die Drama. Axiom* 1 and S.

35

We infer that an unbounded operator is discontinuous at every point of its domain of definition and cannot be defined in the whole H. In the sequel we shall have to deal with sequences of operators {An,n = 0 , 1 , 2 , . . . } . We say that such a sequence converges strongly or weakly towards an operator A if Mm \\(An - A)VL\\ = 0 V u e U r , (3.1.12) n—*oo

or lim (v, Anu) = (v, Au)

Vu € Dx .

Vu e X ,

(3.1.13)

n—+oo

respectively, where D^ is a common domain of definition of the operators An and A. Of course, (3.1.12) implies (3.1.13). We are now going to introduce the notions of an adjoint, symmetric (hermitian), essentially self-adjoint and self-adjoint operator which play an important role in quantum field theory and in our considerations. Consider («, Au) for v 6 M arbitrary but fixed and Vu e DA • Assume that for this particular choice of v we have \(v,Au)\ 0 and is independent of u. Then (w, Au) is a linear bounded (continuous) functional over DA ■ According to the Theorem of Riesz (v,Au) = (wi,u),

vi e M .

If DA is dense ui is defined uniquely, otherwise we may add to wj any vector X ^ 0 for which (x, «) = 0. We assume DA = )( and define the (hermitian) adjoint operator as follows: A'v — vi . If DA is dense, A^ exsits and is a linear operator defined for each v € DAt for which (3.1.14) holds (N.B.: DAt may consist of the zero vector only). For further developments we need the notion of a closed operator; if {u„ £ DA , u„ -» u, -Au„ -+ t>}

Introduction to Symmetry and Supertymmetry

36

=> {u e DA , Au = v) then A is a closed operator, noted A = A. Notice t h a t this does not imply that A is continuous at u. An operator A although not closed can sometimes be extended to a closed one. If an operator is closable and everywhere defined in the Hilbert space it is also bounded. Notice t h a t there are examples of unbounded operators defined everywhere in the Hilbert space. One can show t h a t A* is always closed, regardless of whether A is closed or not. If DAi is dense then A^ exist and is an (closed) extension of A, DAn

2 DA

and A = Au One can show that A^

on

DA .

is the smallest closed extension of A A = A"

(DA = DA„

2 DA) ■

The necessary and sufficient condition for DAi to be dense is the existence of A. We call an operator A symmetric (hermitian) if (v, Au) = (Av, u)

Vu, v S DA ;

in other words Au = A*u

Vu E DA .

Then DA,

2 DA

(3.1.15)

and DAi is dense. Consequently A^ = A always exists. We conclude that a symmetric operator is always closed. As A^ is the smallest extension of A we have DA1 2 DA1,

= DADDA

.

(3.1.16)

We say t h a t A is self-adjoint if DA = DAt{=

DAn

= DA)

(3.1.17)

Scene and Subject of the Drama. Axiomt 1 and S.

37

or essentially self-adjoint if DA = DA,(=DAu)DDA.

(3.1.18)

An essentially self-adjoint operator A has a unique self-adjoint extension to A. The self-adjoint as well as essentially self-adjoint operators are important in physics as they represent in an abstract way the measurable physical quantities. The reason for that is that the self-adjoint operator admits a unique spectral decomposition. In other words, such an operator has uniquely defined real eigenvalues, which are either discrete or continuous, as well as eigenfunctions which correspond to these eigenvalues; these eigenfunctions either are vectors in the Hilbert space (for discrete eigenvalues) or are vector-valued distributions (for continuous spectrum). Then the mean value

H of (3.1.1) is always real and uniquely defined. This is just what we expect for measurable quantities. To get a uniquely defined spectral decomposition it is enough to have an essentially self-adjoint operator as it can be uniquely extended to a self-adjoint operator by closing it. This is, in general, not the case with symmetric op­ erators. It can happen that either they cannot be extended to a self-adjoint operator and therefore cannot be spectrally decomposed or the extension is sometimes not unique and then the spectral decomposition depends on the details of the procedure of extension. This makes a physical interpretation difficult or even impossible. Notice that an unbounded self-adjoint operator does not essentially differ from a bounded self-adjoint operator which is defined in the whole Hilbert space as we may cut the spectrum of the unbounded operator in pieces. To end this section let us examine as an example, the case of the operator A = -t— dx encountered in quantum mechanics. First we show that it is not defined in the whole Hilbert space; take u(x) = x~* , 0 < e < - ,

0< x < 1 ;

Introduction to Symmetry and Sapertymmetry

38

we have

-It

/ \u\2dx= Jo

I Jo

x~'

l-2e o

l-2e

< oo

but

f \Au\2dx = e2 [ Jo

x-2l*+1Ux

=

Jo

—x-1-2']1 l + 2e lo

+ 00

l + 2e The operator A is defined on the dense set DA consisting of all absolutely continuous functions u(x) in the closed interval [0,1] with the property u(0) = u ( l ) = 0 and is symmetric. This operator has infinitely many self-adjoint extensions Aa = A\ where D^a consists of all absolutely continuous functions u(x) in the closed interval [0,1] with the property u(0) = exp { i a } u ( l ) ,

a - real number .

2

A side remark: for L (—00,00) there is a unique self-adjoint extension of —id/dx, but for L 2 (0,oo) there is none (e.g. for a radial coordinate in a spherical coordinate system —id/dr cannot be made a canonically conjugate self-adjoint momentum). For more detailed presentation of elements of the functional analysis see for example Ref. 2. 3.2. S u b j e c t of t h e D r a m a 3.2.1. Axiom 2 By now we have learnt something about the scene of the drama. It is time to get acquainted with the subject of this drama. Here is Axiom 2: In the 4-dimensional Minkowski space we are given a finite number of operator-valued distributions *

V

,

where e"" = - e " " = e*"

*,„ = 0,1,2,3,

Principle of Relativity. Carnality. Axioms 3, 4 and 5.

51

3

e>° = 2 Re (ay) ,e>k = 2 £ V * ' Im (a,)

(4.1.16a)

are the parameters, (4.1.16b)

1=1

(aoj)AB

=

l

-Wi)AB ,

j, k =1,2,3,

(4.1.16c)

are generators and ( 1 when (jkl) is an even permutation \ S]ki = 1—1 when (jkl) is an odd permutation > . ^ 0 otherwise. J

(4.1.17)

Here again (a^,,)^ are quantities which act in the spinor space as generators of the group SL(2,C) and so have definite transformation properties. On the other hand they can, as 2-dimensional matrices, be written as linear combina­ tions in the elements of the basis. These linear combinations have, in specially chosen representation, the form given by the formulae (4.1.16). Incidentally, the generators of the SU(2) group appearing in (4.1.15) can be chosen as -i

-a3',

j=

1,2,3,

(4.1.15b)

[W^-'t^-

(4.1.15c)

with the Lie-Cartan relations

The commutation relations of the generators, the Lie-Cart;an relations (4.1.13b) for the SL(2,C) group, can be found from (4.1.16) v.'ith the help of the commutation relations of the Pauli matrices; we get kxA>°V»']

=

i(Vx^a^

+ f ^ C x " ~ Ix!*17^

~

IXv^xn)

(4.1.18)

where r/A/, is given by (2.1.2b). Relations (4.1.18) yield explicitly the structure constants of the algebra. As mentioned before the structure constants remain the same for all representations of the group while the form of the generators varies.

Introduction to Symmetry and Supertymmetry

52

To get the Lie-Cartan relations in an abstract form which is independent of the representation we have to replace crA/J related to the 2-dimensional rep­ resentation by abstract symbols for these generators, say, S^; then [Sxx, Spv] = iirix^Sxp

+ rfklMSxv - rjxllSXl/

- rj A t / 5 x ^) .

(4.1.19)

We may formulate the Lie algebra of SL(2,C) in a different way so as to exhibit the similarity between the infinitesimal transformations of the SL(2,C) group (4.1.14) and that of the SU(2) group (4.1.15a) with complex parameters as mentioned before briefly. It turns out that we may express it as a direct sum of Lie algebras of two 5(7(2) groups. Indeed, we have J, = ±(Mt - iN,) ,

1=1,2,3,

^ = i(M, + ^ ) , i

3

M

£ 3kS

>= 2 £

'

^-

Ni = 50/ ,

(4.1.20)

where we used the abstract form of the generators, like in (4.1.19). Such a choice of generators is admissible as each linear combination of the generators can be used as a new generator corresponding to a different parametrization. The Lie-Cartan relations now read 3

[J3;Jk\ = tJ2eJk'J'

•

(4.1.21a)

,

(4.1.21b)

3

[Ks,Kk\ = iJ2ejk'Kl 1=1

[J3;Kk]

= 0.

(4.1.21c)

Formulae (4.1.21a) and (4.1.21b) are exactly the Lie-Cartan relations for the SU(2) group, given by (4.1.15c). The infinitesimal transformation reads 3

l+l-e>"'Sli„

+ o(e) = l + iJ2rilJ' 1=1

3

+*^21n'Kl

+ o{r,,rj) .

(4.1.22)

1=1

For every finite-dimensional representation Jj as well as Ki are hermitian, and the parameters are complex.

Principle of Relativity. Causality. Axioms S, 4 and 5.

53

The 2-dimensional representation of SL(2,C) given by the A matrices is not the only one possible. The other one is given, up to an equivalence, by -(°^)AB

= -{"*»)A* ~ {o^A*

(4.1.23a)

A^B=AA*.

(4.1.23b)

and

The inequivalence between the latter and the original one is easy to show by using the relations (4.1.16c) as well as saj£-1

= -a3' ,

j = l, 2, 3,

(4.1.24)

where e and a3 are given by (4.1.3) and (4.1.11). There are no other inequivalent representations beyond these two. To distinguish between these two different mappings and two spinor spaces induced by them we shall label the latter by dotted indices as indicated in (4.1.23); we follow here the notation introduced by van der Waerden. 7 For the undotted 2-dimensional representation (rep J,)AB

= \{al)AB

(rep Ki)AB

,

= 0 ;

(4.1.25a)

for the dotted one (rep Jt)AB

= 0,

(rep K,)A*

= ifea'e"1)^ ■

(4.1.25b)

Further on we shall denote the undotted 2-dimensional representation as well as its representation space and the corresponding generators by P ' ? ' 0 ' and ajil'

= A, Vu and v e X .

(4.1.65)

Let us inspect the strong limit; we have \\{U - [ / ' H | 2 = (v, (C/t - U'i)(U -

U')v)

= 2(v, v) - (Uv, U'v) - {v, U'^Uv) — 0 using the assumed unitarity of U' and U as well as (4.1.65). To make the notation economic we shall denote the generators of the U(a, A) group with the same letters as used for abstract group namely P\ and Sf,^. From the unitarity of U(a, A) as well as from the Stone's Theorem formu­ lated below, it follows that P\ and S ^ are self-adjoint operators in H.

Principle of Relativity. Causality. Axioms 3, 4 and 5.

67

Stone's Theorem: Every one-parameter group U{t), —oo < t < oo, of unitary transformations for which (n,U(t)v),

u.we/f,

is a continuous function of t admits the spectral representation

eixtdE(X) = eiGt ,

U(t) = f J — oo

where E(\) operator

are the spectral projection operators of an (essentially) self-adjoint +oo

/

XdE(X) . -oo

We also have t—o V

t

I

Thus for an arbitrary one-parameter group chosen out of the 10-parameter spinorial group, Stone's Theorem assures that the generator of this subgroup is a self-adjoint operator. From the structure of the spinorial group we can also infer that the gener­ ators are unbounded. We are going to show that all of them are defined on a common dense domain. The proof is good for every Lie group represented in a HUbert space by unitary operators. 1 6 The reason for reproducing it here is t h a t it is short and rather simple and is very important as far as our physical considerations are concerned. Let us call the abstract Lie group §, and its elements A (g) £ Q, where g stands for the parameters of the group. Let us take any infinitely differentiable function with compact support f(g) = f(A(g)) defined on the group. The compactness of the support of }{g) makes it, for example, possible to in­ terchange the differentiation and integration signs. The unitary representation of this group in X we shall call U(g) = UA(g)). Thus for each u G H we have U{g)ueX

.

Using f(g) we may construct a vector u / 6 M, viz. u} = jf(g)U(g)udg

(4.1.66)

68

Introduction to Symmetry and Supertymmetry

where the integration is over the group and dg is the left invariant Haar mea­ sure. The vectors uj span a linear space. For each generator / of the group

9, exp itl,

—oo < t < oo ,

(4.1.67)

represents a one-parameter subgroup of Q. Then from Stone's Theorem, it follows that G=-;^*7(exp,;t0|t=o,

(4-1.68)

where G denotes the representation of / in )i. The action of (4.1.67) upon u / , defined by the formula (4.1.66) is U(exp {itl})uj

= f f(g)U(exp

{itI))U{g)udg

= [ f(g)U(exP {itI}A(g))udg . J

(4.1.69)

Let us perform on the right hand side of (4.1.69) a change of variables A(g') = exp {itI)A{g)

.

If we set / ( e x p {-itI}A(g'))

=

h(t,g')

we get from (4.1.69) f/(exp {itl})us

= j

fj{t, g')U(g')udg'

(4.1.70)

as dg = dg'. We perform on (4.1.70) a differentiation with respect to t. Notice that jtfi(t,g)\t=o

=

ifi(g)

is again an infinitely differentiable function with compact support in g. By virtue of (4.1.68) we get from (4.1.70) Guf = / fi{g)U(g)udg Obviously uj 6 DQV f.

= u/; G M .

69

Principle of Relativity. Causality. Axiomt S, 4 and 5.

It can easily be shown that

by taking for f(g) real positive functions of very small support around g = 0 normalized to / f(g)dg = 1. Thus Do is a common dense domain for all generators G = G* . It can be shown t h a t in case of the proper spinorial group the common dense domain is £>rj, contained in the domain of definition of the field operators Z?. 16 How the domain looks becomes clear after we get acquainted with Axioms 4 and 6. To get a classification of irreducible representations of ^ o w ^ have to refer to the observation made in Sec. 4.1.2 that irreducible representations of the Lorentz group [)(J>K\ JiK — 0, | , 1 , . . . , are labelled according to the eigen3

3

values of the operators (see (4.1.26)) Yl J?

an
"

= -\e^vxS^P*

,

(4.1.73)

70

Introduction to Symmetry and Supertymmetry

is the so-called Pauli-Lubanski vector. W2 as well as W^ are self-adjoint oper­ ators. The eigenvalues of W2 are -M25(5 + l),

5 = 0,^,1,... ,

(4.1.72b)

where /z2 is the eigenvalue of (4.1.71). For y. = 0 all eigenvalues of W2 vanish identically and therefore cannot be used any longer for labelling the irreducible representations. In this case (/u = 0) the Pauli-Lubanski vector W^ is proportional to P M . Then one can use this proportionality factor, instead of W2, for labelling the irreducible representations for (i = 0. This scalar quantity, called helicity, is defined by the relation W„ = HP„ = P^H

(4.1.74a)

valid when acting in M^=o, the Hilbert space characterized by /z2 = 0. It is invariant under the Poincare transformations; the eigenvalues of the helicity are fc = 0, ± i , ± 1

(4.1.74b)

We may conclude from these considerations that we have to make a clear-cut distinction between the two cases, \i ^ 0 and \i = 0. We shall learn later that the differences between both cases are deep-rooted. To summarize: in case \i ^ 0 the operators P2 and W2 commute with the whole algebra spanned by P^ and S^, n, u = 0 , 1 , 2, 3; the numbers (\H\,S)

(4.1.75a)

related to the eigenvalues of (4.1.71) and (4.1.72) respectively can be used to label the irreducible representations of ^3 0 - In case fj, = 0 the irreducible representations are labelled by (fj. = 0,h) .

(4.1.75b)

For later use we list some properties of W^ valid for arbitrary fi, viz. WMP* = 0 \W„PV\

(4.1.76a)

= Q

(4.1.76b)

and finally [WM1 W»\ = ' W " " ^

■

(4.1.76c)

Principle of Relativity. Causality. Axioms S, 4 and 5.

71

Notice t h a t there is no contradiction between (4.1.74) and (4.1.76). To exploit our knowledge about the van der Waerden notation let us write Pp and Slll/ as spinorial tensors. We have PAB = WABP*

= (P+)AB

>

(4-1-77*)

P" = \(°nA*PAB> MABedb

(4.1.77b)

+ (M+)di)eAB = K J ^ K J B D ^ MAB

= -i(^u)ABS^

,

,

(4.1.78a)

o»» =

M = 0,l,2,3,

is a projection operator in X projecting states belonging to eigenvalues of P M < Pp. The generators P M are unbounded, self-adjoint operators with the common dense domain of definition (see Sec. 4.1.7); they commute with each other on this domain. We interpret Po as the energy operator of the physical system and Py, j = 1, 2, 3, as components of the momentum operator. Prom this physical interpretation follows the important mathematical re­ strictive requirement. If we denote the eigenvalues of Py, with p^ we have p e V+

(4.2.2)

where V + is the closed future light cone in Minkowski space. With our inter­ pretation of P M it follows from (4.2.2) that the energy of the system is bounded from below (no negative energy states) Po > 0

(4.2.3)

We may assign to the operator (4.1.71) P2 = P02 - Pi - Pi - Pi

(4.2.4a)

the meaning of a mass squared operator (Einstein's famous relation). If we denote the eigenvalues of this operator by fj.2 then the constraint (4.2.2) entails ix2 > 0 .

(4.2.4b)

We assume further that there exists a nondegenerate eigenstate fl G D, | | 0 | | = 1 of Po belonging to the lowest eigenvalue, viz. P0n = 0 .

(4.2.5)

This state we shall call the vacuum. It is not, in general, the Fock vacuum! In other words, the vacuum is unique up to a numerical complex phase factor.

Principle of Relativity. Caatality. Axioms 3, 4 orxi 5.

77

Axiom 4 in its weaker version is applicable in most of the quantum field theories. The same Axiom in its stronger form is not always applicable as it contains an additional statement about the particle interpretation of the theory. Axiom 4 — strong form: We have to supplement the former weaker assumption by the following additional requirement: the spectrum of the mass squared operator P2 consists in addition to the zero eigenvalue corresponding to the vacuum state fi out of a finite number of discrete eigenvalues rrif >Q,i= 1 , . . . , k, corresponding to the eigenstates describing the q u a n t u m state of one particle of type i and mass |m,|, and a continuous spectrum sup(m?) < 4 i n f ( m 2 ) < fi2 < oo ,

(4.2.6a)

corresponding to the eigenstates describing the states of more t h a n one particle. In what follows we are going to use Axiom 4 in its stronger formulation. The assumption about the existence of the one-particle states is very essential for our further investigations and has far-reaching implications. For simplicity reasons we shall restrict ourselves to the case when all par­ ticles under consideration have the same mass \mi\ = |m2| = . . . = |m*.| = m ,

(4.2.6b)

which does not affect the essence of our condiseration with one exception,(see Sec. 7.1.4). The same is true as far as our second technical assumption about the continuous spectrum is concerned, namely 4 m 2 < n2 < oo .

(4.2.6c)

We shall learn soon (Sec. 4.2.3) that the vacuum state has no mass; then, taking into account (4.2.6) we may write P2 = m2E{p2

= m 2 ) + f°°

fdE(n2)

(4.2.7a)

J4mJ

where E(p2 = m2)U = Hm>

(4.2.7b)

Introduction to Symmetry

78

and Super symmetry

is the space of eigenstates corresponding to the eigenvalue rn2 and E(p2)X

(4.2.7c)

the space of eigenstates of pM where p 2 < /i 2 . We have also /•CO

+ E(p2 = m2) + /

1=E0

dE(ti2)

where 1 is a unit operator in the Hilbert space and projection operator upon the vacuum state. 4.2.2.

(4.2.7d)

EQ = H > < 0 is the

Some explanatory remarks

With the interpretation given in Axiom 4 to the generators of translations, P M , and taking into account the local isomorphism between S Z ( 2 , C ) and the Lorentz group we may also assign physical meaning to the generators of the SL(2,C) or the Lorentz group. We interpret 523 = M i , S31 = Mi, Sn = M3 as the components of the angular momentum of the system (encompassing the specific angular momentum and the spin) and S0j=Njt

j = 1,2,3

as components of the boost. We may regard the Pauli-Lubanski vector (4.1.73) as the spin vector opera­ tor and the eigenvalues of W2 for the massive case, m 2 > 0, given by (4.1.72b) as the square of the maximal spin of the system multiplied by the mass squared; S is just the maximal spin value, S = 0, | , 1 , . . . . To see that let us inspect the case of a rest particle of mass m ^ O and spin S. According to (4.1.72a) 3

-W2u(p

= 0,a) = m2J2

M 2 u ( p = 0, cr) = m 2 5 ( 5 ' + l ) u ( p = 0,a) , (4.2.8a)

t=i

where u(p,cr) is a vector-valued distribution in )i describing a particle of mo­ mentum p , energy po = + V P 2 + rn2 and spin projection a; taking into account (4.1.73) we also have Wou(p = 0,(T) = 0 , (4.2.8b) {W1±iW2)u(p W3u(p

= 0,a) = \m\[S^a){S±a+l)]2u(p

= 0,cr) = |m|cru(p = 0,(T),

= 0),a±l)

a = - 5 , -S + 1 , . . . , +S .

, (4.2.8c) (4.2.8d)

Principle of Relativity. Causality. Aziomt S, 4 and 5.

79

It is clear t h a t the projection of the spin is not Lorentz invariant and, a fortiori, not Poincare invariant, while the spin value S does not depend on the choice of the frame of reference. One reads off (4.2.8) that for resting particles the action of W}- is the same as t h a t of |m|A/y. Since P M is a time-like vector and consequently y? > 0 (see (4.2.4b)) we conclude t h a t W^ is a space-like vector and -IjfSiS

+ 1) < 0 .

(4.2.9)

For massless particles we learnt in Sec. 4.1.7 t h a t the Pauli-Lubanski vector is parallel to the energy-momentum vector. It turned out that the eigenvalues of the operator appearing as a proportionality factor H in (4.1.74), the helicity can be used for labelling the irreducible representations of ?$Q. The notion of helicity is close to t h a t of spin projection in massive case. In massless case, however, there are only two different projections ±|fe|, irrespective of how large \h\ actually is; they do not mix when the Lorentz frame is changed. 4.2.3.

Properties of the vacuum. The set DQ.

The vacuum is defined by (4.2.5) as the unique state of lowest energy be­ longing to D. From (4.2.5) and (4.1.63b), it follows that (P0u, Pitt) = {Piu, P0tt) = 0

Vu e Dn

or less rigorous

p 0 (pn) = p,(p 0 n)=o. Taking into account the uniqueness of the vacuum we get P,fi = A,ft ,

Xi a real number, i = 1,2,3 .

Then 3

»=i

If we have A; ^ 0 this would violate (4.2.2) as well as (4.2.4b). Thus p2n = o

(4.2.10)

as well as p n = 0,

1 = 1,2,3.

(4.2.11)

80

Introduction to Symmetry and Supertymmetry

It is interesting to note that the reverse is also t r u e ; 1 9 from (4.2.11) follows (4.2.5) and (4.2.10). To show that we make use of the formula 2 0 e x p { ^ 5 1 0 } P i e x p { - ^ 5 1 0 } = P i cosh 0 - P o s i n h 0 = f(P)

,

(4.2.12a)

which follows easily from (4.1.63a). From (4.2.12a) we obtain exp{i6S10}P?

fnexp{i6S10}

=

as well as for a = o iOSio j a P i iacosh$-Pi — ias'inh 0-Po elVblfi"*&Sio . e *"e ' = e *e

(4.2.12b)

Let us assume JaPi

1> = 1>

then from (4.2.12b) we have (ip, exp{idSio}xp)

= (ip, exp{—iasinh6Po}

exp{i9Sio}i())

.

For b sinh 6 ,

b real ,

and S - » 0 w e get (rP, VO = (V, e x p { - t i P o M ■

(4.2.12c)

Relation (4.2.12c) holds for each b and implies exp{-ibP0}ip

= i> + V>x

where (V>, i>±) = 0. But (Po is self-adjoint)

lle-^VH3 = U\\2 + ||^x||2 = W 3 and therefore exp{-i'oP0}V< = V> •

(4.2.12d)

We have \im-Uibp°-l)^

= P0il> .

6-+0 10

By virtue of (4.2.12d) this eventually entails P0\p = 0 or ip = Af2, A a number, as asserted.

81

Principle of Relativity. Causality. Axioms S, 4 and 5.

Since Q e D, we may apply the field operators . Actually we have S^Q

= 0 .

(4.2.13)

This can be shown in a similar way as the implication (4.2.11) => (4.2.5), taking into account the Lie-Cartan relations (4.1.63). Conversely, it can be shown t h a t if there is a unique vector which is annihilated by the operators S/U, then this vector is vacuum state. Prom (4.2.5), (4.2.11) and (4.2.13) follows immediately that fl is Poincare invariant, i.e. its mass, spin, energy or momentum is zero. This is in agreement with the physical intuition: the vacuum should be the same for each observer and should be void of any physical characteristics used in the description of matter. Prom the very definition of DQ as well as from the transformation properties of fl and the fields under the Poincare transformation follows that U(CL,A)DQ=DQ

4.3.

Causality

4.3.1.

Axiom 5

.

(4.2.14)

Let us consider any two field operators appearing in the theory, say,