Quantization of Gauge Systems 9780691213866

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Quantization of (~Jauge gystems

Published by Princeton University Press in association with the Centro de Estudios Cientificos de Santiago

Quantization of (j^auge gystems Marc Henneaux and Claudio Teitelboim

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Copyright © 1992 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 All Rights Reserved Library of Congress Cataloging-in-Publication Data Henneaux, Marc. Quantization of gauge Systems / Marc Henneaux and Claudio Teitelboim. p. cm. Includes bibliographical references and index. ISBN0-691-08775-X ISBN 0-691-03769-8 (pbk.) 1. Gauge fields (Physics) 2. Quantum theory. I. Teitelboim, Claudio. II. Title. QC793.3.F5H46 1992 530.T435—dc20 92-11585 Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America 10 9 8 7 6 5 4 3 2

The amount of theoretical work one has to cover before being able to solve problems of real practical value is rather large, but this circumstance is an inevitable consequence of the fundamental part played by transformation theory and is likely to become more pronounced in the theoretical physics of the future. — P.A.M. Dirac

(from the preface to the first edition of The Principles of Quantum Mechanics, Oxford, 1930)

CONTENTS

Preface Acknowledgments Notations

xxiii xxv xxvii

Chapter One. Constrained Hamiltonian Systems

3

1.1. Gauge Invariance—Constraints 1.1.1. The Lagrangian as a Starting Point: Primary Constraints 1.1.2. Conditions on the Constraint Functions 1.1.3. The Canonical Hamiltonian 1.1.4. Action Principle in Hamiltonian Form 1.1.5. Secondary Constraints 1.1.6. Weak and Strong Equations 1.1.7. Restrictions on the Lagrange Multipliers 1.1.8. Irreducible and Reducible Cases 1.1.9. Total Hamiltonian 1.1.10. First-Class and Second-Class Functions

3

First-Class Constraints as Generators of Gauge Transformations 1.2.1. Transformations That Do Not Change the Physical State. Gauge Transformations. 1.2.2. A Counterexample to the Dirac Conjecture 1.2.3. The Extended Hamiltonian 1.2.4. Extended Action Principle

4 6 9 11 12 13 13 14 15 15

1.2.

1.3.

Second-Class Constraints: The Dirac Bracket

16 16 19 20 21 21

viii

Contents

1.3.1. 1.3.2. 1.3.3. 1.3.4.

Separation of First-Class and Second-Class Constraints Treatment of Second-Class Constraints: An Example Dirac Bracket Reducible First-Class and Second-Class Constraints

21 22 23 25

1.4. Gauge Fixation—Independent Degrees of Freedom 1.4.1. Canonical Gauges 1.4.2. Counting of Degrees of Freedom 1.4.3. Do All Second-Class Constraints Arise from Gauge Fixation?

27 27 29

1.5. Gauge-Invariant Functions 1.5.1. Functions on the Constraint Surface 1.5.2. Classical Observables 1.5.3. Algebraic Characterization of the Observables 1.5.4. Gauge-Invariant Extensions

32 32 33 34 34

1.6. Examples 1.6.1. System with n Generations of Constraints 1.6.2. L = 0 a n d L = - ^ E O f ) 2

35 35 36

1.6.3.

More on the Consistency Algorithm

31

37

Appendix l.A. Global proof of G « 0 => G = gj(t>3

40

Exercises

41

Chapter Two. Geometry of the Constraint Surface

48

2.1. Induced Two-Form on the Constraint Surface 2.1.1. An Analogy: Surfaces in Minkowski Space 2.1.2. Geometry of Phase Space (Symplectic Geometry) 2.1.3. Induced Two-Form

49 49 49 50

2.2. First-Class Constraint Surface 2.2.1. Rank of Induced Two-Form 2.2.2. Null Surfaces and Gauge Orbits 2.2.3. Reduced Phase Space

52 52 53 54

2.3. Second-Class Constraints 2.3.1. Rank of Induced Two-Form 2.3.2. Dirac Bracket Revisited 2.3.3. Solving the Constraints inside the Action

55 55 56 58

2.4. Mixed Case

60

Contents

ix

Appendix 2.A. More on the Structure of the Reduced Phase Space

60

Exercises

63

Chapter Three. Gauge Invariance of the Action

65

3.1. Structure of the Gauge Symmetries 3.1.1. Notations 3.1.2. Gauge Transformations 3.1.3. Noether Identities 3.1.4. Gauge Group—Gauge Algebra 3.1.5. Trivial Gauge Transformations 3.1.6. Independent Noether Identities 3.1.7. Generating Sets 3.1.8. "Open Algebras" 3.1.9. Reducible Generating Sets 3.1.10. Relation between Different Generating Sets 3.1.11. Generating Sets and Gauge Orbits

66 66 67 68 69 69 71 71 72 73 74 74

3.2. Gauge Transformations of the Extended Action 3.2.1. Algebra of the Constraints 3.2.2. Gauge Transformations 3.2.3. Another Generating Set 3.2.4. Gauge Transformations as Canonical Transformations 3.2.5. Open and Closed Algebras 3.2.6. Reducible First-Class Constraints 3.2.7. Conclusions

75 75 76 77

Gauge Transformations of the Original Lagrangian Action 3.3.1. Gauge Symmetries of ST and SL 3.3.2. Proof of the Dirac Conjecture under Simplifying Assumptions 3.3.3. Lagrangian Form of the Gauge Transformations—Basic Equations 3.3.4. Solution of the Basic Equations 3.3.5. Lagrange Multiplier Dependence of Gauge Transformations 3.3.6. Gauge Invariance and Degree of Freedom Count 3.3.7. Total and Extended Hamiltonians Compared and Contrasted

78 79 80 82

3.3.

3.4. Noncanonical Gauges 3.4.1. Derivative Gauges

82 82 82 85 86 88 89 90 91 91

x

Contents 3.4.2. 3.4.3.

Multiplier Gauges Reducible Gauge Transformations—Redundant Gauge Conditions

Exercises

93 93 94

Chapter Four. Generally Covariant Systems

102

4.1.

102

Introduction

4.2. Time as a Canonical Variable—Zero Hamiltonian 4.2.1. Parametrized Systems 4.2.2. Zero Hamiltonian 4.2.3. Parametrization and Explicit Time Dependence

103 103 104 104

4.3. Time Reparametrization Invariance 4.3.1. Form of Gauge Transformations 4.3.2. Must the Hamiltonian Be Zero for a Generally Covariant System? 4.3.3. Simple Example of a Generally Covariant System with a Nonzero Hamiltonian

105 105

106

4.4. "True Dynamics" versus Gauge Transformations 4.4.1. Interpretation of the Formalism 4.4.2. Reduced Phase Space

107 107 108

Exercises

109

105

Chapter Five. First-Class Constraints: Further Developments

112

5.1.

112

Preliminaries and Notations

5.2. Abelianization of Constraints 5.2.1. Ambiguity in the Description of the Constraint Surface 5.2.2. Abelianization Theorem 5.3. Exterior Derivative Operator along the Gauge Orbits ("Longitudinal Derivative") 5.3.1. Definition of Longitudinal Derivative 5.3.2. Longitudinal Cohomology 5.3.3. Representation of Longitudinal Derivative in the Irreducible Case 5.3.4. Representation of Longitudinal Derivative in the Reducible Case

113 113 115 117 117 120 120 121

Contents 5.3.5.

Phase Space Characterization of Longitudinal Forms

xi 122

5.4. Hamilton-Jacobi Theory 5.4.1. Unconstrained Systems—Complete Integrals 5.4.2. Unconstrained Systems—Incomplete Integrals 5.4.3. Constrained Systems 5.4.4. Gauge Invariance of the Hamilton-Jacobi Solutions 5.4.5. Hamilton Principal Function

123 123 124 126

Exercises

130

128 129

Chapter Six. Fermi Degrees of Freedom: Classical Mechanics over a Grassmann Algebra

134

6.1. Fermions and Anticommuting c-Numbers

135

6.2. Formal Properties of Anticommuting c-Numbers 6.2.1. Grassmann Algebra 6.2.2. Superfunctions 6.2.3. Grassmann parity 6.2.4. Complex Conjugation 6.3. Changes of Variables 6.3.1. Invertible Matrices 6.3.2. Invertible Changes of Variables

136 136 138 139 140 140 140 141

6.4.

Canonical Formalism in the Presence of Odd Variables

143

6.5. Generalized Poisson Bracket 6.5.1. Definition 6.5.2. Properties of the Generalized Poisson brackets 6.5.3. Algebra of Superfunctions over Phase Space as the Central Object in Grassmann Mechanics

144 144 146

6.6. Physical Fermions Need First-Order Equations 6.6.1. A Simple Model System 6.6.2. Negative Norm States Generic for Nondegenerate Fermionic Lagrangians 6.6.3. Supersymmetry

148 148

6.7. Geometry of Phase Space in the Anticommuting Case 6.7.1. Phase Space 6.7.2. Supersymplectic Geometry

150 150 151

Exercises

151

147

149 150

xii

Contents

Chapter Seven. Constrained Systems with Fermi Variables 7.1. Odd-Dimensional Phase Space 7.1.1. Example 7.1.2. Boundary Term in Action Principle 7.1.3. Alternative Boundary Conditions in the Hamiltonian Variational Principle for Bosonic Variables

156 157 157 158 160

7.2. Incorporation of Appropriate Sign Factors 7.2.1. Gauge Transformations 7.2.2. Gauge Orbits—Exterior Derivative along the Gauge Orbits

161 161 162

Exercises

163

Chapter Eight. Graded Differential Algebras— Algebraic Structure of the BRST Symmetry

165

8.1. Introduction—Ghosts

165

8.2. Graded Differential Algebras 8.2.1. Supercommutative Algebras 8.2.2. Examples 8.2.3. Graded Lie Algebra of Graded Derivations 8.2.4. Gradings 8.2.5. Ideals 8.2.6. Differentials-Cohomology Algebras 8.2.7. Contracting Homotopy 8.2.8. Cohomology for the Lie Algebra of Derivations 8.2.9. Differential modulo 6 8.3. Resolution 8.3.1. Definition 8.3.2. Example

166 166 167 168 169 170 171 172 172 173 174 174 175

8.4. Elements of Homological Perturbation Theory 8.4.1. Main Theorem 8.4.2. Proof of the Main Theorem: (i) Existence of s 8.4.3. Proof of the Main Theorem: (ii) Evaluation oiHk(s) 8.4.4. Comments

177 177 178 179 181

8.5. Geometric Application: The BRST Construction in Brief 8.5.1. Introduction

181 181

Contents 8.5.2. 8.5.3. 8.5.4.

Geometric Ingredients BRST Differential Canonical Action of s

xiii 182 183 183

Exercises

184

Chapter Nine. BRST Construction in the Irreducible Case

187

9.1. Koszul-Tate Resolution 9.1.1. Definition 9.1.2. Homology of 6

187 187 189

9.2. Extended Phase Space 9.2.1. Ghosts and Longitudinal d 9.2.2. Bracket Structure—Ghost Number 9.2.3. 8 and d in the Extended Phase Space

189 189 190 191

9.3.

Bringing 6 and d Together: The BRST Symmetry as a Canonical Transformation 9.3.1. BRST Generator 9.3.2. Existence of the BRST Generator 9.3.3. The BRST Generator Is Unique up to Canonical Transformations

9.4. The 9.4.1. 9.4.2. 9.4.3. 9.4.4. 9.5.

BRST Generator in Simple Cases—Rank Abelian Constraints Constraints that Close According to a Group Higher Order Structure Functions Rank

Conclusions

192 192 193 195 196 196 196 197 197 198

Appendix 9.A. Proof of Theorem 9.1 (Homology of 6) 9.A.I. ^-Covering of Phase Space 9.A.2. Homology of 6 on O, at Positive Antighost Number 9.A.3. Homology of 6 on Va at Positive Antighost Number 9.A.4. Homology of 8

198 198

200 201

Exercises

201

Chapter Ten. BRST Construction in the Reducible Case

205

10.1. The Simplest Example

205

199

xiv

Contents

10.2. Description of Reducible Theories 10.2.1. First-Order Reducibility Functions 10.2.2. Completeness in Terms of Strong Equalities 10.2.3. Higher Order Reducibility Functions 10.2.4. Ambiguity in the Reducibility Functions 10.2.5. Canonical Form 10.3. The Koszul-Tate Differential 10.3.1. Nontrivial Cycles and How to Kill Them 10.3.2. Homology of S 10.4. More on the Longitudinal Exterior Differential 10.4.1. Problem with the Definition of the Extended Phase Space 10.4.2. The Longitudinal Differential 10.4.3. Auxiliary Differential A 10.4.4. Auxiliary Grading 10.4.5. The Differential D 10.4.6. Cohomology of D 10.4.7. Conclusions 10.5. BRST Transformation 10.5.1. Extended Phase Space 10.5.2. Combining 6 with D 10.5.3. Equations Determining the BRST Generator 10.5.4. Existence of the BRST Generator 10.5.5. Uniqueness of the BRST Generator

207 208 209 210 212 213

10.6. Conclusions

228

Appendix 10.A.I. 10.A.2. 10. A.3. 10.A.4.

228 228 229 230 231

10.A. Proofs of Theorems 10.1 through 10.4 ^-Covering of Phase Space Proof of Theorem 10.1 Proofs of Theorems 10.2 and 10.3 Proof of Theorem 10.4

213 213 216 216 216 217 218 219 220 221 221 222 222 223 223 225 226

Exercises

232

Chapter Eleven. Dynamics of the Ghosts— Gauge-Fixed Action

234

11.1. BRST Cohomology and the Poisson Bracket 11.1.1. BRST Observables 11.1.2. What Is the Meaning of the Higher Cohomological Groups Hk(s), k > 0? 11.1.3. Ghost Transformation Law under Global Symmetries

234 234 236 237

Contents 11.2. Ghost Dynamics; Gauge Fixing. The BRST Function as the Generator of a Symmetry 11.2.1. BRST-Invariant Hamiltonians 11.2.2. BRST Symmetry—Gauge-Fixed Action 11.2.3. Comments

xv 238 238 239 240

11.3. Nonminimal Solutions 11.3.1. Nonminimal Sector 11.3.2. The Lagrange Multipliers as Canonical Variables 11.3.3. The Faddeev-Popov Action 11.3.4. Lagrangian Form of the BRST Symmetry—The BRST Generator as a Noether Charge 11.3.5. Hamilton Principal Function and Ghosts

241 241 242 244

Exercises

249

Chapter Twelve. The BRST Transformation in Field Theory

253

12.1. Local Functionals and Nonintegrated Densities

254

12.2. Local Completeness and Regularity Conditions 12.2.1. Hamiltonian Definition of a Local Gauge Theory 12.2.2. Regularity Conditions 12.2.2a. Local Completeness of the Constraint Functions 12.2.2b. Zero Is a Regular Value of the Map Defined by the Constraint Functions 12.2.3. Local Completeness of the Reducibility Functions

259 259 260

12.3. Locality of the BRST Charge 12.3.1. Homology of S modulo dkjk as the Central Issue in the Problem of the Spacetime Locality of the BRST Formalism 12.3.2. Proof of Theorem 12.5: (i) Local Homology of 6 12.3.3. Proof of Theorem 12.5: (ii) Homology of 8 modulo dk j k 12.3.4. Locality of the Gauge-Fixed Action

263 263 265

Exercises

269

Chapter Thirteen. Quantum Mechanics of Constrained Systems: Standard Operator Methods

272

13.1. Quantization of Second-Class Constraints 13.1.1. An Example 13.1.2. Correspondence Rules in the General Case 13.1.3. Difficulties

273 273 273 274

246 247

260 260 262

267 269

xvi

Contents

13.2. Reduced Phase Space Quantization of First-Class Constraints 13.2.1. Description of the Method 13.2.2. Gauge Conditions 13.2.3. Difficulties

275 275 276 277

13.3. Dirac Quantization of First-Class Constraints 13.3.1. Formal Aspects 13.3.2. Anomalies 13.3.3. Generally Covariant Systems 13.3.4. Scalar Product 13.3.5. A Different Derivation of the Physical Condition 13.3.6. Projected Kernel of Gauge-Invariant Operators

277 277 279 280 281 283 283

13.4. Dirac-Fock Quantization of First-Class Constraints 13.4.1. Definition 13.4.2. Physical Subspace 13.4.3. Conclusions

286 286 288 290

Exercises

291

Chapter Fourteen. B R S T Operator Method— Quantum B R S T Cohomology

296

14.1. General Features 14.1.1. States and Operators 14.1.2. Ghost Number 14.1.3. Physical State Condition 14.1.4. Quantum BRST Cohomology 14.1.5. Anomalies

296 296 297 299 300 301

14.2. Analysis of Quantum BRST Cohomology: General Theorems 14.2.1. Jordan Canonical Form of the BRST Charge: Operator Cohomology versus State Cohomology 14.2.1a. State Cohomology 14.2.1b. Operator Cohomology 14.2.1c. Lefschetz Trace Formula 14.2.2. Duality Formula for the Operator Cohomology 14.2.3. (Pseudo-)Unitary Representations of the BRST-Ghost Number Algebra 14.2.4. Duality Formula for the State Cohomology 14.2.5. Physical States and Ghost Number 14.2.6. No Negative Norm State Criterion 14.3. Time Evolution

302 302 302 303 304 305 306 309 309 310 311

Contents 14.3.1. Schrodinger Equation 14.3.2. Unitarity in the Physical Subspace

xvii 311 312

14.4. BRST Quantization in the Fock Representation 14.4.1. BRST Charge and Ghost Number Operator 14.4.2. Quartet Mechanism 14.4.3. Comments

313 313 314 315

14.5. BRST Quantization and Solutions of the Constraint Equations Ga\ip) = 0 14.5.1. Quantum Constraints and Ordering of $7 14.5.2. Redefinitions of the Constraints 14.5.3. BRST Cohomology at Ghost Number ±m/2 14.5.4. Forming Ghost Number Zero States 14.5.5. BRST Formalism and Projected Kernels

317 317 318 319 322 323

Exercises

326

Chapter Fifteen. Path Integral for Unconstrained Systems

333

15.1. Path Integral Method of Bose Systems— Basic Features 15.1.1. Path Integral as a Kernel 15.1.2. Comments 15.1.3. Quantum Averages of Functionals 15.1.4. Equations of Motion—Schwinger-Dyson Equations 15.1.5. Stationary Phase Method—Lagrangian Path Integral

334 334 336 338 340

343

15.2. Path Integral in the Holomorphic Representation (Bose Systems) 15.2.1. Definition of Holomorphic Representation 15.2.2. Path Integral

346 346 348

15.3. Path Integral for Systems with Indefinite Metric 15.3.1. Introduction 15.3.2. Coordinate Representation 15.3.3. Path Integral in the Coordinate Representation 15.3.4. Holomorphic Representation 15.3.5. Path Integral in the Holomorphic Representation

349 349 349 351 352 354

15.4. Path Integral for Fermions 15.4.1. Path Integral in the Holomorphic Representation

355 355

xviii

Contents

15.4.2. Path Integral for the Weyl Symbol of the Evolution Operator 15.4.2a. Action Principle 15.4.2b. Weyl Correspondence Rule 15.4.2c. Path Integral Representation of the Evolution Operator 15.4.3. Example: Spin-i in a Magnetic Field 15.4.4. Ghost Transition Amplitude

356 357 357 359 360 362

15.5. A First Bite at the Antifield Formalism 15.5.1. Koszul-Tate Differential Associated with the Stationary Surface 15.5.2. Antibracket 15.5.3. Schwinger-Dyson Operator 15.5.4. Geometric Interpretation of A and of the Antibracket 15.5.5. The Antibracket Does Not Define a Measure

364

Exercises

373

Chapter Sixteen. Path Integral for Constrained Systems

380

16.1. Path Integral for Second-Class Constraints 16.1.1. Derivation of the Path Integral 16.1.2. Difficulties

381 381 382

16.2. Reduced Phase Space Path Integral 16.2.1. Derivation of the Path Integral 16.2.2. Faddeev Formula 16.2.3. Gauge Independence of Path Integral for a Parametrized System Illustrated. Equivalence of the Gauges t = r and t — 0 16.2.3a. Reduced Phase Space Transition Amplitude as a Reduced Phase Space Path Integral 16.2.3b. Canonical Gauge Conditions 16.2.3c. Gauge t = 0 16.2.3d. Gauge t