Towards Infrared Finite S-matrix in Quantum Field Theory (Springer Theses) 9811630445, 9789811630446

Table of contents :
Supervisor’s Foreword
Parts of this thesis have been published in the following journal articles:
Acknowledgements
Contents
1 Introduction and Summary
1.1 Summary of the Author's Results
1.2 Organization of This Thesis
References
2 Asymptotic Symmetry
2.1 Physical Symmetry and Trivial Symmetry
2.2 Classical Asymptotic Symmetry
2.3 Asymptotic Structure of Minkowski Spacetime and Useful Coordinates
2.4 Classical Asymptotic Charge
2.5 Classical Charge Conservation Associated with the Asymptotic Symmetry
2.5.1 The Asymptotic Charge Conservation and Memory Effect at Leading Order
2.5.2 Asymptotic Charge Conservation and Memory Effect at Subleading Order
2.6 Asymptotic Symmetry in Quantum Field Theory
2.7 Asymptotic Symmetry in Canonical Quantization
2.8 Charges Associated with Asymptotic Symmetry in QED
2.9 Asymptotic Symmetry in BRST Formalism
2.9.1 BRST Quantization in QED and the Correct BRST Charge
2.9.2 Large r Expansion of Fields in QED
2.9.3 Asymptotic BRST Condition and Asymptotic Symmetry
References
3 Infrared Divergences of S-Matrix
3.1 S-Matrix
3.2 Conventional S-Matrix
3.3 Leading Soft Theorem
3.4 Subleading Soft Photon Theorem
3.5 Infrared Divergences in S-Matrix
References
4 Equivalence Between Soft Theorem and Asymptotic Symmetry
4.1 Soft Photon Theorem Asymptotic Charge Conservation, at Leading Order
4.2 Soft Photon Theorem Asymptotic Charge Conservation, at Subleading Order
4.2.1 Hard Part of the Subleading Charges
References
5 Towards the IR Finite S-Matrix
5.1 Motivations for Seeking IR Finite S-Matrix
5.2 Chung's Dressed State
5.3 Derivation of the Faddeev-Kulish Dressed State
5.4 Problem on the Gauge Invariance of Faddeev …
5.5 Gauge Invariant Asymptotic States
5.6 Interpretation of the Faddeev-Kulish Dresses
5.6.1 Coulomb Potential by Point Charges in the Asymptotic Region
5.6.2 Coulomb Potential from Dressed States with iε Prescription
5.7 Asymptotic Symmetry in the Dressed State Formalism
References
6 Conclusion and Further Discussion
6.1 Softness of Dresses and Infrared Scales
6.2 Extension to Other Theories
6.3 Other Future Directions
References
Appendix A Asymptotic Behavior of the Residual Gauge Parameter
Appendix B Electromagnetic Fields in Null Infinity
B.1 Electromagnetic Fields by Uniformly Moving Charges
B.2 Computation of Memories
Appendix C Asymptotic Expansion of Radiation Fields
Appendix D Gauss Law Constraint in Canonical Quantization of QED
Appendix E Asymptotic Behaviors of the Massive Particles
Appendix F Integration of Jnm and Non-negativity of Aβα
Appendix G Derivation of Eq. (4.40摥映數爠eflinkeq:subspshardspsh34.404)
Appendix H Particle Trajectory Under the Coulomb Force and Phase Factor in FK Dressed State
H.1 Particle Trajectory Under the Coulomb Force
H.2 Stationary Trajectory of Phase Factor in FK Dressed State

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Springer Theses Recognizing Outstanding Ph.D. Research

Hayato Hirai

Towards Infrared Finite S-matrix in Quantum Field Theory

Springer Theses Recognizing Outstanding Ph.D. Research

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More information about this series at http://www.springer.com/series/8790

Hayato Hirai

Towards Infrared Finite S-matrix in Quantum Field Theory Doctoral Thesis accepted by Osaka University, Osaka, Japan

Author Dr. Hayato Hirai Natural Science Education National Institute of Technology, Kisarazu College Kisarazu, Japan

Supervisor Prof. Satoshi Yamaguchi Department of Physics Osaka University Osaka, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-16-3044-6 ISBN 978-981-16-3045-3 (eBook) https://doi.org/10.1007/978-981-16-3045-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

It is my great pleasure to introduce the work of Dr. Hayato Hirai, now published in this Springer Thesis series. His work is on the infrared problems in the quantum field theory. The quantum field theory, which describes the physics of elementary particles, is a very successful theory. The Standard Model is formulated by a quantum field theory and describes matters and forces other than gravity very well. A good example of the success of this theory is that the anomalous magnetic moment of the electron agrees between theory and experiment to more than 8 significant digits. This is one of the best agreements between theory and experiment in physics. However, there are still some conceptually poorly understood aspects of the quantum field theory. One of them is the infrared problem. In a quantum field theory that includes a massless field, there is a divergence from the infrared region, and the S-matrix elements, which quantify the scattering, are not well defined. This is also a problem in quantum electrodynamics, which appears in the early stage of the history of the quantum field theory, and is one of the most successful examples. A solution to this problem in standard textbooks is called inclusive formalism, which brings up the “resolution of the detector”. In this formalism, although we cannot define the S-matrix elements, we can define the inclusive scattering cross section, which is an observable quantity. Although this has convinced many people, some people still think it is important to define the S-matrix elements in some way, and this has been studied for a long time. Dr. Hirai recognizes the importance of an infrared finite S-matrix and has done these works. In this thesis, he makes two important contributions towards the construction of an infrared finite S-matrix. One is to show that what is called “asymptotic symmetry” is a physical symmetry, and furthermore to show that the conservation law for asymptotic symmetry is equivalent to a theorem called “the soft photon theorem”. Asymptotic symmetry is a subtle “symmetry” that at first glance appears to be unphysical, i.e., a symmetry in which the corresponding conserved quantity is always zero regardless of the state. Therefore, its treatment has been controversial. In this work, using a standard formulation called the BRST formalism, he and collaborators have fully clarified that the asymptotic v

vi

Supervisor’s Foreword

symmetry is a physical symmetry. Furthermore, they have developed the relation between the sub-leading soft photon theorem and this symmetry. These results have had a great impact on the field. The other one is that the asymptotic state of electrons proposed by Faddeev and Kulish is shown to be a physical state, using the BRST formalism. Chung, Faddeev, Kulish, and other people were not completely satisfied with inclusive formalism. They have tried to change the concept of asymptotic states and obtain an S-matrix without infrared divergence. They have constructed a class of states, now called “Faddeev-Kulish states”. However, since the BRST formalism was not well developed at that time, the Faddeev-Kulish states were not considered to be “physical states”. Therefore, people were lost in trying to make them physical. Dr. Hirai and collaborators have fully clarified that Faddeev-Kulish states are physical states using the now-standard method, the BRST formalism. This is a major step in constructing the S-matrix theory without infrared divergence. It will lead to future comparisons with inclusive formalism and experimental verification, which will have a significant impact on the field. Osaka, Japan November 2020

Prof. Satoshi Yamaguchi

Parts of this thesis have been published in the following journal articles: HS1. H. Hirai and S. Sugishita, “Conservation Laws from Asymptotic Symmetry and Subleading Charges in QED”, Journal of High Energy Physics 07 (2018) 122. HS2. H. Hirai and S. Sugishita, “Dressed states from gauge invariance”, Journal of High Energy Physics 06 (2019) 023. Parts of Chaps. 2, 4 and Appendices A, B, C, E, G are reprinted from [HS1], and parts of Chaps. 5, 6 are reprinted from [HS2], with revisions and with permission.

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Acknowledgements

First and foremost, I would like to thank my advisor Satoshi Yamaguchi for his support during my Ph.D. program. I have always been influenced by his very deep understanding of physics and also how to explain complicated concepts in a simple and clear way. I also thank him for allowing me to work freely on the topics that I am interested in and for always supporting my research enormously. Second, I am very grateful to my collaborator, Sotaro Sugishita. This thesis is largely based on the two joint papers with him. I have always enjoyed discussing with him and I have learned so many things through the joint research with him. I would also like to express my appreciation to Kotaro Tamaoka for many discussions and his kind support in the last few years. Third, I would like to thank Masaharu Aoki, Minoru Tanaka, Satoshi Yamaguchi, Shinya Kanemura, and Tetsuya Onogi for reading this thesis carefully and giving me useful comments. I am also deeply grateful to all the members of the Particle Physics Theory Group at Osaka University for providing me with a good environment during my Ph.D. program. I acknowledge the financial support from the JSPS fellowship for the past year. Last but not least, I would like to express my gratitude to my family, in particular my parents, for their constant support and encouragement.

ix

Contents

1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Summary of the Author’s Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5 6 7

2 Asymptotic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Physical Symmetry and Trivial Symmetry . . . . . . . . . . . . . . . . . . . . . . 2.2 Classical Asymptotic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Asymptotic Structure of Minkowski Spacetime and Useful Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Classical Asymptotic Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Classical Charge Conservation Associated with the Asymptotic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The Asymptotic Charge Conservation and Memory Effect at Leading Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Asymptotic Charge Conservation and Memory Effect at Subleading Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Asymptotic Symmetry in Quantum Field Theory . . . . . . . . . . . . . . . . 2.7 Asymptotic Symmetry in Canonical Quantization . . . . . . . . . . . . . . . 2.8 Charges Associated with Asymptotic Symmetry in QED . . . . . . . . . 2.9 Asymptotic Symmetry in BRST Formalism . . . . . . . . . . . . . . . . . . . . 2.9.1 BRST Quantization in QED and the Correct BRST Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Large r Expansion of Fields in QED . . . . . . . . . . . . . . . . . . . . 2.9.3 Asymptotic BRST Condition and Asymptotic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 12 18 21 22 26 29 29 31 33 34 39 41 45

xi

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Contents

3 Infrared Divergences of S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Conventional S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Leading Soft Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Subleading Soft Photon Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Infrared Divergences in S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 52 54 57 60

4 Equivalence Between Soft Theorem and Asymptotic Symmetry . . . . . 4.1 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Leading Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Subleading Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Hard Part of the Subleading Charges . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 64 66 68

5 Towards the IR Finite S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Motivations for Seeking IR Finite S-Matrix . . . . . . . . . . . . . . . . . . . . 5.2 Chung’s Dressed State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Derivation of the Faddeev-Kulish Dressed State . . . . . . . . . . . . . . . . . 5.4 Problem on the Gauge Invariance of Faddeev-Kulish Dressed State and Our Approach to this Problem . . . . . . . . . . . . . . . . . . . . . . . 5.5 Gauge Invariant Asymptotic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Interpretation of the Faddeev-Kulish Dresses . . . . . . . . . . . . . . . . . . . 5.6.1 Coulomb Potential by Point Charges in the Asymptotic Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Coulomb Potential from Dressed States with i Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Asymptotic Symmetry in the Dressed State Formalism . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72 74

83 86 89

6 Conclusion and Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Softness of Dresses and Infrared Scales . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Extension to Other Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Other Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 92 94 94 95

Appendix A: Asymptotic Behavior of the Residual Gauge Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

78 80 82 83

Appendix B: Electromagnetic Fields in Null Infinity . . . . . . . . . . . . . . . . . . 101 Appendix C: Asymptotic Expansion of Radiation Fields . . . . . . . . . . . . . . 107 Appendix D: Gauss Law Constraint in Canonical Quantization of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Appendix E: Asymptotic Behaviors of the Massive Particles . . . . . . . . . . . 111

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Appendix F: Integration of Jnm and Non-negativity of Aβα . . . . . . . . . . . . 115 Appendix G: Derivation of Eq. (4.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Appendix H: Particle Trajectory Under the Coulomb Force and Phase Factor in FK Dressed State . . . . . . . . . . . . . . . . . . 123

Chapter 1

Introduction and Summary

This thesis is devoted to a better understanding of infrared structures of scattering theory in gauge theories, in particular, quantum electrodynamics (QED) in fourdimensional Minkowski spacetime. The investigation of scattering phenomena in the gauge theories describing our world at low energy, such as QED, has a long history since the early nineteenth century. The comparison between theoretical predictions and experimental data of scattering cross-sections has been one of the main sources of ideas in developing the models explaining our world. Nowadays the Standard Model of particle physics has demonstrated tremendous successes in explaining the data provided by collider experiments with great accuracy, at least with the current experimental resolution. However, the infrared dynamics in the gauge theories involving long-range forces has recently turned out to be worth reinvestigating, which was first triggered by the discovery of the asymptotic symmetry in Maxwell and Yang–Mills theory coupled to massless charged matters [1]. Moreover, the discovery of these new symmetries led to the discoveries of a triangle equivalence of seemingly unrelated following three subjects that concern the infrared dynamics in QED, QCD, gravity: asymptotic symmetry, soft theorem, and memory effect. This equivalence, called infrared triangle, is one of the main subjects in this thesis. Asymptotic symmetry transformations, which I will define precisely later, refers to a gauge transformation which changes boundary conditions on asymptotic spacetime regions while keeping physically reasonable fall-offs behaviors of gauge fields. Although asymptotic symmetry is local symmetry, this is physical symmetry. The history of the asymptotic symmetry analysis goes back to the seminal work in gravity by Bondi, van der Burg, Metzner and Sachs (BMS) [2, 3] in 1962. They found the infinite-dimensional subgroup of diffeomorphisms of asymptotically flat spacetime that act non-trivially on the boundary data, which is now called the BMS group. On the other hand, as already mentioned, the asymptotic symmetries in gauge theories in asymptotically flat spacetime were revealed recently [1, 4–6]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3_1

1

2

1 Introduction and Summary

The soft theorems describe the universal property of scattering amplitudes with external soft particles i.e. massless particles whose energies are much less than the energies of external charged matters. For example, the soft photon theorem [7–14] in QED simply says that the scattering amplitude of the process α → β, say Mβα , and the one with an additional external soft photon of momentum k μ = (ω, k), say Mβα (k), are proportional to each other as Mβα (k) = Mβα

N  ηn en (k) · pn + O(ω0 ), p · k n n

(1.1)

where en and pn are the charge and the four-momentum of n-th particle in the initial state α and the final state β, μ (k) is the polarization of the soft photon, and ηn is a sign factor which takes +1 for particles in β and −1 for particles in α. The factor of proportionality, called the (leading) soft factor, diverges as the energy of soft photon tends to zero, since it is order O(ω−1 ). This fact reflects one of the important property of infrared dynamics; the slight acceleration of a charged particle results in the radiation of infinite number of low energy photons. The memory effect is concerned with the detection of waves coming from faraway sources. It has been investigated mainly as a mechanism for the observation of gravitational waves, called the gravitational memory effect. It originates in a proposal in 1974 by Zel’dovich and Polnarev [15], and developed by many others [16–26]. The gravitational memory effect claims that the passage of gravitational waves coming from faraway sources produces a permanent displacement of the metric and results in a permanent displacement of relative position of freely falling particles (viewed as detectors).1 Although the gravitational memory effect has a long history, the electromagnetic analog of the memory effect was first studied recently [23, 29, 30]. The electromagnetic memory effect is a phenomenon: the total net charge that has passed through at an local angle is given by a permanent displacement of the gauge field at the corresponding angle, instead of the metric in gravity. The infrared (IR) triangle reveals the surprising fact that the above three subjects, describing seemingly different aspects in infrared (long distance) physics, are just different perspective of one subject. The equivalence of those was first discussed in the Yang–Mills theory [1], and extended to gravity [31–33] and also to QED [4–6]. The equivalence has also been extended to higher orders of the soft expansion (e.g. [34–37]), to higher spacetime dimensions (e.g. [38, 39]), and also to other theories (e.g. [40–42]) by many others.2 The IR triangles are not just about the mathematical equivalence among the three things already known because the IR triangles are universal equivalences valid for many theories including massless particles and there were a few theories in which all corners of the triangle were fully understood. In fact, the insight from the viewpoint of the infrared triangle has led to the many discoveries of new corners in many different theories in the last several years (for example, 1 Detection

of the memory effect at LISA [27] and at LIGO [28] has been proposed recently. have just referred to relatively old references here because there are too many. We will refer to the references directly related to our works in the later chapters.

2 We

1 Introduction and Summary

3

see [34–37, 42]). Furthermore, the application goes beyond just finding the new corners of the triangles. The investigation of asymptotic symmetries had led to many interesting applications to various directions: flat space holography [43–46], black hole information paradox [33, 47, 48], and IR finite S-matrix [49, 50]. In particular the last one, construction of IR finite S-matrix in gauge theories, is one of our main motivation for pursuing the infrared physics. The study of infrared dynamics involving the soft particles has played the important role in quantum field theory (QFT) and collider physics. As suggested by the soft theorems in QED and (perturbative) gravity, the emissions of infinite number of soft photon and graviton are inevitable in any scattering process of charged particles. In other words, charged particles should be accompanied by the cloud of soft photons and gravitons in any scattering process. As a result, the S-matrix elements with a definite number of soft particles have infrared divergences in QED and gravity. Therefore, such S-matrix is not a well-defined object. This problem is sometimes referred as the infrared problem. There are two possible prescriptions for curing this problem: the inclusive formalism and the dressed state formalism. Here I will review the two prescriptions and their problems very briefly in the following. The first one, the inclusive formalism, has been used as a standard prescription. with emitted soft particles indistinguishable by a detector. In this formalism, observables are inclusive transition probabilities that are given by sum over all the transition probabilities for processes indistinguishable by a detector. Because neutral particles of energies lower than the detector’s energy resolution cannot be measured, all the transition probabilities for processes that differ only in the emissions of such soft particles are summed over in inclusive ones. As is well known, the inclusive transition probabilities give finite results (at least for charged particles with definite momenta) and have been consistent with the experimental data so far. However, the S-matrix remains ill-defined in this formalism. Again, the origin of this problem would be the fact that charged particles are not accompanied by any clouds of soft particles. More concretely, we use free states, which are not surrounded by any cloud of soft particles, as the asymptotic states of charged particles. This seems unnatural or even illegal because the interactions mediated by photons and gravitons are long-range forces that cannot be ignored even when the charged particles are very far apart from each other. The second one, the dressed state formalism, is a main subject in this thesis. This formalism tries to cure the S-matrix of infrared divergences by using the modified asymptotic states surrounded by clouds of soft particles, called dressed states. In other words, this formalism tries to compute the inclusive quantity at the level of S-matrix, instead of the transition probability. Our hope is that the S-matrix with properly constructed dressed states will give IR finite results and describes the scattering processes with hopefully higher precision. A dressed state was first introduced in the work [51] by Chung in 1965. The dressed state, which we call Chung’s dressed state, for a single electron with momentum p μ = (E p , p) is given by

4

1 Introduction and Summary

   1   3 k|S (l) (k)|2 exp 3 k S (l) (k)a (l)† (k) b† ( p)|0 | pC = exp − d d 2 l=1,2 λ l=1,2 λ (1.2) with S (l) (k) =

ep ·  (l) (k) k·p

(1.3)

where k μ = (ωk , k) is a four-momentum of a photon and μ(l) (k)(l = 1, 2) is the 3 k is the Lorentz invariant transverse polarization vectors of a photon. The measure d 3k ≡ measures for the integration of the spatial momentum which is defined as d d3k , and λ is the photon mass introduced as an IR cutoff that is taken to zero at (2π)3 2ωk the end of the calculation. a (l)† (k) and b† ( p) are the creation operators for a photon and an electron, respectively, and |0 is a Fock vacuum. The electron in Chung’s dressed state is surrounded by the coherent photon cloud. In [51], the elements of Dyson’s S-matrix operator for the dressed states instead of conventional Fock states are computed and it is shown that all the IR divergences are canceled out at all orders of perturbative expansion in the S-matrix. This is a rather surprising observation. However, the function S (l) (k) for k > 0 can be deformed rather freely as long as the (l) integral in (1.2) is convergent as k → ∞ and S (l) (k) → ep·k· p(k) as k → 0. This is because Chung’s dressed states were given just by requiring to cancel out the infrared divergences. Therefore, these states do not have the ability to predict real scattering processes due to the ambiguity of the hard momentum part of the dressing. Also, there was no clear reason to use such a state except for the infrared divergence to disappear. After the Chung’s discovery, Faddeev and Kulish in 1970 derived another version of dressed states by solving the infrared QED dynamics [52]. The dressed state, nowadays called the Faddeev-Kulish (FK) dressed states, is given by | p1 , · · · , p N  F K = lim e R(t) ei (t,ts ) | p1 , · · · , p N  t→±∞

(1.4)

with     pμ  i p·k t −i p·k t  3 3k aμ (k)e E p − aμ† (k)e E p , R(t) ≡ e d p ρ(p) d p·k 2  e p·q |t| 3 q : ρ(p)ρ(q) :  (t, ts ) = − sgn(t) ln , d 3 p d 2 4 4π ts ( p · q) − m

(1.5) (1.6)

where eρ(p) ≡ e(b† (p)b(p) − d † (p)d(p)) is the charge density operator for electrons and positrons,3 and | p1 , · · · , p N  is a Fock state of electrons and positrons with the momenta p1 , · · · , p N . If we extract the soft momentum region k ∼ 0 for the dressing operator (1.5), the operator takes the form b(†) ( p) and d (†) ( p) is the annihilation (creation) operator for an electron and a positron, respectively.

3

1 Introduction and Summary

Rsoft i

p·k

5

    3 ∼ e d p ρ(p)

 pμ

d 3k aμ (k) − aμ† (k) , 3 (2ω) p · k (2π ) soft

(1.7)

t

because e E p ∼ 1 at k ∼ 0. This reproduces the cloud of soft photons used in Chung’s dressed state (1.2). Once this approximation is justified, the IR finiteness of the Smatrix for the FK dressed states follows from the proof by Chung. However, we need more careful analysis for the complete understanding of the contributions from the dressing factors. In addition, the contribution of the hard momentum region (k > 0) of the photon cloud (1.5) to physical observables is still not clear. Futhermore, Faddeev and Kulish argued in [52] that the dressed state in (1.4) is not gauge invariant because the state does not satisfy the Gupta-Bleuler condition. In order to make the state gauge invariant, they modified the dressed state by adding new terms in the dressing factor (1.5). However, such ad-hoc modification seemed unnatural because the dressed state (1.4) was derived from the QED dynamics. These problems in the dressed formalism should be resolved towards the complete formulation of the IR finite S-matrix theory. Recently, one important relation between the IR divergences in S-matrix and the asymptotic symmetry was pointed out in [49]. In [49], it was argued that the appearance of IR divergences in the conventional S-matrix is a consequence of the conservation of the charges associated with the asymptotic symmetries. This relation strongly suggests that the asymptotic symmetry is one of the key ingredients for the construction of IR finite S-matrix.

1.1 Summary of the Author’s Results We present the following results in this thesis: (I) Asymptotic symmetries give charges that are conserved between the asymptotic future and past. [in Chap. 2] We rederive charges associated with asymptotic symmetries in QED defined on the asymptotic Cauchy slices. We show that the charges can have nontrivial values for physical solutions and the charges are conserved between the asymptotic future and past Cauchy slices by analyzing the asymptotic behaviors of classical fields and confirming that the contribution from the spatial infinity to the charge conservation vanishes. (II) Asymptotic symmetries are physical symmetries in the canonical quantization and also in the BRST quantization. [in Chap. 2] We show that the charges that generate the asymptotic symmetries on past/future infinities act on the physical Hilbert space nontrivially both in the canonical quantization and in the BRST quantization, although the symmetries are local symmetries. (III) Equivalence between the soft photon theorem and the asymptotic symmetry at subleading order. [in Chaps. 2 and 4] We find that the subleading photon theorem is equivalent to a charge conser-

6

1 Introduction and Summary

vation in massive scaler QED. We also find the relation between the conserved charge and the subleading component of the charge of asymptotic symmetry. (IV) The Faddeev–Kulish dressed state is gauge invariant. [in Chap. 5] We drive the BRST condition incorporating the asymptotic interaction and show that the FK dressed state satisfies the condition. (V) The Faddeev–Kulish dressed state carries the asymptotic charges for the classical free charged particles with their relativistic Coulomb fields. [in Chap. 5] We show that the dressing operator in FK dressed state carries asymptotic charge, and the eigenvalue is given by the classical asymptotic charge for the classical free charged particles with their relativistic Coulomb fields. This result leads to the quantum analog of the electromagnetic memory effect. These results are mainly based on the following two author’s works in collaboration with Sotaro Sugishita: • Hirai and Sugishita [53], “Conservation Laws from Asymptotic Symmetry and Subleading Charges in QED”, JHEP 07 (2018) 122. • Hirai and Sugishita [54], “Dressed states from gauge invariance”, JHEP 06 (2019) 023. The results (I), (II), (III) are based on [53] and the results (IV), (V) are based on [54].

1.2 Organization of This Thesis In the first three chapter, the fundamental results about the asymptotic symmetries, the soft theorems, and the memory effects are shown and their relationships are also discussed. In Chap. 2, we first explain the definition of the physical symmetry and the trivial symmetry, and then show that the asymptotic symmetry in QED is physical symmetry both in the classical theory and in the quantum theory. In Chap. 3, we review the definition of S-matrix, the soft photon theorem and the infrared divergence. Chapter 4 starts with the review of the equivalence between the asymptotic symmetry and the leading soft photon theorem. After that, we derive the charge whose conservation law is equivalent to the subleading soft photon theorem. In Chap. 5, we treat the construction and the properties of the asymptotic states for IR finite S-matrix and also discuss its relation to asymptotic symmetry. We conclude with further discussion in Chap. 6.

References

7

References 1. Strominger A (2014) Asymptotic symmetries of Yang–Mills theory. JHEP 07:151 arXiv:1308.0589 [hep-th] 2. Bondi H, van der Burg MGJ, Metzner AWK (1962) Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems. Proc Roy Soc Lond A 269:21–52 3. Sachs RK (1962) Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times. Proc Roy Soc Lond A 270:103–126 4. He T, Mitra P, Porfyriadis AP, Strominger A (2014) New symmetries of Massless QED. JHEP 10:112 arXiv:1407.3789 [hep-th] 5. Kapec D, Pate M, Strominger A (2017) New symmetries of QED. Adv Theor Math Phys 21:1769–1785 arXiv:1506.02906 [hep-th] 6. Campiglia M, Laddha A (2015) Asymptotic symmetries of QED and Weinberg’s soft photon theorem. JHEP 07:115 arXiv:1505.05346 [hep-th] 7. Bloch F, Nordsieck A (1937) Note on the radiation field of the electron. Phys Rev 52:54–59 8. Low FE (1954) Scattering of light of very low frequency by systems of spin 1/2. Phys Rev 96:1428–1432 9. Gell-Mann M, Goldberger ML (1954) Scattering of low-energy photons by particles of spin 1/2. Phys Rev 96:1433–1438 10. Low FE (1958) Bremsstrahlung of very low-energy quanta in elementary particle collisions. Phys Rev 110:974–977 11. Kazes E (1959) Generalized current conservation and low energy limit of photon interactions, Il Nuovo Cimento (1955–1965) 13(6) 1226–1239. https://doi.org/10.1007/BF02725129 12. Yennie DR, Frautschi SC, Suura H (1961) The infrared divergence phenomena and high-energy processes. Annals Phys 13:379–452 13. Weinberg S (1965) Infrared photons and gravitons. Phys Rev 140:B516–B524 14. Burnett TH, Kroll NM (1968) Extension of the low soft photon theorem. Phys Rev Lett 20:86 15. Zel’dovich YB, Polnarev AG (1974) Radiation of gravitational waves by a cluster of superdense stars. Soviet Astronomy 51 16. Braginsky VB, Grishchuk LP (1985) Kinematic resonance and memory effect in free mass gravitational antennas. Sov Phys JETP 62:427–430. [Zh Eksp Teor Fiz 89, 744 (1985)] 17. Braginsky VB, Thorne KS (1987) Gravitational-wave bursts with memory and experimental prospects. Nature 327 18. Christodoulou D (1991) Nonlinear nature of gravitation and gravitational wave experiments. Phys Rev Lett 67:1486–1489 19. Wiseman AG, Will CM (1991) Christodoulou’s nonlinear gravitational wave memory: evaluation in the quadrupole approximation. Phys Rev D 44(10):R2945–R2949 20. Blanchet L, Damour T (1992) Hereditary effects in gravitational radiation. Phys Rev D 46:4304–4319 21. Thorne KS (1992) Gravitational-wave bursts with memory: the Christodoulou effect. Phys Rev D 45(2):520–524 22. Favata M (2010) The gravitational-wave memory effect. Class Quant Grav 27. arXiv:1003.3486 [gr-qc] 23. Tolish A, Wald RM (2014) Retarded fields of null particles and the memory effect. Phys Rev D 89(6). arXiv:1401.5831 [gr-qc] 24. Tolish A, Bieri L, Garfinkle D, Wald RM (2014) Examination of a simple example of gravitational wave memory. Phys Rev D 90(4). arXiv:1405.6396 [gr-qc] 25. Winicour J (2014) Global aspects of radiation memory. Class Quant Grav 31. arXiv:1407.0259 [gr-qc] 26. Zhang PM, Duval C, Gibbons GW, Horvathy PA (2017) The memory effect for plane gravitational waves 27. Favata M (2009) Post-newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries. Phys Rev D 80(2). http://dx.doi.org/10.1103/PhysRevD.80. 024002

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1 Introduction and Summary

28. Lasky PD, Thrane E, Levin Y, Blackman J, Chen Y (2016) Detecting gravitational-wave memory with ligo: implications of gw150914. Phys Rev Lett 117(6). http://dx.doi.org/10.1103/ PhysRevLett.117.061102 29. Bieri L, Garfinkle D (2013) An electromagnetic analogue of gravitational wave memory. Class Quant Grav 30. arXiv:1307.5098 [gr-qc] 30. Susskind L, Electromagnetic memory. arXiv:1507.02584 [hep-th] 31. Strominger A (2014) On BMS invariance of gravitational scattering. JHEP 07:152 arXiv:1312.2229 [hep-th] 32. He T, Lysov V, Mitra P, Strominger A (2015) BMS supertranslations and Weinberg’s soft graviton theorem. JHEP 05:151 arXiv:1401.7026 [hep-th] 33. Strominger A, Zhiboedov A (2016) Gravitational memory, BMS supertranslations and soft theorems. JHEP 01:086 arXiv:1411.5745 [hep-th] 34. Cachazo F, Strominger A, Evidence for a new soft graviton theorem. arXiv:1404.4091 [hep-th] 35. Lysov V, Pasterski S, Strominger A (2014) Low’s Subleading soft theorem as a symmetry of QED. Phys Rev Lett 113(11). arXiv:1407.3814 [hep-th] 36. Campiglia M, Laddha A (2016) Subleading soft photons and large gauge transformations. JHEP 11:012 arXiv:1605.09677 [hep-th] 37. Campiglia M, Laddha A (2017) Sub-subleading soft gravitons: new symmetries of quantum gravity? Phys Lett B 764:218–221 arXiv:1605.09094 [gr-qc] 38. Kapec D, Lysov V, Strominger A (2017) Asymptotic symmetries of Massless QED in even dimensions. Adv Theor Math Phys 21:1747–1767 arXiv:1412.2763 [hep-th] 39. Kapec D, Lysov V, Pasterski S, Strominger A (2017) Higher-dimensional supertranslations and Weinberg’s soft graviton theorem. Ann Math Sci Appl 02:69–94 arXiv:1502.07644 [gr-qc] 40. Geyer Y, Lipstein AE, Mason L (2015) Ambitwistor strings at null infinity and (subleading) soft limits. Class Quant Grav 32(5). arXiv:1406.1462 [hep-th] 41. Adamo T, Casali E, Skinner D (2014) Perturbative gravity at null infinity. Class Quant Grav 31(22). arXiv:1405.5122 [hep-th] 42. Dumitrescu TT, He T, Mitra P, Strominger A, Infinite-dimensional fermionic symmetry in supersymmetric gauge theories. arXiv:1511.07429 [hep-th] 43. Bagchi A, Basu R, Kakkar A, Mehra A (2016) Flat holography: aspects of the dual field theory. JHEP 12:147 arXiv:1609.06203 [hep-th] 44. Pasterski S, Shao S-H, Strominger A (2017) Flat space amplitudes and conformal symmetry of the celestial sphere. Phys Rev D 96(6). arXiv:1701.00049 [hep-th] 45. Pasterski S, Shao S-H (2017) Conformal basis for flat space amplitudes. Phys Rev D 96(6). arXiv:1705.01027 [hep-th] 46. Pasterski S, Shao S-H, Strominger A (2017) Gluon Amplitudes as 2d Conformal Correlators. Phys Rev D 96(8). arXiv:1706.03917 [hep-th] 47. Hawking SW, Perry MJ, Strominger A (2017) Superrotation charge and supertranslation hair on black holes. JHEP 05:161 arXiv:1611.09175 [hep-th] 48. Hawking SW, Perry MJ, Strominger A (2016) Soft Hair on Black Holes. Phys Rev Lett 116(23). arXiv:1601.00921 [hep-th] 49. Kapec D, Perry M, Raclariu A-M, Strominger A (2017) Infrared Divergences in QED, Revisited. Phys Rev D 96(8). arXiv:1705.04311 [hep-th] 50. Choi S, Akhoury R (2018) BMS supertranslation symmetry implies Faddeev–Kulish amplitudes. JHEP 02:171 arXiv:1712.04551 [hep-th] 51. Chung V (1965) Infrared divergence in quantum electrodynamics. Phys Rev 140:B1110–B1122 52. Kulish PP, Faddeev LD (1970) Asymptotic conditions and infrared divergences in quantum electrodynamics. Theor Math Phys 4:745. [Teor Mat Fiz 4:153 (1970)] 53. Hirai H, Sugishita S (2018) Conservation laws from asymptotic symmetry and subleading charges in QED. JHEP 07:122 arXiv:1805.05651 [hep-th] 54. Hirai H, Sugishita S (2019) Dressed states from gauge invariance. JHEP 06:023 arXiv:1901.09935 [hep-th]

Chapter 2

Asymptotic Symmetry

2.1 Physical Symmetry and Trivial Symmetry Symmetry is one of the most fundamental concepts in physics. Why are symmetries so important? Mostly, it is because symmetries constrain the property of dynamics or characterize the states of the system under consideration. For classical systems, the existence of a symmetry gives a conserved charge as a consequence of the Noether’s theorem. The conservation law of the charge constrains the dynamics of the observables in the system. For example, a time translation symmetry in a system of multiple particles constrains the motion of the particles in such a manner that the total energy of the particles do not change during the time evolution. Therefore the physical solutions can be characterized by the total energy. Also in quantum systems, a symmetry of Hamiltonian gives a conserved charge and it constrains the observables of the system. More generally, the constraint from a symmetry is given by the Ward-Takahashi identity. For example, if a system has a translation symmetry, the correlation functions are restricted to the functions that are invariant under the translation of the positions of fields. Once we recognize a symmetry in a system, it tells us nontrivial information of the observables and states in the system. In this sense, it would be natural to call a transformation a physical symmetry if it gives a nontrivial constraint to the theory. According to the definition, most of gauge (local) transformations are not physical symmetries both in classical theories and in quantum theories. Before explaining the reason for it, we first review some basic things about symmetry in classical theories. Suppose that a system has a local conserved current J μ . By integrating ∂μ J μ = 0 over a closed spacetime region B with boundary  ( = ∂ B), we can define a charge Q() as   d V ∂μ J μ = d Sμ J μ . (2.1) Q() ≡ B



© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3_2

9

10

2 Asymptotic Symmetry Asymptotic Symmetry

This charge is trivially zero because of ∂μ J μ = 0. Since we can freely change the spacetime region B in (2.1), the charge Q() does not depend on the closed surface , i.e. Q() = Q(  )

(2.2)

holds for any closed surface   . The conventional conservation of charges between different time slices follows from (2.1) by taking  as follows. Take  as a closed surface composed of the three surfaces,  F ,  R ,  I .  F and  I are the spatial surfaces at t = t F and t = t I with the radius 0 ≤ r ≤ R, respectively.  R is the timelike surface at r = R with t I ≤ t ≤ t F . Then the conservation of charge in (2.1) gives 0 = Q() = Q( F ) + Q( R ) − Q( I ),

(2.3)

 with Q(a ) = a d Sμ J μ where we choose d S μ as the future-directed surface element when the surface is spacelike. In particular, Q( R ) is given by 



tF

Q( R ) = R 2

dt

ˆ d 2  J r (t, R, ),

(2.4)

tI

ˆ is a two-dimensional angular coordinate. Here, we usually drop this terms where  by assuming that the amplitude of the current falls off rapidly enough as r → ∞. Under the assumption of lim R→∞ Q( R ) = 0, (2.3) gives Q(t F ) = Q(t I )

(2.5)

where Q(t F ) = lim R→∞ Q( F ) and Q(t I ) = lim R→∞ Q( I ) are the charges on the time slices at t = t F and t = t I , respectively. This is the charge conservation between two different times.

2.2 Classical Asymptotic Symmetry Let us define the classical asymptotic symmetry group (cASG) as 1 cASG =

classical allowed gauge transformations . classical trivial gauge transformations

(2.6)

Here, the classical allowed gauge transformations are any gauge transformations whose charges (generators) are well-defined and take nontrivial values. The classical trivial gauge transformations are the ones whose charges (generators) are zero. Under 1 Here,

A/B is the quotient group of A by B.

2.2 Classical Asymptotic Symmetry

11

this definition, we will see that the nontrivial charges have one-to-one correspondence with the asymptotic symmetries. Let’s see what happens for the conservation of charges for gauge symmetries. Consider QED for concreteness. The Lagrangian is 1 ¯ / − m ψψ, L Q E D = − Fμν F μν + i ψ¯ Dψ 4

(2.7)

with Dψ / = γ μ (∂μ ψ + ie Aμ ψ). The gauge symmetry is the following local U (1) transformation, ¯ ¯ → e−ie(x) ψ(x) ψ(x) → eie(x) ψ(x) , ψ(x) , Aμ (x) → Aμ (x) + ∂μ (x). (2.8) with an arbitrary function (x). The Noether current for this transformation is μ

J μ = F μν ∂ν  + jmat ,

(2.9)

μ ¯ 0 ψ is the matter current density for U (1) global symmetry. The where jmat = ψγ local current conservation ∂μ J μ = 0 is guaranteed by the equation of motion μ ν ) and the conservation of the matter current (∂μ jmat = 0). If we (∂μ F μν = − jmat set (x) to be constant, the the above current reduces to the U (1) global current, since ∂ν  vanishes. Using the equation of motion, we can write the current in the form of total derivative as

J μ = ∂ν (F μν ).

(2.10)

Then the charge on a time slice  is given by  Q[] =



d 3  μ Jμ (x)

  0 d 3 x ∂(x) · E(x) + (x) jmat (x)     = d 3 x∂ν F 0ν (x)(x) = lim d 2  r 2 F 0r (x)(x) .

=

r →∞

(2.11) (2.12)

Here, we know that the radial component of the electric field, F 0r (x), generally falls off like r −2 so as to have a finite amount of the charge. Therefore, if we choose (x) such that it falls off enough rapidly as r → ∞, the charge vanishes, Q[] = 0, for any physical configurations. In this case, the conservation law is trivial in the sense that all physical solutions trivially satisfy the constraint. (In other words, we cannot use the charge to characterize physical solutions.) We call such gauge transformations trivial (or small) gauge transformations. In contrast, if (x) behaves like r 0 at spatial infinity, Q[] can have nontrivial finite value. We call such transformations large gauge transformations. The global symmetry is a special case of the large gauge

12

2 Asymptotic Symmetry Asymptotic Symmetry

symmetry because the charge in (2.12) reduces to the global U (1) charge in the case of  =constant. In (2.12), we can see that the charge depends on the gauge parameter only by the value at R = ∞. Therefore, two large gauge transformations generated by two gauge parameters with the same values at R = ∞ are identified as the same asymptotic symmetry transformation, even though they have different values at R < ∞. This is the reason why we take the quotient by trivial gauge transformations in (2.6). We have seen that the charge in (2.12) can have a nontrivial value for a large gauge parameter . However, it does not necessarily lead to the usual charge conservation between arbitrary different time slices. As already mentioned, Q( R ) = 0 with R = ∞ must vanish in order for the usual charge conservation (2.5) to hold. In the case of global U (1) transformation, Q( R ) in (2.4) with R = ∞ for the current (2.9) is given by  lim Q( R ) = lim R

R→∞

R→∞



tF

2

dt tI

  r d 2  F r ν ∂ν  + jmat  =0.

(2.13)

r Here, we have used lim R→∞ jmat = 0 because the matter current is localized at the position of the charged particles, and also ∂ν  = 0 for the global transformation. Thus the conservation law,

Q(t F ) = Q(t I ) ,

(2.14)

holds for the global symmetry. However, for the general large gauge transformations, we should be careful about the charge conservation laws because Q( R ) = 0 as R → ∞ becomes nontrivial condition. From now, we will study the charge conservation between infinitely past and future Cauchy slices because we will use the asymptotic symmetry to study the properties of S-matrix that is defined as the transition amplitude between two asymptotic states on infinite past and future. In order to do that, we will review the asymptotic structure of Minkowski spacetime in the next section and study the charge conservation between asymptotic regions in Sect. 2.4.

2.3 Asymptotic Structure of Minkowski Spacetime and Useful Coordinates In this section, we review the asymptotic structures of Minkowski spacetime and introduce several kinds of coordinates that are useful to study the dynamics of massive and massless fields at the asymptotic past and future. To talk about the asymptotic regions, introducing the Penrose diagram is very useful. Because the Minkowski spacetime has infinite volume, the asymptotic regions like t → ±∞ surface can not be visualized easily. Therefore, it is useful to scale the spacetime to some spacetime with finite volume so that we can draw the entire spacetime diagram on my thesis.

2.3 Asymptotic Structure of Minkowski Spacetime and Useful Coordinates

13

i+

t

u

I+ + I−

r

i0 − I+

v

I−

i− Fig. 2.1 The left diagram is the usual Minkowski spacetime diagram, while the right diagram is the Penrose diagram of the Minkowski space. Every point in both diagrams corresponds to S 2 (except for r = 0), since the angular directions are not represented. In both diagrams, the blue lines represent the trajectories of massive point particles, while the red lines represent the trajectories of massless point particles. The grey regions represent the effective regions where the scatterings occur

The Penrose diagram of a spacetime is a diagram of a spacetime that is obtained by scaling the original spacetime in a way that the scaled spacetime has finite volume and all light-rays propagate at ±45 degree in the diagram. The Penrose diagram of the Minkowski spacetime is given by the right figure in Fig. 2.1. The red lines correspond to the trajectories of massless point particles (or the trajectories of the wave fronts of massless waves) and green lines correspond to the trajectory of a massive point particle (or of the wave front of massive wave). There are five kinds of asymptotic regions that have different qualitative features as follows; • future timelike infinity (i + ): place where massive particles reach as t → ∞. • future null infinity (I+ ): place where massless particles reach as t → ∞. • spacelike infinity (i 0 ): place where any particles can not reach and only long range forces, like Coulomb force and gravitational force, can exist. • past null infinity (I− ): place where massless particles reach as t → −∞. • past timelike infinity (i − ): place where massive particles reach as t → −∞. In other words, the past/future timelike infinity is the Cauchy slice at the infinite past/future for the massive particles, and the past/future null infinity is the Cauchy slice at the infinite past/future for the massless particles. The standard Minkowski coordinates are given by ds 2 = −dT 2 + d R 2 + R 2 γ AB d A d B ,

(2.15)

14

2 Asymptotic Symmetry Asymptotic Symmetry

where  A (A = 1, 2) can be any coordinate of a unit two-dimensional sphere with a metric γ AB . This coordinate are not useful ones to analyze the physics around the asymptotic regions due to the following reasons. First, both i ± and I± are the region at t = ±∞, so we can not deal with them separately in the constant-T surface in (2.15). Secondly, I± can not be parametrized properly by T and R because both R and T are infinite on I+ . Therefore we will use the following coordinates to study the physics around each asymptotic region. • The coordinates for i ± ·(τ, ρ,  A ) coordinates The Minkowski line element of the coordinates (τ, ρ,  A ) is given by ds 2 = −dτ 2 + τ 2

 dρ 2 2 A B , + ρ γ d d AB 1 + ρ2

(2.16)

Theses coordinates are useful for focusing on the physics around the timelike infinity i + . These can be obtained by the following coordinate transformation from the Minkowski coordinate in (2.15), τ 2 = T 2 − R2 , ρ = √

R T2

− R2

.

(2.17)

For τ 2 > 0, the constant-τ hypersurfaces are three-dimensional hyperbolic surfaces H3 (or Euclidean AdS3 ). For τ 2 < 0, the hypersurfaces are three-dimensional de Sitter space dS3 . For later convenience, we introduce the following notation for τ > 0, ds 2 = −dτ 2 + τ 2 h αβ dσ α dσ β ,

(2.18)

where σ α = (ρ,  A ) are the coordinates of the unit three-dimensional hyperbolic space H3 with the line element h αβ dσ α dσ β =

dρ 2 + ρ 2 γ AB d A d B . 1 + ρ2

(2.19)

The reason why these coordinates are useful is the following. Solving (2.17) inversely for τ > 0, we have

T = ±τ 1 + ρ 2 , R = τρ .

(2.20)

The second equation in (2.17) can be rewritten as  R=±

ρ2 T , 1 + ρ2

(2.21)

2.3 Asymptotic Structure of Minkowski Spacetime and Useful Coordinates

15

i+

t

τ = constant (τ 2 > 0) ρ = constant

ρ=∞

I+

τ

τ

ρ=∞ ρ=∞

ρ=0

ρ=0

r ρ=∞

I− ρ=∞

i− Fig. 2.2 The left diagram is the usual Minkowski spacetime diagram, while the right diagram is the Penrose diagram of the Minkowski space. In both diagram, the blue lines represent the constant-ρ surfaces, while the red lines represent the constant-τ surfaces

where 0 ≤



ρ2 1+ρ 2

≤ 1. This means that the constant-ρ line corresponds to the tra ρ2 jectory of a massive particle with the constant velocity v = ± 1+ρ 2 , which is the asymptotic trajectory of massive particle as |t| → ∞ with the velocity. In other words, ρ has one-to-one correspondence to the asymptotic trajectory of a massive particle, therefore ρ is a natural coordinate for parametrizing the spacetime region that massive particles reach as t → ∞. In (2.20), we can easily see that τ → ∞ with the fixed ρ corresponds to T → ∞. Therefore τ = ∞ surface spanned by (ρ,  A ) corresponds to the future timelike infinity. The constant-τ hypersurfaces and the constant-ρ hypersurfaces are presented in Fig. 2.2. The surface element on the constant-τ hypersurface is given by √ τ 3ρ2 γ di + ≡ dρd 2  det(τ 2 h αβ ) = dρd 2 

. 1 + ρ2

(2.22)

We define an infinitesimal vector field di + μ as +

di + μ = n (iμ ) di + with n μ(i +

+

)

= δμτ ,

(2.23)

where n (iμ ) is defined as the unit vector orthogonal to the constant-τ surface. The metric in the matrix representation is given by

16

2 Asymptotic Symmetry Asymptotic Symmetry

⎛

gμν

⎛ ⎞ −1 0 −1 0 0 2 μν τ2 ⎝ ⎠ = ⎝ 0 1+ρ = , g 0 0 1+ρ 2 τ2 0 0 0 0 τ 2 ρ 2 γ AB

⎞ 0 ⎠ . 0 1 AB γ τ 2ρ2

(2.24)

The nonzero components of the Christoffel symbols2 are given by τ 1 ρ ρ ρ ,  τAB = τρ 2 γ AB , τρ = , ρρ = , (2.25) 2 1+ρ τ 1 + ρ2 1 1 A A = −(1 + ρ 2 )ργ AB , τAB = δ AB , ρAB = δ AB ,  AB = ˜ AB , (2.26) τ ρ

τ =− ρρ ρ

 AB

A is the Christoffel symbols on unit S 2 defined as where ˜ AB 



A  AB ≡

1 A B  γ (∂ A γ B  B + ∂ B γ B  A − ∂ B  γ AB ) . 2

(2.27)

• The coordinates for I± To study the asymptotic behavior of massless fields in Minkowski spacetime, it is useful to introduce the retarded coordinate u and the advanced coordinate v, u=T−R, v=T+R.

(2.28)

The constant-u line corresponds to R(T ) = T − u ,

(2.29)

which is the outgoing trajectory of a massless particle with R(0) = −u. Therefore, the retarded coordinate u is a natural coordinate that spans the future null infinity I+ . Similarly, The constant-v line corresponds to R(T ) = −T + v ,

(2.30)

which is the incoming trajectory of a massless particle with R(0) = v. Then the advanced coordinate v spans the past null infinity I− (Fig. 2.3). We will use the following coordinates to study the physics around I± . ·(u, r,  A ) coordinates. The line element is given by ds 2 = −du 2 − 2dudr + r 2 γ AB d A d B .

2∇

βV

α

ρ

α V ρ, ∇ V = ∂ V +  V . = ∂β V α + ρβ β α β α αβ ρ

(2.31)

2.3 Asymptotic Structure of Minkowski Spacetime and Useful Coordinates

i+

i+

u = constant r = constant

u=∞

u

17

v = constant r = constant

I+

I+

r=∞

r=∞

r=0

u = −∞

0

i

r=0

i0

v=∞

r=∞

r=∞

v

I−

I−

v = −∞

i−

i−

Fig. 2.3 Both diagrams are the Penrose diagram of the Minkowski spacetime. The red lines represent the constant-u surfaces in the left diagram and constant-v surfaces in the right diagram. In both diagrams, the constant-r surfaces are represented by the blue lines

These coordinates are related to the Minkowski coordinates in (2.15) by the coordinate transformations, u = T − R , r = R . The matrix representation of the metric (2.31) is given by ⎛

gμν

⎛ ⎞ ⎞ −1 −1 0 0 −1 0 0 ⎠ , g μν = ⎝ −1 1 0 ⎠ . = ⎝ −1 0 2 0 0 r γ AB 0 0 r −2 γ AB

(2.32)

The nonzero components of the Christoffel symbols are given by  uAB = r γ AB , rAB = −r γ AB , rAB =

1 A A A δ B ,  AB = ˜ AB . r

(2.33)

The future null infinity (I+ ) is parametrized by (u,  A ) with r = ∞. The infinitesi√ mal surface element of the constant-r surface is given by dI+ ≡ r 2 γ dud 2 . We define a infinitesimal vector field dI+ μ as +

dI+ μ ≡ n μ(I ) dI+ with n μ(I where n μ(I

+

)

+

)

= δrμ ,

is a unit normal vector orthogonal to the constant-r surface.

(2.34)

18

2 Asymptotic Symmetry Asymptotic Symmetry

·(v, r,  A ) coordinates. The metric is give by ds 2 = −dv2 + 2dvdr + r 2 γ AB d A d B .

(2.35)

These coordinates are related to the Minkowski coordinates in (2.15) by the coordinate transformations, v = T + R , r = R . The matrix representation of the metric is given by ⎛

gμν

⎛ ⎞ ⎞ −1 1 0 01 0 0 ⎠ . = ⎝ 1 0 0 ⎠ , g μν = ⎝ 1 1 2 −2 AB 0 0 r γ AB 00r γ

(2.36)

The past null infinity (I− ) is parametrized by (v,  A ) with r = ∞. The infinitesimal √ surface element of the constant-r surface is given by dI− ≡ r 2 γ dvd 2 . We define a infinitesimal vector field dI− μ as −

dI− μ ≡ n μ(I ) dI− with n μ(I

−

)

= δrμ ,

(2.37)

−

where n μ(I ) is a unit normal vector orthogonal to the constant-r surface. − + Finally, we introduce the two places I+ − and I+ (see Fig. 2.1). The I− is defined as the infinite past of future null infinity which is parametrized by  A with u = −∞, r = ∞, and the I− + is defined as the infinite future of past null infinity which − 0 is parametrized by  A with v = +∞, r = ∞. Note that I+ − , I+ , and i are not the + same places. For example, I− can be reached by first taking the limit r → ∞ with fixed u and then taking the limit u → −∞, whereas i 0 can be reached by taking the limit r → ∞ with fixed T . In fact, we will see that the Coulomb field at I+ − and are not the same but those are generally related to each other, and the the one at I− + relation is important for the conservation of charges associated with the asymptotic symmetry.

2.4 Classical Asymptotic Charge In this section, we study the charge associated with the classical asymptotic symmetry on the asymptotic Cauchy slices, i + ∪ I+ and i − ∪ I− , in QED. We first study the charge on future infinity, which is given by integrating the local gauge current in (2.9) on i + ∪ I+ : Q + [] ≡ Q +H [] + Q + S [] , where

(2.38)

2.4 Classical Asymptotic Charge

Q +H [] ≡ Q+ S [] ≡

19

 i

+

 μ  di + μ F μν ∂ν  +  jmat ,

(2.39)

dI+ μ F μν ∂ν  .

(2.40)

I+

In (2.40), we have dropped the matter current term in Q + S [] because the matter current does not reach the null infinity. We call Q +H [] the hard charge and Q + S [] the soft charge. Plugging (2.23) into (2.39), we obtain Q +H [] =

 i+

  τ . di + F τ ν ∂ν  +  jmat

(2.41)

Plugging (2.37) into (2.40), we also obtain Q+ S []



dI+ F r ν ∂ν     √ = lim dud 2  γ r 2 F r u ∂u  + F r B ∂ B  . =

I+

r →∞

(2.42)

In the remaining part of this section, we express the soft charge by gauge fields with a covariant gauge fixing for later convenience. In order to do that, we need to study the fall-off behaviors of field strength for physical solutions. By the “physical solutions”, we mean that the field strength is a solution of the Maxwell equations and carries nonzero but finite amounts of conserved charges e.g. energy, angular momentum, and electric charge. The electromagnetic energy momentum tensor is given by T

μν

  −1 1 μν ρλ μρ ν F F ρ − g F Fρλ . = 2π 4

(2.43)

The electromagnetic energy flux on I+ is then given by  I+

dI+ μ T μν δ νu = lim

r →∞

= lim

r →∞

 

√ dud 2  γ r 2 T ru √ γ AB dud 2  γ [Fu A Fu B − Fr A Fu B ] . 2π

(2.44)

To have a finite amount of the energy flux, the field strength should fall off as γ AB [Fu A Fu B − Fr A Fu B ] ∼ O(1) as r → ∞ . The total momentum flux of the radial direction on I+ is then given by

(2.45)

20

2 Asymptotic Symmetry Asymptotic Symmetry

 I+



√ dud 2  γ r 2 T rr r →∞    γ AB √ r2 Fur Fr u + 2 Fr A Fr B . = lim dud 2  γ r →∞ 4π r

dI+ μ T μν δ νr = lim

(2.46)

To have finite amount of the momentum flux, the field strength should fall off as Fur ∼ O(r −1 ) , Fr A ∼ O(1) .

(2.47)

Moreover, in order to have finite amount of electric charge, the radial component of electric field should fall off as r −2 , F tr = F ur = −Fur ∼ O(r −2 ) .

(2.48)

Here, we rewrite the above fall-off conditions for field strengths as the ones for gauge fields. We expand gauge fields by power series of r as Aμ (u, r, ) =

∞  A(n) μ (u, ) n=0

rn

.

(2.49)

From now, we work in the Lorenz gauge by imposing ∇ μ Aμ = 0 . In this gauge, there are residual gauge transformations gauge parameter (x) satisfying

(2.50) 3

that are generated by the

(x) = 0 .

(2.51)

This condition is written as follows in (u, r,  A ) coordinates, √  1 0 = ∇μ Aμ = √ ∂μ g Aμ g  √ B  1 1  = ∂u Au + 2 ∂r r 2 Ar + √ ∂ B γA r γ 2 γ AB = −∂u Ar + ∂r (Ar − Au ) + (Ar − Au ) + 2 D A A B , r r

(2.52)

where we have used the metric given in (2.32). Plugging (2.49) into the above constraint, we have

3

The residual gauge transformations are defined as gauge transformations such that the transformed fields satisfy the gauge fixing condition.

2.4 Classical Asymptotic Charge

21

∂u Ar(0) = 0 , −∂u Ar(1) + 2(Ar(0) − A(0) u ) = 0.

(2.53)

Moreover, plugging (2.49) into (2.48), we have ∂u Ar(0) = 0 , ∂u Ar(1) = 0 .

(2.54)

Combining (2.53) and (2.54), we find ∂u Ar(0) = 0 , ∂u Ar(1) = 0 , Ar(0) − A(0) u =0.

(2.55)

Further constraints do not come from the fall-off conditions (2.45) and (2.47). Now we express the soft charge defined in (2.42) in term of the gauge fields. Under the fall-off conditions of (2.55), the field strengths can be expanded as γ AB γ AB − F = (F ) (∂u A B − ∂ B Au − ∂r A B + ∂ B Ar ) u B r B r2 r2 γ AB −3 = 2 ∂u A(0) (2.56) B + O(r ) , r

Fr A =

where the second and the third terms cancel each other because of the third condition in (2.55). The residual gauge parameter can be expanded as (A.13): (u, r, ) =  (0) () +

2r u log |u|

2r

S2  (0) () + O(r −1 ).

(2.57)

See Appendix A for the detail of the expansion. Then inserting (2.56) and (2.57) into the soft charge (2.42), we find Q+ S [] =



√ dud 2  γ γ AB ∂ A  (0) ∂u A(0) B .

(2.58)

This is the asymptotic charge on I+ . This expression coincides with the soft charge that was first introduced in [1, 2].

2.5 Classical Charge Conservation Associated with the Asymptotic Symmetry In this section, we illustrate the existence of an infinite number of asymptotically conserved charges associated with large gauge transformations in the classical electromagnetism. (Precise meaning of “asymptotically conserved charges” will be given μ , which is later.) We represent the matter current for massive charged particles by jmat μ the source in Maxwell’s equation ∂ν F νμ = − jmat . Here, we assume that the charged particles behave as free particles except in a small region where they scatter, and

22

2 Asymptotic Symmetry Asymptotic Symmetry

we ignore the back-reaction. We also impose the initial condition that there is no radiation before the scattering of charged particles just for simplicity. The Noether current for the gauge transformation with gauge parameter (x) is given by (2.9), which is μ

J μ = F μν ∂ν  + jmat .

(2.59)

In order to study whether Q( R ) = 0 at R = ∞ vanishes or not, we first consider a closed surface with a finite area in Minkowski spacetime and then take the limit that makes the area infinitely large. Consider the integration of the current conservation equation ∂μ J μ = 0 over the region represented in Fig. 2.4. The region is parametrized by two parameters T and U , and has five boundaries,  f , i , + , − , 0 . The two surfeces,  f and i , are time slices at the distant future and past, t = ±(T + U ) with 0 ≤ r ≤ T − U . 0 is a timelike surface r = const. = T + U with −T + U < t < T − U . ± are null surfaces where ±t + r = const. = 2T . We set the parameter T μ vanishes on ± and 0 . We also to be so large that the massive matter current jmat set U to be so large that any electromagnetic radiation coming from the scattering region goes through the null surface + . Later, we take the limit first T → ∞ and then U → ∞ such that ± becomes the future and the past null infinities I± , where I+ (I− ) is parametrized by the retarded (advanced) time u = t − r (v = t + r ) and the angular coordinates  A . The ordering of the limits, T → ∞ before U → ∞, is crucial for our derivation of the memory effect formulae. The integration of the current conservation proves that the sum of the surface integral of the current over each boundary vanishes:  0=

d V ∂μ J μ = Q f + Q + + Q 0 − Q − − Q i

(2.60)

with  Qa ≡

a

d S μ Jμ , (a = f, i, 0, +, −) ,

(2.61)

where we choose the surface element d S μ to be future-directed when the surface is spacelike or null.

2.5.1 The Asymptotic Charge Conservation and Memory Effect at Leading Order We now see that Q 0 , which is defined on 0 as  Q0 =

T −U

√ dt d 2  γ r 2 J r |r =T +U ,

−T +U

(2.62)

2.5 Classical Charge Conservation Associated with the Asymptotic Symmetry

23

Fig. 2.4 The region where we consider the current conservation. The directions along the twodimensional sphere S 2 are suppressed in this figure. The blue lines represent trajectories of massive charged particles, which scatter at a small region. The red line represents a direction of the radiation emitted from this scattering. The region is parametrized by two parameters T and U . The parameter T is so large that all of the given massive particles go through the surface i and  f , and U is also so large that any radiation coming from the scattering region passes through + . (This figure is reused from [3] licensed under CC-BY 4.0.)

vanishes in the limit T → ∞ without the limit U → ∞. As the current can be represented as a total derivative J r = ∂μ (F r μ ), Q 0 splits into the integrations over two spheres on the future and the past boundaries of 0 as  Q0 =

√ d 2  γ r 2 Ftr |t=T −U,r =T +U −



√ d 2  γ r 2 Ftr |t=−T +U,r =T +U . (2.63)

Since any radiation cannot reach 0 due to our choice of the region, the field strength Ftr is the Coulombic electric field created by free-moving charged particles before the scattering. The behavior of the fields are analyzed in Appendix B. The leading term of Ftr in the large T expansion is O(T −2 ), and the leading terms are represented as lim r 2 Ftr (t, r, )|t=T −U,r =T +U ≡ Ftr+(2) () ,

(2.64)

lim r 2 Ftr (t, r, )|t=−T +U,r =T +U ≡ Ftr−(2) () .

(2.65)

T →∞ T →∞

The important facts are that they are independent of U and they satisfy the antipodal matching condition (see Appendix B.1) ,

24

2 Asymptotic Symmetry Asymptotic Symmetry

¯ . Ftr+(2) () = Ftr−(2) ()

(2.66)

¯ = ( ¯ 1,  ¯ 2 ) is the antipodal angle of  = (1 , 2 ).4 We thus obtain where   lim Q 0 =

T →∞

√ d  γ Ftr+(2) () (0) () −



2

√ ¯ d 2  γ Ftr−(2) () (0) ()

(2.67)

¯ near I− . where we used the fact that (x) approaches  (0) () near I+ and  (0) () Therefore, due to the matching condition (2.66), Q 0 vanishes in the limit T → ∞: 

√ ¯ (0) () = 0 . d 2  γ [Ftr+(2) () − Ftr−(2) ()]

lim Q 0 =

T →∞

(2.68)

Thus in the limit T → ∞, the conservation Eq. (2.60) indicates lim (Q f + Q + ) = lim (Q i + Q − ) .

T →∞

T →∞

(2.69)

The Q f and Q i are given by  Qf =

T −U

√ t dr d 2  γ r 2 (F ti ∂i  + jmat )|t=T +U ,

0

 Qi =

(2.70)

T −U

√ t dr d 2  γ r 2 (F ti ∂i  + jmat )|t=−T −U .

0

(2.71)

The field strength F ti in the above integrands is the Coulombic electric field created by free-moving charged particles, since the radiation does not reach  f and i . Thus Q f and Q i are the usual gauge charges for free-moving charged particles and their Coulomb-like potential in the three-dimensional ball with radius T − U at time t = ±(T + U ). In particular, if  is a constant, Q f and Q i equal to the total electric charges (×). Equation (2.69) just means that such gauge charges do not conserve for the nontrivial large gauge parameters (x) unless we include the contributions from the radiation Q ± .5 Let us call the conservation laws (2.69) “asymptotic conservation”, since it only holds in the limit T → ∞. Since we can take arbitrary functions  (0) () on unit two-sphere, we have an infinite number of the asymptotic conserved charges. Using the retarded time u = t − r , we can write Q + as  Q+ =

4 The

2U

√ dud 2  γ r 2 (F r u ∂u  + F r A ∂ A )|r =T −u/2 .

(2.72)

−2U

importance of this condition in the context of the asymptotic symmetry was first emphasized in [4] (see also [5] for the detailed explanation). 5 A similar statement was given for the case without massive charged fields in [6].

2.5 Classical Charge Conservation Associated with the Asymptotic Symmetry

25

In the large T limit, the integration region becomes the subregion −2U ≤ u ≤ 2U in I+ . Using the condition (2.55) and expansion (2.57), we obtain  lim Q + =

T →∞

2U

√ (0) dud 2  γ γ AB ∂u A(0) B ∂A

−2U √ (0) (0) = d 2  γ γ AB [A(0) . B (u = 2U ) − A B (u = −2U )]∂ A 

(2.73)

On the other hand, for our setup or our initial condition, there is no radiation at − , and thus lim T →∞ Q − = 0. Therefore, Eq. (2.60) gives  −

√ (0) (0) = lim (Q f − Q i ) . d 2  γ γ AB [A(0) B (u = 2U ) − A B (u = −2U )]∂ A  T →∞

(2.74) This equation holds for any function  (0) () on the two-sphere. It implies that the + shift of A(0) B on I is related to the change of the gauge charges between Q f and Q i .6 In particular, when we perform the partial integration for the angular integral, (2.74) becomes  √ (0) d 2  γ γ AB  (0) [∂ A A(0) B (u = 2U ) − ∂ A A B (u = −2U )] = lim (Q f − Q i ) T →∞

(2.75) Moreover, if we define the local electric charge at timelike past and future infinity, ρi () and ρ f () for every angle , by  Qi ≡ Qf ≡



√ d 2  γ  (0) ρi () ,

(2.76)

√ d 2  γ  (0) ρ f () ,

(2.77)

we find (0) γ AB [∂ A A(0) B (u = 2U ) − ∂ A A B (u = −2U )] = ρ f () − ρi () .

(2.78)

This result means that the total charge that has passed through the angle  after all + particles have left is memorized by the shift of γ AB ∂ A A(0) B on I . This is called the electromagnetic memory effect formula [7–9]. Note that the shift is gauge invariant although γ AB ∂ A A(0) B is not. Thus we have found the equivalence between charge conservation associated with the asymptotic symmetry and the electromagnetic memory effect.

6 The

-independent part of the shift cannot be determined from this formula.

26

2 Asymptotic Symmetry Asymptotic Symmetry

If we take the limit U → ∞, the charge from the radiation (2.73) becomes the soft charge given in (2.58),  ≡− Q lead,+ S

√ (0) dud 2  γ γ AB ∂u A(0) B ∂A .

(2.79)

I+

Although lim T →∞ Q − vanishes in our setup, if we consider the general situation where electromagnetic radiation exists initially, limU →∞ lim T →∞ Q − becomes the defined on I− . The Q f and Q i become the so called hard charges soft charge Q lead,− S lead,± in the limit. Note that the hard charges include not only matter currents but QH also the contributions from the Coulombic electric field produced by the charged matters. We will study the expression of those charges concretely in Sects. 2.8 and 2.9 (for example see (2.111) and (2.197).) In the limit U → ∞, we thus obtain + Q lead,+ = Q lead,− + Q lead,− . Q lead,+ S H S H

(2.80)

We note again that the conservation laws (2.80) come from the fact that the charge on spacelike infinity, lim T →∞ Q 0 , vanishes due to the antipodal matching condition. Since the current J μ is a total derivative, the total asymptotic charge Q lead,+ + Q lead,+ S H + is equal to the integral on two-sphere at the past boundary of I at u = −∞ as  Q lead,+ S

+

Q lead,+ H

=−

I+ −

√ d 2  γ Ftr+(2) () (0) () .

(2.81)

Similarly, Q lead,− + Q lead,− also equals to the integral on two-sphere at the future S H − boundary of I as  + Q lead,− =− Q lead,− S H

I− +

√ ¯ , d 2  γ Ftr+(2) () (0) ()

(2.82)

which is equal to (2.81). We also remark that the surface ± ∪  f /i becomes the Cauchy surface in the limit U → ∞ after T → ∞. Therefore, Q lead,± + Q lead,± is S H actually a charge defined on the asymptotic Cauchy surface.

2.5.2 Asymptotic Charge Conservation and Memory Effect at Subleading Order We have seen that the current conservation equation (2.60) leads to the formula of the leading memory effect (2.74) in the large T limit because the charge from spacelike infinity, Q 0 , vanishes in the limit due to the antipodal matching. The subleading

2.5 Classical Charge Conservation Associated with the Asymptotic Symmetry

27

memory effect can also be obtained by considering the corrections in the large T expansion. We first consider the correction to Q + defined as Eq. (2.72). By using the formula (A.13), we can expand the gauge parameter (x) on + in the large T limit as (u, r = T − u/2, ) =  (0) () −

u log 2|u|T S2  (0) () + O(T −1 ) , 2T

(2.83)

where S2 is the Laplacian on the unit two-sphere. Note that the correction to  (0) () starts from O(T −1 log T ). In Appendix C, we give general expansions of radiation fields which are compatible with this large gauge parameters. Using the expansions (2.83), (C.1), (C.2) and (C.3), we find that the first correction to Q + is of order O(T −1 log T ), and takes the form  Q+ = −

√ log log log T BA + O(T −1 ) , dud 2  γ ∂u A(0) ∂ A  (0) − (Q + + Q + ) B γ T −2U (2.84) 2U

where the first term is just the leading soft charge (2.73), and the second term is the log log first correction. Here, Q + and Q + are given by log

Q+ =

1 2



1 =− 2 log

2U

√ BA dud 2  γ u∂u A(0) ∂ A S2  (0) B γ

−2U  2U

√ dud 2  γ  (0) u∂u S2 ∇ B A(0) B ,

 √  (1) S2  (0) dud 2  γ Ar(1) + ∇ B A(0) + 2C u B −2U   (1) (0) ∂ , + 2γ AB ∂u C (1) − ∂ C  A u B A

Q+ = −

1 2

(2.85)

−2U  2U

(2.86)

where ∇ B denotes the covariant derivative on the unit two-sphere w.r.t. the metric γ AB , and the derivative with the upper index is defined as ∇ A ≡ γ AB ∇ B . The definition of Cu(1) , Ar(1) and C (1) A are given in (C.1), (C.2) and (C.3). On the spatially distant surface 0 , the leading part of the charge Q 0 in the large T expansion is O(T −1 ) as shown in Eq. (B.10) in Appendix B.1, and it does not have any O(T −1 log T ) term. Thus the coefficient of T −1 log T is also conserved without the contribution from 0 like the leading memory. The contributions of the future and past timelike infinities are extracted as log

T 2 d Q f,i . T →∞ log T dT

Q f,i = lim

Then we obtain the subleading memory effect formula (with finite U ) as

(2.87)

28

2 Asymptotic Symmetry Asymptotic Symmetry

1 − 2



2U

√ log log log dud 2  γ  (0) u∂u S2 ∇ B A(0) B = −Q + − Q f + Q i .

(2.88)

−2U

log

In the limit U → ∞, the subleading radiation charge Q + becomes log

Q sub,+ ≡ lim Q + = − S U →∞

1 2



√ dud 2  γ  (0) u∂u S2 ∇ B A(0) B ,

(2.89)

I+

which agrees, up to a numerical factor, with the electric-type subleading soft charge in [10], and it is shown that the charge also agrees with that in [11] by taking their vector field Y A on unit two-sphere as Y A ∝ ∇ A  (0) . We also represent the other contributions in the limit U → ∞ including the initial radiation on − by7 log

Q sub,− ≡ lim Q − , S U →∞

log

log

Q sub,+ ≡ lim (Q + + Q f ) , H U →∞

log

log

Q sub,− ≡ lim (Q − + Q i ) . H U →∞

(2.90) We then obtain the subleading charge conservation + Q sub,+ = Q sub,− + Q sub,− . Q sub,+ S H S H

(2.91)

Unlike the leading memory effect, this equation does not directly relate a shift of the radiation field to the change of the configuration of hard charges. Nevertheless, − Q sub,− is known for causing a permanent the existence of a nonzero shift Q sub,+ S S displacement of the position of a probe charged particle [12]. Thus this is the memory effect. It can also be detectable as a permanent change of the direction of the spin of a heavy probe particle with the magnetic moment [13]. Before closing this section, we briefly comment on the sub-subleading order. We have seen that the charge Q 0 on spacelike infinity is O(T −1 ), which is the same order as the sub-subleading corrections to Q a (a = +, −, f, i). Therefore, we conclude that there is no asymptotic conservation at the sub-subleading order without including the contribution from Q 0 (see also similar discussions in [10, 14]).8 This conclusion is probably related to the fact that there is no sub-subleading soft photon theorem in QED as we will see in Sect. 3.4 (see also [15] for the discussion that the soft expansion of amplitudes is not associated with a universal soft factor at the subsubleading order).

7 As

log

log

shown in Appendix B.2, Q + generally diverges in the limit U → ∞, and Q f also does. log

log

Nevertheless, the combination (Q + + Q f ) is finite as far as we know. 8 The

arguments that there is no subsubleading charge were given in [10, 14]. However, the reason seems to be different from ours. Their reason is that the sub-subleading charge has an inevitable divergence. On the other hand, from our construction, the charges for the large gauge transformations are “conserved” if we take account of the contributions from spacelike infinity.

2.6 Asymptotic Symmetry in Quantum Field Theory

29

2.6 Asymptotic Symmetry in Quantum Field Theory In this section, we argue about the asymptotic symmetry and its charge in quantum field theories. Here, we define the asymptotic symmetry transformations (AST) in QFT as AST =

allowed gauge transformations . trivial gauge transformations

(2.92)

The allowed gauge transformations are any gauge transformations generated by the well-defined charge. The allowed gauge transformations are the ones generated by the charge that acts trivially on any physical states in the given theory. In classical cases, the charge associated with the gauge symmetry was just given by integrating the local gauge current on a Cauchy slice. In contrast, the procedure become subtle in quantum theories due to the following reason. In quantum case, we first need to quantize the theory. In the case of gauge theories, the quantization becomes subtle due to the existence of gauge (redundant) degrees of freedom, and we need to fix them to quantize the theory. There are two well-known procedures for the quantization of gauge theories: the canonical quantization and the BRST quantization. In both procedures, we need to add some new terms to the original Lagrangian to eliminate the gauge degrees of freedom. More concretely, we change the Lagrangian by introducing Lagrangian multipliers in the canonical quantization. In the BRST quantization, we add BRST exact terms. After the gauge fixing, the gauge symmetries of the original Lagrangian are no longer symmetries and there is no Noether current of gauge symmetries. Therefore we need different ways to find the conserved charge associated with the asymptotic symmetries. We explain asymptotic symmetries and their charges in QED, first in the canonical quantization in the next section, and in BRST quantization in Sect. 2.9.

2.7 Asymptotic Symmetry in Canonical Quantization Consider QED again for concreteness. The Lagrangian for QED (2.7) has the following two first class constraints (see Appendix D for the derivation), φ0 ≡ 0 (x) = 0, φ1 ≡ ∂i  (x) − i

0 jmat (x)

(2.93) = 0,

(2.94)

where 0 = −F 00 = 0 and i = −F 0i = −E i are the canonical conjugate variables for A0 and Ai , respectively. In quantum theory, the first class constraints are imposed on any physical state |ψ as

30

2 Asymptotic Symmetry Asymptotic Symmetry

0 (x)|ψ = 0,   0 ∂i E i (x) − jmat (x) |ψ = 0.

(2.95) (2.96)

These conditions make the physical states gauge invariant because the first class constraints are the generators of gauge transformations. The second constraint (2.96) is called the Gauss law constraint. Multiplying (2.96) by (x) and integrating over space, we get 

  0 d 3 x (x) ∂i E i (x) − jmat (x) |ψ (2.97)     0 = lim (x) |ψ , d 2  r 2 (x)E r (x)|ψ − d 3 x ∂i (x)E i (x) + (x) jmat

0=

r →∞

(2.98) where we have performed a partial integration in the second equality. Thus we find for any physical state |ψ ,  Q[]|ψ = lim

r →∞

d 2  r 2 (x)E r (x)|ψ ,

(2.99)

where Q[] was defined in (2.12). The trivial (small) gauge transformations are the gauge transformations such that (x) falls off so rapidly that the righthand side in (2.99) vanishes: Q[]|ψ = 0 .

(2.100)

The trivial gauge transformation actually does not act on anything physical. (We could call it “do-nothing transformations.”) Therefore it is not the physical transformation. Once the physical Hilbert space is restricted to an invariant subspace for the trivial gauge transformations, physical observables are also restricted to the set of invariant operators under the transformations due to Elitzur’s theorem. Therefore, the WardTakahashi identity for the trivial gauge transformation is

Q[](· · · ) = 0,

(2.101)

where (· · · ) can be any physical observables. All physical observables trivially satisfy this equation by definition, so the Ward-Takahashi identity actually constraints nothing. On the contrary, the large gauge transformations can act on physical states by (2.99) nontrivially as  Q[]|ψ = lim

r →∞

d 2  r 2 (x)E r (x)|ψ = 0 .

(2.102)

2.7 Asymptotic Symmetry in Canonical Quantization

31

Thus the physical states can be characterized by these charges. Note that we have the charges for each choice of the large gauge parameter . However these charge operators do not act on any observable O with a finite spatial support because 

ψ|[Q[], O]|ψ = lim

r →∞

d 2  r 2 (x) ψ|[E r (x), O]|ψ = 0 ,

(2.103)

where we have used the causality for observables in the second equality. Therefore, these charges can not be changed by acting any quasilocal operators. (See [16] for the rigorous proof.) Thus the charges associated with the large gauge transformations decompose the Hilbert space into superselection sectors. In the next section, we give more systematic definition for the asymptotic symmetries.

2.8 Charges Associated with Asymptotic Symmetry in QED The Gauss law constraint in (2.96) can be written in (τ, ρ,  A ) coordinates defined in (2.16) as   0 |ψ 0 = ∇i E i − jmat   1 ρ A 0 = ∇ρ E + 2 2 ∇ A E − jmat |ψ , τ ρ

(2.104) (2.105)

where we used the metric in (2.24) and ∇ A E A ≡ γ AB ∇ A E B . The Gauss law constraint in (2.96) can be written in the retarded coordinates as   0 |ψ 0 = ∇i E i − jmat   1 r A 0 = ∇r E + 2 ∇ A E − jmat |ψ , r

(2.106) (2.107)

where we have used E 0 = g0μ E μ = −E u in the final equality. Multiplying the Gauss law constraint by (x) and integrating it over I+ , we have   1 ρ A 0 d   ∇ρ E + 2 2 ∇ A E − jmat |ψ 0 = lim τ →∞ i + τ ρ    1 3 r A + + lim d I  ∇r E + 2 ∇ A E |ψ r →∞ I+ r 

3

i+

(2.108)

where we have dropped the matter current term in the second term because the massive matter current asymptotically goes to the time-like infinity and can not reach to the null infinity. Performing the partial integrations and using ∇r E r = ρ ρ ∂r E r , ∇ρ E ρ = ∂ρ E ρ + τ1 E τ + 1+ρ 2 E , we obtain

32

2 Asymptotic Symmetry Asymptotic Symmetry

Q + []|ψ √   ρ 2 γ () 3 2 = lim lim τ  Eρ d 

ρ→∞ τ →∞ 1 + ρ2   



2 2 r 2 2 r + lim lim d  γ () r  E − lim lim d  γ () r  E |ψ , u→∞ r →∞

u→−∞ r →∞

(2.109) where Q + [] is defined as Q + [] ≡ Q +H [] + Q + S []

(2.110)

with   1 0 , d 3 i + (∂ρ )E ρ + 2 2 (∂ A )E A +  jmat τ ρ i+    1 3 r A + (∂ . Q+ [] ≡ d  )E + (∂ )E I r A S r2 I+

Q +H [] ≡



(2.111) (2.112)

This is the charge that generates the asymptotic symmetry with the gauge parameter  on the future asymptotic infinity, which is the quantum analog of the classical asymptotic charge given in (2.38). In the right hand side of (2.109), the first term and the second term must be canceled out with each other because the boundary of i + and the boundary of I+ at u = ∞ are the same spacetime surfaces, and the outward vectors orthogonal to each surface have opposite direction. Therefore, (2.109) becomes 

Q + []|ψ = − lim lim d 2  γ () r 2  E r |ψ u→−∞ r →∞ 

=− d 2  γ () r 2  + E r |ψ , (2.113) I+ −

where we have defined  + () as  + () ≡ lim lim (u, r, ) , u→−∞ r →∞

(2.114)

It is important that the action of the asymptotic charge on physical states, given by (2.113), only involves  + (). The charges with the gauge parameter  + () = 0 act on any physical state trivially and the charges with the gauge parameter  + () = 0 act on physical states nontrivially in general. Therefore, The latter ones are the physical asymptotic charges and the transformations generated by the charges are the asymptotic symmetries. In the case of  = constant, the asymptotic charge in (2.110) reduces to

2.8 Charges Associated with Asymptotic Symmetry in QED

Q + [] =  lim

33



τ →∞ i +

0 di + jmat ,

(2.115)

which is the usual global U (1) charge. The right hand side in (2.113) also reduces to the total electric charge, since there exist only Coulomb forces at I+ −. Repeating the same argument for the asymptotic past infinity, we have 

d 2  γ () r 2 E r |ψ Q − []|ψ = lim lim v→∞ r →∞ 

= d 2  γ () r 2 E r |ψ , I− +

(2.116)

where the asymptotic charge on the past infinity is given by Q − [] ≡ Q −H [] + Q − S []

(2.117)

with   1 0 , d 3 i − (∂ρ )E ρ + 2 2 (∂ A )E A +  jmat τ ρ i−    1 3 r A − (∂ . Q− [] ≡ d  )E + (∂ )E I r A S r2 I−

Q −H [] ≡



(2.118) (2.119)

The conservation law for these asymptotic charges is  

β|Q + []S|α + β|T Q 0 []S |α = β|S Q − []|α

(2.120)

where Q 0 [] is the charge defined on i + . Here, we assume that the antipodal matching condition also holds in QED and the second term in the left hand side of (2.121) vanishes. We will see that this assumption has a support if we consider the dressed state formalism in Sect. 5.7. The asymptotic charges then conserve between the future infinity and the past infinity:

β|Q + []S|α = β|S Q − []|α .

(2.121)

We will see that this conservation law is equivalent to the soft photon theorem in the next chapter.

2.9 Asymptotic Symmetry in BRST Formalism We have seen why the large gauge transformations are physical symmetries in the canonical quantization. However, we often quantize gauge theories by the BRST quantization rather than the canonical quantization, especially for computing S-

34

2 Asymptotic Symmetry Asymptotic Symmetry

matrixes. In this section, we argue that the asymptotic symmetry in QED is physical symmetry based on the BRST formalism, although they are naïvely parts of the gauge symmetry. We first review the covariant quantization of QED in the BRST formalism in Sect. 2.9.1. Next, we look at the asymptotic behaviors of gauge fields and ghost fields in Sect. 2.9.2. Finally, we see that the charges associated with asymptotic symmetry act on physical Hilbert space nontrivially under the BRST condition.

2.9.1 BRST Quantization in QED and the Correct BRST Charge In this subsection we review the covariant quantization of massive scalar QED9 in the BRST formalism,10 and discuss the symmetries. The Lagrangian in the Feynman gauge is given by L Q E D = L E M + Lmatter + LG F + L F P ,

(2.122)

where 1 1 1 L E M = − Fμν F μν , Lmatter = − Dμ φ † Dμ φ − m 2 φ † φ , 4 2 2  1 μ 2 μ L G F = − ∇μ A , L F P = i∂ c∂ ¯ μ c. 2

(2.123) (2.124)

Here, φ is a massive charged scalar field11 with charge e where Dμ φ = ∂μ φ − ie Aμ φ, and c, c¯ are ghost fields.12 LG F + L F P 13 is a BRST exact term, which is added to the original Lagrangian to eliminate the gauge symmetry. The equation of motions are given by Aμ = − j μ , c = 0 , c¯ = 0 .

(2.125)

The canonical conjugate fields for each Aμ , c, c¯ are given by

9 Although

the arguments of this section can be straightforwardly applied to any kind of charged particles such as massive or massless fermions and scalars, we explicitly write the expressions of a massive scalar because we use them in Sect. 4.1. 10 In QED, the ghost sector is completely decoupled. Nevertheless, we use the BRST formalism to discuss what the physical states are. 11 In this paper, we consider only a single species of charge for simplicity. The generalization to many species is straightforward. 12 Both c and c¯ are Hermitian fields. This Hermicity is required to make the Lagrangian Hermite. 13 We already integrated out the Nakanishi-Lautrup field in L GF.

2.9 Asymptotic Symmetry in BRST Formalism

∂L ∂L = −∇μ Aμ , i = = −F 0i , 0 ∂(∂0 A ) ∂(∂0 Ai ) ∂L ∂L = ¯ π¯ (c) = = −i∂0 c, = i∂0 c. ∂(∂0 c) ∂(∂0 c) ¯

35

0 =

(2.126)

π(c)

(2.127)

The Hamiltonian is then given by free part H0 and the other interacting part V : H = H0 + V ,

(2.128)

H0 = HE M + Hmatter + Hghost ,

(2.129)

where

with14  1 1 μ i 0 ij , HE M = d x μ  + (∂i 0 )A + (∂i i )A + Fi j F 2 4  Hghost = i d 3 x(π¯ (c) π(c) − ∂i c∂ ¯ i c), 



3

(2.130) (2.131)

and Hmatter is the free part of the Hamiltonian of matter fields. The equal-time (anti)commutation relations for the gauge fields and the ghost fields are given by ¯ π¯ (c) (y)} = iδ 3 (x − y). [Aμ (x), ν (y)] = iημν δ 3 (x − y) , {c(x), π(c) (y)} = {c(x), (2.132) In the interaction picture, the equation of motions for them are given by free ones; AμI (x) = 0. The general solutions of the equation of motions that can be expanded by the Fourier expansions are given by    3 k a (k)eikx + a † (k)e−ikx , AμI (x) = d μ μ    I 3 k c(k)eikx + c† (k)e−ikx , c (x) = d    ikx 3 k c(k)e c¯ I (x) = d ¯ + c¯† (k)e−ikx ,

(2.133)

3 k is the Lorentz invariant measures where k μ is massless on-shell momenta and d d3k 3k ≡ . The for the integration of the spatial momentum which is defied as d (2π)3 2ωk mode expansions of the above operators in the Schrödinger picture can be given by

14 H

E M given by (2.130) is different from the Hamiltonian obtained in a canonical way from Lagrangian (2.122) by a total derivative term. We have eliminated the boundary term, and then this H E M commutes with the BRST charge without a boundary term. This difference is not important except in Sect. 5.7.

36

2 Asymptotic Symmetry Asymptotic Symmetry

taking t = ts in the above expansion. The conjugate fields in (2.126) are then given by  0I (x) = −i



iI (x) = −i 

  3 k k μ a (k)eikx − k μ a † (k)e−ikx , d μ μ

(2.134)

 3 k (k a (k) + ωa (k))eikx − (k a † (k) + ωa † (k))e−ikx , d i 0 i i 0 i (2.135)

 d k 1 I ikx π(c) c(k)e ¯ (x) = − − c¯† (k)e−ikx , 3 (2π ) 2  3  k 1 d I c(k)eikx − c† (k)e−ikx , π¯ (c) (x) = 3 (2π ) 2 3

(2.136) (2.137)

The commutation relations (2.132) lead to the following equal-time (anti) commutation relations between annihilation and creation operators: [aμ (k), aν† (k )] = (2π )3 (2ωk )ημν δ (3) (k − k ) , {c(k), c¯† (k )} = i(2ω)(2π )3 δ 3 (k − k ),

(2.138)

{c(k), ¯ c† (k )} = −i(2ω)(2π )3 δ 3 (k − k ).

(2.139) The Hilbert space is just the Fock space HFock . This Lagrangian has two kinds of symmetries: the residual gauge symmetry and the BRST symmetry [17, 18] . The residual gauge symmetry is δg φ(x) = ie(x)φ(x) , δg φ † (x) = −ie(x)φ † (x) , δg Aμ (x) = ∂μ (x), (2.140) with (x) = 0.

(2.141)

These residual symmetries are usually regarded as “gauge” redundancies, but we will argue in the next Sect. 2.9.2 that a part of them associated with the large gauge parameters (A.4) are physical symmetries, which impose the nontrivial constraints on the S-matrices as the Ward-Takahashi identities. Using the Lagrangian (2.122), we have the Noether current associated with (2.140); μ

μ

jμ (x) = jS (x) + j H (x),

(2.142)

where (2.143) jSμ (x) = ∇ν (x)F μν (x) + ∇ μ (x)∇ν Aν (x) ,  μ †  μ μ μ † μ j H (x) = (x) jmat (x) , with jmat (x) = ie D φ (x) φ(x) − φ (x) D φ(x) . (2.144)

2.9 Asymptotic Symmetry in BRST Formalism

37

The current is different from (2.59) due to the gauge fixing term. The charge that generates the above symmetry on a spatial surface  is given by  Q[] =



d S μ jμ (x) .

(2.145)

The asymptotic charge on the future infinity is then given by Q + [] = Q i + [] + Q I+ []

(2.146)

with  Q i + [] =

i+

d 3 i + μ jμ (x) , Q I+ [] =

 I+

μ

d 3 I+ μ jS (x) .

(2.147)

On the other hand, the BRST symmetry (in the case of scaler matter) is given by δg φ(x) = iec(x)φ(x) , δg φ † (x) = −iec(x)φ † (x) , δg Aμ (x) = ∂μ c(x) . (2.148) We have the charge that generates the BRST symmetry called the BRST charge. The BRST charge on a Minkowski time slice (x 0 = t) is given by  Q B R ST = =



  d 3 x −(∂μ c)μ + cj 0   d 3 x i π¯ (c) 0 − (∂i c)i + cj 0 .

(2.149)

It is important that the last term involving the matter current is needed to generate the BRST transformation (2.148) correctly, in particular, [i Q B R ST , φ] = iecφ,

(2.150)

although the term has been sometimes missed. We will see that the term of cj 0 becomes important for the gauge invariance of the dressed state in Sect. 5.5. This BRST charge commutes with the total Hamiltonian and the matter current [Q B R ST , H ] = 0 , [Q B R ST , j μ (x)] = 0.

(2.151)

The BRST charge on general Cauchy slice  is given by  Q B R ST =



  d Sμ (∂0 c)g μ0 ∇ν Aν + (∂i c)F μi + cj μ .

(2.152)

When we take  as the usual time slice (t = const.) and substitute (2.133) into (2.149), the BRST charge is expressed as

38

2 Asymptotic Symmetry Asymptotic Symmetry

  3 k c(k){k μ a † (k) + e−iωt j˜0I (t, −k)} + (h.c.) , Q IB R ST (t) = − d μ

(2.153)

where j˜0I (t, k) is the Fourier transformation of j 0I (t, x) defined as j˜μI (t, k) =



d 3 x e−ik·x jμI (t, x).

(2.154)

In order to ensure that transition amplitudes are independent of the choice of gauge fixing, we need to impose the BRST condition [19] on any physical state|ψ as Q B R ST |ψ = 0 ,

(2.155)

where Q B R ST (= Q IB R ST (ts )) is the BRST charge in the Schrödinger picture. We will see in Sect. 5.5 that the BRST condition for the past asymptotic state |ψ 0 for |ψ 15 is given by lim Q IB R ST (t)|ψ 0 = 0 .

t→−∞

(2.156)

We usually assume that the ghost sector of physical Hilbert space is always the vacuum state that is annihilated by c(k), since the ghost fields are completely decoupled from the other fields in QED. Moreover, in the conventional scattering theory, it is usually assumed that the QED interaction can be ignored in the past and future infinities. If we adopt these assumptions, the BRST condition for the photon sector gives the Gupta-Bleuler condition [20, 21] for the free Maxwell theory: k μ aμ (k) |ψ = 0 ,

(2.157)

which ensures that the longitudinal modes of gauge fields do not contribute to the dynamics of QED. We will call this condition the free Gupta-Bleuler condition. We will see in the next section that the free Gupta-Bleuler condition is modified once the effect of QED interaction in the asymptotic regions is incorporated. The subspace annihilated by Q B R ST is called the BRST closed subspace and represented by Hclosed . Hclosed is not the same as the physical Hilbert space from the following reason. For any state |ξ ∈ HFock , Q B R ST |ξ is included in Hclosed because of Q 2B R ST = 0. The set of such states is called the BRST exact subspace, which is defined as Hexact ≡ {Q B R ST |ξ | |ξ ∈ HFock } .

(2.158)

All the states in Hexact are orthogonal to any state  in Hclosed because of the BRST condition (2.155). Therefore, two closed states, ψ  , |ψ , that differs by a BRST exact state as 15 The

definition of the asymptotic state is explained in Chap. 3.

2.9 Asymptotic Symmetry in BRST Formalism

  ψ = |ψ + iλQ B R ST |ξ ,

39

(2.159)

have the same inner products with all the physical states. It means that such two states are physically indistinguishable. The true physical space is obtained by identifying such equivalent states and thus given by the cohomology of Q B R ST : H phys ≡ Hclosed /Hexact .

(2.160)

Therefore, the BRST transformation,   |ψ → ψ  = eiλQ B RST |ψ = |ψ + iλQ B R ST |ψ ,

(2.161)

is a trivial transformation, since it just transforms a physical state into an identical physical state. In order for observable to be independent of the above BRST transformation, any physical operator O acting on H phys must satisfy the BRST invariant condition, i.e.,   δ B R ST O = iλQ B R ST , O = 0 .

(2.162)

We have observed that the BRST symmetry is not physical symmetry but trivial symmetry because the BRST charge acts trivially on all physical states and observables.

2.9.2 Large r Expansion of Fields in QED In this subsection, we consider the asymptotic behaviors of gauge fields to study the expression of the BRST charge and the charge associated with large gauge symmetry on the future and past infinities. To investigate the asymptotic behaviors near future null infinity I+ , we use the retarded coordinates (u, r,  A ). Near the asymptotic region I+ , the radiation fields Aμ (x) would be well approximated by the free field which has the free wave expansion (2.133). The free field can be expressed in (u, r, θ, ϕ) coordinates as 

 d3k ˆ ˆ + (h.c.) aμ (k)e−iωk u−iωk r (1−k·x) (2π )3 2ωk  ∞  π  2π  1 = dω ω dθ sin θ dϕ aμ (k)e−iωk u−iωk r (1−cos θ) + (h.c.) , k k 16π 3 0 0 0

Aμ (x) =

(2.163)

where we have used the polar coordinates (ωk , θ, ϕ) for k and performed the integration of ϕ in the second line. Here we use the saddle point approximation at large r with fixed u. The saddle point in θ space is given by 0=

∂ (ωk u − ωk r (1 − cos θ )) = −ωk r sin θ ⇒ θ = 0, π . ∂θ

(2.164)

40

2 Asymptotic Symmetry Asymptotic Symmetry

At the saddle point at θ = π , the phase factor in (2.163) becomes ωk u − 2ωk r . In this case, the phase factor diverges as r → ∞ and the integration over ωk is expected to be zero due to the Riemann-Lebesgue Lemma. The fact that the saddle point at large r is given at θ = 0 justifies the intuitive picture that every wave can be seen as coming from the center of space when we observe the waves at r = ∞. Then by expanding the integrand in (2.163) around θ = 0 and we find      −iω u π  1  ∞ (0) −iωk r 2 /2 + (h.c.) k a Aμ (x) = A¯ μ + dω ω x ˆ e dθθe ω μ k k k 8π 2 α=± 0 0   + O r −2  ∞   i (0) = A¯ μ − dω aμ (ω x)e ˆ −iωu − (h.c.) + O(r −2 ) , (2.165) 2 8π r 0

where A¯ (0) μ is the exact zero mode which is given by 1 ω→0 16π 3

A¯ (0) μ ≡ lim



  d 2 k γ (k ) ωk aμ (k) + (h.c.) .

(2.166)

We have separated the zero mode from other parts because the saddle point approximation at large r is not valid for ωk = 0 mode in (2.163).16 Accordingly, each component of the gauge field in the (u, r,  B ) coordinates is obtained as follows: ∂xμ A(1) u (u, ) Aμ (x) = A¯ (0) + O(r −2 log r ), + u ∂u r ∂xμ A(1) (u, ) Aμ (x) = A¯ r(0) + r + O(r −2 log r ), Ar (x) = ∂r r ∂xμ (0) −1 A B (x) = Aμ (x) = r A(−1) log r ) , B () + A B (u, ) + O(r ∂ B Au (x) =

(2.167) (2.168) (2.169)

with (−1)

AB

(0)

() = ∂ B xˆ i A¯ i

(0)

(0)

(0)

(0)

(0)

, Au = At , Ar = At + xˆ i Ai  ∞   dω au (ω x)e ˆ −iωu − (h.c.) , 0  ∞   dω ar (ω x)e ˆ −iωk u − (h.c.) , 0  ∞   dω a B (ω x)e ˆ −iωu − (h.c.) ,

i (1) Au (u, ) = − 2 8π i (1) Ar (u, ) = − 2 8π i (0) A B (u, ) = − 2 8π 0

,

(2.170)

where each annihilation operator is defined as (0) One may consider A¯ μ = 0, since it contains limω→0 ωk aμ (k). However, we should be careful because aμ (k) is not a c-number but an operator. In fact, aμ (k) gives a factor of order ω1k when it acts on out-states as we can see in the leading soft photon theorem (3.54). In this case, limω→0 ωk aμ (k) gives an order-one contribution in the S-matrix.

16

2.9 Asymptotic Symmetry in BRST Formalism

41

au (k) = at (k) , ar (k) = q μ aμ (k) , a B (k) = ∂ B xˆ i ai (k) , q μ ≡ (1, x) ˆ . (2.171) Note that the first term in (2.169) diverge as r → ∞. However, it gives the finite contribution when we compute the gauge (BRST) invariant operator like Fr B . We now see that the leading O(1) components A(0) B constitute the Cauchy data on the future null infinity. In other words, the two components correspond to the two physical degrees of freedom of photons. We now consider the fall-off condition for the ghost field. Since the ghost field satisfies the free equation of motion c(x) = 0,

(2.172)

the saddle point approximation at large r for (2.133) can be used in the same way for the gauge fields. We then obtain c(u, r, ) = c(0) +

c(1) (u, ) + O(r −2 log r ) , r

(2.173)

where 1 ω→0 16π 3

c(0) = lim

c(1) (u, ) = −



π 0

i 8π 2

d 2 k γ (k ) [ωk c(k) + (h.c.)] ,



∞

  dωk c(ωk x)e ˆ −iωk u − (h.c.) + O(r −2 log r ) .

(2.174) (2.175)

0

2.9.3 Asymptotic BRST Condition and Asymptotic Symmetry Now we study the BRST charge on the future null infinity. Writing the BRST charge (2.152) on the future null infinity, we obtain (+) (+) Q (+) B R ST = Q B R ST,H + Q B R ST,S .

(2.176)

Here, Q (+) B R ST,H is given by Q (+) B R ST,H =

 i+

 μ  di + μ (∂0 c)g μ0 ∇ν Aν + (∂i c)F μi + cjmat .

(2.177)

If we adopt the assumption that the QED interaction can be ignored in the past and future infinities, any electromagnetic field does not exist in i + . Then we can drop the first term and second terms in (2.177) and we obtain  τ Q (+) = di + cjmat . (2.178) B R ST,H i+

42

2 Asymptotic Symmetry Asymptotic Symmetry

Also, Q (+) B R ST,S in (2.176) is given by Q (+) B R ST,S

 = =

I

+

I+

 μ  dI+ μ (∂0 c)g μ0 ∇ν Aν + (∂i c)F μi + cjmat   dI+ ∂u c∇ν Aν + (∂i c)F ri ,

(2.179)

where we have dropped the term cj μ in the first line since the matter fields do no reach I+ and have also used dI+ μ = dI+ δrμ and the metric given in (2.32) in the second line. Here, ∇ν Aν and (∂i c)F ri can be written as 2 γ AB ∇ν Aν = −∂u Ar + ∂r (Ar − Au ) + (Ar − Au ) + 2 D A A B , r r AB γ (∂i c)F ri = g iμ (∂i c)F rμ = 2 ∂ A c (Fr B − Fu B ) . r

(2.180) (2.181)

Inserting above ones with the expansions (2.167)-(2.168) and (2.173) into (2.179), we have 

   Q (+) = dud 2  γ () ∂u c(1) ∂u Ar(1) + 2(Ar(0) − A(0) u ) B R ST,S 

  1 = ˆ r† (ω x) ˆ + c† (ω x)a ˆ r (ω x) ˆ dωd 2  γ ()ω2 c(ω x)a 2 64π 

  1 (0) (0) − (A − A ) dωd 2  γ () lim c(ω x) ˆ ˆ + c† (ω x) r u 2 ω→0 4π (2.182) The BRST condition (2.155) for the photon sector is then equivalent to   |ψ = 0 . ∂u Ar(+)(1) |ψ = 0 , Ar(0) − A(0) u

(2.183)

where Ar(+)(1) is the part of Ar(1) which includes the annihilation operators. The first condition is equivalent to ˆ |ψ = ωq μ aμ (ω x) ˆ |ψ , 0 = ar (ω x)

(2.184)

which is the free Gupta-Bleuler condition (2.157) with the identification of the photon ˆ momentum with q μ (= (1, x)). The BRST condition for the matter sector is given by 0 = Q (+) B R ST,H |ψ =

 i+

τ |ψ , di + cjmat

(2.185)

2.9 Asymptotic Symmetry in BRST Formalism

43

where Q (+) B R ST,H was given in (2.178). Now we focus on the falloff behaviors of c τ near i + . By (A.19), the ghost field c at i + is given by and jmat cH3 (ρ, ) ≡ lim c(τ, ρ, ) = c(0) . τ →∞

(2.186)

τ The fall-off behavior of jmat is given by τ jmat (τ, ρ, ) =

j τ (3) (ρ, ) + O(τ −3−δ ) , (δ > 0) . τ3

(2.187)

This fall-off behavior is required to have a finite amount of electric charge because the global U (1) charge can be written as  Q=

i+



τ di + jmat = lim

τ →∞ H 3

√ τ d 3 σ τ 3 h jmat .

(2.188)

The fall-off behavior (2.187) can be derived by the saddle point approximation and the j τ (3) can be determined (see Appendix E). Then plugging (2.186) and (2.187) into (2.185), we obtain the BRST condition for the matter sector as  √ τ (3) |ψ |ψ . = (2.189) 0 = Q (+) d 3 σ h c(0) jmat B R ST,H H3

τ (3) |ψ = 0. HowTo satisfy this condition, we have two choices: c(0) |ψ = 0 or jmat ever, the latter condition is too strong because it restricts all the physical states onto the sector with zero U (1) charge. We thus impose

c(0) |ψ = 0 ,

(2.190)

on the physical Hilbert space and the matter sector is not restricted by any nontrivial BRST condition. Under these conditions, we can see that the charge associated with the large gauge transformation acts on physical states in nontrivial ways as follows. First, the charge associated with the large gauge transformation was given by (2.146), which can be written as Q + [] = Q i + [] + Q + I+ []

(2.191)

with  Q [] = i+

d i+

i+

τ  jmat

 , Q [] = I+

I+

  dI+ ∂ r ∇ν Aν + (∂μ )F r μ . (2.192)

Let us rewrite the expansion of the gauge parameter (2.57) as

44

2 Asymptotic Symmetry Asymptotic Symmetry

1 (u, r, ) =  (0) () +  (1) + O(r −2 ) , r 

(2.193)



where  (1) takes the form of  (1) =  (1) (u, ) log r +  (1) (u, ). Plugging (2.180) and (2.181) into (2.192), with the expansions (2.167)-(2.169), (2.173), and (2.193), we obtain  √ τ (3) Q i + [] = d 3 σ h H3 jmat (2.194) H3 

    (0) Q I+ [] = + γ AB ∂ A  (Fr B − Fu B ) dud 2  γ () 2∂u  (1) A(0) u − Ar I+

(2.195) where H3 (σ ) is the gauge parameter on i + represented as (A.18). With the BRST condition (2.183), Q (+) [] acts on the physical Hilbert space as   lead,+ (0) (0) [ ] + Q [ ] |ψ Q (+) [] |ψ = Q lead,+ H S

(2.196)

where  √ τ (3) (0) + Q lead,+ [ ] = Q [] = d 3 σ h H3 jmat , i H H3 

(0) Q lead,+ [ ] = − dud 2  γ () γ AB ∂ A  (0) ∂u A(0) S B .

(2.197) (2.198)

I+

We thus have obtained the leading hard charge Q lead,+ [ (0) ] and the leading soft H lead,+ (0) charge Q S [ ], which were first introduced in [1, 2, 22]. We also have found in the BRST formalism that the charge associated with the large gauge symmetry acts on physical states nontrivially and the charge can be represented on the physical Hilbert space in the same form as the asymptotic charge in the classical case (2.58). Therefore, the asymptotic symmetry is also physical symmetry in the BRST formalism and gives the quantum version of the classical asymptotic charge conservation. Performing the integration over u in (2.198) gives the delta function of ω, then the soft charge can be expressed as17 (2.170) [ (0) ] = Q lead,+ S

1 lim 8π ω→0





ˆ + ωa †B (ω x) ˆ . d 2  γ () γ AB ∂ A  (0) ωa B (ω x) (2.199)

We will use this expression when we show the equivalence between the leading soft theorem and the asymptotic charge conservation in Sect. 4.1.

17 We

use

∞ 0

f (ω)δ(ω) =

1 2

limω→0 f (ω).

References

45

References 1. He T, Mitra P, Porfyriadis AP, Strominger A (2014) New symmetries of Massless QED. JHEP 10:112 arXiv:1407.3789 [hep-th] 2. Campiglia M, Laddha A (2015) Asymptotic symmetries of QED and Weinberg’s soft photon theorem. JHEP 07:115 arXiv:1505.05346 [hep-th] 3. Hirai H, Sugishita S (2018) Conservation laws from asymptotic symmetry and subleading charges in QED. JHEP 07:122 arXiv:1805.05651 [hep-th] 4. Strominger A (2014) On BMS invariance of gravitational scattering. JHEP 07:152 arXiv:1312.2229 [hep-th] 5. Strominger A, Lectures on the infrared structure of gravity and gauge theory. arXiv:1703.05448 [hep-th] 6. Hamada Y, Seo M-S, Shiu G (2018) Electromagnetic duality and the electric memory effect. JHEP 02:046 arXiv:1711.09968 [hep-th] 7. Bieri L, Chen P, Yau S-T (2012) The electromagnetic christodoulou memory effect and its application to neutron star binary mergers. Class Quant Grav 29. arXiv:1110.0410 [astroph.CO] 8. Bieri L, Garfinkle D (2013) An electromagnetic analogue of gravitational wave memory. Class Quant Grav 30. arXiv:1307.5098 [gr-qc] 9. Susskind L, Electromagnetic memory. arXiv:1507.02584 [hep-th] 10. Campiglia M, Laddha A (2016) Subleading soft photons and large gauge transformations. JHEP 11:012 arXiv:1605.09677 [hep-th] 11. Lysov V, Pasterski S, Strominger A (2014) Low’s subleading soft theorem as a symmetry of QED. Phys Rev Lett 113(11). arXiv:1407.3814 [hep-th] 12. Mao P, Ouyang H, Wu J-B, Wu X (2017) New electromagnetic memories and soft photon theorems. Phys Rev D 95(12). arXiv:1703.06588 [hep-th] 13. Hamada Y, Sugishita S (2018) Notes on the gravitational, electromagnetic and axion memory effects. JHEP 07:017 arXiv:1803.00738 [hep-th] 14. Conde E, Mao P (2017) Remarks on asymptotic symmetries and the subleading soft photon theorem. Phys Rev D 95(2). arXiv:1605.09731 [hep-th] 15. Hamada Y, Shiu G, An infinite set of soft theorems in gauge/gravity theories as Ward–Takahashi identities. arXiv:1801.05528 [hep-th] 16. Strocchi F (1974) Wightman AS, Proof of the charge superselection rule in local relativistic quantum field theory. J Math Phys 15:2198–2224. [Erratum: J Math Phys 17, 1930 (1976)] 17. Becchi C, Rouet A, Stora R (1976) Renormalization of Gauge theories. Annals Phys 98:287– 321 18. Tyutin IV, Gauge invariance in field theory and statistical physics in operator formalism. arXiv:0812.0580 [hep-th] 19. Kugo T, Ojima I (1978) Manifestly covariant canonical formulation of yang-mills field theories: physical state subsidiary conditions and physical S matrix unitarity. Phys Lett 73B:459–462 20. Gupta SN (1950) Theory of longitudinal photons in quantum electrodynamics. Proc Phys Soc A 63:681–691 21. Bleuler K (1950) A New method of treatment of the longitudinal and scalar photons. Helv Phys Acta 23:567–586 22. Kapec D, Pate M, Strominger A (2017) New symmetries of QED. Adv Theor Math Phys 21:1769–1785 arXiv:1506.02906 [hep-th]

Chapter 3

Infrared Divergences of S-Matrix

3.1 S-Matrix The S-matrix is the most fundamental quantity in scattering theories in quantum mechanics. In this section, we review how to construct the S-matrix in general. First of all, in order to construct physical quantities which can be compared with experimental outputs, we should ask what we are supposed to observe in the scattering experiment of elementary particles. In typical scattering experiments, the basic situation is the following. We first prepare some spatially separated incoming particles at a initial time t = ti . After that, the particles approach to each other and scatter to various directions via the interactions between them. Finally, the particle detectors located far enough away from where the scattering occurred, measure the momentum of each outgoing particle at final time t = t f . By repeating the same scattering event, we construct the probability distribution of momentum of each outgoing particle for a given initial state. In the real experiments, the momenta of particles are basically determined by observing the trajectory of each particle under an homogeneous magnetic field. The solution of the classical equation of motion for a point particle with the electric charge e gives p = e R B where R is the curvature radius of the trajectory and B is the strength of the magnetic field, therefore observing the trajectories of each particle leads to determining the momentum p of each particle. In these sequential procedures, there are important assumptions about the dynamics. A crucial assumption is the below one. Assumption I Asymptotically well localized wave; The wave functions of incoming and scattered particles, enough before or after the collision, are localized enough in space and far way from each other so that the interaction between them can be neglected effectively. When we consider how to measure the momentum of particles in the real scattering experiment, we would need a stronger assumption,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3_3

47

48

3 Infrared Divergences of S-Matrix

Assumption II Asymptotically classical point particle; The wave function of incoming and scattered particles, enough before or after the collision, are well localized in both spatial and momentum space (i.e., uncertainty between position and momentum is enough small when compared to the resolution scale of the detector used in the experiment), and its expectation values of position and momentum obey the classical equation of motions. The Assumption I is needed to observe the quantum wave effectively as a collection of individual particles in whicheach particle has the dispersion relation of a single particle in the form of E p = m 2 + p2 . Assumption II justifies the classical mechanism to determine the momentum of the particles which was already mentioned above. On the other hand, the dynamics of elementary particles is described by quantum field theory in which the fundamental dynamical degrees of freedoms are fields. The fields behave as waves and of course they are not identical to the particles that satisfy the above assumptions. Therefore, we should specify the states describing such particles. The physical system can be specified by all possible states and their dynamics, in other words, the Hilbert space and the Hamiltonian of the system, (H, H ). We then usually define particles as the eigenstates of the Hamiltonian H and spatial momentum operators P with eigenvalues p μ = (E α , pα ), H |α = E α |α , P|α = pα |α ,

(3.1)

where α stands for all the internal parameters characterizing the state: total momentum, relative momentum of different d.o.f, spins, charges, etc. However, the quantum states with definite momentum are not spatially localized waves, rather spread out all over the space due to the uncertainty principle. Therefore, they are not the particles observed in the scattering experiments because they do not satisfy the assumption I. Moreover, they do not change at all during the time evolution because the time evolution is generated by U (t) = e−i H t and it just gives the irrelevant over all phase ei Eα t to |α. Therefore, they are not the states describing the scattering processes that generally involve complicated time evolutions. To satisfy the assumption, the wave must be the wave packet that is the superposition of many waves with different momentum(i.e., wave length). We write the state of wave packets as  |f ≡

dα f (α)|α .

(3.2)

where f (α) is a smooth function of α. In Schrödinger picture, the time evolution of this state is given by  | f, t = U (t, ts )| f  =

dαe−i Eα (t−ts ) f (α)|α .

(3.3)

3.1 S-Matrix

49

where ts is an arbitrary finite time at which the Schrödinger operators are defined. Assumption I is equivalent to assuming the dynamics such that the effect of interaction in U (t) becomes weaker as the wave packets become well spatially separated from each other as t → ±∞. Let us express Uas (t) as a time evolution operator that generates the effective time evolution with the weaker interaction after a long time has passed since the collision and each wave packet become well separated. Let us define the asymptotic Hamiltonian Has (t) as the Hamiltonian which generates Uas (t) by i∂t Uas (t) = Has (t)Uas (t), and the asymptotic Hilbert space Has as a Hilbert space within which Uas (t) can act. Then Assumption I means the existence of the in-state | f in and the out-state | f out such that they approach the well-separated wave packets evolved by Uas (t, ts ) as t → ±∞ :  lim U (t, ts )| f in = lim (3.4) dα f (α)Uas (t, ts )|αas , t→−∞ t→−∞  lim U (t, ts )| f out = lim (3.5) dβ f (β)Uas (t, ts )|βas , t→+∞

t→+∞

where |αas ∈ Has . By (3.2) and (3.3), the in-state and out-state can be written in terms of the asymptotic states as 



dα f (α)|αin = lim U (t, ts )Uas (t, ts ) dα f (α)|αas , t→−∞  dβ f (β)|βout = lim U † (t, ts )Uas (t, ts ) dβ f (β)|βas †



t→∞

(3.6) (3.7)

For brevity, we formally represent (3.6) and (3.7) as |αin = lim (ti )|αas , |βout = lim (t f )|βas ti →−∞

t f →∞

(3.8)

where (t) is the Møller operator1 defined as (t) ≡ U † (t, ts )Uas (t, ts ) .

(3.9)

What we observe as incoming and outgoing particles in a scattering experiments are supposed to be the asymptotic wave packets in the form of  lim | f, ti in ≡ lim

t→−∞

t→−∞

lim |g, t f out ≡ lim

t→∞

t→∞



dα f (α)Uas (t, ts )|αas , dβg(β)Uas (t, ts )|βas .

(3.10) (3.11)

The transition amplitude for the process f → g is then given by 1 Usually the Møller operator is defined as lim

t→∞ (t). However, let us call (t) Møller operator.

50

3 Infrared Divergences of S-Matrix

 out g|

f in =

 dβ

dαg(β)∗ f (α)out β|αin .

(3.12)

Therefore, the S-matrix defined as Sβα ≡ out β| αin

(3.13)

is the fundamental constitute of the transition amplitudes of scattering particles. Defining the S-operator as S ≡ lim

lim † (t f )(ti ) = lim

t f →∞ ti →−∞

† lim Uas (t f , ts )U (t f , ts )Uas (ti , ts ) ,

t f →∞ ti →−∞

(3.14) we can write the S-matrix in terms of the asymptotic states as Sβα ≡ out β| αin =

as β|S|αas

.

(3.15)

3.2 Conventional S-Matrix In the conventional scattering theory for the QED and the low energy gravity in four dimensional Minkowski spacetime, we make an important assumption that the interaction between charged particles can be neglected in asymptotic regions. This means that the asymptotic dynamics is generated by the free Hamiltonian H0 , namely Uas (t, ts ) = U0 (t, ts ) = e−i H0 (t−ts ) ,

(3.16)

and the asymptotic Hilbert space is the Fock space H0 . The incoming particle and the outgoing particle in scattering experiment are assumed to be the wave packet evolved by the free dynamics generated by H0 . In this case, the in-state and out-state in (3.8) can be expressed as |αin = lim (ti )|α0 , |βout = lim (t f )|β0 , ti →−∞

t f →∞

(3.17)

where |α0 and |β0 are the eigenstates of H0 and the Møller operator is given by (t) = U † (t, ts )U0 (t, ts ) = ei H (t−ts ) e−i H0 (t−ts ) .

(3.18)

The S-matrix is defined as Sβα = out β| αin = 0 β|S|α0 , where S-operator is

(3.19)

3.2 Conventional S-Matrix

S = lim

51

lim † (t f )(ti ) = lim

t f →∞ ti →−∞

lim ei H0 (t f −ts ) e−i H (t f −ti ) e−i H0 (ti −ts ) .

t f →∞ ti →−∞

(3.20) If we define





S(t, t ) ≡ ei H0 (t−t ) e−i H (t−t ) e−i H0 (t−t ) ,

(3.21)

and divide the Hamiltonian into two parts as H = H0 + V , the time derivative of S(t, t ) is then given by i

∂ S(t, ti ) = V I (t)S(t, ti ) ∂t

(3.22)

where V I (t) is the interaction operator in the interaction picture, V I (t) ≡ U0 (t, ts )−1 V U0 (t, ts ).

(3.23)

The solution of (3.22) can be represented as the Dyson series [1]   t  S(t, t ) = T exp −i ds V I (s) .

(3.24)

t

where the symbol T represents the time-ordered product. Thus the S-operator is given by   S = T exp −i

+∞ −∞

dt V I (t )

 .

(3.25)

However, this S-matrix is not well defined due to the infrared(IR) divergences. We will see in Sect. 3.5 that when computing the S-matrix elements by the standard perturbation theory, we encounter the IR divergences. This problem is sometimes referred to as infrared problem. There are two possible prescriptions for this problem; the inclusive formalism and the dressed state formalism. The inclusive formalism has been the conventional prescription as seen in QFT textbooks such as [2, 3]. In this formalism, we give up the well-defined S-matrix and compute IR finite inclusive cross section. On the other hand, we try to construct the well-defined S-matrix in the dressed state formalism by seeking the proper asymptotic states which are not the free wave packets. The dressed state formalism is the main subject in Chap. 5.

52

3 Infrared Divergences of S-Matrix

3.3 Leading Soft Theorem In this subsection, we briefly explain the the leading soft photon theorem in QED, which captures a universal property of infrared dynamics of the theory. We first consider the amplitudes for the emission of a single soft photon. Let us define the scattering amplitude Mβα of the process α → β as Sβα = 0 β|S|α0

    ≡ δ(α − β) − 2iπ δ 3 pβ − pα δ E β − E α Mβα .

(3.26)

Since we are interested in the case of α = β, we can drop the first term. The amplitude Mβα can be computed by the usual Feynman diagram in the standard perturbation theory. Let us also write Mβα (k) as the amplitude with an additional external soft photon with the outgoing momentum k μ = (ω, k) and the polarization mode μλ (k), 0 β|aλ (k)S|α0

    ≡ −2iπ δ 3 pβ − pα δ E β − E α Mβα (k) ,

(3.27)

where aλ (k) is an annihilation operator of the soft photon. Since aλ (k) can be decomμ posed as aλ (k) = λ∗ μ (k)a (k), Mβα (k) can also be decomposed as μ

Mβα (k) = λ∗ μ (k)Mβα (k) .

(3.28)

   μ  (k)S|α0 ≡ −2iπ δ pβ − pα δ E β − E α Mβα (k) .

(3.29)

μ

where Mβα (k) is defined as 0 β|a

μ

μ

To compute Mβα (k), we just need to attach one external soft photon line to the Feynman diagram associated with Mβα . Consider attaching one external soft photon line to the external line of a charged particle with charge e, momentum p and spinor mode u( p, σ ), then it gives an additional internal line of the charged particle with momentum p + k together with a new vertex, as in Figure 3.1. Such a new contribution is given by u( ¯ p, σ )(−ieγ μ )

i( p/ + k) / +m i p/ + m = u( ¯ p, σ )(−ieγ μ ) + O(ω0 ) 2 2 ( p + k) + m − i 2 p · k − i (3.30) ep μ u( ¯ p, σ ) + O(ω0 ) , = (3.31) p · k − i

where we have used E p  ωk 2 and the on-shell conditions for p, k in the first equality, ¯ p, σ ) and u( ¯ p, σ )γ μ u( p, σ ) = 2δσ σ p μ in and also used i p/ + m = σ u( p, σ )u( would not need p μ  k μ for the following reason. When we calculate the S-matrix element, (3.30) is contracted with another spinor and it gives a scalar function, like

2 We

3.3 Leading Soft Theorem Fig. 3.1 This is the Feynman diagram for the scattering process in which the one of the external charged particle emits a soft photon with momentum k

53

β pn

k

pn + k

α the second equality. If we attach the soft photon line to an incoming charged particle of momentum p, the internal line of charged particle after emitting the soft photon has the momentum p − k, instead of p + k in the previous case, the new contribution is given by (−ieγ μ )

ep μ i( p/ − k) / +m u( p, σ ) = u( p, σ ) + O(ω0 ) . ( p − k)2 + m 2 − i − p · k − i

(3.32)

On the other hand, attaching the soft photon line to the internal line of a charged particle gives an additional term, (−ieγ μ )

i( p/ ± k) / +m i( p/ ± k) / +m = (−ieγ μ ) 2 , 2 2 2 ( p + k) + m − i p + m ± 2 p · k − i

(3.33)

where p is the off-shell momentum of the internal charged particle. This kind of terms does not give the additional contributions of order O(ω−1 ) to the amplitude beacuse the denominator in (3.33) does not have a pole at ω = 0, called a soft pole. Therefore, the amplitude for emitting a single soft photon with the momentum k and the polarization μ (k) in the process α → β is given by

i( p / +k/ )+m u( ¯ p, σ )(−ieγ μ ) ( p+k)2 +m 2 −i · · · u( p , σ ). In this scalar function, the contribution coming from k/ can be neglected if m  ωk . (More precise condition is E min  ωk where E min ≡ min{E p1 , · · · , E p N }).

54

3 Infrared Divergences of S-Matrix μ

Mβα (k) = Mβα



μ

n∈{α,β}

ηn en pn + O(ω0 ), pn · k − iηn

(3.34)

where en and pn are the charge and the four-momentum of n-th particle in the initial state α and the final state β, and ηn is a sign factor which takes +1 for particles in β and −1 for particles in α. The result (3.34) is called the leading soft photon theorem (for a single soft photon). The factor of proportionality, called (leading) soft factor, diverges as the energy of soft photon goes to zero because it is of order ω−1 . This fact reflects one of the important property of infrared dynamics; the slight acceleration of a charged particle results in the radiation of infinite number of low energy photons. Another important property is that the leading soft photon theorem is universal in the sense that the soft factor is independent of the details of scattering process and only depends on the information of the initial and final states. After some technical calculations, the leading soft theorem can be generalized to the case μ1 ···μ N (k1 , · · · , k N ) for emissions of multiple soft photons [4]. The amplitudes Mβα for emitting N number of soft photon with momentum k1 , · · · , k N and polarization indices μ1 , · · · , μ N in the process α → β is given by μ ···μ Mβα1 N (k1 , · · ·

, k N ) = Mβα

N r =1

⎛ ⎝

 n∈{α,β}

⎞ μ ηn en pn r ⎠ + O(ω0 ) . pn · kr − iηn

(3.35)

In the next section, we will study the subleading term.

3.4 Subleading Soft Photon Theorem In this section, we review the derivation of the subleading soft theorem based on [5]. The scattering amplitude of a single soft photon with momentum k and N charged scalar particles with the momentum pn and the charge en (n = 1, · · · N ) can be expanded in terms of soft photon energy as M μ (k; p1 , · · · , p N ) =

N  n=1

μ

ηn en pn M N ( p1 , · · · , pn + k, · · · , p N ) + B μ (k; p1 , · · · , p N ) . pn · k − iηn (3.36)

We have used the leading soft photon theorem (3.34) to the first term, and second terms is the subleading term which is order O(ωk0 ). We now use the Ward-Takahashi identity for U (1) gauge symmetry (for example see subsec. 10.5 in [3]): 0 = kμ M μ (k; p1 , · · · , p N ) .

(3.37)

3.4 Subleading Soft Photon Theorem

55

Plugging (3.36) into (3.37), we have 0=

N 

ηn en M N ( p1 , · · · , pn + k, · · · , p N ) + kμ B μ (k; p1 , · · · , p N ) .

(3.38)

n=1

If we set ωk = 0, the above reduces to 0=

 N 

 ηn en

M N ( p1 , · · · , p N ) .

(3.39)

n=1

N This ensures the conservation of the total electric charge, n=1 ηn en = 0. Under the assumption that M N ( p1 , · · · , pn + k, · · · , p N ) and B μ (k; p1 , · · · , p N ) are analytic around k μ = 0, we expand them around k μ = 0 in (3.38), and we obtain 0=

  ∂ ηn en M N ( p1 , · · · , pn , · · · , p N ) + kμ M N ( p1 , · · · , pn , · · · , p N ) ∂ pnμ n=1

N 

+ kμ B μ (k = 0; p1 , · · · , p N ) + O(ωk2 ) =

N  n=1

ηn en kμ

(3.40)

∂ M N ( p1 · · · , p N ) + kμ B μ (k = 0; p1 , · · · , p N ) + O(ωk2 ) , ∂ pnμ (3.41)

where we have used (3.39) in the second equality. Thus we have kμ B μ (0; p1 , · · · , p N ) = −

N  n=1

ηn en kμ

∂ M N ( p1 , · · · , p N ) . ∂ pnμ

(3.42)

By solving this constraint, B μ (0; p1 , · · · , p N ) can be written as B μ (0; p1 , · · · , p N ) = −

N  n=1

ηn en

∂ M N ( p1 , · · · , p N ) + C μ ( p1 , · · · , p N ) , ∂ pnμ (3.43)

where C μ ( p1 , · · · , p N ) is a function that satisfies kμ C μ ( p1 , · · · , p N ) = 0. However, such C μ ( p1 , · · · , p N ) actually does not exist because it is just a local (analytic) function of p1 , · · · , p N . Plugging this into (3.36) gives

56

3 Infrared Divergences of S-Matrix

M μ (k; p1 , · · · , p N ) =

N  n=1

−

μ

ηn en pn M N ( p1 , · · · , pn + k, · · · , p N ) pn · k − iηn

N 

ηn en

n=1

=

N  n=1

∂ M N ( p1 , · · · , p N ) + O(ωk ) ∂ pnμ μ

(3.44) νμ

 ηn en kν Jn ηn en pn M N ( p1 , · · · , p N ) − i M N ( p1 , · · · , p N ) + O(ωk ) pn · k − iηn pn · k − iηn N

n=1

(3.45) where Jnνμ

 ≡i

pnμ

∂ ∂ − pnν μ ν ∂ pn ∂ pn

 (3.46)

is the orbital angular momentum operator of the n-th charged particle. Multiplying both the sides of (3.44) by the polarization vector μλ (k) of the soft photon, we thus find the Low-Burnett-Kroll-Goldberger-Gell-Mann (or shortly Low’s) soft photon theorem [6–9]:   ωk β|aλ (k)S|α = Jλ(0) + Jλ(1) β|S|α + O( ) , m

(3.47)

where Jλ(0) =

N  n=1

μ

ηn en

μλ∗ pn

pn · k − iηn

, Jλ(1) = −i

N  n=1

νμ

en

μλ∗ kν Jn

pn · k − iηn

.

(3.48)

It is important that the subleading soft factor Jλ(1) is also universal in the sense that it only depends on the information of the external particles, or in other words, it is independent of the detail of the scattering processes. If we expand (3.38) up to the second order, we obtain ∂2 ∂ Bμ 1 ei kμ kν M N ( p1 , . . . , pn ) + k μ k ν (0; k1 , . . . , kn ) = 0 . (3.49) 2 i=1 ∂ piμ ∂ piν ∂kν N

This can be rewritten as  μ  ∂B ∂2 ∂ Bν ei kμ kν M N ( p1 , . . . , pn ) + k μ k ν + (0; k1 , . . . , kn ) = 0 . ∂ piμ ∂ piν ∂kν ∂kμ i=1

N 

(3.50)

3.4 Subleading Soft Photon Theorem

57 ν

Therefore, the only symmetric part of the ∂∂kBμ (0; k1 , . . . , kn ) can be fixed by the gauge invariance. It implies that the antisymmetric part depends on the detail of the scattering. In this sense, the universal sub-subleading soft photon theorem does not exist. Now let us rewrite the Low’s soft photon theorem (3.47) for later convenience. Applying limω→0 (1 + ω∂ω ) to the both sides of (3.47) and using the properties below, (1 + ω∂ω )ω−1 = 0 , (1 + ω∂ω )1 = 1 , lim (1 + ω∂ω )O(ω) = 0 , ω→0

(3.51)

we have ˆ = Jλ(1) β|S|α . lim β|(1 + ω∂ω )a B (ω x)S|α

ω→0

(3.52)

This is what we call the subleading soft photon theorem in scaler QED. As we have already seen in Sect. 2.9.2, in the mode expansion of the gauge fields Aμ (x), the mode with k μ = ωq μ (= ω(1, x)) ˆ only remains on the null infinity. In this case, the transverse modes of polarization vectors can be chosen as μ

ˆ = ∂B q μ . B (x)

(3.53)

With these bases, the leading soft photon theorem can be rewritten as 

 ek pk · ∂ B xˆ  ek pk · ∂ B xˆ − lim 0out| ωa B (ω x)S ˆ ω→0 pk · q pk · q k∈out k∈in

 0out| S,

(3.54)

where we omit the i factors for brevity. Then the subleading soft photon theorem can also be rewritten as lim 0out| (1 + ω∂ω )a B (ω x)S ˆ −i

ω→0

k  ek q μ JμB k∈in,out

pk · q

0out| S.

(3.55)

3.5 Infrared Divergences in S-Matrix The radiative loop corrections involving soft particles give universal contributions to scattering amplitudes. Here, we see that the loop diagrams involving virtual soft photons cause infrared divergences. We now calculate all the orders of loop-corrections involving virtual soft particles in the process α → β. Such loop diagrams can be given by attaching additional photon propagators to the Feynman diagram of α → β. If we attach at least one of the ends of a photon propagator to internal line of a charged particle, such diagrams do not have infrared divergences. This is because such attachments do not give the new

58

3 Infrared Divergences of S-Matrix

internal line with a soft pole, as already mentioned in the below of (3.33). (We will see it more concretely later.) Therefore, we focus on the diagrams with the loops of virtual soft photons exchanged only among the external charged particles. We evaluate the leading infrared contributions of such loops by using the soft photon theorem in (3.35). Since the theorem is valid only when every soft photon momentum k1 , · · · , k N is smaller than the minimum of the energy of the charged particles in the initial and final states, p1 , · · · , p N , we introduce an energy scale   min{E p1 , · · · , E p N }. The upper bound cut-off  is just a convenient dividing scale chosen low enough to justify the soft photon theorem and then a soft photon is defined as the one with energy smaller than . We also introduce the low bound energy scale λ of photon momentum to regulate the infrared divergent integrals. The dependence of λ must be removed at the end of calculations by taking λ → 0. We can evaluate the diagram with additional M photon loops connecting the external charged particles from a given amplitude Mβα as follows. In (3.35) with N = 2M, we first make M pairs among 2M soft factors and set the soft momenta in s-th pair to be the same momentum, say ks . Secondly, we multiply the amplitude with N photon propagators, N −iημs μ

s

s=1

ks2 − i

,

(3.56)

and contract photon polarizations indices and integrate over the soft photon fourmomenta. Then each pair gives the contribution,  en em ηn ηm Jnm ≡

λ≤|k|≤

d 4k −ien em ηn ηm pn · pm   , 4 2 (2π ) k − i [ pn · k − iηn ] [− pm · k − iηm ] (3.57)

where pn and pm are the momenta of the two charged particles exchanging the virtual soft photons in each pair. Summing over all the contributions from all the possible combinations of the pairs, the effect of adding N virtual soft photon loops to the diagram of Mβα ends up with multiplying Mβα by a factor ⎡ ⎤N 1 ⎣  en em ηn ηm Jnm ⎦ N !2 N n,m∈{α,β}

(3.58)

where N !2 N is the symmetric factor of the diagram. This symmetric factor comes from the fact that rearranging N virtual photon lines (N ! combinations) and exchanging the two ends of the each line (2 N combinations) are duplicated by a part of rearλ for a process ranging the vertices. Summing over N , we find that the amplitude Mβα including any number of photon loops with momenta |k| ≥ λ is given by

3.5 Infrared Divergences in S-Matrix

59

⎤  1  = Mβα exp ⎣ en em ηn ηm Jnm ⎦ 2 n,m∈{α,β} ⎡

λ Mβα

(3.59)

 is the amplitude including any number of virtual photons only with the where Mβα momenta greater than . Note that the contributions of virtual soft photons do not depend on the details of the scattering process because of the universality of the soft theorem. Dimensional analysis tells us that Jnm in (3.57) has a logarithmic divergence as λ → 0 3 . In fact, performing the integration (see Appendix F for the details) gives

Jnm = −

      λ iδηn ηm 1 1 + βnm λ ln + ln ln 8π 2 βnm 1 − βnm  4πβnm 

(3.60)

where βnm is the relativistic relative velocity of particles n and m:  βnm ≡

1−

m4 . ( p n · p m )2

(3.61)

Then the matrix element can be written as λ Mβα =



λ 

 Aβ,α /2

 ei Pβα Mβα

(3.62)

where   en em ηn ηm 1 + βnm , ln 8π 2 βnm 1 − βnm n,m∈{α,β}    en em ηn ηm λ . ≡ δη n η m ln 8πβ  nm n,m∈{α,β}

Aβ,α = − Pβα



(3.63) (3.64)

Because of the non-negativity of Aβ,α shown in [10] (see the proof in Appendix F), most matrix elements go to zero as λ → 0 except in the case where the initial state and the final state are identical [10]. This is what we call infrared suppression. This is the result of summing up all the orders of perturbative series. If we calculate the amplitude up to a finite order of the perturbation, the equation (3.62) means the divergence of the amplitude. On the other hand, Pβα gives an infrared-divergent phase factor for the matrix elements. We do not usually pay attention to the phase factor because it drops out when we take the absolute value of the matrix element with definite momenta so as to compute the transition rate.

3 Dimensional

analysis also tells us that attaching an end point of a virtual soft photon line to an internal line of a charged particle gives no infrared divergent contributions.

60

3 Infrared Divergences of S-Matrix

References 1. Dyson FJ (1949) The S matrix in quantum electrodynamics. Phys Rev 75:1736–1755. https:// doi.org/10.1103/PhysRev.75.1736 2. Peskin ME, Schroeder DV (1995) An Introduction to quantum field theory. Addison-Wesley, Reading, USA 3. Weinberg S (2005) The quantum theory of fields. vol. 1: foundations. Cambridge University Press, Cambridge 4. Weinberg S (1965) Infrared photons and gravitons. Phys Rev 140:B516–B524. https://doi.org/ 10.1103/PhysRev.140.B516 5. Bern Z, Davies S, Di Vecchia P, Nohle J (2014) Low-energy behavior of gluons and gravitons from gauge invariance. Phys Rev D90(8):084035. https://doi.org/10.1103/PhysRevD.90. 084035, arXiv:1406.6987 [hep-th] 6. Low FE (1954) Scattering of light of very low frequency by systems of spin 1/2. Phys Rev 96:1428–1432. https://doi.org/10.1103/PhysRev.96.1428 7. Gell-Mann M, Goldberger ML (1954) Scattering of low-energy photons by particles of spin 1/2. Phys Rev 96:1433–1438. https://doi.org/10.1103/PhysRev.96.1433 8. Low FE (1958) Bremsstrahlung of very low-energy quanta in elementary particle collisions. Phys Rev 110:974–977. https://doi.org/10.1103/PhysRev.110.974 9. Burnett TH, Kroll NM (1968) Extension of the low soft photon theorem. Phys Rev Lett 20:86. https://doi.org/10.1103/PhysRevLett.20.86 10. Carney D, Chaurette L, Neuenfeld D, Semenoff G (2018) On the need for soft dressing. JHEP 09:121. https://doi.org/10.1007/JHEP09(2018)121, arXiv:1803.02370 [hep-th]

Chapter 4

Equivalence Between Soft Theorem and Asymptotic Symmetry

As we have already explained in Chap. 1, the infrared triangle equivalence refers to the mathematical equivalence among the asymptotic symmetry, the soft theorem, and the memory effect. The equivalence of those was first discussed in the Yang-Mills theory [1], and extended to gravity [2–4] and also to QED [5–7]. The equivalence has been also extended to higher orders of the soft expansion, to higher spacetime dimensions, and also to other theories by many others. In particular for QED, it is known in [5, 6] that the Ward-Takahashi identities for the large transformations result in the leading soft photon theorem. The similar analysis for the subleading soft theorem was done in [8–10] for massless scalar QED. In [9, 10] it was found that the symmetries are nothing but the large gauge transformations.1 In [11], we extended the discussions to massive scalar QED, and obtained the expression of the charges associated with the subleading soft theorem. Since we have already seen the equivalence between the charge conservation associated with the asymptotic symmetry and the electromagnetic memory effect, we focus on the equivalence between the asymptotic symmetry and the soft theorem. First, we review the equivalence between the leading soft photon theorem and the asymptotic symmetry in Sect. 4.1 based on [11].2 We then proceed to the case of

1 The

large gauge transformations are slightly different in two papers [9, 10]. In [9], the gauge parameter is O(r ) at I + , and thus the generator is divergent but includes the subleading finite part, which is relevant to the subleading soft theorem. On the other hand, in [10], it is shown that the subleading part of O(1) gauge parameter is related to the subleading soft theorem. Our argument is similar to the latter, although the gauge fixing condition is different. 2 The equivalence between the asymptotic symmetry and the leading soft theorem was first derived by using the specific angular coordinates in [5, 6], and the rederivation without choosing specific angular coordinates can be seen in [11]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3_4

61

62

4 Equivalence Between Soft Theorem and Asymptotic Symmetry

subleading order in the next section. We show that the subleading soft photon theorem are also equivalent to the conservation of the charges related to the asymptotic symmetry in massive QED, mainly based on [8, 11].

4.1 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Leading Order In this section, we review the equivalence between the leading soft photon theorem and the asymptotic symmetry based on [11]. For simplicity, we concentrate on future infinities and omit the analysis for past infinities because it can be done just by repeating the similar analysis. The charge associated with the asymptotic symmetry defined on I + is given by (2.196); [ (0) ] Q (lead,+) [] |ψ = Q +H [ (0) ] + Q lead,+ S

(4.1)

where  √ τ (3) (0) + [] = Q lead,+ [ ] = Q d 3 σ h H3 jmat , i H H3   (0) Q lead,+ [ ] = − dud 2  γ () ∂ A  (0) ∂u A(0) S B .

(4.2) (4.3)

I+

Here, H3 (σ ) is a limit of large gauge parameter (x) on i + given by (A.18), which is defined as3   H3 (σ ) ≡ lim (τ, ρ, ) = d 2  γ ( ) G H3 (σ ;  ) (0) ( ) (4.4) τ →∞

with G H3 (σ ;  ) =

1   2 . 4π − 1 + ρ 2 + ρ q( ˆ  ) · x() ˆ

(4.5)

τ (3) Besides, jmat (σ ) in (4.2) is defined as τ (3) τ (σ ) ≡ lim τ 3 : jmat (τ, σ ) : , jmat τ →∞

3 In fact, in the coordinates (τ, ρ, σ ), Green’s function (A.6) does not depend on τ . Hence, 

(τ, σ ).

(4.6)

H3 (σ )

=

4.1 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Leading Order

63

which is the leading coefficient at large τ of the matter current with the normal ordering. It is given by τ (3) (σ ) = jmat

 em 2  † b (p)b(p) − d † (p)d(p) |p=mρ x() . ˆ 3 2(2π )

(4.7)

See Appendix E for our convention of the quantization of the scalar field and the derivation of the above result. From this expression, one can easily find that the hard charge operator Q lead,+ [ (0) ] acts on an asymptotic future state 0 out| containing H ˜ k ) and charge ek as4 charged particles with momenta pk = mρk yˆ ( 0 out|

Q lead,+ [ (0) ] = H



˜ k ) 0 out| . ek H3 (ρk , 

(4.8)

k∈out

Similarly, the past hard charge Q lead,− [ (0) ] acts on asymptotic past states as H Q lead,− [ (0) ] |in0 = H



˜ k ) |in0 . ek H3 (ρk , 

(4.9)

k∈in

As explained in Sect. 2.5, the leading charges associated with the large gauge transformations are “asymptotically conserved”. Therefore, we should have the following Ward-Takahashi identity for the physical S-matrix: 0 out|





 |in0 = 0. Q lead,+ S − S Q lead,− + Q lead,+ + Q lead,− S H S H

(4.10)

We now show that the conservation law (4.10) is equivalent to the leading soft photon theorem (3.54); 

 ek pk · ∂ B xˆ  ek pk · ∂ B xˆ − ˆ |in0 = lim 0 out| ωa B (ω x)S ω→0 pk · q pk · q k∈out k∈in

0 out| S

|in0 , (4.11)

ˆ where q μ = (1, x). Integrating the l.h.s. of the soft theorem (4.11) w.r.t. the direction x() ˆ of the momentum of the soft photon, we obtain a S-matrix element with the insertion of soft charge (2.199) as 

√ ˆ |in0 d 2  γ γ AB ∂ A  (0) 0 out| ωa B (ω x)S

 |in0 . = 0 out| Q lead,+ S − S Q lead,− S S

1 ω→0 4π lim

4 yˆ () ˜

(4.12)

˜ A , and ek is +e is a unit three-dimensional vector parametrized by spherical coordinates  for particles and −e for antiparticles.

64

4 Equivalence Between Soft Theorem and Asymptotic Symmetry

We can show Eq. (4.12) by using the property, ˆ |in0 = − lim 0 out| Sωa †B (ω x) ˆ |in0 lim 0 out| ωa B (ω x)S

ω→0

ω→0

and Eqs. (2.170), (4.2), and equates (4.12) with 1 4π



∞ 0

f (ω)δ(ω) =

1 2

(4.13)

limω→0 f (ω). The soft theorem (4.11)

 ηk ek pk · ∂ B xˆ √ d 2  γ γ AB ∂ A  (0) 0 out| S |in0 , pk · q k

(4.14)

where we have introduced the symbol ηk which is +1 (−1) for k ∈ out (k ∈ in). Performing a partial integration and using the formula   pk · ∂ B x() ˆ ˜ k ; ) − 1 with pk ≡ mρk yˆ ( ˜ k) , ∇ A γ AB = 4π G H3 (ρk ,  pk · q() (4.15) we then have   ηk ek pk · ∂ B xˆ 1 √ d 2  γ γ AB ∂ A  (0) 4π pk · q k     1 2 √ (0) ˜ ηk ek d  γ  () G H3 (ρk , k ; ) − =− 4π k    1  √ ˜ k) + =− ηk ek H3 (ρk ,  ηk ek d 2  γ  (0) . 4π k k Since

 k

(4.16)

ηk ek = 0 due to the total electric charge conservation, we finally obtain

(4.12) = −



 ˜ k ) 0 out| S |in0 = − 0 out| Q lead,+ S − S Q lead,− |in0 ηk ek H3 (ρk ,  H H

(4.17)

k

where we have used (4.8) and (4.9). Therefore, we have confirmed that we can obtain the Ward-Takahashi identity (4.10) from the soft theorem (4.11), and vice versa because (4.10) holds for any  (0) .

4.2 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Subleading Order Like the leading soft theorem (4.11), the subleading soft photon theorem gives the following relation between an amplitude containing a soft photon and an amplitude without the soft photon,

4.2 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Subleading Order

lim 0 out| (1 + ω∂ω )a B (ω x)S ˆ |in0 = S B(sub) 0 out| S |in0

ω→0

65

(4.18)

with S B(sub) ≡ −i

k  ek q μ JμB k

pk · q

,

(4.19)

where the sum in S B(sub) is taken for all of the incoming and outgoing charged particles k is the total angular momentum operator of k-th particle which are labeled by k, and Jμν (with momentum pk and charge ek ) defined as  k Jμν

= −i

∂ ∂ pkμ ν − pkν μ ∂ pk ∂ pk

 ,

(4.20)

and q μ = (1, x) ˆ represents the direction of four momentum of the soft photon (k μ = μ ωk q ). On the other hand, we have already seen that the asymptotic symmetry leads to the subleading asymptotic charge conservation (2.91): 0 out|



 sub,− sub,+

sub,−

|in0 = 0. Q sub,+ S − S Q + Q + Q S H S H

(4.21)

As noted below Eq. (2.89), the soft part is given by Q sub,+ S

1 =− 2



√ dud 2  γ  (0) u∂u S2 ∇ B A(0) B ,

(4.22)

I+

and by using Eq. (2.170), we can write as =− Q sub,+ S

i lim 16π ω→0



  √ ˆ − a †B (ω x)) ˆ . d 2  γ S2  (0) ∇ B (1 + ω∂ω )(a B (ω x) (4.23)

The soft part of the past charge, Q sub,− , also takes the same expression. Hence, S † contains (1 + ω∂ )[a (ω x) ˆ − a (ω x)] ˆ which corresponds to the subleading Q sub,± ω B S B soft photons. Using (4.23), we can rewrite the subleading soft theorem (4.18) as sub,+ S 0 out| (Q S

=−

1 8π



− S Q sub,− ) |in0 S   μ i  2 √ (0) B ek q ∂ B xˆ k J ∇ d  γ S 2  0 out| S |in0 . pk · q μi k

exist such that the r.h.s. of (4.24) is equal to If operators Q sub,± H

(4.24)

66

4 Equivalence Between Soft Theorem and Asymptotic Symmetry

− 0 out| (Q sub,+ S − S Q sub,− ) |in0 , H H

(4.25)

then we can establish the equivalence between the subleading soft theorem and the subleading charge conservation (4.21). were obtained [8–10], where the For massless QED, such hard operators Q sub,± H operators are defined on the future and past null infinities I ± . What we want to do is for massive charged particles. This is the subject to obtain the expression of Q sub,± H of the next subsection.

4.2.1 Hard Part of the Subleading Charges Unlike the case of the massless QED, Q sub,± is an operator acting on the asymptotic H states of massive particles living on on the timelike infinities i ± . Thus like the leading case (4.2), it should be expressed as an integral over three-dimensional hyperbolic space H3 with gauge parameter H3 (σ ). We now obtain such an expression for the . future part Q sub,+ H ˜ A ) as p μ = (E p , p yˆ ()) ˜ First, let usparametrize an on-shell momentum by ( p,  2 2 where E p = p + m and yˆ · yˆ = 1. With this parametrization, the angular momentum operators are expressed as   E p AB ˜ i  γ˜ (∂ A yˆ )∂ B , J0i = i yˆ i E p ∂ p + p   i ˜ j Ji j = −i yˆ (∂ A yˆ ) − yˆ j (∂˜ A yˆ i ) γ˜ AB ∂ B , where ∂˜ A ≡

(4.26) (4.27)

∂ ˜A ∂

is the derivative with respect to the direction of on-shell momentum of massive particle. Here, γ˜ AB is the inverse of the induced metric γ˜ AB ≡ (∂˜ A yˆ ) · i ± . Note that if we parametrize the on-shell momentum (∂˜ B yˆ ) on a unit sphere in  ˜ E p = m 1 + ρ 2 , the angular momentum operators can also be as p = mρy(), represented as  1 AB ˜ i ˜ yˆ ∂ρ + γ˜ (∂ A yˆ ) ∂ B , J0i = i 1 + ρ   Ji j = −i yˆ i (∂˜ A yˆ j ) − yˆ j (∂˜ A yˆ i ) γ˜ AB ∂˜ B . 



ρ2

i

(4.28) (4.29)

We then define the following operator Q sub B () with an angular index B as Q sub B () =

e 2



d3 p q μ ()∂ B xˆ i () p · q() (2π )3 2E p

× [(Jμi b† (p))b(p) − b† (p)(Jμi b(p)) − (Jμi d † (p))d(p) + d † (p)(Jμi d(p))].

(4.30)

4.2 Soft Photon Theorem ⇔ Asymptotic Charge Conservation, at Subleading Order

67

One can confirm, by performing some partial integrations,5 that the first term and the second term in (4.30) are the same when they act on the physical states. The third term and the forth term are also the same. Accordingly, one can find that Q sub B acts on the 1-particle state as q μ ()∂ B xˆ i () k Jμi | pk  , pk · q() q μ ()∂ B xˆ i () k pk | Q sub () = −e Jμi pk | . k B pk · q() Q sub B () | pk  = ek

(4.31) (4.32)

Therefore, if one defines the hard charge operator as Q sub,± =− H

1 8π



√ d 2  γ S2  (0) ∇ B Q sub B () ,

(4.33)

it satisfies the desired property sub,+ S 0 out| (Q H

− S Q sub,− ) |in0 H    μ i  1 2 √ (0) B ek q ∂ B xˆ k J = ∇ d  γ S2  0 out| S |in0 . 8π pk · q μi k

(4.34)

Next, we now express Q sub B in terms of the local matter current of charged particles in the asymptotic region i + . The matter current jμmat asymptotically decays as O(τ −3 ) with τ -dependent oscillations. Assuming that the charged scalar is free in the asymptotic region, one can extract τ -independent finite parts of jμmat (see Appendix E) as 

 1 2 ∂ + 1 τ 3 : jαmat (τ, σ˜ ) : (4.35) τ τ →∞ 4m 2   iem = ∂α b† (p)b(p) − b† (p)∂α b(p) − ∂α d † (p)d(p) + d † (p)∂α d(p) |p=mρ yˆ () ˜ , 4(2π )3

Iαmat (σ˜ ) ≡ lim

(4.36)

˜ A ) are the coordinates on H3 . In addition, using Eq. (4.36) and where σ˜ α = (ρ,  also Eqs. (4.28), (4.29), one can obtain the following equations: [(J0i b† (p))b(p) − b† (p)(J0i b(p)) − (J0i d † (p))d(p) + d † (p)(J0i d(p))]|p=mρ yˆ () ˜ 3 4(2π ) 1 ˜ + γ˜ AB ∂˜ A yˆ i I Bmat (ρ, )] ˜ , = 1 + ρ 2 [ yˆ i Iρmat (ρ, ) (4.37) em ρ [(Ji j b† (p))b(p) − b† (p)(Ji j b(p)) − (Ji j d † (p))d(p) + d † (p)(Ji j d(p))]|p=mρ yˆ () ˜

5 These

partial integrations involve not only the creation and annihilation operators but also soft factors and the integration measure.

68

4 Equivalence Between Soft Theorem and Asymptotic Symmetry

=−

4(2π )3 i ˜ j ˜ ( yˆ ∂ A yˆ − yˆ j ∂˜ A yˆ i )γ˜ AB I Bmat (ρ, ). em

(4.38)

From these equations, (4.30) can be rewritten as  Q sub B () = +

1 q ·Y



 d 3 σ˜ h˜

H3



˜ mat 1 + ρ 2 ∂ B x() ˆ · yˆ () ˜ Iρ (ρ, ) q ·Y

 1 + ρ2 ˜ ∂ B xˆ · ∂˜ A yˆ − (xˆ · yˆ )(∂ B xˆ · ∂˜ A yˆ ) + (xˆ · ∂˜ A yˆ )(∂ B xˆ · yˆ ) γ˜ AC ICmat (ρ, ) , ρ

(4.39)  2 ˜ √ρ where d 3 σ˜ h˜ = dρd 2 

1+ρ 2



 ˜ γ˜ and Y μ = ( 1 + ρ 2 , ρ yˆ ()).

Therefore, the hard charge Q sub,+ can be expressed in terms of the asymptotic H matter current Iαmat by inserting (4.39) into (4.33). However, the expression seems to is given by an integral over S 2 with parameter function be unnatural because Q sub,+ H sub,+ (0)  , not H3 . Since Q H should be related to the generator of subleading part of the large gauge transformation acting on massive particles, Q sub B () should be written as an integral over the surface at timelike infinity H3 with parameter function H3 on that surface, like the leading case (4.2). In fact, after some computations (see in such an integral as follows: Appendix G), one can express Q sub,+ H Q sub,+ = H

   √ 1 + ρ 2  2 αβ (h) (h) 1 (h) ρ h (∇α ∇ρ H3 )Iβmat + 2ρh αβ (∇α H3 )Iβmat , d3σ h 2 H3 ρ

(4.40)

where ∇α(h) denotes the covariant derivative compatible with the metric h αβ on H3 . The obtained charge is written as an integral over the three-dimensional hyperbolic space at timelike infinity H3 , and the integrand takes a local form and contains the components of matter current Iαmat which is defined in (4.35). This form of the hard charge is similar to the one in the massless case [8–10].

References 1. Strominger A (2014) Asymptotic symmetries of Yang-Mills theory. JHEP 07:151. arXiv:1308.0589 [hep-th] 2. Strominger A (2014) On BMS invariance of gravitational scattering. JHEP 07:152. arXiv:1312.2229 [hep-th] 3. He T, Lysov V, Mitra P, Strominger A (2015) BMS supertranslations and Weinberg’s soft graviton theorem. JHEP 05:151. arXiv:1401.7026 [hep-th] 4. Strominger A, Zhiboedov A (2016) Gravitational memory, BMS supertranslations and soft theorems. JHEP 01:086. arXiv:1411.5745 [hep-th] 5. He T, Mitra P, Porfyriadis AP, Strominger A (2014) New symmetries of massless QED. JHEP 10:112. arXiv:1407.3789 [hep-th] 6. Campiglia M, Laddha A (2015) Asymptotic symmetries of QED and Weinberg’s soft photon theorem. JHEP 07:115. arXiv:1505.05346 [hep-th]

References

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7. Kapec D, Pate M, Strominger A (2017) New symmetries of QED. Adv Theor Math Phys 21:1769–1785. arXiv:1506.02906 [hep-th] 8. Lysov V, Pasterski S, Strominger A (2014) Low’s subleading soft theorem as a symmetry of QED. Phys Rev Lett 113(11):111601. arXiv:1407.3814 [hep-th] 9. Campiglia M, Laddha A (2016) Subleading soft photons and large gauge transformations. JHEP 11:012. arXiv:1605.09677 [hep-th] 10. Conde E, Mao P (2017) Remarks on asymptotic symmetries and the subleading soft photon theorem. Phys Rev D95(2):021701. arXiv:1605.09731 [hep-th] 11. Hirai H, Sugishita S (2018) Conservation laws from asymptotic symmetry and subleading charges in QED. JHEP 07:122. arXiv:1805.05651 [hep-th]

Chapter 5

Towards the IR Finite S-Matrix

5.1 Motivations for Seeking IR Finite S-Matrix As we have already seen in Chap. 3, the conventional S-matrix is not well defined due to the infrared divergences, and this infrared problem can be avoided by computing only the inclusive transition probability. However, the fact that the inclusive amplitudes are IR finite does not mean that the infrared problem is perfectly resolved because the S-matrix is still not well-defined in the inclusive formalism. Moreover, there are several points which seem unnatural or even problematic in the inclusive formalism. The first point is about the assumption that the effect of the interaction becomes zero at the large time limit and the asymptotic states are just the Fock states with the free dynamics. This assumption may be too strong because the interaction mediated by massless particles create infinitely long-ranged force between charged particles. In fact, it is known in quantum mechanics that the asymptotic state with the Coulomb potential is not a free state. Based on the principle of quantum theories that all physically possible history are superposed at the level of wave function, it would be natural to expect that the asymptotic state is not a free state but a superposition of charged states surrounded by emitted soft particles in all possible ways. The dressed state formalism is the approach to the infrared problem based on the expectation mentioned above. This formalism tries to construct the IR finite S-matrix by using the asymptotic states surrounded by soft bosonic particles. The history of the dressed state formalism goes back to the 60’s. In 1965, Chung first proposed a asymptotic state of electrons surrounded by a cloud of coherent soft photon and showed that the infrared divergences are canceled out in all order perturbation theory at the level of S-matrix if we employ the state as the asymptotic state of the standard S-matrix [1]. We call the dressed state Chung’s dressed state. The scattering theory with the asymptotic state dressed by the coherent soft cloud was developed by others in [2–6]. In particular, Faddeev and Kulish [6] in 1970 proposed another version of dressed states, which is the same as Chung’s dressed states except for oscillating phase

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3_5

71

72

5 Towards the IR Finite S-Matrix

factors, by solving the infrared QED dynamics. The dressed states are nowadays called the Faddeev-Kulish (FK) dressed states. Although the dressed state formalism was proposed many years ago, it has recently been reconsidered in the connection with the asymptotic symmetry (see, e.g., [7– 13]). It was pointed out in [9] that the vanishing of the amplitudes due to the infrared suppression explained in Sect. 3.5 is related to the asymptotic symmetry of QED. More concretely, they argued that the initial and the final states used in the conventional S-matrix generally belong to different sectors with respect to the asymptotic symmetry. Therefore, the amplitude between them should vanish, otherwise it breaks the conservation law, and we need dressed states in order to obtain non-vanishing amplitudes. Moreover, it was argued in [12, 14, 15] that the inclusive and the dressed state formalism yields different answer to the cross section if the incoming state is a wave packet and the difference is related to the conservation of asymptotic charge. Motivated by these facts, we investigated the dressed state formalism in [16]. In particular, we revisited the gauge invariance of the dressed state. We argued that there is a problem on the gauge invariant condition in [6], and resolved the problem. In our method, the dressed states are obtained just from the appropriate gauge invariant condition. We will also discuss the i prescription for the dressed states. In addition, the relation between the dressed state formalism and the asymptotic symmetry is also developed. The outline of this chapter is as follows. We first briefly review Chung’s dressed states and the IR finiteness of S-matrix for the states in Sect. 5.2. We then review the derivation of the FK dressed state based on [6]. Section 5.4 explains the problem of the gauge invariance for the FK dressed state and we resolve the problem in Sect. 5.5 based on [16]. In Sect. 5.6, the physical interpretation of the FK dressed state is discussed. In Sect. 5.7, we reveal the relation between the FK dressed state and the asymptotic symmetry.

5.2 Chung’s Dressed State In [1], Chung introduced the following state, which we call Chung’s dressed state; | p1 , . . . , pn C         3 k S (l) (k)a (l)† (k) − S (l)∗ (k)a (l) (k) | p , . . . , p  d 3 p ρ( p) d = exp n 0 1 l=1,2

λ

(5.1) with S (l) (k) =

ep ·  (l) (k) , k·p

(5.2)

5.2 Chung’s Dressed State

73

+

+ e

e

e

e

e

e

+ e

e

Fig. 5.1 The tree level and its radiative corrections up to one-loop order under the usual Feynman rule. The soft photon contributions in the one-loop integrals create infrared divergences

where | p1 , . . . , pn 0 is an N -particle Fock state associated with the free Hamiltonian. Here, λ is the infrared cutoff on photon momentum space which we will take the limit λ → 0 in the end of the calculations, and  is an arbitrary constant that has the dimension of energy. Roughly speaking, these states describe the states of electrons and positrons with the superpositions of no soft photon state, one soft photon state, two soft photons state, . . .. In other words, Chung’s dressed states are the states of electrons and positrons surrounded by a cloud of indefinite number of soft photons. It has been known that the new “S-matrix” given by  C  p1 , . . . ,

pn |S| p1 , . . . , pm C

(5.3)

where S is the Dyson’s S-operator (3.25), has no infrared divergence in the all orders of perturbation series in QED. The structure for cancelling out the infrared divergences is as follows. For concreteness, let us consider a single electron scattering under a external potential. The tree level diagram and the radiative corrections by virtual photons to the diagram up to the second order (e2 ) in the usual Feynman rule are represented in Fig. 5.1. These one-loop virtual photon integrals give infrared divergences as we have seen in Sect. 3.5. However, in the new S-matrix in (5.3), there are creation and annihilation operators of soft photons in the cloud of soft photons in the asymptotic state (5.1). Therefore, the creation and annihilation operators of soft photons in the cloud can also be Wick-contracted with the creation operators in the interaction vertexes, and such Wick-contractions give new one-loop corrections represented in Sect. 5.2. These new diagrams describe the processes of the emissions or absorptions of real soft photons. In [1], Chung showed that the infrared divergences coming from the new diagrams cancel out the usual infrared divergences at any order of perturbation series for any QED processes. More precisely, the λ dependence coming from Chung’s dressed state (5.1) cancels out the λ dependence in the form A /2 of λ β,α in the usual S-matrix (3.62).1 The parameter  in (5.1) is not fixed in this discussion because it is not relevant for the IR finiteness. We will comment this arbitrariness in Sect. 6.1.

1 The

infrared divergent phase factor in in the usual S-matrix (3.62) is not eliminated in the new S-matrix (5.3).

74

5 Towards the IR Finite S-Matrix e

e

+

+ e

e e

e

+

e

+ e

e

e

Fig. 5.2 The new one-loop corrections coming due to exchange of soft photons between a vertex and a cloud or a cloud and another cloud

5.3 Derivation of the Faddeev-Kulish Dressed State In this section, we review the derivation of the FK dressed state based on [6]. The QED Hamiltonian is given by  H = H0 + V with V = −e

μ

d 3 x Aμ (x) jmat (t, x) .

(5.4)

Here, H0 is the free Hamiltonian given by (2.129) with the matter free Hamiltonian  Hmatter =



d 3 x ψ −iγ i ∂i + m ψ ,

(5.5)

where ψ(x) and ψ(x) are the standard Dirac and conjugate Dirac fields. These fields can be expanded as ψ(x) =



  d 3 p bs (p)u s (p)ei px + ds† (p)vs (p)e−i px ,

(5.6)

  d 3 p ds (p)v s (p)ei px + bs† (p)u s (p)e−i px ,

(5.7)

s=±

ψ(x) =

 s=±

5.3 Derivation of the Faddeev-Kulish Dressed State

75

where  d 3 p is the Lorentz invariant measure defined by  d 3 p ≡ (2π)d 3 p2E p . Here bs (p), ds (p) (, bs† (p), ds† (p)) are the annihilation (creation) operators of the electron and the positron with the commutation relations; 3

    bs (p), bs† (q) = (2π )3 2E p δ 3 (p − q) δss  , ds (p), ds† (q) = (2π )3 2E p δ 3 (p − q) δss  , others = 0 .

(5.8)

The spinors and barred spinors satisfy u¯ s  (p)γ μ u s (p) = 2 p μ δs  s , v¯s  (p)γ μ vs (p) = 2 p μ δs  s .

(5.9)

The mode expansion of the gauge fields is given in (2.133). The matter local current μ jmat (t, x) is given by μ

jmat (t, x) =: ψ(x)γ μ ψ(x) :

(5.10)

where : O : is the normal ordering of the operator O. Plugging the mode expansions (5.6) and (2.133) into the interaction operator in (5.4), we have V I (t) = −

   s=± s  =±

 d3x

3k d



 d3 p



  † (k)e−ikx  d 3 q aμ (k)eikx + aμ

  : ds (p)v s (p)ei px + bs† (p)u s (p)e−i px γ μ bs  (q)u s  (q)eiq x + ds† (q)vs  (q)e−iq x : 

(5.11) Here we focus on the phase factor of each term. The spatial integration produces the delta function in each term. For example, in the terms that involve bs† (p)bs  (q), the phase factor is given by



exp −i E p − E q ± ωk t δ 3 (p − q ± k)

    = exp −i p 2 − m 2 − (p ± k)2 − m 2 ± ωk t δ 3 (p − q ± k) .

(5.12) (5.13)

For large time t → ±∞, the interaction operator acts on the asymptotic state in the S-matrix and becomes c-number. The phase highly oscillates for large t and so one may expect that the integration over k in (5.11) gives zero as t → ±∞ if the integrand is a smooth function. However, when the soft photon creation or annihilation operator with bs† (p)bs (q) or ds† (p)ds (q) acts on the asymptotic state, it corresponds to the Feynman diagram in which a charged external line emits a photon and gives a singular contribution by the soft factor as we have already seen in the soft theorem (3.47). Thus we expect that for large t, the leading contribution in the QED interaction comes from the terms with bs† (p)bs (q) or ds† (p)ds (q) multiplied by the photon creation and the annihilation operators in (5.11). By extracting only such terms from the the QED interaction, we obtain

76

5 Towards the IR Finite S-Matrix

 VasI (t)

=

3k d



 d 3 p pμ  i Ep·kp t −i Ep·kp t † a ρ(p) , (k)e + a (k)e μ μ (2π )3 E p

(5.14)

where ρ(p) is the electric charge density given by   e † †  (q) − ds (p)d  (q) b (p)b . s s s (2π )3 2E p s=±

ρ(p) = −

(5.15)

To get the expression (5.14), we have used (5.9) and 

p 2 − m2 −



(p ± k)2 − m 2 ± ωk = ±

 2 ωk p·k . +O Ep |p|2

(5.16)

If we define μ (t, −k) jas

 ≡

d3 p p μ i p·k t ρ(p) e Ep , (2π )3 Ep

(5.17)

we can express (5.14) as 

 3 k a (k)e−iωk t + a † (−k)eiωk t j μ (t, −k) , d μ μ as

VasI (t) =

(5.18)

which we call the asymptotic interaction. The expression of (5.17) in the position space is   d 3 k ik·x μ d3 p d 3 k p μ ik·(x− Epp t) e j (t, k) = e ρ(p) as (2π )3 (2π )3 (2π )3 E p    d 3k p pμ 3 = x − ρ(p) δ t . (5.19) (2π )3 Ep Ep

μ jas (t, x) ≡



This expression is the same as the classical current for the point particle with the constant velocity v = Epp . Using this current, we can express (5.18) in the position space as  μ (t, x) , VasI (t) = − d 3 x Aμ (x) jas

(5.20)

We then expect that the time-evolution operator of the asymptotic state is given by i

∂ Uas (t, ts ) = HasI (t)Uas (t, ts ) , ∂t

(5.21)

5.3 Derivation of the Faddeev-Kulish Dressed State

77

where Has (t) = H0 + VasI (t). If we write Uas (t, ts ) as Uas (t, ts ) = e−i H0 (t−ts ) Z (t, ts ) ,

(5.22)

∂ Z as (t, ts ) = Vas(I ) (t)Z as (t, ts ) . ∂t

(5.23)

(5.21) is equivalent to i The solution is given by   t  Z as (t, ts ) ≡ T exp −i dt  VasI (t  ) .

(5.24)

ts

Then Uas (t, ts ) is given by   t  dt  VasI (t  ) . Uas (t, ts ) = e−i H0 (t−ts ) T exp i

(5.25)

ts

Furthermore, since the commutator [VasI (t1 ), VasI (t2 )] commutes with VasI (t) for any t, we obtain Uas (t, ts ) = U0 (t, ts ) e−i

  t1 t  I  1 t I I tsdt Vas (t ) − 2 tsdt1 ts dt2 [Vas (t1 ),Vas (t2 )]

e

,

(5.26)

and by performing the t-integral, we have 

t

−i ts

dt  VasI (t  ) = R(t) − R(ts )

(5.27)

with R(t) ≡

    pμ  i p·k t −i p·k t 3k aμ (k)e E p − aμ† (k)e E p . e  d 3 p ρ(p) d p·k

(5.28)

The exponent including the commutator [VasI (t1 ), VasI (t2 )] in (5.26) is a c-number function, and we represent it as i(t, ts ) where (t, ts ) ≡

i 2



t



dt1 ts

ts

t1

dt2 [VasI (t1 ), VasI (t2 )] .

(5.29)

Plugging (5.14) into (5.29) and performing the integration, we find (t, ts ) = −

e2 4π



p·q |t| 3 q ρ(p)ρ(q)   sgn(t) ln , d 3 p d ts ( p · q)2 − m 4

(5.30)

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5 Towards the IR Finite S-Matrix

In Appendix H, it is shown that the stationary point of this phase at the large time limit leads to the classical trajectories of point charged particles under the relativistic Coulomb forces. In [6], R(ts ) in (5.27) was deleted by a requirement for an initial condition. Under this assumption, The Møller operator defined in (3.9) is given by2 as (ti ) = U † (t, ts )Uas (t, ts ) = ei H (t−ts ) e−i H0 (t−ts ) e R(t) ei(t,ts ) .

(5.31)

The S-operator defined in (3.14) is then given by S= =

lim

as (t f )† as (ti )

lim

   tf  e−R(t f ) e−i(t f ,ts ) T exp −i dt  V I (t  ) e R(ti ) ei(ti ,ts ) .

t f →∞, ti →−∞

t f →∞, ti →−∞

ti

(5.32) We also assume that the asymptotic Hilbert space Has is the Fock space H Fock . As a result, this S-matrix differs from the usual Dyson’s one (3.25) only in the dressing factors e R and ei . Thus if we formally introduce a dressed Hilbert space H F K as H F K = lim e R(t) ei(t,ts ) H Fock , t→−∞

(5.33)

the S-matrix on H F K is given by the usual one (3.25).3 The FK dressing operator (5.28) is the same as the dressing operator in Chung’s ±i

p·k

t

dressed state (5.1) up to the phase factor e E p . If we focus on the k μ = 0 mode in the FK dressing operator, it is exactly the same as Chung’s one. Thus the FK dressed state is believed to give IR finite S-matrix based on the Chung’s proof. In Sect. 6.1, we will comment on a subtlety of the proof of IR finiteness.

5.4 Problem on the Gauge Invariance of Faddeev-Kulish Dressed State and Our Approach to this Problem We have to restrict H F K to the subspace by imposing a gauge invariant condition in order to make sure that the physical observables are independent of the gauge choice. However, the treatment for the gauge invariance in [6] seems inappropriate.

have renamed the Møller operator to as (t) to distinguish the conventional one (3.18). μ F K , the notion of particles for charged fields is still valid because jcl (t, x) in the dressing R(t) factor e is a diagonal operator on the Fock space. However, the standard interpretation of photons on the Fock space seems to be lost because the dressing factor excites an infinite number of photons. As we will see in Sect. 6.1, the energy of the excited photons by the dressing factor is soft in the limit t → ±∞. Hence, the particle notion for hard photons may be valid.

2 We

3 Even on H

5.4 Problem on the Gauge Invariance of Faddeev …

79

The free Gupta-Bleuler condition (2.157) was imposed on H F K as the physical state condition, i.e., any physical state, |ψ ∈ H F K , was required to satisfy k μ aμ (k) |ψ = 0 for any k.

(5.34)

However, we can easily see that the FK state does not satisfy the condition because k μ aμ (k) e R(t) ei(t,ts ) |α0 = 0 .

(5.35)

In [6], the dressing operator R in (5.28) was modified by introducing a null vector cμ (k) satisfying kμ cμ = 1 in order toto satisfy (5.34). More concretely, the dressing pμ pμ operator was altered by shifting the coefficient p·k in (5.28) to p·k − cμ . We will see that the artificial vector cμ is not needed if we impose an appropriate gauge invariant condition. Our claim is that the contributions of long-range interactions should also be incorporated into the gauge invariant condition, as the FK dressed states are obtained by taking account of such interactions. The free GuptaBleuler condition (5.34) is not adequate for the dressed states. In the next section, we will present the appropriate condition. Furthermore, we will show that the dressed Hilbert space can be obtained just by requiring the gauge invariant condition. In our approach, it turns out that we do not need to solve the dynamics of the asymptotic Hamiltonian Has as we reviewed in Sect. 5.3. In fact, although the Dyson’s S-matrix (3.25) is not a good operator on the usual Fock space H Fock , it may be well-defined on the dressed space H F K .4 The asymptotic Hamiltonian Has is just an approach to deriving the dressing factor e R(t) . We think that the using the gauge invariant condition incorporating the asymptotic interaction is a simpler approach to obtaining the factor, and the interpretation is clear. The condition essentially just says that if there is a charged particle, there should exist electromagnetic fields around it by Gauss’s law. The fields around the charge indeed make up the dress. In Sect. 5.6, as a support of this interpretation and also another justification that we do not have to introduce cμ , we will discuss the meaning of the original dressing operator R(t) in Eq. (5.28). As shown in [17], the dressing factor for a charged particle with momentum p corresponds to the Liénard-Wiechert potential for the uniformly moving charge with momentum p. We will reconfirm this fact especially taking care of the i prescription. Besides, our method allows a variety of dressing factors, and H F K given by (5.33) is just one of them. We will see that in our gauge invariant condition, the physical Hilbert space Has on which Dyson’s S-matrix (3.25) acts takes the form Has = e Ras H f r ee ,

4 In

(5.36)

this paper, we do not take care of problems at ultraviolet regions. We assume that they can be resolved by a standard renormalization procedure.

80

5 Towards the IR Finite S-Matrix

where e Ras is a dressing factor, and H f r ee is a subspace of the Fock space H Fock such that the free Gupta-Bleuler condition is satisfied (k μ aμ (k) |ψ = 0, |ψ ∈ H f r ee ). The operator Ras can be R(t) + i(t, ts ), but not necessarily. We will discuss the relation between the ambiguity of dressing and the asymptotic symmetry of QED in the Sect. 5.7.

5.5 Gauge Invariant Asymptotic States The BRST condition (2.155) is imposed on the physical Hilbert space H phys as Q sB R ST |αin = 0 , |αin ∈ H phys ,

(5.37)

where Q sB R ST and |αin are the BRST operator in (2.149) and the in-state in Schrödinger picture defined at ts . Using (3.17), we have the BRST condition for an asymptotic state |β0 as 0 = Q sB R ST |βin = lim Q sB R ST (ti ) |β0 , ti →−∞

(5.38)

where (ti ) is the Møller operator defined in (3.18). Since Q sB R ST commutes with the exact Hamiltonian H s , we have Q sB R ST (t) = U (ts , t)Q sB R ST U0 (t, ts ) = (t)Q IB R ST (t),

(5.39)

where Q IB R ST (t) is the BRST operator in the interaction picture (2.153): Q IB R ST (t) ≡ U0 (t, ts )−1 Q sB R ST U0 (t, ts )    3 k c(k){k μ a † (k) + e−iωt j˜0I (t, −k)} + (h.c.) . = − d μ

(5.40)

Therefore, |β0 satisfies lim Q IB R ST (ti ) |β0 = 0.

ti →−∞

(5.41)

By restricting the ghost-sector to the ghost-vacuum, this condition becomes  lim

ti →−∞

 k μ aμ (k) + eiωti j˜0I (ti , k) |β0 = 0.

(5.42)

This means that the states satisfying the free Gupta-Bleuler condition k μ aμ (k) |ψ = 0 are generally not the physical asymptotic states. Thus the charged 1-particle states in the standard Fock space, such as b† (p) |0, cannot be the asymptotic physical states if there is the interaction.

5.5 Gauge Invariant Asymptotic States

81

We will show below that the states satisfying the condition (5.42) are dressed ˜ such that states. In fact, if there is an anti-Hermitian operator R(t) ˜ ˜ = −eiωt j˜0I (t, k), [ j˜0I (t, k), R(t)] = 0, [k μ aμ (k), R(t)]

(5.43)

˜ then the states in e R(t) H f r ee are annihilated by k μ aμ (k) + eiωt j˜0I (t, k) where H f r ee is a subspace of H Fock satisfying the free Gupta-Bleuler condition. Thus the Hilbert space satisfying (5.42) is given by ˜

lim e R(ti ) H f r ee .

ti →−∞

(5.44)

There are various choices of the dressing operator satisfying (5.43). One example is ˜ = R(t)



3k d

 1  iωt ˜0I e j (t, k)k˜ μ aμ† (k) − e−iωt j˜0I (t, −k)k˜ μ aμ (k) , 2 2ω

(5.45)

where k˜ μ = (ω, −k). ˜ Although one may use such a dressing operator R(t), we can simplify it by recognizing that the current operator j 0I can be approximated in the asymptotic 0 given by (5.19). For the regions (t ∼ ±∞) by the classical current operator jas time component of the current, we can straightforwardly obtain (see Appendix E for details)5 0 lim j 0I (t, x) = lim jas (t, x).

τ →±∞

τ →±∞

(5.47)

Therefore, we can rewrite the condition (5.42) as lim

ti →−∞

  0 k μ aμ (k) + eiωti j˜as (ti , k) |β0 = 0.

(5.48)

ˆ k) as For later convenience, we represent the operator in (5.48) by G(t, 0 ˆ k) ≡ k μ aμ (k) + eiωt j˜as (t, k). G(t,

(5.49)

Noting that the momentum representation of the classical current operator is given −i

p·k

t

−i p·k t

by (5.17), and a trivial equation e E p = eiωt e E p , we can easily confirm that the Faddeev-Kulish dressing operator R(t) in (5.28) satisfies

5 Other

components of the current also satisfy similar equations: i lim j if Ir ee (t, x) = lim jas (t, x).

τ →±∞

τ →±∞

Here, the subscript f r ee means that the current is that of the free theory.

(5.46)

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5 Towards the IR Finite S-Matrix

ˆ k)e R(t) = e R(t) k μ aμ (k). G(t,

(5.50)

Thus an asymptotic physical Hilbert space satisfying (5.48) is given by lim e R(ti ) H f r ee .

ti →−∞

(5.51)

ˆ k) and R(t),  is not Since the phase operator  in (5.29) commutes with G(t, relevant for the gauge invariance (5.48). Therefore, the Faddeev-Kulish dressed space H F K in (5.33) is gauge invariant without introducing a vector cμ , if we restrict H Fock to the subspace H f r ee . Besides the phase operator, there are other choices of the dressing operator Ras (t) satisfying ˆ k)e Ras (t) = e Ras (t) k μ aμ (k). G(t,

(5.52)

One example of Ras (t) other than the Faddeev-Kulish dressing operator (5.28) is 0 in (5.45). Then we can define another asymptotic obtained by replacing j˜0I with j˜as physical Hilbert space: lim e Ras (ti ) H f r ee ,

ti →−∞

(5.53)

which is a solution of the gauge invariant condition (5.48). Although the question that what types of dressing operators cancel the IR divergences in the S-matrix is beyond the scope of this paper, we will discuss in Sect. 5.7 that the existence of many choices is natural from the viewpoint of asymptotic symmetry.

5.6 Interpretation of the Faddeev-Kulish Dresses It is shown in [17] that the Faddeev-Kulish dressing factor for a charged particle with momentum p μ corresponds to the classical Liénard-Wiechert potential around the particle. This fact supports our statement that Gauss’s law requires the dressing factor. In this section, we will reconfirm this fact with taking care of the i prescription, and see that we should use different prescriptions for initial and final states, which might be useful for the explicit computation of scattering amplitudes.

5.6 Interpretation of the Faddeev-Kulish Dresses

83

5.6.1 Coulomb Potential by Point Charges in the Asymptotic Region Here, we will recall the expression of the electromagnetic potential created by a charged point particle with momentum p μ . The classical equation of motion for the gauge field in the Lorenz gauge is given by  Aμ (x) = − jμ (x) , jμ (x) = e p

∞

dτ −∞

dyμ (τ ) 4 δ (x − y(τ )), dτ

(5.54)

p

where yμ (τ ) = mμ τ = Eμp t is the trajectory of the charged particle, which is supposed to pass through the origin at t = 0. The position at t = 0 is not relevant when we consider the asymptotic region.6 With the use of the retarded Green’s function for the Klein-Gordon equation,  G r et (x) = −

1 d 4k eik·x , (2π )4 (k 0 − ω + i)(k 0 + ω + i)

(5.55)

the general solutions of (5.54) are given by  Aμ (x) = Ain μ (x) +

d 4 x  G r et (x − x  ) jμ (x  )

   t 

p  pμ d3k  1 −iω(t−t  ) iω(t−t  ) − (t−t  ) ik· x− E p t e e dt − e e Ep (2π )3 −∞ 2ω



   pμ pμ ik· x− Epp t −ik· x− Epp t  3k . = Ain (x) − e d + (5.56) e e μ p · k + i p · k − i

= Ain μ (x) + ie

where Ain μ (x) is the incoming free wave, which is specified at t → −∞, and the second term is the Liénard-Wiechert potential created by the particle with momentum p μ and charge e. We represent this second term by Arμet (x; p) as  Arμet (x; p) ≡ −e

3k d



 x− Epp t

pμ ik· e p · k + i

+

pμ −ik· e p · k − i

x− Epp t



. (5.57)

5.6.2 Coulomb Potential from Dressed States with i Prescription Let’s consider a dressed state of a single incoming electron with momentum p μ defined by the position at t = 0 can contribute to subleading orders, and it was shown in [18] that the position is important for the subleading memory effect.

6 However,

84

5 Towards the IR Finite S-Matrix

|| p(t) ≡ e Rin (t) b† (p) |0 ,

(5.58)

where Rin (t) is an operator dressing the incoming single particle state. The gauge field in the interaction picture can be written as  AμI (x) =

3 k a (k)eik·x + a † (k)e−ik·x . d μ μ

(5.59)

Then we demand that its expectation value for the above dressed state7 match the classical gauge field (5.57) created by a charged point particle with momentum p μ as  p(t)||AμI (x)|| p(t) 





pμ pμ ik· x− Epp t −ik· x− Epp t 3k e e = −e d  p(t)|| p(t). + p · k + i p · k − i (5.60) We can easily check that the following dressing operator satisfies the above condition,  Rin (t) = e



 3 3k d p ρ(p) d

 pμ pμ i p·k t −i p·k t aμ (k)e E p − aμ† (k)e E p . p · k − i p · k + i (5.61)

This operator matches the dressing operator (5.28) up to the i insertion. How to insert i in the dressing operator is determined by how the initial condition of gauge fields is specified. Thus the dressed states stand for the states of (anti-)electrons surrounded by relativistic Coulomb fields created by themselves. We have considered a single charged particle state (5.58). The generalization to multi-particle states is trivial, and the expectation value of Aμ is given by the superposition of the Coulomb field created by each particle. In other words, in the dressed state, the charged particles are properly dressed by electromagnetic fields in the asymptotic region where the particles move at almost constant velocities. This result is natural because our dressed states are obtained by solving the BRST (gauge invariant) condition without ignoring the interaction in the asymptotic regions. We also would like to comment that this expectation value changes if we modify the dressing operator by introducing a vector cμ as in [6]. This is another reason for thinking that such a modification is unnatural. † Note also that Rin is anti-Hermitian (Rin = −Rin ). Thus the dressing factor e−Rin is 8 unitary.

7 More

precisely, we should use a wave-packet, since the state (5.58) is not normalized. we write e−Rin in the normal ordering, the normalization factor has an IR divergence if we set  = 0. Thus it is often said (see, e.g., [6]) that the dressing factor is not a unitary operator on the Fock space in a rigorous sense. However, it does not matter if we keep  nonzero. After computing IR finite physical quantities, we can take  to 0.

8 If

5.6 Interpretation of the Faddeev-Kulish Dresses

85

Similarly, we can fix the i prescription for the dressing operator Rout (t) for outgoing states. We consider a dressed outgoing state out  p(t)||

≡ 0| b(p)e−Rout (t) ,

(5.62)

and require that the expectation value of AμI (x) agree with the advanced potential for the point particle, which is given by  Aadv μ (x; p) = −e

3k d



 x− Epp t

pμ ik· e p · k − i

+

pμ −ik· e p · k + i

x− Epp t



. (5.63)

The requirement I out  p(t)||Aμ (x)|| p(t)out

= Aadv μ (x; p) out  p(t)|| p(t)out

(5.64)

can be satisfied by the following dressing operator  Rout (t) = e



 3 3k d p ρ(p) d

 pμ pμ i p·k t −i p·k t aμ (k)e E p − aμ† (k)e E p . p · k + i p · k − i (5.65)

Thus the sign of i terms is opposite to that in the initial dressing operator Rin given † = −Rout ), and the dressing factor by (5.61).9 This Rout is also anti-Hermitian (Rout −Rout is thus unitary. e The unitarity of the dressing factors, e Rin and e−Rout , guarantees that the asymptotic Hilbert space is positive definite. The asymptotic dressed states are given by acting on the states satisfying the free Gupta-Bleuler condition with the unitary dressing factors. The dressed states thus have a positive norm, because the states satisfying the free Gupta-Bleuler condition are positive definite and any unitary transformation preserves the positive definiteness. Here, we also give a formal proof of the unitarity of S-matrix. Including the dressing factors, the S-matrix acting on the Fock space takes the form (up to phase operators) S=

lim

t f →∞,ti →−∞

S(t f , ti ) with S(t f , ti ) = e−Rout (t f ) S0 (t f , ti )e Rin (ti ) ,

(5.66)

where S0 denotes the usual (finite time) S-matrix:   tf  dt  V I (t  ) = U0† (t f , ts )U (t f , ti )U0 (ti , ts ). S0 (t f , ti ) = T exp −i

(5.67)

ti

The unitarity of S0 (t f , ti ) simply follows from the expression of Eq. (5.67). Since Rin and Rout are anti-Hermitian, we can show the unitarity of S(t f , ti ) as 9 This

difference between the i prescriptions for initial states and for final states may be related to the prescription used to define in-out and in-in propagators in nonstationary spacetime [19].

86

5 Towards the IR Finite S-Matrix †

†

S † (t f , ti )S(t f , ti ) = e Rin (ti ) S0† (t f , ti )e−Rout (t f ) e−Rout (t f ) S0 (t f , ti )e Rin (ti ) = e−Rin (ti ) S0† (t f , ti )e Rout (t f ) e−Rout (t f ) S0 (t f , ti )e Rin (ti ) = e−Rin (ti ) S0† (t f , ti )S0 (t f , ti )e Rin (ti ) = 1.

(5.68)

Therefore, the S-matrix is unitary.

5.7 Asymptotic Symmetry in the Dressed State Formalism We now discuss the relation between the asymptotic symmetry and the dressed states in QED (see also [7–13] for related discussions). We will show that the FaddeevKulish dressed states carry the charges associated with the asymptotic symmetry, and investigate the conservation law of the asymptotic charges for the S-matrix in the dressed state formalism. The charge associated with the asymptotic symmetry was given by  s [] Q as

=

 d 3 x −0s ∂0  − is ∂i  + j 0s  ,

(5.69)

where the gauge parameter (x) satisfies  = 0 and the script s refers to the s Schrödinger picture. This charge Q as [] is BRST exact up to the boundary term:     s s Q as [] = − d 3 x ∂i (is ) + Q sB R ST , d 3 x(−c¯s ∂0  + iπ(c) ) .

(5.70)

Therefore, if the gauge parameter (x) is the large gauge parameter, the charge does not vanish in the asymptotic regions as we have already seen. All of the asymptotic charges commute with the BRST charge: s , Q sB R ST ] = 0 , [Q as

(5.71)

and they commute with the Hamiltonian H s up to the BRST exact term10 :

10 This

is the reason why we adopted the Hamiltonian (2.130). As mentioned in Footnote 14, s s = H s − d 3 x ∂ (s Ais + the canonical Hamiltonian Hcan has extra boundary terms: Hcan i 0  s , d 3 x ∂ (s Ais + is A0s )] = is A0s ). The boundary terms affect the commutator (5.72) as [Q as i 0    3 −i d x ∂i (0s ∂i  + is ∂0 ) = {Q sB RST , −i d 3 x ∂i (c¯s ∂i )} − i d 3 x ∂i (is ∂0 ). Since ∂0  = O(r −1 ) at r → ∞, we can neglect the effect of boundary terms if the radial component of the electric field operator, xˆ i i , decays as O(r −2 ). This condition is probably satisfied for physical scattering states in a reasonable setup.

5.7 Asymptotic Symmetry in the Dressed State Formalism

  s [Q as , H s ] = −i d 3 x ∂i  ∂i 0s + ∂0  (∂i is + j 0s )   

s 3 s s = Q B R ST , −i d x ∂i ∂i c¯ + i∂0  π(c) .

87

(5.72)

Therefore, the spectrum of the physical Hilbert space is infinitely degenerated. I in the interacThis fact naturally leads us to classify the asymptotic states by Q as I tion picture. We now see how Q as acts on the initial dressing operator Rin (t) given I in the interaction picture takes in Eq. (5.61). As in (5.70), the asymptotic charge Q as the following form up to the BRST exact part:   I [] = − d 3 x ∂i (i I ) = − d 3 x [iI ∂ i  + (∂i i I )]. Q as

(5.73)

The commutator of iI and Rin (t) is given by [iI (t, x), Rin (t)]    E p ki − ωpi −ik·(x− Epp t) ik·(x− Epp t) 3k e , −e = ie  d 3 p ρ(p) d p·k

(5.74)

where we have set  = 0 because the integrand is not singular at k = 0. On the other hand, the classical electric field for the classical configuration Arμet (x; p) given by (5.57) is computed as  ∂0 Ari et (x; p) − ∂i Ar0et (x; p) = ie

 E p ki − ωpi −ik·(x− Epp t) ik·(x− Epp t) 3k e , d −e p·k (5.75)

where we have also set  = 0. Hence, one can say that the commutator of i I and Rin (t) is given by the “classical operator” which represents the classical LiénardWiechert electric field as  I [i (t, x), Rin (t)] =  d 3 p ρ(p)[∂0 Ari et (x; p) − ∂i Ar0et (x; p)] ≡ F0icl (x). (5.76) Similarly, the commutator of ∂i i I and Rin (t) is given by the classical current as  0 d 3 p ρ(p) δ 3 (x − pt/E p ) = − jas (x). [∂i i I (t, x), Rin (t)] = −e 

(5.77)

I [] in (5.73) acts on e Rin as Therefore, the asymptotic charge Q as

 I [], e Rin ] = e Rin [Q as

0 ]. d 3 x [Fcl0i ∂i  + jas

(5.78)

88

5 Towards the IR Finite S-Matrix

The integral  cl [] ≡ Q as

0 d 3 x [Fcl0i ∂i  + jas ]

(5.79)

is in fact the asymptotic charge operator on the Fock space of charged particles. In the limit t → ±∞, the eigenvalues agree with the classical leading hard charges (2.41). The leading hard charges are the contributions to the asymptotic charges from uniformly moving charged particles and their Coulomb-like electric fields. Therefore, Eq. (5.78) represents that the charged Fock particles with the dressing operator (5.28) carry the asymptotic charges for the classical free charged particles with their Liénard-Wiechert electric fields. This result is natural because the dressing corresponds to creating the Liénard-Wiechert potential as we have seen in Sect. 5.6. The result also suggests that the antipodal matching condition (2.66) also holds at quantum level. At the classical level, the conservation of asymptotic charges lead to the electromagnetic memory effect. Let us see the implication at the quantum level. To make our discussion simple, we suppose that the radiation sector is given by eigenstates of asymptotic charges at t = ±∞; that is, we consider the asymptotic states |in  and out | such that they contain only transverse photons and satisfy I,− I,+ Q as [] |in  = in [ 0 ] |in  , out | Q as [] = out | out [ 0 ],

(5.80)

I,± I [] = limt→±∞ Q as [], and in , out are arbitrary (c-number) functionwhere Q as 0 als of  to which  asymptotically approaches. We then prepare the following dressed states by exciting charged particles on |in  and out | as † ˆ in ˆ out e−Rout (t=+∞) , |in = e Rin (t=−∞)  |in  , out| = out | 

(5.81)

† ˆ in where  is an arbitrary product of creation operators b† , d † of charged particles ˆ and out is any product of annihilation operators b, d. The asymptotic symmetry implies I,+ I,− out| (Q as S0 − S0 Q as ) |in = 0,

(5.82)

where S0 is given by (5.67) with limits t f → ∞, ti → −∞. From (5.78) and a similar computation for e−Rout , we have I,− I,+ |in = (Q −H + in ) |in , out| Q as = out| (Q +H + out ). Q as

(5.83)

† ˆ in |0 and Here, Q −H and Q +H represent the hard charge eigenvalues for the states  ˆ 0| out respectively as

5.7 Asymptotic Symmetry in the Dressed State Formalism



89



 † † cl cl ˆ in ˆ in ˆ out lim Q as ˆ out Q +H . (5.84) |0 = Q −H  |0 , 0|  lim Q as  = 0| 

t→−∞

t→∞

Thus (5.82) becomes (Q +H + out − Q −H − in ) out| S0 |in = 0.

(5.85)

This means that the S-matrix elements can take non-zero values only when the asymptotic charges are conserved, Q +H + out = Q −H + in ,

(5.86)

between the out states and the in states [9]. We would also be able to interpret it as the quantum analog of the classical memory effect. In a scattering event, if the hard charges are not conserved Q +H = Q −H , there should be a change in the radiation sector |in  → |out  so that (5.86) holds for any  0 . Conversely, a change in the radiation sector, out − in , is memorized in the change of the hard charges Q +H − Q −H . We here comment on the possibility of other dressing operators. The standard I 11 . Roughly speaking, the eigenstates in Fock vacuum is not the eigenstate of Q as (5.80) would consist of clouds of soft photons without charged particles. However, the Faddeev-Kulish dressing operator R(t) in (5.28) makes a photon cloud only when there are charged particles. Thus we need other dressing operators than FaddeevKulish’s in order to prepare eigenstates (5.80). As we have already argued in Sect. 5.5, the dressing operators are not uniquely fixed from the gauge invariant condition. We think that this variety is related to the asymptotic symmetry, and leave it for a future work to classify gauge invariant dressed states in terms of the asymptotic charges.

References 1. Chung V (1965) Infrared divergence in quantum electrodynamics. Phys Rev 140:B1110–B1122 2. Kibble TWB (1968) Coherent soft-photon states and infrared divergences. i. Classical currents. J Math Phys 9(2):315–324. https://doi.org/10.1063/1.1664582 3. Kibble TWB (1968) Coherent soft-photon states and infrared divergences. ii. Mass-shell singularities of green’s functions. Phys Rev 173:1527–1535 4. Kibble TWB (1968) Coherent soft-photon states and infrared divergences. iii. Asymptotic states and reduction formulas. Phys Rev 174:1882–1901 5. Kibble TWB (1968) Coherent soft-photon states and infrared divergences. iv. The scattering operator. Phys Rev 175:1624–1640 6. Kulish PP, Faddeev LD (1970) Asymptotic conditions and infrared divergences in quantum electrodynamics. Theor Math Phys 4:745 [Teor Mat Fiz 4, 153 (1970)] 7. Mirbabayi M, Porrati M (2016) Dressed hard states and black hole soft hair. Phys Rev Lett 117(21):211301. arXiv:1607.03120 [hep-th] asymptotic symmetries for general gauge parameters  are spontaneously broken in the standard Fock vacuum [20]. 11 The

90

5 Towards the IR Finite S-Matrix

8. Gabai B, Sever A (2016) Large gauge symmetries and asymptotic states in QED. JHEP 12:095. arXiv:1607.08599 [hep-th] 9. Kapec D, Perry M, Raclariu A-M, Strominger A (2017) Infrared divergences in QED, revisited. Phys Rev D 96(8):085002. arXiv:1705.04311 [hep-th] 10. Choi S, Kol U, Akhoury R (2018) Asymptotic dynamics in perturbative quantum gravity and BMS supertranslations. JHEP 01:142. arXiv:1708.05717 [hep-th] 11. Choi S, Akhoury R (2018) BMS supertranslation symmetry implies Faddeev-Kulish amplitudes. JHEP 02:171. arXiv:1712.04551 [hep-th] 12. Carney D, Chaurette L, Neuenfeld D, Semenoff G (2018) On the need for soft dressing. JHEP 09:121. arXiv:1803.02370 [hep-th] 13. Neuenfeld D, Infrared-safe scattering without photon vacuum transitions and time-dependent decoherence. arXiv:1810.11477 [hep-th] 14. Carney D, Chaurette L, Neuenfeld D, Semenoff GW (2017) Infrared quantum information. Phys Rev Lett 119(18):180502. arXiv:1706.03782 [hep-th] 15. Carney D, Chaurette L, Neuenfeld D, Semenoff GW (2018) Dressed infrared quantum information. Phys Rev D97(2):025007. arXiv:1710.02531 [hep-th] 16. Hirai H, Sugishita S (2019) Dressed states from gauge invariance. JHEP 06:023. arXiv:1901.09935 [hep-th] 17. Bagan E, Lavelle M, McMullan D (2000) Charges from dressed matter: construction. Ann Phys 282:471–502. arXiv:hep-ph/9909257 [hep-ph] 18. Hamada Y, Sugishita S (2018) Notes on the gravitational, electromagnetic and axion memory effects. JHEP 07:017. arXiv:1803.00738 [hep-th] 19. Fukuma M, Sugishita S, Sakatani Y (2013) Propagators in de Sitter space. Phys Rev D88(2):024041. arXiv:1301.7352 [hep-th] 20. He T, Mitra P, Porfyriadis AP, Strominger A (2014) New symmetries of massless QED. JHEP 10:112. arXiv:1407.3789 [hep-th]

Chapter 6

Conclusion and Further Discussion

In this thesis we have studied the universal infrared structures of scatterings physics towards the construction of the infrared finite S-matrix. Because asymptotic symmetries are the physical symmetries that constrains the S-matrix, we have studied the physical effects of the asymptotic symmetries in various ways. We have also investigated the constraints from gauge symmetry and asymptotic symmetry to possible dressed states for the infrared finite S-matrix. In the following, we summarize the results of each chapter and discuss possible future directions. In Chap. 2, we have derived the classical charges associated with the asymptotic symmetry up to the subleading order. We have also shown that the contribution from the charge in the spatial infinity to the conservation law vanishes and therefore the charges are conserved between the asymptotic future and past time slices. We have then derived the quantum leading charge associated with the asymptotic symmetry in the BRST formalism and shown that the charge has the same expression as the one of the leading classical charge if the BRST condition is imposed on the physical Hilbert space. In Chap. 3, we have reviewed the S-matrix theory, the soft photon theorems, and the infrared divergences of the conventional S-matrix. In Chap. 4, we have derived the subleading charge conservation law from the subleading photon theorem in massive scaler QED. In Chap. 5, we have shown that the Faddeev-Kulish dressed states can be obtained just from the gauge-invariant condition without solving the asymptotic dynamics. In addition, we have shown the possibility of other types of gauge-invariant dressed states. We have also shown that the Faddeev-Kulish dressing operator carry the charges associated with the asymptotic symmetry and the S-matrix obeys the conservation law of the charges. In the analysis in Chaps. 2 and 4, we have assumed that massive particles are the free particles in the asymptotic region. Therefore the contributions from electromagnetic potential created by the massive charged particles to the hard charges have been neglected in the analysis. However, the hard charges should contain such contributions because the effect of long-range interactions would not be ignored even in the asymptotic region. In fact, the asymptotic charge carried by the dressed charged particles (5.78) contains not only the contribution from the charge density of massive particles but also the contribution from the electromagnetic potential created by the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3_6

91

92

6 Conclusion and Further Discussion

charged particles. Once the asymptotic interactions in the asymptotic regions are taken into consideration, the subleading charge and also the subleading soft theorems would be modified. Therefore it would be interesting to study the subleading charge of asymptotic symmetry, the subleading soft theorems, and their relations in the dressed state formalism because they may give some important constraints to the IR finite S-matrix. We close this chapter with further discussion and comments on future directions.

6.1 Softness of Dresses and Infrared Scales In the derivation of the FK dressed state, there were several assumptions and subtle points. We assumed that the interaction in the dynamics of the asymptotic state is generated by the asymptotic interaction in (5.18).1 We also assumed that there exists a vacuum state and then the asymptotic Hilbert space can be constructed on the vacuum state by acting the creation operators. However the asymptotic interaction is time-dependent even in the Schrödinger picture. The time-dependence itself would be natural because the short range modes in the QED interaction among charged particles are decoupled as the distance of the wave packets of the charged particles increases in the time evolution. However, the notion of “particle” and vacuum states become subtle due to the time-dependence because stable states do not exist for the time-dependent Hamiltonian. The infrared finiteness of the FK dressed state is based on Chung’s analysis reviewed in Sect. 5.2. The argument is as follows. If we extract soft momentum region k ∼ 0 for the dressing operator (5.28) in the FK dressed state, the operator takes the form     pμ  d 3k aμ (k) − aμ† (k) , e  d 3 p ρ(p) (6.1) Rsoft ∼ 3 soft (2π ) (2ω) p · k i

p·k

p·k

t

t

because e E p ∼ 1 at k ∼ 0. Roughly, this is the dressing operator in Chung’s dressed state (5.1). Since ω > 0 region is not relevant to the proof of the IR finiteness, the behavior of the dressing operator was specified only at ω → 0 in [1] However, we should pay attention to the contribution of the nonzero soft momentum in (5.28) because it would affect the physical observables. We can make a rough argument that the contribution vanishes in the limit |t| → ∞ as follows. First, note ˆ can be zero only when ω = 0 because p μ is an onthat p · k = −ω(E p − p · k) shell momentum of a massive particle (E p > |p|). Then owing to the oscillating i

factor e E p , the contributions from nonzero momenta (ω > 0) can be ignored in the limit t → ±∞. This statement can be made more rigorous with the use of inserted dressing operator Eq. (5.61) or Eq. (5.65). We use the following identity as 1 The

time component of the asymptotic current (5.19) can be derived more precisely as in (E.10).

6.1 Softness of Dresses and Infrared Scales

93

a distribution: eiαt = ∓iπ δ(α). →0 t→±∞ α ± i lim lim

(6.2)

From this identity, we have  3   iπ   d k pμ  3 aμ (k) + aμ† (k) δ(ω), lim lim Rin (t) = − e d p ρ(p) →0 t→−∞ 2 (2π )3 p · k (6.3)   3 μ  iπ   d k p  lim lim Rout (t) = aμ (k) + aμ† (k) δ(ω). e d 3 p ρ(p) 3 t→∞ →0 2 (2π ) p · k (6.4) Therefore, we can say that the only soft photons constitute the dresses in the asymptotic limit t → ±∞. However, the above limit is dangerous or even nonsense in the following sense. First, the large time limit for the FK-type S-matrix is given by lim

t f →∞, ti →−∞

e

−Rout (t f )





T exp −i

tf







dt V (t ) e Rin (ti ) . I

(6.5)

ti

Thus we should first compute the finite time conventional S-matrix element and then take the limits t f → ∞ and ti → −∞ with the dressing factors. In addition, the phase operator such as (5.29) might be needed to make the S-matrix well-defined because the conventional S-matrix in (3.62) suffers from an infinitely oscillating phase factor. Secondly, t f and ti are finite in the real scattering experiments. It may be needed to introduce the infrared scale 1/t with t = ti , t f in the infrared finite S-matrix. The soft expansion used to derive the soft theorem (3.34) is valid for ω  m (ω: soft photon energy, m: electron mass.) In this sense, it is expected that the “soft region” in the dressing operator (5.28) is ω  m. Moreover, the energy range of the soft photons that effectively contribute in the dressing factor to be roughly restricted to ω  1/t 1 exp(i p·k t). For these infrared scales, we usually have due to the oscillating factor p·k Ep the hierarchy 1/t  m in the real experiment.2 In order to clean up the above issues and construct the proper IR finite S-matrix, we need to further study the physical Hilbert space of asymptotic state at the appropriate large time limit.

example, the time for moving 1 µm at the speed of light gives the infrared energy scale c/10−6 = 2 × 10−7 MeV , which is much smaller than the electron mass m e = 0.5 MeV. 2 For

94

6 Conclusion and Further Discussion

6.2 Extension to Other Theories It is important to extend our analysis to other theories. In the perturbative gravity, there are soft graviton theorems up to sub-subleading order [2–8]. Our analysis about the derivation of asymptotic charge conservation and its relations to soft theorems would be extended to gravity. It may be more interesting to work in the (dynamical) black hole backgrounds and study the implications to black hole information paradox[9, 10]. The dressed state formalism for the perturbative gravity was also developed in [11] (see also [12, 13]). However, a tensor cμν , which is an analog of a vector cμ in [14], was introduced by imposing a free “gauge invariant” condition which is a gravitational counterpart of the free Gupta-Bleuler condition (5.34). As in QED, we should impose an appropriate physical condition on the physical Hilbert space, and we expect that the tensor cμν is unnecessary. We also believe that the extension to perturbative QCD (pQCD) is an important direction because the theory has rich and complicated infrared structures. However the construction of infrared S-matrix in pQCD seems more difficult than QED and perturbative gravity in several senses. For example, extracting the physical degrees of freedom by solving the gauge invariant condition is difficult in QCD. Moreover, extracting the effect of IR divergences in S-matrix elements are also difficult in pQCD, whereas IR divergences can be factorized as an exponential of the IR divergent one-loop result in QED or perturbative gravity [2]. The above two difficulties essentially come from the noncommutativity of the gauge fields (color matrices).

6.3 Other Future Directions We would like to comment on other future directions. Mandelstam developed a manifestly gauge-independent formalism of gauge theories [15, 16]. In the formalism, the dynamical variables of QED are the field strength Fμν and path-dependent charged fields such as φ(x; ) ≡ e−ie

x

dξ μ Aμ (ξ )

φ(x).

(6.6)

Such fields attached with Wilson lines are also considered in the context of the bulk reconstruction in the AdS/CFT correspondence (see e.g. [17–19]). A similarity between Mandelstam’s formalism and the dressed state formalism was discussed in [20].3 However, the dressing operator constructed in [20] has the additional terms depending on the choice of the path . Thus the dressing operator seems not to be related directly to Faddeev-Kulish’s one (5.28). Furthermore, we should also investigate the relation to the asymptotic symmetry. As explained in [23] for gravitational 3 The relation between the gauge invariant charged field introduced in [21] and the FK dressed state

was recently revealed in [22].

6.3 Other Future Directions

95

theories in AdS, the operators like (6.6) are transformed under the asymptotic symmetry, and the behavior of the path near the asymptotic boundary is important in determining the transformation law of the symmetry. In [15, 16], the behavior of

near the asymptotic region was not specified. Thus it is interesting to understand more precisely the relations among Mandelstam’s formalism, the dressed state formalism and the asymptotic symmetry. We hope to come back to these issues in the non-asymptotic future.

References 1. Chung V (1965) Infrared divergence in quantum electrodynamics. Phys Rev 140:B1110–B1122 2. Weinberg S (1965) Infrared photons and gravitons. Phys Rev 140:B516–B524 3. Gross DJ, Jackiw R (1968) Low-energy theorem for graviton scattering. Phys Rev 166:1287– 1292 4. Jackiw R (1968) Low-energy theorems for massless Bosons: photons and gravitons. Phys Rev 168:1623–1633 5. White CD (2011) Factorization properties of soft graviton amplitudes. JHEP 05:060. arXiv:1103.2981 [hep-th] 6. Cachazo F, Strominger A, Evidence for a new soft graviton theorem. arXiv:1404.4091 [hep-th] 7. Zlotnikov M (2014) Sub-sub-leading soft-graviton theorem in arbitrary dimension. JHEP 10:148. arXiv:1407.5936 [hep-th] 8. Kalousios C, Rojas F (2015) Next to subleading soft-graviton theorem in arbitrary dimensions. JHEP 01:107. arXiv:1407.5982 [hep-th] 9. Hawking SW, Perry MJ, Strominger A (2016) Soft hair on black holes. Phys Rev Lett 116(23):231301. arXiv:1601.00921 [hep-th] 10. Chu C-S, Koyama Y (2018) Soft hair of dynamical black hole and hawking radiation. JHEP 04:056. arXiv:1801.03658 [hep-th] 11. Ware J, Saotome R, Akhoury R (2013) Construction of an asymptotic S matrix for perturbative quantum gravity. JHEP 10:159. arXiv:1308.6285 [hep-th] 12. Choi S, Kol U, Akhoury R (2018) Asymptotic dynamics in perturbative quantum gravity and BMS supertranslations. JHEP 01:142. arXiv:1708.05717 [hep-th] 13. Choi S, Akhoury R (2018) BMS supertranslation symmetry implies Faddeev-Kulish amplitudes. JHEP 02:171. arXiv:1712.04551 [hep-th] 14. Kulish PP, Faddeev LD (1970) Asymptotic conditions and infrared divergences in quantum electrodynamics. Theor Math Phys 4:745. [Teor Mat Fiz 4, 153 (1970)] 15. Mandelstam S (1962) Quantum electrodynamics without potentials. Ann Phys 19:1–24 16. Mandelstam S (1968) Feynman rules for electromagnetic and Yang-Mills fields from the gauge independent field theoretic formalism. Phys Rev 175:1580–1623. [327 (1968)] 17. Heemskerk I (2012) Construction of bulk fields with gauge redundancy. JHEP 09:106. arXiv:1201.3666 [hep-th] 18. Kabat D, Lifschytz G (2013) CFT representation of interacting bulk gauge fields in AdS. Phys Rev D87(8):086004. arXiv:1212.3788 [hep-th] 19. Harlow D (2016) Wormholes, emergent gauge fields, and the weak gravity conjecture. JHEP 01:122. arXiv:1510.07911 [hep-th] 20. Jakob R, Stefanis NG (1991) Path dependent phase factors and the infrared problem in QED. Ann Phys 210:112–136 21. Dirac PAM (1955) Gauge invariant formulation of quantum electrodynamics. Can J Phys 33:650

96

6 Conclusion and Further Discussion

22. Kapec D, Perry M, Raclariu A-M, Strominger A (2017) Infrared divergences in QED, revisited. Phys. Rev D96(8):085002. arXiv:1705.04311 [hep-th] 23. Harlow D, Ooguri H, Symmetries in quantum field theory and quantum gravity. arXiv:1810.05338 [hep-th]

Appendix A

Asymptotic Behavior of the Residual Gauge Parameter

The residual gauge transformations in Lorenz (Feynman) gauge are generated by the gauge parameter (x) satisfying ∇μ ∇ μ  = 0. In the retarded (u, r, ) coordinates, the condition can be written as   2 1 2 (A.1) 0 = ∂r − 2∂u ∂r + (−∂u + ∂r ) + 2  S 2  . r r We assume that the gauge parameter is O(1) at large r and we expand it as  = (0) (u, ) + O(r −1 ) .

(A.2)

Inserting this expansion to (A.1), we have ∂u  (0) = 0. This means that the leading boundary condition at r → ∞ is given by u-independent function, and the residual gauge parameter can expanded as  = (0) () + O(r −1 ) .

(A.3)

The solution can be expressed by the following integral form [1, 2]:  (x) =

 d 2  γ () G(x; ) (0) () ,

(A.4)

where G(x;  ) is the Green function which satisfies G(x;  ) = δ 2 (xˆ − q) ˆ . G(x;  ) = 0 , lim r →∞

(A.5)

u fixed

This Green function is given by

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

97

98

Appendix A: Asymptotic Behavior of the Residual Gauge Parameter

G(x;  ) = −

1 x μ xμ with q μ = (1, q( ˆ  )), 4π (q μ xμ )2

(A.6)

where q() ˆ is a three-dimensional unit vector parametrized by  A .1 The first property in (A.5) is easily checked as   2x ν 1 2x μ xμ ν ∂ν − q 4π (q · x)2 (q · x)3   8 1 4 4 6x μ xμ 2 = 0. =− − − + q 4π (q · x)2 (q · x)2 (q · x)2 (q · x)4

G(x;  ) = −

(A.7)

The second property is also shown as follows. In (u, r, ) coordinate, the Green function can be written as t2 − r2 1 x μ xμ 1 = −   4π (q μ xμ )2 4π t − r qˆ · xˆ 2 u(u + 2r ) 1 =−  2 . 4π u + r (1 − xˆ · q) ˆ

G(u, r, ;  ) = −

(A.8)

We then easily find G(x, q) ˆ = − lim lim r →∞ r →∞ u fixed

u fixed

2ur 1   = 0 for qˆ · xˆ = 1 , 4π r 2 1 − qˆ · xˆ 2

(A.9)

and 

   1 u(u + 2r ) d 2  γ ()G(x; ) = − d 2  γ ()  2 4π u + r (1 − qˆ · x) ˆ  u(u + 2r ) 1 1 =− d(cos θ ) 2 (u + r (1 − cos θ ))2 −1  1  1 u(u + 2r )  = 1. = (A.10) 2 r (u + r (1 − x))  −1

The Green function thus satisfies the second condition in (A.5). For the Green function (A.8) in the limit that r → ∞ with u = t − r fixed, we have

1 More

precisely, Green’s function is defined as

x μ xμ x μ xμ 1 ˜ G(x; ) = − 8π limε→0 (q μ x −iε) 2 + (q μ x +iε)2 . μ

μ

Appendix A: Asymptotic Behavior of the Residual Gauge Parameter



99

 d 2  γ ( )G(u, r, ;  )Y m ( )

= Y m () +

( + 1)u log |u| + s u 2r Y m () + O(r −1−ε ) , 2r

(A.11)

where Y m () are the spherical harmonics, and the coefficients s are2 ⎛  /2 n/2 n   1  (−1) j (2 − 2 j)! 1 s = c −2 j with cn = −1 + (−1)n + n ⎝4 −2 j!( − j)!( − 2 j)! k 2 j=0

k=1

k=1

⎞ 1⎠ . k

(A.12) As a result, the large gauge parameter (u, r, ) has the following large-r expansion, (u, r, ) =  (0) () +

2r u log |u|

2r

S2  (0) () + O(r −1 ).

(A.13)

Similarly, in the limit that r → ∞ with v = t + r fixed, it is expanded as ¯ − (v, r, ) =  (0) ()

2r v log |v|

2r

¯ + O(r −1 ), S2  (0) ()

(A.14)

¯ is the antipodal angle of . Therefore, if we define the coefficients of large-r where  expansion of (x) as log r + O(r −1 ), r log r ¯ +  (log,−) (v, ) + O(r −1 ), (x) =  (0) () lim r →∞,v:fixed r lim

r →∞,u:fixed

(x) =  (0) () +  (log,+) (u, )

(A.15) (A.16)

¯ holds. then  (log,+) (u = −2U, ) =  (log,−) (v = 2U, ) Now we study the behavior of the residual gauge parameter on timelike infinity i + . In (τ, ρ,  A ) coordinates, the Green function can be written as G(τ, ρ, ;  ) = −

1 1

2 . 4π  2 1 + ρ − ρ xˆ · qˆ

(A.17)

Note that G(τ, ρ, ;  ) have turned out to be independent of τ . It means that the residual gauge parameter (A.4) is independent of τ . Then the residual gauge parameter at i + is given by

2c

0

= 0, c1 = 2.

100

Appendix A: Asymptotic Behavior of the Residual Gauge Parameter

 H3 (σ ) ≡ lim (τ, ρ, ) = τ →∞

 d 2  γ ( ) G H3 (σ ;  ) (0) ( )

(A.18)

with G H3 (σ ;  ) ≡ G(τ, ρ, ;  ). If we take  (0) ( ) =  (0) (=constant), it reduces to H3 (σ ) =  (0) .

(A.19)

References 1. Campiglia M, Laddha A (2015) Asymptotic symmetries of QED and Weinberg’s soft photon theorem. JHEP 07:115. arXiv:1505.05346 [hep-th] 2. Strominger A, Lectures on the infrared structure of gravity and gauge theory. arXiv:1703.05448 [hep-th]

Appendix B

Electromagnetic Fields in Null Infinity

B.1 Electromagnetic Fields by Uniformly Moving Charges Consider a point particle with a charge e that moves at a constant velocity v. The trajectory of the particle is given by x = x0 + v(t − t0 ) ,

(B.1)

where x0 is a position of the particle at t = 0. The gauge potential produced by the charge in the Lorenz gauge is A0 (x) =

ev e , A(x) = , 4π (x) 4π (x)

(B.2)

where (x) =



(1 − |v|2 )(|x − x0 |2 − [vˆ · (x − x0 )]2 ) + (vˆ · (x − x0 ) − |v|(t − t0 ))2 (B.3)

with vˆ =

v . |v|

(B.4)

At the point t = T − U, r = T + U with large T , the electric flux Ftr is expanded as Ftr (t, r, )|t=T −U,r =T +U =−

e(1 − |v|2 ) e(1 − |v|2 ) f (U, ; v, t0 , x0 ) + + O(T −4 ) 2T 2 4T 3 4π(1 − v · x()) ˆ 4π(1 − v · x()) ˆ

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

(B.5)

101

102

Appendix B: Electromagnetic Fields in Null Infinity

with f (U, ; v, t0 , x0 ) ≡2U (1 − |v|2 − 2|v⊥ |2 ) + [1 − v · x() ˆ − 3(1 − |v|2 )]x0 · x() ˆ ˆ − |v⊥ |2 ]t0 , + 3[1 − v · x()]v ˆ · x0 + [2(1 − |v|2 ) − 2(1 − v · x())

(B.6) ˆ x(). ˆ At the point t = −T + U, r = T + U with large where v⊥ ≡ v − [v · x()] T , the electric flux Ftr is Ftr (t, r, )|t=−T +U,r =T +U =−

e(1 − |v|2 ) e(1 − |v|2 ) f (U, ; −v, −t0 , x0 ) + + O(T −4 ). (B.7) 2T 2 4T 3 4π(1 + v · x()) ˆ 4π(1 + v · x()) ˆ

By looking at the leading order terms of Eqs. (B.5) and (B.7), we thus find ¯ , Ftr+(2) () = Ftr−(2) ()

(B.8)

¯ denotes the antipodal point of . This is the antipodal matching condition where  in Eq. (2.66). Thus if we have initially charges en moving as (n) x(n) = x(n) 0 + vn (t − t0 ) ,

(B.9)

Q 0 given by Eq. (2.63) is computed as Q0 =

 en (1 − |vn |2 )  n



2π T

 d 2  γ ()

 (0) () 4 (1 − vn · x()) ˆ

(n) [1 − vn · x() ˆ − 3(1 − |vn |2 )]x(n) ˆ + 3[1 − vn · x()]v ˆ n · x0 0 · x()  +[2(1 − |vn |2 ) − 2(1 − vn · x()) ˆ − |vn⊥ |2 ]t0(n) + O(T −2 ) . (B.10)

Therefore, we have confirmed that the log T /T term doesn’t appear in Q 0 at least in this setup.

B.2 Computation of Memories Here, we check the memory effect formulae (2.74) and (2.88) for a concrete example. We consider the following trajectory of a charged particle with charge e such as it first rests at x0 and moves with a constant velocity v after a time t0 : x = x0 + (t − t0 )v(t − t0 ).

(B.11)

Appendix B: Electromagnetic Fields in Null Infinity

103 μ

We represent the matter current for this trajectory by jmat , which is the source in μ Maxwell’s equation ∂ν F νμ = − jmat . The retarded electromagnetic field created by this particle is written in the Lorenz gauge ∂μ Aμ = 0 as e e + (−|x − x0 | + t − t0 ) , 4π |x − x0 | 4π (x) (B.12) ev , (B.13) A(x) = (−|x − x0 | + t − t0 ) 4π (x) A0 (x) = (|x − x0 | − t + t0 )

where (x) is given by Eq. (B.3). We first consider the charge Q f . It is given by  Qf =

 √  d 2  γ r 2 F tr  |t=T +U,r =T −U .

(B.14)

At t = T + U, r = T − U with large T , electric field F tr is expanded as F tr |t=T +U,r =T −U =

e(1 − |v|2 ) + O(T −3 ) . 4π(1 − v · x) ˆ 2T 2

(B.15)

Since our gauge parameter has the expansion as Eq. (2.83), Q f is expanded as  e(1 − |v|2 )  (0) √ Qf = d 2 γ 4π (1 − v · x) ˆ 2 2  U log T e(1 − |v| ) S2  (0) √ + + O(T −1 ) . d 2 γ T 4π (1 − v · x) ˆ 2

(B.16)

Thus we have  e(1 − |v|2 )  (0) √ lim Q f [ ] = , (B.17) d 2 γ T →∞ 4π (1 − v · x) ˆ 2  U e(1 − |v|2 ) S2  (0) √ log = −U lim Q f [S2  (0) ] . Q f [ (0) ] = − d 2 γ T →∞ 4π (1 − v · x) ˆ 2 (B.18) (0)

log

log

log

Note that Q f diverges in the limit U → ∞ although Q f + Q + is finite as we will see later. Similarly, the charge Q i is computed as  e √ d 2  γ  (0) , T →∞ 4π  Ue √ log (0) Q i [ ] = − d 2  γ S2  (0) = 0 . 4π lim Q i [ (0) ] =

(B.19) (B.20)

104

Appendix B: Electromagnetic Fields in Null Infinity

Next, we compute the future null infinity charge lim T →∞ Q + given by (2.73). Since the angular components of the gauge field is expanded as A B (x) = (u + x0 · xˆ − t0 )

ev · ∂ B xˆ + O(T −1 ) , 4π(1 − v · x) ˆ

(B.21)

the charge is given by  lim Q + =

T →∞

√ d 2  γ  (0) γ AB ∇ A



 ev · ∂ B xˆ . 4π(1 − v · x) ˆ

(B.22)

With the use of the formula   v · ∂ B xˆ −2v · xˆ |v|2 − (v · x) ˆ 2 1 − |v|2 = + =1− , (B.23) γ AB ∇ A 2 1 − v · xˆ 1 − v · xˆ (1 − v · x) ˆ (1 − v · x) ˆ 2 the charge has the form lim Q + =

T →∞

e 4π



e(1 − |v|2 ) √ d 2  γ  (0) − 4π



√ d2 γ

 (0) = − lim (Q f − Q i ) . T →∞ (1 − v · x) ˆ 2

(B.24) This certainly agrees with the leading memory effect (2.74). log log Finally, we compute the subleading charges Q + and Q + . Since we now have ∂u A(0) B = δ(u + x0 · xˆ − t0 )

ev · ∂ B xˆ , 4π(1 − v · x) ˆ

(B.25)

log

the charge Q + given by Eq. (2.85) is computed as log

 1 ev · ∂ B xˆ √ d 2  γ γ AB (x0 · xˆ − t0 ) ∇ A S2  (0) 2 4π(1 − v · x) ˆ      x0 · v − (v · x)(x ˆ 0 · x) ˆ e 1 − |v|2 √ + d 2  γ (x0 · xˆ − t0 ) 1 − S2  (0) . = 8π (1 − v · x) ˆ 2 1 − v · xˆ

Q+ = −

(B.26) 5 Note that this does not depend on U . As shown in [1], this charge is related to the soft factor in the subleading soft photon theorem. The momentum of the  charged particle is initially p μ = m(1, 0) and finally p μ = ω(1, v) with ω = m/ 1 − |v|2 . The angular momentum is initially J μν = x0μ p ν − x0ν p μ and finally J μν = x0μ p ν − ˆ = −q · p  , p B = ωv · ∂ B x, x0ν p μ . They read p u = m = −q · p, p u = ω(1 − v · x) u u  u u ˆ J B = − p x0 · ∂ B x and J B = −(x0 · xˆ − t0 ) p B − p x0 · ∂ B x where q μ = (1, x). Using them, we have

Appendix B: Electromagnetic Fields in Null Infinity

(x0 · xˆ − t0 )

105

  J uB 1 J uB v · ∂ B xˆ = . − 1 − v · xˆ r q · p q·p

(B.27)

log

Thus Q + can also be written as log

Q+ = −

e 8π



√ d 2  γ lim

r →∞



r J u A r JuA − q · p q·p



∇ A S2  (0) .

(B.28)

log

The charge Q + given by (2.86) is computed as follows. The radial component Ar in (u, r, ) coordinates is Ar = −

e + O(r −2 ), 4πr

(B.29)

and we thus have Ar(1) = −e/(4π ), which does not contribute to the charge Q + because   e √ 2 √ (1) (0) (B.30) d  γ A r S 2  = − d 2  γ S2  (0) = 0 . 4π log

We also have Cu(1) = 0, C (1) A = 0 and   e 1 − |v|2 (u + x0 · xˆ − t0 ) 1 − 4π (1 − v · x) ˆ 2   e ˆ 0 · x) ˆ x0 · v − (v · x)(x δ(u + x0 · xˆ − t0 ) . + 4π 1 − v · xˆ

∇ B A(0) B =

(B.31)

Therefore, log

   e 1 − |v|2 √ S2  (0) d 2  γ (2U + x0 · xˆ − t0 ) 1 − 8π (1 − v · x) ˆ 2    ˆ 0 · x) ˆ e x0 · v − (v · x)(x 2 √ S2  (0) , − (B.32) d  γ 8π 1 − v · xˆ

Q+ = −

and for any  (0) (), we have log

log

   e 1 − |v|2 √ S2  (0) d 2  γ (x0 · xˆ − t0 ) 1 − 8π (1 − v · x) ˆ 2    ˆ 0 · x) ˆ e √ x0 · v − (v · x)(x S2  (0) − (B.33) d 2 γ 8π 1 − v · xˆ

Q+ + Q f = −

log

= −Q + . This is the subleading memory effect.

(B.34)

106

Appendix B: Electromagnetic Fields in Null Infinity

Reference 1. Hamada Y, Sugishita S (2018) Notes on the gravitational, electromagnetic and axion memory effects. JHEP 07:017. arXiv:1803.00738 [hep-th]

Appendix C

Asymptotic Expansion of Radiation Fields

In this appendix, we investigate the large-r expansion of radiation fields in the Lorenz gauge ∂μ Aμ = 0. We suppose that gauge fields are generally expanded as follows3 : log |u| log |u| 1 1 2r 2r Cu(1) (u, ) + A(1) Cu(2) (u, ) + 2 A(2) (u, ) + · · · , (C.1) u (u, ) + r r r2 r u |u| log 2r (2) 1 1 Ar = Ar(1) (u, ) + Cr (u, ) + 2 Ar(2) (u, ) + · · · , (C.2) r r2 r log |u| 1 2r C B(1) (u, ) + A(1) A B = A(0) (u, ) + · · · . (C.3) B (u, ) + r r B Au =

Inserting the above expansions into the Lorenz gauge condition 2 1 −∂u Ar + ∂r (−Au + Ar ) + (−Au + Ar ) + 2 ∇ B A B = 0 , r r

(C.4)

we find ∂u Ar(1) = 0 , Cu(1) + ∂u Cr(2) = 0 , 1 (1) B (0) − ∂u Ar(2) − Cr(2) + Cu(1) − A(1) u + Ar + ∇ A B = 0 , u

(C.5) (C.6)

where ∇ B is the covariant derivative associated with the two-sphere metric γ AB , and ∇ B = γ B A∇A. Equation (2.86) is obtained by expanding F r B and F r u in (2.72) with (C.1), (C.2) and (C.3). With the use of the above expansion, F r B and F r u are computed as 3 The

expansion is more general than that in [1], because we allow log r terms like the gauge parameter (x) [see (2.83)]. In particular, log r terms with coefficients Cu(1) , Cr(2) and C B(1) are included because these terms are generated by the large gauge transformation with the gauge parameter (2.83). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

107

108

Appendix C: Asymptotic Expansion of Radiation Fields

γ BC ∂C (Au − Ar ) r2 |u|  log 2r B A  1 (1) (1) = − 2 γ BC ∂u AC(0) − ∂ + O(r −3 ) , γ C − ∂ C u A u A r r3  1  (1) + O(r −2−ε ) , = ∂u Ar − ∂r Au = 2 Ar(1) + ∇ B A(0) B + 2C u r

F r B = −∂u A B + ∂r A B +

Fru

(C.7) (C.8)

and the above expansions lead to (2.86). The free equations of motion Aμ = 0 in the Lorenz gauge can be written in the retarded coordinates as   2 1 ∂r2 − 2∂u ∂r + (−∂u + ∂r ) + 2 S2 Au = 0 , (C.9) r r   2 1 2 2 ∂r2 − 2∂u ∂r + (−∂u + ∂r ) + 2 S2 Ar − 2 (−Au + Ar ) − 3 ∇ B A B = 0 , r r r r (C.10)  2  1 2 ∂r − 2∂u ∂r A B + 2 S2 A B + ∂ B (−Au + Ar ) = 0 . (C.11) r r Inserting the expansions (C.1), (C.2) and (C.3), we obtain   1 1 (2) 1 (1) 1 (1) (1) ∂u Cu(1) = 0, ∂u Cu(2) = − S2 Cu(1) , ∂u A(2) , u + C u = − C u + S2 C u − Au 2 u 2 2

(C.12)

∂u Cr(2) ∂u C B(1)

1 1 (1) (1) B (0) = −Cu(1) , ∂u Ar(2) + Cr(2) = Cu(1) − A(1) u + Ar − S2 Ar + ∇ A B , (C.13) u 2 1 (1) 1 (0) (1) (1) (1) = ∂ B Cu(1) , ∂u A(1) (C.14) B + C B = −∂ B (C u − Au + Ar ) − S2 A B . u 2

Solving the above equations with the condition (C.5) and (C.6), we find that Ar(1) is a constant.

Reference 1. Strominger A, Lectures on the infrared structure of gravity and gauge theory. arXiv:1703.05448 [hep-th]

Appendix D

Gauss Law Constraint in Canonical Quantization of QED

In this appendix, we briefly review how the Gauss law constraint appears in the context of the canonical quantization of QED. The following argument holds without specifying the concrete metric. We consider the following QED Lagrangian, 1 L = − Fμν F μν + Aμ j μ + Lmat , 4

(D.1)

Fμν = ∂μ Aν − ∂ν Aμ ,

(D.2)

where

and Lmat is the kinetic term of the matter sector and j μ is the current of global U (1) charge. Here we try to quantize the gauge fields in the covariant manner. The canonical conjugate fields are given by μ =

∂L = −F 0μ . ∂(∂0 Aμ )

(D.3)

Here we find a primary constraint, 0 = 0 .

(D.4)

The Poisson brackets are given by [Aμ (x), ν (y)] P = iδμν δ 3 (x − y) , μ

ν

[Aμ (x), Aν (y)] P = [ (x),  (y)] P = 0 ,

(D.5) (D.6)

where x 0 = y 0 . We define a Hamiltonian H on a time slice  as © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

109

110

Appendix D: Gauss Law Constraint in Canonical Quantization of QED



  d 3  k A˙ k − L    1 1 3 k 0k ij μ = d   (F0k + ∇k A0 ) + F0k F + Fi j F − Aμ j 2 4    1 1 3 k k ij μ = d  ,  k − ∇k  A0 + Fi j F − Aμ j 2 4

H=

(D.7)

where we have performed a partial integration and dropped out the boundary term  d Sk k A0 in the final line. To impose the primary constraint (D.4), we define a new  by adding a new degree of freedom λ(x) as a Lagrangian multiplier: Hamiltonian H ≡ H + H

 d 3  λ(x)0 (x) .

(D.8)

Because the constraint (D.4) must hold during the time evolution, we have a following secondary constraint as a consistency condition, ] P = ∇k k + j 0 . ˙ 0 = [0 , H 0 = i

(D.9)

Thus we find (the local) Gauss law: −∇k F 0k + j 0 = 0 ,

(D.10)

as a secondary constraint. The consistency condition for the Gauss law constraint is given by ] P = ∇i ∇k F ik , 0 = [−∇k F 0k + j 0 , H

(D.11)

which is trivially satisfied by the antisymmetry of the indices of F ik . Therefore, further constraints do not come up. We have the Poisson bracket relation for the constraints, [−∇k F 0k + j 0 , 0 ] P = 0 .

(D.12)

This means that the both constraints, (D.4) and (D.10), are the first class constraints, which generate the flows along the gauge orbits. When we quantize the theory, we need to impose the first class constraints on any physical state |ψ as 0 |ψ = 0 ,   −∇k F 0k + j 0 |ψ = 0 ,

(D.13) (D.14)

to make the physical states gauge invariant. The constraint (D.14) is called the Gauss law constraint.

Appendix E

Asymptotic Behaviors of the Massive Particles

In this appendix, we show the concrete expressions of the matter current of a massive scalar in the asymptotic regions. A free massive complex scalar φ(x) can be expressed as  φ(x) =

 d3 p  b(p)ei px + d † (p)e−i px , (2π )3 2E p

(E.1)

where b(p) and d(p) are the annihilation operators for particles and antiparticles, respectively. The nonzero commutation relations of the creation and annihilation operators are given by [b(p), b† (p  )] = [d(p), d † (p  )] = (2π )3 (2E p )δ (3) (p − p  ) .

(E.2)

All massive particles go to the future timelike infinity i + , not the null infinity in the asymptotic future time. When we work around the timelike infinity, it is convenient to use the coordinates (2.16): τ 2 = t2 − r2 , ρ = √

r

.

(E.3)

ds 2 = −dτ 2 + τ 2 h αβ dσ α dσ β ,

(E.4)

t2

− r2

The Minkowski line element then takes the form

where σ α = (ρ,  A ) are coordinates of the unit three-dimensional hyperbolic space H3 with the line element h αβ dσ α dσ β =

dρ 2 + ρ 2 γ AB d A d B . 1 + ρ2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

(E.5)

111

112

Appendix E: Asymptotic Behaviors of the Massive Particles

In the large τ limit (τ → +∞), using the saddle point approximation [1], we can obtain the asymptotic form of the scalar field as φ(τ, ρ, ) =

√  3 m  b(p)e−imτ −3πi/4 + d † (p)eimτ +3πi/4 |p=mρ x() + O(τ − 2 −ε ). ˆ 2(2πτ )3/2

(E.6) Therefore, in the asymptotic region, φ(τ, ρ, ) only creates (or annihilates) the (anti-)particle with localized momentum,  p = mρ x() ˆ , E p = m 1 + ρ2

(E.7)

at the leading order. Then if we ignore the interaction near the timelike infinity, the global U(1) current of the massive charged scalar with the normal ordering is given by   jμmat (τ, ρ, ) = ie : ∂μ φ † (x)φ(x) − φ † (x)∂μ φ(x) : =

jμ(3) (τ, ρ, ) τ3

+ O(τ −3−ε ),

(E.8) (E.9)

where  em 2  † b b − d †d , (E.10) 3 2(2π)     iem  † ∂ρ b b − b† ∂ρ b + i b† ∂ρ d † e2imτ − ∂ρ b de−2imτ − (b ↔ d) , jρ(3) (τ, ρ, ) = 4(2π)3 jτ(3) (τ, ρ, ) = jτ(3) (σ ) = −

(E.11)     iem  j A(3) (τ, ρ, ) = ∂ A b† b − b† ∂ A b + i b† ∂ A d † e2imτ − ∂ A b de−2imτ − (b ↔ d) . 4(2π)3

(E.12) Here, we have represented b = b(mρ x()), ˆ d = d(mρ x()) ˆ for brevity. Then one can extract the diagonal parts from the jρ(3) and j A(3) by multiplying the projection operator 4m1 2 (∂τ2 + 4m 2 ), −i(2π )3 2 (∂τ + 4m 2 ) jρ(3) , em 3 −i(2π )3 2 ∂ A b† b − b† ∂ A b − ∂ A d † d + d † ∂ A d = (∂τ + 4m 2 ) j A(3) . em 3

∂ρ b† b − b† ∂ρ b − ∂ρ d † d + d † ∂ρ d =

Since 4m1 2 (∂τ2 + 4m 2 ) jρ(3) and resent them by Iαmat (σ ) as

1 (∂ 2 4m 2 τ

(E.13) (E.14)

+ 4m 2 ) j A(3) are independent of τ , we can rep-

Appendix E: Asymptotic Behaviors of the Massive Particles

113



 1 2 ∂ + 1 τ 3 jαmat (τ, σ ) τ →∞ 4m 2 τ  iem  † ∂α b b − b† ∂α b − ∂α d † d + d † ∂α d . = 3 4(2π )

Iαmat (σ ) ≡ lim

(E.15) (E.16)

Reference 1. Campiglia M, Laddha A (2015) Asymptotic symmetries of QED and Weinberg’s soft photon theorem. JHEP 07:115. arXiv:1505.05346 [hep-th]

Appendix F

Integration of Jnm and Non-negativity of Aβα

In this appendix, we perform the integration of Jnm in (3.57) based on [1] and also show the positivity of Aβα in (3.63). The proof of the positivity of Aβα is based on [2]. Jnm is defined in (3.57) as  Jnm ≡

λ≤|k|≤

d 4k −i pn · pm   . (2π )4 k 2 − i ( pn · k − iηn ) (− pm · k − iηm )

(F.1)

We first perform the integration of k 0 in this integral. The integrant has poles at k 0 = |k| − i ,

k 0 = −|k| + i ,

k = vn · k − iηn  , k = vm · k + iηm  , 0

0

(F.2) (F.3)

where vn ≡ pn / pn0 . We can calculate the integral over k 0 by picking up the contributions from a subset of the above four poles as follows. In the case of ηn = −ηm = ±1, we can close the contour in either upper half or lower half k 0 -plane to avoid the contributions from poles at k 0 = vn · k − iηn  , vm · k + iηm  and then pick up a single residue at either k 0 = |k| − i or −|k| + i. In this case, we have   d 3k pn · pm  =  3 λ≤|k|≤ (2π ) (2|k| − i)( pn · k − iηn ) (− pm · k − iηm ) k 0 =|k|−i    λ d 3k pn · pm 1 anm . = ln = (F.4) ˆ pm · k) ˆ 2 λ≤|k|≤ (2π )3 |k|3 ( pn · k)(  

Jnm

In the final line, we have defined anm ≡

1 16π 3



d 2 kˆ

pn · pm ˆ pm · k) ˆ ( pn · k)(

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

(F.5)

115

116

Appendix F: Integration of Jnm and Non-negativity of Aβα

where kˆ μ ≡ k μ /|k| and d 2 kˆ is the integral measure on unit two-dimensional sphere. In the case of ηn = ηm = ±1, both the upper half and the lower half planes have two poles, so we close the contour in either the upper half or the lower half plane and pick up those two residues. In this case, we have  Jnm =

λ≤|k|≤

d 3k (2π )3



  pn · pm  (2|k| − i)( pn · k − iηn ) (− pm · k − iηm ) k 0 =|k|−i    pn · pm    +  2  k − i − pn0 (− pm · k − iηm )  0 k =vn ·k−iηn 

 λ ηn ηm en em (anm + ibnm ) = ln  

(F.6)

where we have defined bnm

  pn · pm   d k|k|  2  k − i pn0 (− pm · k − iηm )  0 k =vn ·k−iηn   i pn · pm 1  . = d 2 kˆ  16π 3 pn0 pm0 ˆ 2 vn · kˆ − vm · kˆ − iηm  1 − (vn · k) −i ≡ 16π 3



2ˆ

2

(F.7)

We have left the i in the last term to make the integral well-defined in the case of m = n. This integral gives a real-valued result because the complex conjugate of bnm with the change of the variable kˆ → −kˆ goes back to itself. Combining the results (F.4) and (F.6), the integration (F.1) is given by  Jnm = ln

 λ (anm + iδηn ηm bnm ) 

(F.8)

By performing the integration in (F.5) and (F.7),4 we obtain anm =

  1 1 + βnm i , bnm = ln , 8π 2 βnm 1 − βnm 4πβnm

(F.9)

where βnm is the relativistic relative velocity of particles n and m:  βnm ≡

1−

m4 . ( p n · p m )2

This result (F.9) leads to the expression in (3.60). Next we show the non-negativity of Aβα in (3.63). Aβα can be written as 4 We

can perform the integrals by using the Feynman parameter representations.

(F.10)

Appendix F: Integration of Jnm and Non-negativity of Aβα



Aβα =

ηn ηm en em anm =

n,m∈{α,β}

1 = 16π 3





1 16π 3

117



d 2 kˆ

n,m∈{α,β}

ˆ · t (k) ˆ , d 2 kˆ t (k)

ηn ηm en em pn · pm ˆ pm · k) ˆ ( pn · k)( (F.11)

where we have defined ˆ ≡ t μ (k)

 ηn en pnμ . pn · kˆ

(F.12)

n∈{α,β}

This vector is orthogonal to kˆ : ˆ · kˆ = t μ (k)



ηn en = 0

(F.13)

n∈{α,β}

where the final equality is ensured by the conservation of the U (1) global charge. ˆ can be expanded by the three orthogonal vectors perpendicular to the Then t μ (k) (1)μ (2)μ null vector kˆ μ . We can choose such three vectors as kˆ μ itself, and kˆ⊥ , kˆ⊥ that are normalized spacial vectors. Then we have ˆ = c0 (k) ˆ kˆ + t μ (k)



(i)μ ˆ kˆ⊥ ci (k)

(F.14)

i=1,2

ˆ c1 (k), ˆ c2 (k). ˆ Putting this decomposition (F.14) with some real coefficients c0 (k), back to (F.11), we find    1 2ˆ ˆ 2 + |c2 (k)| ˆ 2 ≥0. (F.15) Aβα = ( k)| d k |c 2 16π 3

References 1. Weinberg S (2005) The quantum theory of fields. Vol. 1: foundations. Cambridge University Press 2. Carney D, Chaurette L, Neuenfeld D, Semenoff GW (2017) Infrared quantum information. Phys Rev Lett 119(18):180502. arXiv:1706.03782 [hep-th]

Appendix G

Derivation of Eq. (4.40)

In this appendix, we explain some details of the computation to derive (4.40).5 Inserting (4.39) into (4.33), Q sub,+ is written as a sum of two parts Q sub,(ρ) and H H sub,(ϕ) QH as follows: sub,(ρ)

Q sub,+ = QH H sub,(ρ)

QH

sub,(ϕ)

QH

≡−



1 8π

sub,(ϕ)

+ QH

,

(G.1)

  ˜ 1 + ρ 2 ∂ B x() ˆ · yˆ () √ ˜ B d2 γ d 3 σ˜ h˜ S2  (0) Iρmat (ρ, )∇ , q ·Y H3











(G.2)

√ ˜ d2 γ d 3 σ˜ h˜ S2  (0) γ˜ C D I Dmat (ρ, ) H3    1 + ρ2 1 B ˜ ˜ ˜ ∂ B xˆ · ∂C yˆ − (xˆ · yˆ )(∂ B xˆ · ∂C yˆ ) + (xˆ · ∂C yˆ )(∂ B xˆ · yˆ ) , ×∇ q ·Y ρ

≡−

1 8π

(G.3)  2 ˜ √ρ where d 3 σ˜ h˜ = dρd 2 

1+ρ 2

sub,(ρ) QH

sub,(ϕ)

QH

1 = 2



γ˜ . We now show that



1 + ρ 2 2 ρρ (h) (h) d σ h ρ h (∇ρ ∇ρ H3 )Iρmat + 2ρh ρρ (∇ρ(h) H3 )Iρmat , ρ H3



3

√

(G.4)



1 + ρ 2 2 AB (h) (h) 1 (h) = d 3σ h ρ h (∇ A ∇ρ H3 )I Bmat + 2ρh AB (∇ A H3 )I Bmat , 2 H3 ρ 

√

(G.5)

where ∇α(h) denotes the covariant derivative compatible with the metric h αβ on H3 . If these (G.4) and (G.5) are obtained, Eq. (4.40) is obvious. In the following calculations, the formulae 5 This

appendix is based on Appendix D in [1]. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

119

120

Appendix G: Derivation of Eq. (4.40)

∂ A xˆ · ∂ B xˆ = γ AB , γ AB ∂ A xˆi ∂ B xˆ j = δi j − xˆi xˆ j , S2 xˆi = −2 xˆi

(G.6)

are useful. We first derive Eq. (G.4). The key equation is  ∇

B

˜ ∂ B x() ˆ · yˆ () ˜ q() · Y (ρ, )

 =

1 4π ˜ ) − , G H3 (ρ, ; ρ ρ

(G.7)

˜ ) was defined by Eq. (4.5). Furthermore, G H3 (ρ, ; ˜ ) satisfies where G H3 (ρ, ; the following property ˜ ) =  ˜ S 2 G H3 (ρ, ; ˜ ),  S 2 G H3 (ρ, ;

(G.8)

˜ ) depends on angle  A only through the inner product x() since G H3 (ρ, ; ˆ · ˜ We thus have yˆ (). 

√ ˜ ) d 2  γ [S2  (0) ()] G H3 (ρ, ;  ˜ S2 d 2 √γ  (0) () G H3 (ρ, ; ˜ ) =  ˜ S2 H3 (ρ, ), ˜ = sub,(ρ)

where H3 was defined by (4.4). By virtue of the above equations, Q H written as sub,(ρ)

QH

=−

1 2



(G.9) can be

    1 + ρ 2 1 √ ˜ ˜ ) − Iρmat (ρ, ) d 3 σ˜ h˜ d 2  γ S2  (0) G H3 (ρ, ; ρ 4π H3

 1 + ρ 2 1 ˜  ˜ S2 H3 (ρ, ) ˜ , Iρmat (ρ, ) =− d 3 σ˜ h˜ 2 H3 ρ 

(G.10) (G.11)

 √ where we have used (G.9) and d 2  γ S2  (0) = 0 in the second equality. In addition, since H3 (σ ) is a solution of the Laplace equation on H3 as H3 H3 (σ ) = 0, it satisfies S2 H3 = −(1 + ρ 2 )ρ 2 ∇ρ(h) ∇ρ(h) H3 − 2(1 + ρ 2 )ρ∇ρ(h) H3 .

(G.12)

Using this equation and noting that h ρρ = 1 + ρ 2 , we can obtain Eq. (G.4). We next consider Eq. (G.5). In (G.3), performing a partial integration, one encounters the following quantity:

Appendix G: Derivation of Eq. (4.40)

 S 2 ∇

A

1 q ·Y



121

 1 + ρ2 ∂ A xˆ · ∂˜C yˆ − (xˆ · yˆ )(∂ A xˆ · ∂˜C yˆ ) + (xˆ · ∂˜C yˆ )(∂ A xˆ · yˆ ) . ρ (G.13)

Performing the derivative, we have  (G.13) = S2

   xˆ · ∂˜C yˆ 2 + ρ − 1 + ρ 2 xˆ · yˆ . (q · Y )2 ρ

(G.14)

Performing the Laplacian, it further becomes    2 xˆ · ∂˜C yˆ 2 2 xˆ · y 1 + ρ ˆ − ρ + (q · Y )4 ρ    1 + ρ2 2 ˜ (h) (h) ˜ (h) ˜ ˜ ∇ρ ∇C G H3 (ρ, ; ) + ∇C G H3 (ρ, ; ) . = −4π ρ ρ (G.15)

(G.14) = −

Therefore, (G.3) can be written as sub,(ϕ)

QH

 1 + ρ 2 ˜ d 3 σ˜ h˜ γ˜ C D I Dmat (ρ, ) ρ H3    2 √ (h) ˜ ) + ∇˜ (h) G H3 (ρ, ; ˜ ) × d 2  γ  (0) ∇ρ(h) ∇˜ C G H3 (ρ, ; ρ C      1 + ρ 2 C D mat 2 1 ˜ ∇ρ(h) ∇˜ (h) H3 (ρ, ) ˜ + ∇˜ (h) H3 (ρ, ) ˜ d 3 σ˜ h˜ γ˜ I D (ρ, ) = C C 2 H3 ρ ρ  

2 √ 1 + ρ C D mat 1 (h) (h) d3σ h h I D (ρ, ) ρ 2 ∇ρ(h) ∇C H3 (ρ, ) + 2ρ∇C H3 (ρ, ) , = 2 H3 ρ =

1 2



(G.16)

where we have just renamed the integration variables and also used γ AB = ρ 2 h AB in the last line. Thus we have obtained Eq. (G.5).

Reference 1. Hirai H, Sugishita S (2018) Conservation laws from asymptotic symmetry and subleading charges in QED. JHEP 07:122. arXiv:1805.05651 [hep-th]

Appendix H

Particle Trajectory Under the Coulomb Force and Phase Factor in FK Dressed State

H.1 Particle Trajectory Under the Coulomb Force The relativistic equation of motion (e.o.m) for a point particle with its trajectory y μ (τ ) with the electric field Fνμ is m

dy ν d 2 yμ =0. − eFνμ 2 dτ dτ

(H.1)

We consider the case where the particle is moving in the static Coulomb potential, Aμ (x) = −

Q 1 δμ0 . 4π r

(H.2)

Then we drive the asymptotic trajectory at t → ±∞ by solving the e.o.m. First, the electric fields is given by Fi0 = ∂ 0 Ai − ∂i A0 =

Q ri . 4π r 3

(H.3)

Then we decompose y μ (τ ) into two parts as y μ (τ ) = y0μ (τ ) + δy μ (τ ) ,

(H.4)

where μ

y0 (τ ) = a μ + v μ τ ,

(H.5) μ

with arbitrary constant vectors a μ and v μ (= p μ /m). y0 (τ ) is the trajectory of the free particle that is the solution of the equation of motion without the Coulomb

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 H. Hirai, Towards Infrared Finite S-matrix in Quantum Field Theory, Springer Theses, https://doi.org/10.1007/978-981-16-3045-3

123

124

Appendix H: Particle Trajectory Under the Coulomb Force …

potential. Since it is expected that the trajectory y(τ ) approaches the free trajectory )| → 0 as τ → ∞. Plugging (H.5) into (H.1), y(τ ) in large-τ region, we assume |δy(τ |y0 (τ )| the time component of e.o.m is given by   eQ 1 r d 2 δy 0 · (v + δ y˙ ) =− m 2 2 dτ 4π r r r=y(τ ) 1 y0 + δy eQ  =− · (v + δ y˙ ) 4π (y0 + δy)2 (y0 + δy)2 =−

|δy(τ )| eQ 1 y0 · v  + O( ) 4π (y0 )2 (y0 )2 |y0 (τ )|

=−

|δy(τ )| eQ 1 ). + O( 4π |v|τ 2 |y0 (τ )|

(H.6)

We can easily solve this equation and the solution is given by δy 0 (τ ) =

eQ ln τ + cτ + d , 4π m|v|

(H.7)

)| → 0 as τ → ∞, c is set with two integration constants c and d. Since we used |δy(τ |y0 (τ )| to be zero and d can also be neglected as τ → ∞. Then we have found the leading solution in asymptotic region,

δy 0 (τ ) =

eQ ln τ . 4π m|v|

(H.8)

The e.o.m of the spatial component is   d 2 δy eQ 1 r 0 0  (v + δ y˙ ) m =− dτ 2 4π r 2 r r=y(τ ) =−

eQ v 0 v |δy(τ )| ). + O( 4π |v|3 τ 2 |y0 (τ )|

(H.9)

The solution is δy(τ ) =

eQv 0 v ln τ , 4π m|v|3

(H.10)

where we have already set the integration constant to zero for the same reason as the solution of time component. Thus we found the asymptotic trajectory of a point particle in the static Coulomb potential: eQ y (τ ) = v τ + 4π m|v|3 μ

μ



|v|2 v v0 v

 ln τ + O(

ln τ ). τ

(H.11)

Appendix H: Particle Trajectory Under the Coulomb Force …

125

The subleading term means that the particle is pulled off to the source if the charges have the opposite signs (eQ > 0) and the particle is repulsed away from the source if the charges have the same signs (eQ < 0). Now we try to find the fully covariant solution of the asymptotic trajectory. Since y μ (τ ) is a covariant four-vector, it should be composed of the covariant vectors existing in our system; the velocity four-vector of the particle v μ and the velocity four-vector of the source particle wμ . In previous case, since we supposed that the source particle with the charge Q is at rest, w μ is given by w μ = (1, 0). Using this vector, we can write |v| and v 0 in Lorentz invariant forms as |v| =



v 2 + (v · w)2 =

 (v · w)2 − 1 ,

v = −v · w . 0

(H.12) (H.13)

With the use of these expressions, the four-vectors appeared in (H.11) can be written as   2  (v · w)2 − 1 |v| = = −(v · w)v μ − w μ . (H.14) v0 v −(v · w)v Plugging (H.12) and (H.14) back into (H.11), we find the covariant solution, y μ (τ ) = v μ τ −

eQ (v · w) v μ + w μ ln τ ).  3/2 ln τ + O( 2 4π (v · w) − 1 τ

(H.15)

We can trivially generalize this solution to multi-particles case. The result is μ yi (τ )

=

μ vi τi

  μ μ ei e j vi · v j vi + v j ln τi − ),  3/2 ln τi + O( 2 4π m τi i j=1( j=i) vi · v j − 1 N 

(H.16)

where τi , yiμ (τ ) and viμ are the proper time, the position and leading velocity fourvector of the i-th particle with the charge ei . Now let us write this spacial trajectory in terms of the reference time t. The time component of (H.16) is t (τ ) =

vi0 τi

  vi · v j vi0 + v 0j ei e j ln τi − ).  3/2 ln τi + O(  2 4π m τi i j=1( j=i) vi · v j − 1 N 

(H.17)

Solving this equation in terms of τi , the proper time for i-th particle then is given by t τi = 0 + vi

  vi · v j vi0 + v 0j ei e j ln t ). 3/2 ln t + O( 0   2 t 4π m v i i j=1( j=i) vi · v j − 1 N 

(H.18)

126

Appendix H: Particle Trajectory Under the Coulomb Force …

Plugging (H.18) into the spatial component of (H.16), we have ⎛

⎞   0 0 vi · v j vi + v j ei e j ⎜ t ⎟ yi (τ ) = vi ⎝ 0 + 3/2 ln t ⎠ 0   2 vi 4π m i vi j=1( j=i) vi · v j − 1   N  vi · v j vi + v j ei e j ln t − )  3/2 ln t + O(  2 4π m i t j=1( j=i) vi · v j − 1 N 

=

vi t+ vi0

N 

v 0j vi − vi0 v j ei e j ln t ). 3/2 ln t + O( 0   2 t 4π m v i i j=1( j=i) vi · v j − 1

(H.19)

H.2 Stationary Trajectory of Phase Factor in FK Dressed State In this section, we consider the wave function with the dressing phase factor (5.30) in the FK state, and show that the stationary point of the phase of the wave function corresponds to the trajectory of electrons and positrons under the Coulomb force (H.19). For simplicity, let us consider the wave packet of two charged particle with the dressing phase factor (5.30) in QED in the one-dimensional space, which is given by  |ψ(t) =

dp1 dp2 f ( p1 , p2 ) ei(t,t0 ) | p1 , s1 ; p2 , s2 ,

(H.20)

where | p1 , s1 ; p2 , s2 = bs†1 (p1 )bs†2 (p2 )|0 , and f ( p1 , p2 ) is a smooth function of p1 , p2 . We consider the wave function in the position space which is given by  ψ(x1 , x2 ) =  ≡

dp1 dp2 f ( p1 , p2 )0|ψ(x1 )ψ(x2 )ei(t,t0 ) | p1 , p2   dp1 dp2 f ( p1 , p2 ) u s1 (p1 )u s2 (p2 ) exp (i F(t; x1 , p1 ; x2 , p2 )) + ( p1 ↔ p2 )

(H.21) where ψ(x) is the free Dirac field operator (5.6) and F(t; x1 , p1 ; x2 , p2 ) is given by

Appendix H: Particle Trajectory Under the Coulomb Force …

127

F(t; x1 , p1 ; x2 , p2 ) = i p1 x + i p2 x2 + i( p1 , p2 ; t, t0 )   ei e j − p1 · p2 t = p10 + p20 (t − t0 ) − p1 · x1 − p2 · x2 + log .   1/2 2 4π ( p1 · p2 ) − m 2 m 2 t0 1 2 (H.22) The stationary phase is then given by  0=

m 21 m 22

p20 p p10 1

 − p2

ei e j ∂ p1 t F(t; x1 , p1 ; x2 , p2 ) = 0 (t − t0 ) − x1 +   log . ∂p1 4π ( p1 · p2 )2 − m 2 m 2 3/2 t0 p1 1 2

(H.23) This solution corresponds to the following particle trajectry,   ei e j m 21 m 22 p20 p1 − p10 p2 p1 t x1 = 0 (t − t0 ) + log  0  3/2 2 t0 p1 4π p1 ( p1 · p2 ) − m 2 m 2 1 2 =

v20 v1 − v10 v2 e1 e2 1 v1 t + ln t + O( ) ,  0 0  3/2 t v1 4π m 1 v1 (v1 · v2 )2 − 1 μ

(H.24) (H.25)

μ

where we have used pi = m i vi in the final equality. We can trivially generalize this trajectory to the trajectory in N charged particle case as xi =

vi t+ vi0

N 

v 0j vi − vi0 v j ei e j 1  3/2 ln t + O( ) . 0   2 t 4π m i vi j=1( j=i) vi · v j − 1

(H.26)

This trajectory is exactly the same as the asymptotic trajectory of point particles with the relativistic Coulomb interactions (H.19). This means that if the wave packet is effectively localized at p1 , · · · , p N in the momentum space ( f (P1 , · · · , PN ) has the peaks at p1 , · · · , p N ), the wave function of the dressed state is effectively localized at the classical trajectories (H.19) of point particles under the Coulomb force at the large time limit.