Non-Perturbative Methods in 2 Dimensional Quantum Field Theory

Non-perturbative methods in

T I

i

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con

Elcio Abdalla M. Cristina B. Abdalla Klaus D. Rothe

World Scientific

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Non-perturbative methods in

2 DIMENSIONAL QUANTUM HELD THEORY

Non-perturbative methods in

2 DIMENSIONAL QUANTUM FIELD THEORY Second Edition

Elcio Abdalla Universidade de Sao Paulo, Brazil

M. Cristina B. Abdalla Universidade Estadual Paulista, Brazil

Klaus D. Rothe Universitat Heidelberg, West Germany

fe World Scientific !•

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NON-PERTURBATIVE METHODS IN TWO-DIMENSIONAL QUANTUM FIELD THEORY (2nd Edition) Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4596-3

Printed in Singapore by Uto-Print

Dedicated to my parents, Annemarie and Kurt, to my wife, Neusa, and to my son, Thomas. Klaus

Preface While other — more phenomenologically motivated — branches of high energy physics, such as Quantum Chromodynamics, the electroweak model of Glashow, Salam and Weinberg, as well as purely group-theoretical studies have by now been extensively portrayed in a large number of research reports and books, the same has not been the case for two-dimensional quantum field theory. This book is intended to fill this gap. Its aim is to give a fairly detailed survey of the developments in twodimensional quantum field theory since the pioneering work of Thirring, without loosing sight of their relevance to the four-dimensional world. Though many of the properties and techniques to be used are peculiar to two-dimensional space-time, the structural richness of the models portrays the complexity to be expected in realistic quantum field theories. Our emphasis therefore lies on the non-perturbative aspects of the two-dimensional models and our tools shall correspondingly be operator and functional methods, although we shall occasionally make also use of Feynman diagram techniques. The present edition represents an extensively revised version of the first edition, and involves major changes and additions, as well as corrections. In particular, the chapter on Conformal Field Theory has been completely rewritten and split into three separate chapters, 16, 17 and 18. We have found it useful to include in Chapter 17 some recent applications to statistical models, and in particular to the Ising model. Chapter 18, on the other hand, is devoted to the application of Conformal Field Theory to two-dimensional gravity, and includes some discussion on Liouville theory. Chapter 4, on functional determinants and heat-kernel methods, has also been reorganized, and a section on the heat-kernel at finite temperature, as well as the application of heat-kernel methods for obtaining asymptotic mass expansions has been added. Some material originally contained in Chapter 11 on Quantum Chromodynamics concerning the calculation of the QCD2 determinant has also been included. In Chapter 7 we further included a discussion on the algebra obeyed by nonlocal charges (Yangian algebra), and Chapter 8 contains an additional section on S-matrices in the presence of boundaries. Chapter 11 on Quantum Chromodynamics has also been extensively rewritten. We have incorporated recent results concerning the decoupled (coset) formulation of QCD2, with much emphasis on the BRST constraints defining the Hilbert space in this formulation, and related questions such as vacuum degeneracy. We have

6

also added a section on QCD2 with massive fermions and the related problem of "screening versus confinement" in the non-Abelian case. A new chapter devoted to the finite temperature Schwinger model has also been added in this edition. Chapter 15 (originally 14) on chiral electrodynamics now also includes a section on the Batalin-Fradkin-Tyutin embedding of this theory into a bonafide gauge theory. Furthermore, we ommitted Appendices H, I, N, O of the first edition in favour of two new ones - Appendices L and M. Except for minor additions or changes, the remaining chapters have remained untouched. The following paragraphs contain a brief summary of the material in this second edition. After two introductory chapters on generalized free fields and their application to the solution of two exactly solvable models, we turn in Chapter 4 to a discussion of heat-kernel techniques for calculating functional determinants at zero and finite temperature. We illustrate these techniques in terms of a number of examples, which shall play an important role in subsequent chapters. This material is to a large extent also relevant to higher dimensions. In Chapter 5 we then turn to a discussion of the Gross-Neveu model (chiral and non-chiral). This model exhibits some interesting properties such as mass transmutation and spontaneous "Kosterlitz-Thouless" type symmetry breakdown. We demonstrate this using the l/A r -expansion as well as operatorial methods. Chapters 6 and 7 deal with the classical and quantum aspects of non-linear sigma models, respectively, where very elegant results have been obtained. The rather impressive results on the exact 5-matrices of a vast number of two-dimensional models could not remain unmentioned as they represent the first examples of non-trivial scattering matrices obeying the principle of minimal analyticity of the sixties. These results are presented in Chapter 8. Chapter 9 prepares the ground for non-Abelian fermion-boson equivalences (bosonization) which shall play a central role in the remaining part of the book. Chapters 10 to 15 deal with twodimensional (Abelian and non-Abelian) gauge theories, including the more recent developments on anomalous chiral gauge theories. In view of the hidden complexity of two-dimensional Quantum Electrodynamics, we have thought it appropriate to start off the discussion on gauge theories with the operator solution of this model. This is done in Chapter 10, where we discuss, both the case of massless and massive fermions using operator bosonization techniques, with emphasis on delicate questions such as "screening versus confinement" of quarks, as well as the existence of charge (soliton) and kink sectors in the bosonized formulation. In chapters 11 to 15 we choose the functional approach. The ordering of chapters 11/12,13 and 14/15 (non-Abelian versus Abelian) may appear unnatural to the reader. The criterion here has been to regard the abelian theory as a special case of the non-Abelian one, with the additional property of being exactly soluble. Chapters 11 and 14 include some recent developments concerning the bosonic (coset) representation of nonAbelian gauge- and chiral-gauge theories in terms of factorized partition functions, subject to BRST constraints defining the physical Hilbert space. Finally, our survey of two-dimensional quantum field theory would be incomplete without an account of the rapidly developing field of research on two-dimensional Conformal Field Theories. This is left to Chapters 16 and 17, discussing the BPZ (Belavin-Polyakov-Zamolodchikov) construction as applied to Abelian and non-Abelian conformal theories, respectively, and to Chapter 18, discussing two-

7 dimensional gravity. Chapter 19 contains our final conclusions. Several appendices deal with notation and technical details. This book is intended for students working on their Ph.D. degree, as well as for post-doctoral research workers wishing to become acquainted with non-perturbative aspects in quantum field theory. Although much of the material in this book is special to two dimensions, the techniques used should prove useful also in the development of techniques applicable in higher dimensions. In particular, the last three chapters of this book will be of direct interest to researchers wanting to work in the field of conformal field theory and strings. It is clear that in view of the scope of this book and the tremendous number of published papers on this subject, it is impossible to do full justice to all authors who have contributed to this area of research. For the same reason, a strongly biased selection of the topics, as well as form of presentation necessarily had to be made. When referring to the original literature much care should be exercised with respect to the conventions and definitions being used. We have therefore summarized the conventions used in this book in Appendices A through C. Further details concerning the main text of the book are reserved for the remaining Appendices D through M. We would like to thank FAPESP (Fundacao de Amparo a Pesquisa do Estado de Sao Paulo), the DA AD (Deutscher Akademischer Austauschdienst) and the "Alexander von Humboldt Stiftung" for their financial support of several exchange visits within the Brazilian-German agreement, which greatly facilitated our collaboration. We also wish to thank, in this respect, CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico), and the European Commission, through the Human Capital and Mobility program. The hospitality of CERN's Theory Division, where part of the work has been carried out, is gratefully acknowledged.

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Contents 1

Introduction

17

2 Free Fields 2.1 Introduction 2.2 Bosonic Free Fields 2.3 Fermionic Free Fields 2.4 Bosonization of Massless Fermions 2.5 The RS-Model 2.6 Conclusions

25 25 25 29 31 35 38

3

41 41 42 45 45 47 48 53 53 55 61

The 3.1 3.2 3.3

3.4 3.5 3.6 4

Thirring model Introduction The Massless Thirring Model The Massive Thirring Model 3.3.1 Equivalence with sine-Gordon equation 3.3.2 Classical conservation laws 3.3.3 Quantum conservation laws Bosonization Revisited 3.4.1 Fermions in terms of bosons The Soliton as a Disorder Parameter Conclusion

Determinants and Heat Kernels 4.1 Introduction 4.2 Functional Determinant, one-loop diagram 4.2.1 Determinants and the Generalized Zeta-Function 4.2.2 One Point Compactification 4.2.3 The associated Dirac operator 4.3 Calculating Seeley Coefficients 4.3.1 The perturbative approach 4.3.2 The Schwinger-DeWitt method 4.3.3 The Fujikawa method 4.4 Computing Functional Determinants 4.4.1 (-function regularization 4.4.2 Proper-time regularization

65 65 66 69 75 78 81 81 83 86 88 88 90

10

CONTENTS 4.4.3 The Fujikawa point of view A Theorem on a one parameter family of factorizable operators . . . The QCD 2 functional determinant Zero-modes 4.7.1 Axial anomaly equation in the presence of zero-modes 4.7.2 Atiyah-Singer Index Theorem 4.8 Ambiguities in Functional Determinants 4.8.1 Ambiguities in the regularization 4.8.2 Dependence on the scale parameter 4.9 Mass expansion in proper-time regularization 4.10 The Finite Temperature Heat Kernel 4.10.1 Scalar field in a static background potential 4.10.2 Scalar field in a static background gauge potential 4.11 Conclusion

91 95 98 101 101 104 107 107 108 Ill 115 117 119 123

5

Self-Interacting fermionic models 5.1 Introduction 5.2 The O(N) Invariant Gross-Neveu Model 5.2.1 Classical conservation laws 5.2.2 Effective potential and /?-function in a ^ expansion 5.2.3 The -^ Expansion: Feynman rules 5.2.4 Leading order S-matrix elements 5.2.5 Quantization of the non-local charge 5.3 Chiral Gross-Neveu Model 5.3.1 Cancellation of infrared singularities 5.3.2 The -^ expansion 5.3.3 Operator formulation 5.3.4 Quantization of non-local charge 5.4 Conclusions and Physical Interpretation

127 127 127 128 129 133 135 138 141 142 144 146 151 152

6

Non-linear a Models: Classical Aspects 6.1 Historical development 6.2 Sigma models and current algebra 6.3 Two-dimensional a models: preliminaries 6.4 Purely Bosonic Non-linear a Models 6.4.1 Formal developments 6.4.2 Dual symmetry and higher conservation laws 6.4.3 An explicit example: the Grassmannians 6.5 Non-linear a Models with Fermions 6.5.1 Definition and properties 6.5.2 Dual symmetry and higher conservation laws 6.5.3 Construction of an explicit example 6.6 Analogies with 4D Gauge Theories 6.7 Concluding Remarks

155 155 156 158 166 166 172 182 184 184 188 195 200 204

4.5 4.6 4.7

CONTENTS 7

8

Non-linear a Models - Quantum Aspects 7.1 Introduction 7.2 Grassmannian Bosonic Models 7.2.1 JJ expansion 7.2.2 Renormalization 7.2.3 Infrared divergencies 7.2.4 Physical interpretation of the results 7.3 Grassmannian Models and Fermions 7.3.1 jjj expansion and Feynman rules 7.3.2 Physical interpretation of the results 7.4 Quantization of Higher Conservation Laws 7.4.1 Purely bosonic sigma models and anomalies 7.4.2 Fermionic interaction and anomaly cancellation 7.5 Algebra of non-local charges 7.5.1 Bosonic 0(iV)-symmetric sigma models 7.6 Non-local charges in the WZNW model 7.7 Perturbative Renormalization 7.7.1 Background Field Method 7.7.2 Parallelizable manifolds; applications to string theory 7.8 Anomalous Non-Linear a Models in four dimensions 7.9 Conclusion Exact S-matrices of 2D Models 8.1 Introduction 8.1.1 Consequences of higher conservation laws 8.1.2 Factorizable S-matrix 8.1.3 Fusion rules 8.1.4 Bound state scattering 8.2 S-matrices and Conservation Laws 8.2.1 SU{N) invariant S-matrices 8.2.2 Sine-Gordon and massive Thirring models 8.2.3 Exact S-matrix for 0(N) symmetry 8.2.4 The ZN invariant S-matrix 8.3 Quantum Non-Local Charges and S-Matrices 8.3.1 S-matrices of purely fermionic models 8.3.2 S-matrices of non-linear sigma models 8.4 Boundary S-matrices 8.5 Further Developments 8.6 Conclusion

9 The 9.1 9.2 9.3

W e s s - Z u m i n o - W i t t e n Theory Introduction Existence of a Critical Point Properties at the Critical Point 9.3.1 The Polyakov-Wiegmann formula 9.3.2 The Affine algebra

11

....

211 211 212 212 218 219 221 222 222 227 233 233 239 242 242 254 257 257 263 265 266 273 273 273 274 278 280 280 280 282 287 288 289 289 293 303 307 308 313 313 315 318 319 320

12

CONTENTS

9.4

9.5

9.3.3 The WZW fields in terms of fermions 322 9.3.4 The Sugawara form of the energy-momentum tensor 323 9.3.5 The non-Abelian bosonization in the operator language . . . 324 Properties off the Critical Point 325 9.4.1 Integrability of the WZNW action 326 9.4.2 On the solution off the critical point 327 9.4.3 Supersymmetric WZW model 329 Conclusion 331

10 QED2: O p e r a t o r A p p r o a c h 10.1 Introduction 10.2 The Massless Schwinger Model 10.2.1 Quantum solution 10.2.2 The Maxwell current 10.2.3 Chiral densities 10.2.4 Vacuum structure 10.2.5 Gauge transformations 10.2.6 Correlation functions and violation of clustering 10.2.7 Absence of charged states (screening) 10.2.8 The quark-antiquark potential 10.2.9 Adding flavour 10.2.10Fractional winding number and the U{1) problem 10.3 The Massive Schwinger Model 10.3.1 Equivalent bosonic formulation 10.3.2 The quantum Dirac equation 10.3.3 Vacuum structure and all that 10.3.4 Screening versus confinement 10.3.5 Adding flavour 10.3.6 Lorentz transformation properties 10.3.7 The MSM as the limit of a massive vector theory 10.4 Conclusion

333 333 335 335 337 340 341 345 348 349 351 353 356 360 360 362 365 366 374 380 383 386

11 Quantum Chromodynamics 11.1 Introduction 11.2 The 1/N expansion: 't Hooft model 11.3 Currents, Green functions and determinants 11.3.1 Tree graph expansion of the current 11.3.2 Recovering the QCD2 effective action 11.3.3 Fermion Green Function 11.4 Local decoupled formulation andBRST constraints 11.4.1 Local decoupled partition function and BRST symmetries 11.4.2 Systematic derivation of the constraints 11.5 Non-local decoupled formulation and BRST constraints

391 391 394 399 400 402 405 408 409 414 417

CONTENTS 11.5.1 Non-local decoupled partition symmetries 11.6 The physical Hilbert space 11.7 The QCD2 vacuum 11.8 Massive two-dimensional QCD 11.9 Screening in two-dimensional QCD ll.lOFurther algebraic aspects 11.11 Conclusions

13 function

and

BRST 417 421 422 425 427 433 434

12 QED2: Functional Approach 12.1 Introduction 12.2 Equivalent Bosonic Action 12.3 Gauge Invariant Correlation Functions 12.3.1 The external field current, and chiral densities 12.4 Vacuum Structure 12.4.1 Chirality of the vacuum 12.5 Why Study Gauge-Invariant Correlators 12.6 Screening versus Confinement 12.7 Quasi-Periodic Boundary Conditions and the #-Vacuum 12.8 Axial anomaly and the Dirac sea 12.9 Functional Representation of Tunneling Amplitudes 12.10 Interpretation of the Result 12.10.1 Zero modes 12.10.2 Calculation of det i Jfi from the anomaly equation 12.11 Eigenvalue Spectrum of the Dirac Operator 12.12 Zero Modes and Boundary-Value Problem 12.12.1 Free Dirac operator and non-local boundary conditions 12.12.2 The little Dirac operator 12.13 The U(l) Problem Revisited 12.14 Conclusion

439 439 440 441 441 442 443 447 448

13 The 13.1 13.2 13.3 13.4 13.5 13.6 13.7

483 483 484 488 490 495 503 506

Finite Temperature Schwinger Model Introduction Heat kernel and Seeley expansion The Atiyah-Singer Index theorem Fermions in an Instanton potential Chiral condensate and symmetry breaking Polyakov loop-operator and screening Conclusion

450 454 456 458 460 462 464 467 468 470 474 479

14

CONTENTS

14 Non-Abelian Chiral G a u g e Theories 509 14.1 Introduction 509 14.2 Anomalies and Cocycles 514 14.2.1 Consistent anomaly 514 14.2.2 More about cocycles 518 14.2.3 Gauss anomaly 520 14.2.4 Relation between consistent and covariant anomaly 521 14.3 Isomorphic Representations of Chiral QCD2 524 14.3.1 Gauge-invariant embedding 525 14.3.2 External Field Ward Identities 527 14.3.3 Construction of the one-Cocycle from the Anomaly 533 14.3.4 Bosonic Action in the GNI and GI Formulation 534 14.3.5 Symmetries of the Model 538 14.3.6 Relation of Source Currents in GNI and GI Formulations . . 540 14.3.7 Poisson Algebra of the Currents 541 14.3.8 Hamiltonian Quantization 545 14.3.9 Fermionization of ai[j4,5] 554 14.3.10 BRST Quantization of GI Formulation 555 14.3.11 Chiral QCD2 in Terms of Chiral Bosons 562 14.4 Constraint Structure from the Fermionic Hamiltonian 567 14.5 Chiral QCD2 in the local decoupled formulation 575 14.5.1 Gauge non-invariant formulation 575 14.5.2 Gauge-invariant formulation 584 14.6 Conclusion 587 15 Chiral Quantum Electrodynamics 15.1 Introduction 15.2 The JR Model 15.3 Quantization in the GNI Formulation 15.3.1 Hamiltonian and constraints 15.3.2 Commutation relations 15.3.3 Current-potential and bosonic representation of fermion field 15.3.4 Energy-momentum tensor 15.3.5 Vector-field two-point function 15.3.6 Fermionic two-point function 15.4 Quantization in the GI Formulation 15.4.1 Hamiltonian and constraints 15.4.2 Implementation of gauge conditions 15.4.3 Isomorphism between GI and GNI formulations: phase space view 15.4.4 WZ term and BFT Hamiltonian embedding 15.4.5 Alternative approach to quantization 15.4.6 Operator solution in Lorentz-type gauges 15.5 Path-Integral Formulation

593 593 594 596 596 598 600 602 603 604 604 604 606 608 611 616 617 618

CONTENTS 15.6 Perturbative Analysis in the Fermionic Formulation 15.6.1 Perturbative analysis in the GNI formulation 15.6.2 Perturbative analysis in the GI formulation 15.7 Anomalous Poisson Brackets Revisited 15.7.1 Operator view of anomalous Poisson brackets 15.7.2 Bjorken-Johnson-Low view of anomalous Poisson brackets 15.7.3 Reconstruction of commutators of the GNI formulation . . 15.8 Chiral QED2 in terms of Chiral Bosons 15.9 Conclusion 16 Conformally Invariant Field Theory 16.1 Introduction 16.2 Conformal transformations and conformal group 16.2.1 Dilatations 16.2.2 The conformal group in D dimensions 16.3 The conformal group in two dimensions 16.3.1 Mobius transformations 16.4 The BPZ construction 16.4.1 Primary and quasi-primary fields 16.4.2 Radial quantization 16.4.3 Descendants of primary fields 16.4.4 Virasoro algebra 16.5 Realization of Conformal Algebra for c < 1 16.6 Superconformal Symmetry 16.7 Conclusion

15

625 625 630 632 633 635 635 638 641 645 645 646 647 647 653 655 659 659 666 671 675 683 688 691

17 Conformal Field Theory with Internal Symmetry 695 17.1 Introduction 695 17.2 Conformal algebra and Ward identities 695 17.3 Realizations of non-Abelian conformal algebra 700 17.3.1 The Wess-Zumino-Witten field 700 17.3.2 The non-Abelian Thirring field at the Critical Point 706 17.4 Coset description of CQFT 711 17.4.1 Coset realization of the FQS minimal unitary series 712 17.4.2 Fermionic coset realization of SU(N)i 713 17.4.3 Fermionic coset realization of FQS series 716 17.4.4 Reduction formula for negative level WZW fields 718 17.5 Critical statistical models 722 17.5.1 Fermionic coset description of the critical Ising model . . . . 722 17.6 Conclusions 730

16

CONTENTS

18 2D gravity a n d s t r i n g r e l a t e d topics 18.1 Introduction 18.2 The Nambu-Goto string 18.3 The effective action of 2D quantum gravity 18.3.1 Uniqueness of the Polyakov action 18.3.2 Quantum Gravity 18.4 The Liouville theory 18.4.1 The classical Liouville theory 18.4.2 The quantum Liouville theory 18.5 Gravity in the light-cone gauge 18.5.1 Canonical quantization and SL(2, R) symmetry 18.5.2 Operator product expansions and Ward identities 18.5.3 Interaction of matter fields with gravity 18.5.4 Two-Dimensional Supergravity 18.6 Conclusion

733 733 734 736 736 738 746 747 750 753 753 759 760 762 768

19 Final R e m a r k s

771

Appendices A

N o t a t i o n (Minkowski Space)

775

B

N o t a t i o n (Euclidean Space)

781

C

F u r t h e r Conventions

785

D

Functional Bosonization of t h e Massive T h i r r i n g M o d e l

789

E

Bosonization of t h e Fermionic Kinetic T e r m

793

F

Classical Integrability in t h e Massive T h i r r i n g M o d e l

795

G

Q u a n t u m Non-Local C h a r g e : Action on A s y m p t o t i c S t a t e s

797

H

S-Matrices

801

I

C o m p l e t e S-matrix of t h e G r o s s - N e v e u M o d e l

805

J

Poisson Brackets a n d C o m m u t a t o r s

809

K

Chiral Bosons

811

L

Axial A n o m a l y from Dispersion Relations

817

M

Loop Expansion in QCD2

821

Index

826

Chapter 1

Introduction The development of Relativistic Quantum Field Theory started in 1932 as a natural extension of Quantum Mechanics to the relativistic domain [1]. The work of Feynman in the late forties provided a powerful tool for the calculation of processes in Quantum Electrodynamics. Second quantization led, however, to new conceptual and technical difficulties. Quantum fields had to be regarded as operator-valued distributions, their local products being ill defined as a result of ultraviolet divergencies, which plagued the higher order computations in perturbation theory. This problem was partially mastered via the techniques of renormalization, and later on completely understood [2]. In the early fifties, there appeared a series of papers concerned with the extraction of general non-perturbative properties of quantum field theory from a perturbative setup. Of particular importance in this respect were the papers of Lehmann, Symanzik and Zimmermann [3] (as well as of Wightman, Haag, and others [4]). The so-called LSZ formalism established the relation between fields and particles in terms of the asymptotic conditions for the interpolating fields. The reduction formula provided the connection between expectation values of fields and 5-matrix elements of particle scattering. From the study of the analytic properties of Feynman diagrams dispersion relations were derived, which could be used to obtain non-perturbative information. These developments were followed by a new axiomatic approach to Quantum Field Theory (QFT), which became known as constructive QFT. Here functional analysis played an important role. Some of the (non-perturbative) results of the LSZ formalism, which had been based on perturbative studies, could thereby be derived from general principles, if the theory has a non-vanishing mass gap. An important consequence of this approach was a theorem connecting spin and statistics. Meanwhile, dynamical calculations in QFT were restricted to perturbation theory. This rendered computations involving strong interactions unreliable, and made information about the bound state spectrum only accessible within approximative non-perturbative — and often non-unitary — schemes (e.g. the Bethe-Salpeter equation in the ladder approximation). As a result, QFT had fallen into stagnation, and even discredit, in the late fifties.

18

Introduction

These difficulties provided, in particular, the motivation for a new approach to the strong interactions, known as S-matrix theory [5], which was to play a dominant role in the sixties. The predictive power of this theory turned out to be very limited, being entirely based on kinematical principles and analyticity, supplemented by the bootstrap idea. An underlying dynamical framework was lacking. Nevertheless, analyticity in the complex angular momentum plane led to the important concept of duality, expressing the possibility of representing a given scattering amplitude as a sum over poles in crossed channels [6]. An explicit realization of these concepts in terms of a remarkable formula proposed by Veneziano [7] led to a new parallel development in the sixties, summarized under the name of dual models. The high-energy behaviour was however found to be incorrectly described in this approach. Moreover, an analysis of the pole structure of higher order corrections required the introduction of a somewhat mysterious new concept, the Pomeron [8]. The ever increasing number of parameters which were required to describe experiments within these schemes, and the resulting loss of predictive power, eventually led physicists to abandon them, and to turn again to QFT. In the meantime, QFT had scored some remarkable successes in the realm of the weak interactions [9]. Moreover, symmetry principles, such as those advocated by Gell-Mann and Ne'eman, had proven powerful in predicting the masses of strongly interacting particles, as well as the existence of new ones (the fi~), without the recourse to dynamical calculations. This situation led to a revival of QFT in the late sixties and in the seventies, when much attention was given to the non-perturbative aspects. Quantum chromodynamics (QCD) had been proposed as the fundamental theory of the strong interactions, but reliable calculations confronting QCD with experimental tests were lacking. The high-energy behaviour of QFT was investigated by means of the renormalization-group and Callan-Symanzik equations, which describe the behaviour of the theory under finite renormalizations of the parameters. As a result it was possible to relate the zero mass limit to the high energy behavior. A momentum dependent, running coupling constant turns out to properly characterize the strength of the interaction: depending on the properties of the so-called /?function, the running coupling constant is sufficiently small for either high or small momenta, legitimating perturbation theory in one of these regions. In the case of non-Abelian gauge theories, perturbation theory turned out to be a good approximation at high energy (asymptotic freedom). Furthermore, classical solutions minimizing the action were shown to play a central role in the (non-perturbative) semi-classical analysis of QFT. Monopole solutions (for Minkowski space), and instanton solutions (for Euclidean space) have been obtained, showing the important role played by the topology of the manifold on which the fields are defined. In this way, more abstract branches of mathematics such as algebraic topology turned out to be significant in the unraveling of the structural properties of gauge theories. Although the above studies have been weighty for revealing a non-trivial and highly interesting structure, exact non-perturbative results for all correlators are only available for specific models, all in two-dimensional space-time. The first such model, discussed by Thirring [12] in 1958, describes the current-current interaction

19 of massless fermions. It provided an example of a completely soluble quantum field theoretic model obeying the general principles of a QFT. The complete quantum solution was given in a classic paper of Klaiber [13], and was shown to satisfy all Wightman axioms. Following the above work, Schwinger [14] obtained an exact solution of Quantum Electrodynamics in 1+1 dimensions (QJSZ^)- A number of interesting properties, such as the nontrivial vacuum structure of this model, were however only later revealed in the work of Lowenstein and Swieca [15], who explored the consequences of the long range Coulomb force for the charge sectors of the theory. This long range force was interpreted as being responsible for the confinement of quarks [16], that is, their occurrence in the form of permanently bound states of quark-antiquarks pairs (baryonic bound states happen to be absent in QED-i). The problem of confinement and the related phenomenon of screening of charge quantum numbers has been extensively studied [17], and has served as a basis for sharpening these concepts appearing also in higher dimensions. The surprisingly rich structure of two-dimensional quantum electrodynamics was found to describe several important features of non-Abelian gauge theories, under investigation in the seventies. As a matter of fact, in the late sixties one learned that the short distance singularities of quantum field theory play a key role in the dynamical structure of the theory [18]. The experimental results on leptonproton scattering at large momentum transfer, required that a realistic theory of the strong interactions be asymptotically free [10, 11]. This promoted QCD to a good candidate for the theory describing strong interactions, since it was shown that no renormalizable theory without non-Abelian gauge fields can be asymptotically free [19]. The vacuum structure and confinement property attributed to QCD± were found to be explicitly realized in two-dimensional QED, which made the theory a very interesting laboratory for studying such questions. Following these exactly solvable quantum field theoretical models, the study of other two-dimensional quantum field theories has played an important role in the development of a non-perturbative understanding of quantum field theory in general. By the end of the seventies a very comprehensive knowledge had been gathered, revealing an unexpected complexity and richness of the non-perturbative structure of relativistic quantum field theories. Several further developments of growing importance in two-dimensional QFT, followed. Classically exactly integrable models, and the quantization of solitons were extensively studied in two dimensions. Such integrable models were generally characterized by the existence of an infinite number of conservation laws [20]. In the cases where these conservation laws survive quantization, the S-matrices and their associated monodromy matrices could be computed exactly [21]. Besides being the first examples of exact S-matrices realizing the ideas of "minimal analyticity" of the sixties, these exact results also play an important role in checking approximation schemes such as the semi-classical approximation, and the 1/TV-expansion, and have important applications in statistical mechanics [22]. Some of the results concerning classical integrability have also been generalized to higher dimensions [23]. In the particular case of the sine-Gordon theory, the exact results for the Smatrix of fundamental fields could be extended to include the scattering of bound

20

Introduction

states and solitons, as well. Moreover, one found an unexpected 0(2) « U(l) symmetry, reflecting the fact that the solitons in the sine-Gordon theory correspond to the fermions of the massive Thirring model. This equivalence, partially conjectured long ago by Skyrme [24], was proven by Coleman [25] at the level of Green functions, and later obtained by the use of operatorial methods [26]. In both versions (bosonic or fermionic) the S-matrices could be exactly computed, and turn out to be identical. In the framework of two-dimensional models, the possibility of writing fermions in terms of bosons (bosonization) has been a powerful tool for obtaining nonperturbative information. One of the features to be stressed in this context is that charge sectors of the fermionic theory are found to correspond to soliton sectors in the purely neutral bosonic theory: dynamical aspects of the fermionic formulation thus turn into topological properties of the bosonic counterpart. The building blocks of the bosonization scheme are the exponentials of bosonic fields, the fermionic and chiral selection rules associated with this composite operator being directly linked to the infrared behavior of the zero mass scalar fields. This leads to a superselection rule [27], which makes the charge sectors appear in a natural way. The bosonization technique becomes cumbersome when applied to non-Abelian theories. The reason is twofold: the symmetry transformations of the fermionic theory are non-local with respect to the fundamental Bose fields, which lie in nonlinear representations of the global symmetry group of the fermions. Significant progress in the direction of non-Abelian bosonization was provided by the work of Polyakov, Wiegmann and Witten [28]. Although these authors discussed the problem in different contexts, they all arrived at an equivalent bosonic action involving the action of the principal sigma model plus a Wess-Zumino term. In two dimensions, fermionic theories were thus found to exhibit a remarkable universality in the bosonic formulation, where the non-linear sigma model and a topological term seem to play a fundamental role. Non-linear sigma models have a long history. Particularly important has been the class of the two-dimensional integrable ones. Their geometrical origin makes them very interesting mathematical objects to be studied in their own right [29]. They have also been shown to share several properties with Yang-Mills theories in four dimensions [30]: at the classical level, both are conformally invariant and present similar geometrical identities as well as non-trivial classical solutions [31] (e.g. instantons in the Euclidean formulation). The non-linear sigma models for symmetric spaces [29], and the Yang-Mills theories for either the self-dual sector, or with extended supersymmetry, share similar integrability properties. When quantized, non-linear sigma models further exhibit features believed to be properties of realistic theories, such as confining long range force generated by quantum fluctuations, when the gauge-group is not simple [32], and dynamical mass generation. These properties make them appealing as toy models for the strong interactions. They are also important the context of string theory, where the D-dimensional target manifold is compactified to four-dimensional space-time [33], the corresponding action being described by a sigma model. The requirement of conformal invariance at the quantum level leads directly to the Einstein equation of general relativity, and predicts its quantum corrections [34].

21 Two-dimensional space-time has further proven to be an excellent laboratory for the study of gauge-anomalies and the consistency of anomalous chiral gauge theories. An unexpected underlying differential geometric significance of such anomalies has thereby been revealed [35]. The exact solubility of two-dimensional chiral QED [36] has played here an important role in opening up a whole new line of developments in the area of chiral gauge theories. The successful quantization of such seemingly inconsistent theories without recourse to an algebraic cancellation of the gaugeanomalies on group-theoretical grounds was of much interest at a time, where the top quark had not yet been found. Although interest in such model studies had levelled off by the end of the seventies, this area of research experienced a remarkable back-come in the eighties, when an almost forgotten acquaintance from the seventies - string theory - experienced itself a dramatic revival as a result of pioneering work due to Green and Schwarz [37]. Ever since, two-dimensional quantum field theory — which had already found applications in statistical mechanics in the past - has become an important subject of elementary particle physics. More recently, it was shown that in two-dimensional quantum field theories, Poincare and scale invariance, alone, imply invariance under the infinite dimensional conformal group in two dimensions [38]. As a result, non-trivial correlators could be exactly computed and were found to be related to solutions of hypergeometric differential equations. The parameters labelling these equations, which are regarded as the critical indices, have been classified and characterize the correlators uniquely. The conformal algebras are realized in terms of the so-called primary fields and their descendents. In Minkowski space, this construction leads naturally to the use of Artin Braids, which relate this problem to the algebraic construction of exact S-matrices, since the star-triangle relations obtained from the infinite local conservation laws have the same structure as the permutation relations of knot theory [39]. The above ideas may be generalized to include the interaction with conformally invariant gravity [40]. In the light-cone-gauge the theory simplifies dramatically, due to a new SL(2, R) symmetry [40, 41]. The critical indices of the theory may be computed from a very simple equation relating them to the critical indices of the theory in flat space. The results have also been generalized to the supersymmetric case [42]. To summarize, two-dimensional models have been an extraordinary laboratory to test ideas in quantum field theory. Thus, the Thirring model provided a realization of an exactly soluble quantum field theory, while the Schwinger and the non-linear sigma models were found to exhibit properties of four-dimensional nonAbelian gauge theories. However, two-dimensional QFT also plays a direct role in the description of physical reality, having applications in string theories, as well as statistical mechanics. In particular, the methods developed in two-dimensional QFT have been used to extract results concerning the critical behavior of models in statistical mechanics, using conformal invariance alone. An extraordinary amount of physically interesting as well as mathematically elegant concepts have emerged from the study of such theories. Beyond their status as a theoretical laboratory, and their applications in string

22

BIBLIOGRAPHY

theories and statistical mechanics, the study of these models has also led to recent developments opening new possibilities for applications of some of the above methods in the study of quantum field theories in higher dimensions. There is a deep relationship between rational conformal invariance in two-dimensional space-time and the Chern-Simons action in three dimensions (which is also equivalent to conformal gravity in three dimensions). The Chern-Simons action proves to be a key element in the generalization of the fermion-boson equivalence to three-dimensional spacetime, and also plays an important role in the discussion of non-Abelian anomalies of chiral gauge theories in any dimension.

Bibliography [1] P.A.M. Dirac, Proc. Roy. Soc. 117 (1928) 610; 118 (1928) 351. [2] F.J. Dyson, Phys. Rev. 75 (1949) 1736; 85 (1952) 631; W. Zimmermann, in Lectures on Elementary Particles and Quantum Field Theory, vol 1, 1970, Brandeis Univ. Summer Institut; G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [3] H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cimento 1 (1955) 205. [4] A.S. Wightman, Phys. Rev. 101 (1956) 860; R. Haag, Dan. Mat. Fys. Medd. 29 (1955) 12; R. Haag and D. Kastler, J. Math. Phys. 5 (1964) 848; G. Kallen, Quantum Electrodynamics, Springer Verlag, Berlin 1972; J. Schwinger, Phys. Rev. Lett. 3 (1959) 296. [5] G.F. Chew, S-Matrix Theory of Strong Interactions, W.A. Benjamin INC. Publishers (1961). [6] R. Dolen, D. Horn and C. Schmid, Phys. Rev. Lett. 19 (1967) 402; Phys. Rev. 166 (1968) 1768. [7] G. Veneziano, Nuovo Cimento 57A (1968) 190. [8] J. Scherk, Rev. Mod. Phys. 47 (1975) 123. [9] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in Elementary Particle Theory, ed. N. Svartholm, Almquist and Wiksells, Stockholm 1968. [10] W. Marciano and H. Pagels, Phys. Rep. C36 (1978) 137. [11] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Commun. Math. Phys. 18 (1970) 227; Springer Tracts in Mod. Phys. 57 (1971) 222; D.J. Gross and F A . Wilczek, Phys. Rev. Lett. 30 (1973) 1346; H.D. Polizer, Phys. Rev. Lett. 30 (1973) 1346; H.D. Politzer, Phys. Rep. 14 (1974) 129. [12] W. Thirring, Ann. of Phys. 3 (1958) 91.

BIBLIOGRAPHY

23

[13] B. Klaiber, in Lectures in Theoretical Physics, Boulder 1967, Gordon and Breach, New York, 1968. [14] J. Schwinger, Phys. Rev. 128 (1962) 2425. [15] J. Lowenstein and J.A. Swieca, Annals of Phys. 68 (1971) 172. [16] A. Casher, J. Kogut and L. Susskind, Phys. Rev. Lett. 31 (1973) 31; Phys. Rev. D10 (1974) 732. [17] H.J. Rothe, K.D. Rothe and J.A. Swieca, Phys. Rev. D15 (1977) 1675. [18] K. Wilson, Phys. Rev. 179 (1969) 1499. [19] S. Coleman and D. Gross, Phys. Rev. Lett. 31 (1973) 851. [20] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207. [21] M. Liischer, Nucl. Phys. B135 (1978) 1; A.B. Zamolodchikov and Al.B. Zamolodchikov, Annals of Phys. 120 (1979) 253; M. Karowski, Phys. Rep. 49 (1979) 229; E. Abdalla, Lee. Notes in Phys. 226 (1984) 140 ed. N. Sanchez and H.J. de Vega; H. J. de Vega, H. Eichenherr and J. M. Maillet, Proc. Meudon and Paris VI, 83/84; Commun. Math. Phys. 92 (1984) 507; Nucl. Phys. B240 (1984) 377; Phys. Lett. 132B (1983) 337. [22] M. Karowski, Nucl. Phys. B300 (1988) 473. [23] L.L. Chau, M.K. Prasad and A. Sinha, Phys. Rev. D23 (1981) 2321; D24 (1981) 1574; E. Witten, Phys. Lett. 77B (1978) 394; Nucl. Phys. B266 (1986) 245; E. Abdalla, M. Forger and M. Jacques, Nucl. Phys. B307 (1988) 198; J. Harnad, J. Hurtubise, M. Legare and S. Shnider, Nucl. Phys. B256 (1985) 609; J. Avan, H.J. de Vega and J.M. Maillet, Phys. Lett. 171B (1986) 255. [24] T.H.R. Skyrme, Proc. Roy. Soc. A260 (1961) 127. [25] S. Coleman, Phys. Rev. D l l (1975) 2088; B. Schroer, Berlin-Prep. 75/5. [26] S. Mandelstam, Phys. Rev. D l l (1975) 3026. [27] J.A. Swieca, Fortschritte der Physik 25 (1977) 303. [28] A.M. Polyakov and P.B. Wiegmann, Phys. Lett. 131B (1983) 121;141B (1984) 223; E. Witten, Commun. Math. Phys. 92 (1984) 455. [29] H. Eichenherr, Nucl. Phys. B146 (1978) 215; E B155 (1979) 544; A. D'Adda, P. di Vecchia and M. Liischer, Nucl. Phys. B146 (1978) 63. [30] A.A. Migdal, Soviet Phys. Jetp 42 (1976) 413,742; A.M. Polyakov, Phys. Lett. 59B (1975) 79. [31] A. Chakrabarti, Introduction to classical solutions of Yang-Mills equations, Rencontre de Rabat, 1978; A. Actor, Rev. Mod. Phys. 51 (1979) 461; A.M. Din, W.J. Zakrzewski, Nucl. Phys. B174 (1980) 397; Phys. Lett. 95B (1980) 419.

24

BIBLIOGRAPHY

[32] E. Abdalla, M. Forger and M. Gomes, Nucl. Phys. B210 (1982) 181. [33] A.A. Tseytlin, Int. J. Mod. Phys. A 5 (1990) 589. [34] C.G. Callan, E.J. Martinec, M.J. Perry and D. Friedan; Nucl. Phys. B262 (1985) 593. [35] L.D. Faddeev, Phys. Lett. 145B (1984) 81; L.D. Faddeev, S.L. Shatashivili, Phys. Lett. 167B (1986) 225; S.L. Shatashvili, Teor. Mat. Fiz. 71 (1987) 40. [36] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 (1985) 1219; H.O. Girotti, H.J. Rothe and K.D. Rothe, Phys. Rev. D33 (1986) 514. [37] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Vol. I and II, Cambridge Monographs on Mathematical Physics (1987). [38] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333. [39] K.H. Rehren and B. Schroer, Phys. Lett. 198B (1987) 480; J. Frohlich, Proceedings of the Cargese School, 1987. [40] A.M. Polyakov, Mod. Phys. Lett. A2 (1987) 893. [41] A.M. Polyakov, Les Houches, 1988; E. Abdalla, M.C.B. Abdalla and A. Zadra, Trieste-Preprint IC/89/56 unpublished. [42] A.M. Polyakov, A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 1213; E. Abdalla, M.C.B. Abdalla and A. Zadra, Mod. Phys. Lett.AA (1989) 849.

Chapter 2

Free Fields 2.1

Introduction

A study of quantum field theory usually starts out with the definition of free fields. In two-dimensional space-time this is not only the rule, but also shows an interesting feature concerning the zero-mass limit of bosonic free fields. Their correlators do not define tempered distributions, due to infrared divergencies, which cannot be tamed in two dimensions [l]-[3]. However, for correlators involving the exponential of free fields the infrared divergencies are shown to cancel, once a superselection rule is imposed [2, 4]. Therefore, charges may be naturally defined for real bosonic fields in two dimensions, a very important issue that shall be tackled in detail in Chapter 3, and also used very often later on. Nonetheless, it is not so trivial to obtain these results from a path integral formulation in the general case, since the equivalence between fermions and bosons is a collective phenomenon. Moreover, we shall see that topological aspects are the major issue in the case of the bosonization procedure, when we analyse the connection between the massive-Thirring model and the sine-Gordon theory in Chapter 3.

2.2

Bosonic Free Fields

Massive real bosonic fields in two-dimensional space-time are defined in terms of the free-field Lagrangian

with the corresponding Klein-Gordon equation (a+m2)(p

=0

.

(2.1)

The solution of (2.1) has the Fourier decomposition

/

Hk1 -j=={a{k)e-ik*+a{kyeik*]

= ^+\x)+^-\x)

,

(2.2)

26

Free Fields

where k0 = \Jk\ + m 2 , and (+), (—) refer to the positive and negative frequency parts. The momentum conjugate to ( x - y ) = 2TT

J 2k°

= —K0

- y)2 + i(x° - y°)e)

(my/-{x

,

(2.4)

while for the commutator-function we have iA{x - y) = [(+) (a;) = _ i - i n ( _ M V + ix°e)

,

(2.8)

4-7T

where fi = e7A and 7 is the Euler constant. It is clear that in the above the two-point function violates the Wightman axiom of positivity [1, 3], due to the term that has been subtracted in (2.7). Indeed, for test functions with support in —x2 > -^, the norm is negative, as we immediately see from (2.8). These arguments however do not preclude the existence of well defined operators obtained from the field 0 leads to charge superselection rules, as we next demonstrate. Consider the two-point function given by (2.4) in the zero-mass limit; we have (0\(+)( x; X) = ~

lim <j /

^ fk

Vk2 + m2

J f o ( n u ' ) — ./

cos kx1-^-

,. fk

f

2n Jn

cos kx1

^p

dy —j==cos(myx1) \/y2 + l

for m and A small, cos(mya;) « 1, and using i f o f m i 1 ) RJ - I n f y i r a 1 ) , we thus have

Dt+'tO^jA) w — L m £ l m x i _ J_ l n (V2 A) 2TT

2

2TT

m'

=

_J_i n ( Ve 7 Aa; i) 2TV

'

.

The ie prescription is chosen in such a way that the integral (2.7) converges. Lorentz invariance implies (2.8).

28

Free Fields

We may however construct well denned operators satisfying the Wightman axioms in terms of Wick ordered exponential of zero-mass fields [8, 4], . e%aip{x).

1

_

e%aifi'-~'

(x)e%aip(+'(:r) and D^+\x)

as

^ + ) ( r ) = i?(+)(o £ ( + ) ( £ ' ) = £(+)(£) + A .

•

(2.24)

Z7T

As we next show, free Dirac fields may be represented in terms of the exponentials of ip and (p.

2.3

Fermionic Free Fields

The massive free spin-| field obeys the Dirac equation (i @ - m)tp = 0 , which can be derived from the Lagrangian C=^{ip-m)ip .

(2.25)

The Fourier decomposition of the fermionic field is given by

/

dk1 -^={b(k)u(k)e-ik*+J(k)v(k)eik*}

,

with

{bHk),b(p)} = {dt(fc),d(P)} = sik1 -pl) 2

We use In (x + ie) — In \x\ + m9(—x).

.

(2.26)

30

Free Fields

In the Weyl-representation of the 7M-matrices, * = ( ' ; ; )

7i=(_°1

;'j

«(*) = ( ;'k-/ £ )) ', v(k) =\

-Vk

,

(2.27)

the Dirac-spinors take the form v J

fVk+

They have the usual property uaup =fcM7M+ m

,

VaVp = k^j^ - m .

In the zero-mass limit we have U(*)

= V^(^J))

, «(*) = v ^ ( _ J ^ 1 ) )

.

(2.28)

In the massless case, due to the separation of right- and left-movers, we can rewrite (2.26) in terms of the spinor u(k) alone, by defining an operator d{k1) as d(k1) = d(k1) for A;1 > 0 , dik1) =-dik1)

ioi k1 < 0 .

Expression (2.26) then takes the form 4>{x) = [ -^tL=u(k)[b(k)e-ik* J v27r2A;u

+${k)eikx]

.

(2.29)

The canonical anticommutator for the ip(x) fields (2.26) reads, for arbitrary times, {^a(x),i/jp(y)}=iS(x-y)ap , where S{x-y) = (i0 + m)A(x-y)

,

and A(x — y) is given by (2.5). Analogous to (2.4)-(2.6) we also define the two-point functions (0\M^p(y)\0)=(i^

+ m)A^(x-y)=S^(x-y)

,

(O\T-ipa{x)^0(y) | 0) = -i{i $> + m)AF{x -y) = iSF(x - y)

.

These functions are well defined in the zero-mass limit. 3 For m = 0 one has SM(x) = --(_J_ 3

0

In the zero-mass limit, we have D(x) = -\6(x2)e(x°)

l

*--" )

•

and (2.8) for D(x)

(2.30)

2.4 Bosonization of Massless Fermions

31

There exist two conserved local currents: with the usual charge conservation is associated the conserved vector current j^x)

= :^(x)j^(x):

,

(2.31)

where the double points indicate normal ordering; the chiral symmetry of the Lagrangian (2.25) in the zero-mass limit implies in turn the conservation of the axial vector current jl{x)

= :^(Z)7M75(Z):= W

,

(2-32)

where we used the relation (see Appendix A) 7^,75 = eM„7". Conservation of charge and pseudo-charge thus implies V

=0 =e " " ^

,

or d±u=0

.

(2.33)

Majorana representation Another useful representation for the gamma matrices is given by 70 =

0 I i

- A 0) •

71

(0 i "V* 0

In terms of this representation, the Dirac equation leads to d+ip2 - mipi = 0 , d-ipi + mip2 = 0 • The spinors u and v defined in (2.26) are now given by

2.4

Bosonization of Massless Fermions

In two dimensions we may always write a vector field in the form U = d»