The Global Approach to Quantum Field Theory 2-Volume Set

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ISBN: 9780198510932 | 0198510934
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Copyright: 5/1/2003

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This new volume takes a complete look at how classical field theory, quantum mechanics and quantum field theory are interrelated. It takes a global approach and discusses the importance of quantization by relating it to different theories such as tree amplitude and conservation laws. There are special chapters devoted to Euclideanization and renormalization, space and time inversion and the closed-time-path formalism.

I CLASSICAL DYNAMICAL THEORY

Fundamentals

(10)

Classical Dynamical Systems

(1)

Space of Histories, Functional Differentiation

(2)

Examples of Simple Functionals. Notational Abbreviations

(3)

Condensed and Supercondensed Notations

(2)

Changing Topology

(1)

Comments on Chapter 1

(1)

Reference

(1)

Dynamics and Invariance Transformations

(17)

Action Functional and Dynamical Equations

(1)

Invariance Transformations

(3)

Commutator of Invariance Transformations

(1)

Gauge Algebras. Gauge Groups. Orbits

(4)

Functional Differential Equation for the Structure Functions

(2)

Further Simplification of Notation

(1)

Physical Observables

(1)

Gauge Groups and Manifest Covariance

(2)

Comments on Chapter 2

(1)

References

(1)

Small Distrubances and Green's Functions

(16)

The Equations of Small Disturbances. Jacobi Fields

(1)

The Structure of i, S,j

(1)

Other Identities Satisfied by the Coefficient Matrices

(1)

Supplementary Conditions

(2)

Retarded and Advanced Green's Functions of F

(1)

Equality of Left and Right Green's Functions

(1)

Reciprocity Relations

(1)

Explicit Structure of F

(2)

Relation between the Green's Functions of J and F

(2)

The Green's Functions Off Shell. Coherent Green's Functions

(2)

Landau Green's Functions

(1)

Transformation Laws for the Green's Functions

(1)

Comments on Chapter 3

(1)

The Peierls Bracket

(14)

Disturbances in Physical Observables

(1)

The Reciprocity Relation for Physical Observables

(1)

Invariance Properties of DAB

(2)

The Peierls Bracket and the Supercommutator Function

(1)

The Bracket Identities

(1)

Standard Canonical Systems. Equivalence of Peierls and Poisson Brackets

(3)

The Wronskian Operator

(2)

The Cauchy Problem for Jacobi Fields

(2)

Comments on Chapter 4

(1)

Reference

(1)

Finite Disturbances. Tree Theorems. Asymptotic Fields

(18)

Differential and Integro-differential Equations for Finite Disturbances

(3)

Iterative Solutions. Tree Functions

(1)

Background Field. Transformation Laws

(2)

Changes in P and k

(1)

Gauge Transformations

(1)

Tree Amplitudes and Tree Theorems

(2)

Tree Theorems When the Configuration Space is not a Vector Space

(1)

Change of Chart

(3)

Asymptotic ``In'' and ``Out'' Fields

(4)

Comments on Chapter 5

(1)

Reference

(1)

Conservation Laws

(39)

Alternative Action Functional

(1)

Internal Sources

(1)

Killing Fields, Currents, and Charges

(2)

The Charge Group

(1)

Flow Invariance of the Charges

(1)

Fundamental Lemma

(2)

A Special Transformation

(2)

The Peierls Bracket (qA, qB). The View from the Remote Past or Future

(3)

The View from Spatial Infinity

(3)

Compact Universes and Linearization Constraints

(1)

Pseudocovariants

(1)

Externally Impressed Fields

(1)

Charges in the Presence of External Fields

(2)

Peierls Bracket for the Charges

(1)

Differences form the Previous Theory

(3)

External Gravitational Field. Diffeomorphism Gup

101

(2)

Stress-energy Density

103

(4)

Killing Vector Fields. The Poincare Group

107

(1)

Energy-momentum Vector and Angular Momentum Tensor

108

(1)

Local Conformal Group

109

(1)

Conformal Killing Vector Fields. The Global Conformal Group

110

(1)

Conformal Charges and their Peierls Brackets

111

(1)

Comments on Chapter 6

112

(1)

References

113

(4)

II THE HEURISTIC ROAD TO QUANTIZATION. THE QUANTUM FORMALISM AND ITS INTERPRETATION

Classical Theory of Measurement

117

(14)

System. Apparatus. Coupling

117

(1)

Apparatus Inertia

118

(2)

Disturbance in the Apparatus

120

(1)

Design of the Coupling. Accuracy of the Measurement

121

(1)

Compensation Term

122

(1)

Stern-Gerlach Experiment

123

(2)

Electric Field Measurement

125

(5)

Comments on Chapter 7

130

(1)

Reference

130

(1)

Quantum Theory of Measurement

131

(14)

Measurement of Two Observables

131

(1)

Mutual Uncertainties

132

(1)

A Universal Uncertainty Principle

133

(1)

The Heuristic Quantization Rule

134

(2)

A Preferred Basis

136

(2)

Relative States. Good Measurements

138

(2)

Many Worlds

140

(1)

Unobservability of the Splits

141

(3)

Comments on Chapter 8

144

(1)

References

144

(1)

Interpretation of the Quantum Formalism I

145

(17)

Reality

145

(1)

Signalling by Permutations

146

(1)

Equal Likelihood

147

(2)

Unequal Probabilities

149

(1)

Irrational Probabilities

150

(2)

Expectation Value. Single System vs. an Ensemble

152

(1)

Relative Frequency Function. The Ensemble Interpretation of Quantum-mechanical Probabilities

153

(3)

Density Operator. Decoherence

156

(3)

Imperfect Measurements

159

(2)

Comments on Chapter 9

161

(1)

Reference

161

(1)

The Schwinger Variational Principle and the Feynman Functional Integral

162

(16)

Transition Amplitudes

162

(1)

The Schwinger Variational Principle

162

(3)

External Sources and Chronological Products

165

(3)

The Operator Dynamical Equations and the Measure Functional

168

(3)

Functional Fourier Analysis. The Feynman Functional Integral

171

(2)

Comments on the Feynman Integral

173

(2)

The Schwinger Variational Principle Revisited

175

(1)

Expressions Involving S,i

176

(1)

Comments on Chapter 10

177

(1)

References

177

(1)

The Quantum Mechanics of Standard Canonical Systems

178

(17)

Problems with the Naive Quantization Rule

178

(1)

The Position Mapping

178

(2)

Vector Operators

180

(3)

The Momentum Operator

183

(1)

Restriction to a Local Chart

184

(2)

Overlapping Charts. Transformation of Coordinates

186

(2)

The Position Representation

188

(1)

The Projection m-form

189

(1)

The Momentum Operator in the Position Representation

190

(2)

The Schrodinger Equation

192

(2)

Remarks on Global Consistency

194

(1)

Comments on Chapter 11

194

(1)

Reference

194

(1)

Interpretation of the Quantum Formalism II

195

(18)

Tracing Out

195

(1)

The Model System

196

(2)

Density Operator

198

(1)

Localization. Sharp Decoherence

199

(2)

Discussion

201

(1)

Coarse Graining. Decoherence Function

202

(1)

Interpretation of the Diagonal Elements

203

(2)

Emergence of Classicality

205

(2)

Many Worlds Again. Prbability as a Phenomenological Concept

207

(2)

Comments on Chapter 12

209

(1)

References

209

(4)

III EVALUATION AND APPROXIMATION OF FEYNMAN FUNCTIONAL INTEGRALS

The Functional Integral for Standard Canonical Systems

213

(21)

The Path Integral

213

(1)

The Point-to-point Amplitude. The Action Function

213

(3)

Formal Computation of det G[x] in the Lagrangian Formalism

216

(1)

Formal Computation of det G[x, p] in the Hamiltonian Formalism

217

(2)

Ambiguity in the Path Integral

219

(1)

Homotopy

220

(1)

The Universal Covering Space. Covering Translations

221

(2)

The Relation of Homotopy to Homology and Cohomology

223

(2)

The Path Integral in C

225

(2)

Fundamental Domains

227

(1)

Partial Amplitudes in C. The Total Amplitude

228

(1)

Combination Law

229

(1)

The Hamiltonian Form of the Path Integral and the Schrodinger Equation

230

(2)

Homotopy in Quantum Field Theory

232

(1)

Comments on Chapter 13

233

(1)

References

233

(1)

Approximation and Evaluation of the Path Integral

234

(17)

The Van Vleck-Morette Determinant

234

(2)

Jacobi Fields and Green's Functions for the Trajectory xc

236

(3)

Determinantal Relations

239

(2)

The Loop Expansion

241

(2)

The WKB Approximation

243

(3)

Other Boundary Conditions

246

(1)

WKB Approximation

247

(3)

Comments on Chapter 14

250

(1)

References

250

(1)

The Nonrelativistic Particle in a Curved Space

251

(24)

From the Hamiltonian Path Integral to the Lagrangian Path Integral

251

(1)

The Nonrelativistic Particle in a Curved Space

252

(1)

Covariant Variation

253

(1)

Covariant Differentiation with Respect to t

254

(1)

The Dynamical Equations

255

(1)

Covariant Functional Differentiation

255

(2)

The Measure for the Lagrangian Path Integral

257

(2)

Computation of HQ

259

(3)

Normalization of the Path Integral

262

(1)

A Two-loop Calculation

263

(2)

Analysis of the Jacobi Field Operator

265

(2)

The Morse Index

267

(1)

Morse Index Theorem

268

(3)

Application to Path Integration

271

(1)

The Generalized Morse Index

272

(2)

Comments on Chapter 15

274

(1)

References

274

(1)

The Heat Kernel

275

(16)

History

275

(1)

Geodesic Normal Coordinates

275

(2)

Coincidence Limits

277

(2)

Caustics

279

(2)

Riemannian Connection. The World Function

281

(2)

Auxiliary Geometrical Quantities. Modified Covariant Derivative

283

(1)

Further Coincidence Limits

284

(1)

Heat-kernel Expansion. Recursion Relations

285

(1)

Comments on Chapter 16

286

(1)

References

287

(4)

IV LINEAR SYSTEMS

Linear Boson Fields in Stationary Backgrounds

291

(29)

The Scalar Field

291

(1)

Stationary Backgrounds

292

(1)

The Field Equations

293

(2)

The Energy

295

(1)

Energy Bounds

296

(1)

Mode Functions

297

(1)

Alternative Representation

298

(2)

Multiple Roots

300

(1)

Wronskian Relations

301

(1)

Zero Roots

302

(2)

Wronskian Relations Again

304

(3)

General Solution. Supercommutator Function. Energy

307

(1)

Matrix Identities

307

(2)

Mode Functions for the Scalar Field

309

(1)

The Massless Scalar Field in a Compact Universe

310

(2)

The Vector Field

312

(3)

Rescaling of Time and Energy. Canonical Form for the Energy

315

(3)

Conformally Invariant Scalar and Vector Fields

318

(1)

Comments on Chapter 17

319

(1)

Quantization of Linear Boson Fields

320

(21)

Green's Functions

320

(1)

Quantization

321

(1)

Super Hilbert and Fock Spaces

322

(5)

Nonstationary Backgrounds and Inequivalent Vacua

327

(1)

Nonuniqueness of ∂/∂x

328

(2)

Vacua Defined by Symmetry Properties

330

(1)

The Feynman Propagator

330

(2)

Feynman Propagators for the Scalar and Vector Fields

332

(3)

When the Energy is Unbounded from Below

335

(2)

When K-1 M is Fully Diagonalizable and has Pure Imaginary Eigenvalues

337

(2)

Comments on Chapter 18

339

(1)

References

340

(1)

Linear Fermion Fields. Stationary Backgrounds

341

(26)

Local Lorentz Frames

341

(1)

The Dirac Matrices

342

(2)

Spin Structures

344

(1)

Space Inversions and Pin Structures

345

(2)

Spinor Fields

347

(2)

The Spin Connection

349

(1)

Generalized Spin or Pin Structures

350

(1)

Real Representations

351

(1)

Lagrange Function. Conformal Invariance. Majorana Representation

351

(2)

Stress-energy Density

353

(2)

Leibniz Rule. Matrix Identities. Canonical Form for the Energy

355

(2)

Energy-momentum Conservation and the Poincare Group

357

(2)

The Dirac Operator

359

(1)

Stationary Backgrounds. A Special Field of Local Lorentz Frames

360

(2)

A Model Stem

362

(1)

Mode Functions

362

(3)

When ---iB is not Positive Definite

365

(1)

Comments on Chapter 19

365

(1)

References

366

(1)

Quantization of Linear Fermion Fields

367

(10)

Green's Functions

367

(1)

Quantization

367

(1)

Fock Space

368

(2)

Hole Theory

370

(2)

The Feynman Propagator

372

(4)

Comments on Chapter 20

376

(1)

Linear Fields in Nonstationary Backgrounds

377

(19)

``In'' and ``Out'' Regions

377

(2)

Bogoliubov Relations

379

(2)

``In'' and ``Out'' Fock Spaces and the S-matrix

381

(1)

Particle Production and Annihilation Amplitudes

382

(3)

One-particle Scattering Amplitudes

385

(1)

Nonsingularity of α

386

(1)

Unitarity. The Vaccum Persistence Amplitude

387

(2)

Unitarily Inequivalent Fock Spaces

389

(2)

Green's Function Representation of eiW. The Feynman Propagator

391

(2)

Functional Integral Representation of eiW

393

(1)

Comments on Chapter 21

394

(1)

References

395

(1)

Linear (or Linearized) Fields Possessing Invariant Flows

396

(13)

The Electromagnetic Field in a Stationary Curved Background

396

(1)

Other Linear or Linearized Fields on Stationary Backgrounds

397

(2)

Mode Functions for F and F

399

(2)

Mode-function Decompositions of G and k

401

(1)

Quantization

402

(1)

Fields on Nonstationary Backgrounds

402

(2)

Green's Function Representation of eiW

404

(2)

Comments on Chapter 22

406

(3)

V NONLINEAR FIELDS

The Effective Action, the S-matrix, and Slavnov-Taylor Identities

409

(47)

Anticipation of Counter Terms

409

(1)

Correlation Functions

410

(1)

The Effective Action

411

(2)

The Relation of Γ to W. The Legendre Transform

413

(1)

The Correlation Functions as Tree Functions

413

(1)

Structure of the Effective Action

414

(2)

The Loop Expansion

416

(3)

Real-valuedness of the Counter Terms. Perturbative Renormalizability

419

(2)

Asymptotic Fields

421

(1)

Asymptotic States

422

(2)

Wave-packet States

424

(2)

The S-matrix and the Scattering Operator

426

(1)

The Lehmann-Symanzik-Zimmermann Theorem

427

(3)

Mode Functions for the Effective Action

430

(2)

The Effective Action as the Generator of Quantum Dynamics

432

(1)

Coherent States as Relative Vacua

433

(3)

Evaluation of (+, rel vac-, rel vac)

436

(2)

Expression of the S-matrix in Terms of Quantum Tree Functions

438

(1)

The Cluster Decomposition Principle

439

(1)

Use of a (Super)Classical Background

440

(2)

Construction of the Relative S-matrix

442

(2)

Structure of the Relative S-matrix

444

(2)

When the Configuration Space is Not a Supervector Space

446

(3)

Relation to the Classical Tree Theorems

449

(1)

Slavnov-Taylor Identities

450

(2)

Current Algebra

452

(1)

Relation to the Effective Action. Finiteness of the Current Operator

453

(1)

Comments on Chapter 23

454

(1)

References

455

(1)

Gauge Theories I. General Formalism

456

(36)

Structure of the Space of Field Histories

456

(1)

Fibre-Adapted Coordinated Patches

457

(2)

A Metric for Φ/G

459

(1)

Vilkovisky's Connection

460

(2)

Properties of Vilkovisky's Connection

462

(1)

The Functional Integral for ``In-Out'' Amplitudes

463

(2)

Properties of the Jacobian J[Y]

465

(2)

A Special Choice for Ω and a New Measure Functional

467

(1)

Explicit Form for the New Measure

468

(2)

Consistency of Eq. (24.80)

470

(1)

Ghosts

471

(2)

Relation to the Functional G. The Batalin---Vilkovisky Equation

473

(1)

Counter Terms, the Measure Functional, and the Quantum BV Equation

474

(2)

The Slavnov Operator and BRST Transformations

476

(1)

Cohomology of the Slavnov Operator

477

(1)

Loop Decomposition of the Measure

478

(2)

The Role of the Measure Functional

480

(2)

The Effective Action

482

(2)

The Zinn-Justin Equation. Proof of Eq. (24.125)

484

(1)

The Yang-Mills Field in Four Dimensions

485

(3)

Rigidity of the Gauge Group

488

(1)

Renomalization Constants

489

(1)

Comments on Chapter 24

490

(1)

References

491

(1)

Gauge Theories II. Background Field Methods. Scattering Theory

492

(21)

Invariants. The Quantum Slavnov Operator

492

(1)

Integrating Out the Ghosts

493

(1)

Reduced Effective Action

494

(1)

Introduction of a Background Field

495

(1)

Gauge Fixing

496

(1)

Loop Expansion of the Reduced Effective Action

496

(2)

Structure of the Reduced Effective Action

498

(2)

The Full Quantum Shell

500

(1)

Alternative Loop Expansions

500

(1)

The S-matrix

501

(2)

Mode Functions for the Effective Action

503

(2)

S-matrix Relative to an Arbitrary Background

505

(2)

Use of the (Super)Classical Background

507

(2)

A Special Phenomenon

509

(1)

Use of Γ[y] in Constructing the S-matrix

510

(1)

Comments on Chapter 25

511

(1)

Reference

512

(1)

Case-I Gauge Theory without Ghosts. Description of Cases II and III

513

Geodesic Normal Fields

513

(1)

A New Effective Action

514

(2)

The Illusory Ghost

516

(2)

The Loop Series

518

(1)

Rules for Differentiating the Measure

519

(1)

Invariance of the Loop Graphs

520

(2)

Renormalization

522

(1)

Difficulties in Applying the LSZ Theorem

523

(1)

Case-II Systems

524

(1)

Uncertain Cohomology of the Slavnov Operator

525

(1)

Decomposition of the Effective Action

525

(1)

Scattering Theory

525

(1)

Case-III Systems

526

(2)

Comments on Chapter 26

528

(1)

Reference

528

Index

Volume 2

VI Tools for Quantum Field Theory. Applications

The Heat Kernel

531

(1)

Formal Representation of the Feynman Propagator in the Nonstationary Case

531

(1)

Schwinger's Representation and the Heat Kernel

532

(1)

Some Technical Details. Coincidence Limits

533

(1)

Geodetic Parallel Displacement

534

(2)

The Cartan-Killing Metric

536

(1)

Application to the Lorentz Group

537

(3)

Traces

540

(1)

The Heat-kernel Expansion

541

(1)

Expansion for the Feynman Propagator

542

(2)

Difficulties in the Nonflat Case

544

(2)

Heat-kernel Representation of eiW

546

(1)

Dimensional Regularization

547

(2)

δW for the Spinor Field

549

(2)

Heat Kernel for the Spinor Field

551

(1)

Dimensional Regularization in the Spinor Case

552

(2)

Zeta Function Regularization

554

(2)

Zeta Function Representation of W

556

(2)

Electromagnetic Background

558

(1)

Comments on Chapter 27

558

(1)

References

559

(1)

Vacuum Currents. Anomalies

560

(1)

Formal Invariances of the Vacuum Persistence Amplitude

560

(1)

The Trace Anomaly

561

(3)

Alternative Derivation of the Trace Anomaly

564

(1)

Conformal Invariance of the Finite Local Terms

565

(1)

Trace Anomaly for the Massless Spinor Field

566

(2)

Chiral Coupling

568

(2)

Traces

570

(2)

Gauge Invariance of the Classical Theory

572

(1)

Vaccum Persistence Amplitude and Heat Kernel

572

(2)

Asymptotic Expansion

574

(1)

The Zeta Function and its Gauge Transformation Law

575

(2)

The Chiral Anomaly

577

(1)

Robustness of the Chiral Anomaly

578

(2)

Computation of the Chiral Anomaly

580

(2)

The Wess-Zumino Consistency Condition

582

(2)

Comments on Chapter 28

584

(1)

References

585

(1)

More Vacuum Phenomena

586

(1)

Particle Production by Weak Backgrounds

586

(1)

Explicit Expressions for Tμν (u~, u)o and jαμ(u~, u)o

587

(2)

Average Number of Particles Producted

589

(3)

Energy Production

592

(1)

No Particle Production by Freely Propagating Linear Background Waves

592

(1)

Expectation Values of the Current Operators

593

(1)

Physical Renormalization of the Current Operators

594

(1)

The Case of the Scalar Field. Lack of Uniqueness when n=4

595

(3)

The Spinor Field. Massless Fields

598

(1)

Conformally Invariant Fields in Conformally Flat Spacetimes

599

(1)

Conformally Invariant Scalar Field

600

(3)

De Sitter and Anti-de Sitter Spacetimes

603

(1)

Einstein Static Spacetime

604

(1)

Alternative Vacua. Rindler Mode Functions

605

(3)

Stress Energy in the Rindler Vacuum

608

(1)

Connection Between Rindler and Minkowski Quanta. Bogoliubov Coefficients

609

(2)

Inverse Bogoliubov Coefficients

611

(1)

Formal Pair Production and Annihilation Amplitudes

612

(1)

Unitary Inequivalence of Minkowski and Rindler Folk Spaces

613

(1)

Thermality of the Minkowski Vacuum Relative to Rindler Fock Space

614

(3)

Properties of the Rindler Gas

617

(1)

Thermal Wightman Function

618

(2)

Detectors

620

(2)

Accelerating Detectors. The Unruh Effect

622

(2)

Comments on Chapter 29

624

(1)

References

624

(1)

Black Hole Vacua. Hawking Radiation

625

(1)

Scalar Field in a Black-hole Metric

625

(2)

Change of Coordinates. Mode Functions

627

(2)

Normalization

629

(1)

Kruskal Coordinates

630

(2)

Extended Mode Functions

632

(2)

Mode Functions in Region II

634

(1)

Mode Functions with ω2 < μ2

635

(1)

Analytic Continuation in ω

635

(2)

Quasi-bound States

637

(1)

Zeroes of α (l, ω)

637

(2)

The Boulware Vacuum

639

(1)

The Hartle-Hawking Vacuum

640

(1)

Properties of the Hartle-Hawking Vacuum

641

(2)

Computation of (Trenμν) for Large r

643

(3)

Reactions of a Monopole Detector to the Hartle-Hawking and Boulware Vacua

646

(1)

Black Holes Formed by Collapse

646

(1)

``In'' Mode Functions

646

(3)

``Out'' Mode Functions

649

(2)

Bogoliubov Coefficients

651

(3)

Pair Production

654

(1)

Wightman Function

655

(1)

Hawking Radiation

656

(1)

Temperature and Entropy of a Black Hole

657

(1)

Black-hole Luminosity

657

(1)

Decay and Ultimate Fate of Black Holes

658

(2)

Comments on Chapter 30

660

(1)

References

660

(2)

The Closed-time-path or ``In-in'' Formalism

662

(1)

Functional Integral for Expectation Values

662

(2)

Properties of the System Described by the Action SY

664

(1)

The Jacobi Field Operator. Mode Functions

665

(2)

Wronskian Relations and Supercommutator Function

667

(1)

Other Green's Functions

668

(2)

Return to the Fields Y+ and Y-

670

(2)

External Sources. Causality

672

(1)

The Measure

673

(1)

The ``In-In'' Effective Action

674

(2)

``In-in'' Formalism for Gauge Theories

676

(1)

Connection 1-form and Horizontal Projection Operator

677

(1)

Vilkovisky's Connection

678

(1)

Geodesic Normal Fields

679

(1)

The Effective Action

680

(1)

Alternative Propagator

681

(1)

The Measure

682

(1)

Discussion

683

(1)

Comments on Chapter 31

684

(1)

References

685

(4)

Special Topics

Euclideanization and Renormalization

689

(1)

Momentum Space

689

(1)

Vertex Functions

690

(1)

Graphical Rules in Momentum Space

691

(1)

Contribution of the Measure in One-loop Graphs

692

(4)

Wick Rotation

696

(1)

Euclideanization

697

(1)

Boundary Conditions

698

(1)

Failure of Euclideanization in the Case of Quantum Gravity

699

(1)

Graphs and Subgraphs

700

(1)

The Pole Operation

701

(1)

Counter Terms

702

(2)

Minimal Subtraction and Finiteness

704

(2)

Comments on Chapter 32

706

(1)

References

706

(1)

Canonical Transformations. Space Inversion and Time Reversal

707

(1)

Infinitesimal Canonical Transformations

707

(1)

Finite Canonical Transformations

708

(1)

Space Inversion

709

(2)

Examples of Systems Invariant Under Space Inversion

711

(1)

Quantum Theory

712

(1)

Pseudoscalar Field

713

(1)

Massive Vector Field in Four Dimensions

714

(1)

Massive Antisymmetric Field in Four Dimensions

714

(1)

Massive Spinor Field in Four Dimensions

714

(3)

Time Reversal

717

(1)

Transformation of the Jacobi Field Operator

718

(2)

Quantum Theory

720

(1)

Chiral Couplings

721

(1)

Time Reversal of Asymptotic Fields

721

(2)

Massive Antisymmetric Field in Four Dimensions

723

(1)

Massive Spinor Field in Four Dimensions

723

(2)

Massless Fields

725

(1)

CP Invariance for Massless Fields

726

(1)

The Relation Between Helicity, Chirality, and Charge

727

(2)

CP Invariance for Massive Fields

729

(1)

The PT Theorem for Even-dimensional Spacetimes

729

(1)

Comments on Chapter 33

729

(2)

Quantum Electrodynamics

731

(1)

Action and Gauge Group

731

(1)

Jacobi Field Operator

732

(1)

One-loop Effective Action

733

(1)

Electrodynamic β-function

734

(1)

Constant Fields

735

(2)

Computation of the Trace

737

(1)

Renormalization. Scattering of Light by Light

738

(2)

Pair Production

740

(2)

Effective Field Equations

742

(3)

Vacuum Polarization

745

(1)

Mass Operator

745

(2)

Anomalous Magnetic Moment

747

(5)

Atomic Level Shifts

752

(2)

Comments on Chapter 34

754

(1)

References

754

(1)

The Yang-Mills and Gravitational Fields

755

(1)

Action and Gauge Group for the Yang-Mills Field

755

(1)

Classical Field Equation and Jacobi Field Operator

756

(1)

The Operators &J and F and the Invariant Metric

757

(1)

Traces of the a2 Coefficients and the One-loop Effective Action

758

(1)

Yang-Mills β-function and Asymptotic Freedom

759

(1)

Stress-energy Density of the Yang-Mills Field

760

(1)

Yang-Mills Trace Anomaly

761

(2)

Constant Fields

763

(3)

Evidence for Confinement

766

(1)

Yang-Mills Charge

767

(2)

Careful Definition of G and Φ

769

(3)

Axial Gauge

772

(1)

Homotopy of Gauge Transformations. Cartan-Maurer Invariant

773

(1)

Evaluation of I ([D])

774

(4)

Chern-Simons Secondary Characteristic

778

(2)

Chern Number

780

(1)

The Total Amplitude. Instantons

781

(2)

Action and Gauge Group for the Gravitational Field

783

(1)

Classical Field Equation and Jacobi Field Operator

784

(1)

The Operators J and F and the Invariant Metric

785

(2)

Traces of the a2 Coefficients and the One-loop Effective Action

787

(1)

One-loop Finiteness

788

(1)

Gravitational Stress-energy Density

789

(1)

Energy, Momentum, and Angular Momentum in the Gravitational Field

790

(2)

Topological Questions

792

(1)

Coupled Yang-Mills and Gravitational Fields

792

(2)

Comments on Chapter 35

794

(1)

References

795

(4)

Examples. Simple Exercises in the use of the Global Formalism

The Nonrelativistic Particle in Flat Space

799

(1)

Action Functional and Green's Functions

799

(1)

Momentum and Energy

800

(1)

The Case Ji = O

801

(1)

The Transition Amplitude in the General Case

802

(3)

The Momentum-to-momentum Transition Amplitude

805

(1)

Variation of the Sources

806

(1)

A Simple Fermi System

807

(1)

Action Function and Green's Functions

807

(1)

Eigenvectors of y

808

(1)

The Energy

808

(1)

A Pure Basis

809

(1)

An Alternative Representation

810

(1)

The Functional Integral Representation of (y, ty1, 1)

811

(3)

Evaluation of the Functional Integral

814

(3)

Other Transition Amplitudes

817

(2)

The Average (Super)Classical Trajectory

819

(1)

Propagator of the Fermi System

820

(2)

A Fermi doublet

822

(1)

Action Function and Green's Functions

822

(1)

A Symmetrical Basis

823

(2)

The Energy

825

(1)

Coherent States

825

(1)

The Coherent-state Representation

826

(2)

Transition Amplitudes

828

(3)

The Stationary Trajectory

831

(2)

Variation of the Sources

833

(2)

Fermi Multiplet

835

(1)

Action Functional and Green's Functions

835

(1)

The Case of Even N

836

(1)

The Case of Odd N

837

(1)

Hamiltonian and Path Integrals

838

(1)

The Fermi Oscillator

839

(1)

Action Functional and Green's Functions

839

(1)

Mode Functions and Hamiltonian

840

(1)

Basis Supervectors

841

(1)

Coherent States

842

(1)

The Functional Integral Representation of (a*, ta1, t1)

843

(2)

Direct Evaluation of the Functional Integral

845

(1)

The Importance of Endpoint Contributions

846

(1)

The Stationary Trajectory as a Matrix Element

847

(1)

The Feynman Propagator

847

(2)

The Bose Oscillator

849

(1)

Action Functional and Green's Functions

849

(1)

Eigenvectors

850

(1)

Mode Functions and Hamiltonian

850

(2)

Energy Eigenvectors

852

(1)

Coherent States

852

(2)

Hamilton-Jacobi Theory

854

(2)

The Point-to-point Amplitude

856

(1)

The Functional Integral Representation (a*, ta1, t1)

857

(1)

The Stationary Path Between Coherent States

858

(1)

The Feynman Propagator

859

(1)

Energy Eigenfunctions

860

(2)

A Fourth-order System

862

(1)

Action Functional and Green's Functions

862

(1)

Wronskian Operator and Mode Functions

863

(1)

Hamiltonian

864

(1)

Wightman Function and Feynman Propagator

865

(1)

Pauli-Villars Regularization

866

(1)

Fourth-order Quantum Gravity

867

(1)

A Model for Ghosts

868

(1)

Action Functional and Green's Functions

868

(1)

Wronskian Operator and Mode Functions

869

(1)

Decomposition of the Dynamical Variables

870

(1)

Energy and Hamiltonian

871

(1)

Use of Ghosts in Gauge Theory

871

(2)

Free Scalar Field in Flat Spacetime

873

(1)

Mode Functions

873

(1)

Green's Functions

874

(2)

Massive Vector Field in Four-dimensional Flat Spacetime

876

(1)

Green's Functions

876

(1)

Mode Functions

877

(3)

Massive Antisymmetric Tensor Field

880

(1)

Action and Field Equation

880

(1)

Minkowski Spacetime

880

(1)

Mode Functions in Four Dimensions

881

(2)

Massive Symmetric Tensor Field in Flat Spacetime

883

(1)

Action and Field Equation

883

(1)

Jacobi Field Operator and Feynman Propagator

884

(1)

Mode Functions in Four Dimensions

885

(1)

Spin

886

(2)

Massive Spinor Field in Flat Spacetime

888

(1)

Action, Field Equation, and Feynman Propagator

888

(1)

Spinor Boosts

888

(1)

Polarization Spinors in Four Dimensions

889

(3)

Mode Functions

892

(2)

Spin

894

(1)

Massive spin-3/2 Field in Flat Spacetime

895

(1)

Action and Field Equation

895

(1)

Jacobi Field Operator and Feynman Propagator

896

(1)

Auxiliary Matrices when n = 4

897

(2)

Mode Functions and Polarization Vector-spinors

899

(1)

Wronskian Operator and Mode-function Relations

900

(1)

Spin

901

(1)

Electromagnetic Field in Flat Spacetime

902

(1)

The Special Character of Masslessness

902

(1)

Invariant-flow Vectors and Supplementary Conditions

902

(1)

Polarization Vectors

903

(1)

The Operator F and its Mode Functions

904

(1)

Spin. Helicity

905

(1)

Massless Symmetric Tensor Field in Flat Spacetime

906

(1)

Invariant-flow Vectors and Supplementary Condition

906

(1)

The Operator F and its Mode Functions

907

(1)

Physical Mode Functions

908

(1)

Spin. Helicity

909

(1)

Massless Spinor Field in Four-dimensional Flat Spacetime

910

(1)

Polarization Spinors

910

(1)

Properties of the Polarization Spinors

911

(2)

Mode Functions

913

(1)

Spin. Helicity

914

(1)

Massless Spin 3/4 Field in Four-dimensional Flat Spacetime

915

(1)

Invariant Flow Vectors and Supplementary Condition

915

(1)

The Operators F

916

(1)

Polarization Vector-spinors

917

(1)

Mode Functions

917

(2)

Spin. Helicity

919

(1)

Renormalization Group and Spontaneous Symmetry Breaking in the λY4 Model

920

(1)

The λY4 Model

920

(1)

Jacobi Field Operator, Heat Kernel, and Effective Action

921

(2)

Renormalization Group

923

(2)

Self-consistency Conditions

925

(2)

Energy Zero Point and Physical Mass

927

(1)

Sponteaneous Symmetry Breaking

928

(2)

The Relativistic Particle in Minkowski spacetime

930

(1)

Action Functional and Dynamical Equations

930

(1)

Gauge Invariance

931

(1)

Green's Functions

931

(1)

The Measure

932

(2)

The Ghost Determinant

934

(1)

The Functional Integral

935

(1)

Consistency Check

936

(2)

Evaluation of the Path Integral

938

(1)

Interpretation of the Path Integral

939

(1)

Change of Variables

940

(2)

Further Reduction of the Functional Integral

942

(1)

Elimination of λ

943

(2)

A Simple Soluble Nonlinear Model

945

(1)

Action Functional and Gauge Invariance

945

(1)

Dynamical Equations and Conserved Quantities

946

(1)

Energy

946

(1)

Jacobi Field Operator and Ghost Operator

947

(3)

Quantization

950

(1)

Dynamics

951

(1)

Quantum Mechanics on a Circle

952

(1)

Lagrangian and Periodicity Conditions

952

(1)

Homotopy

953

(1)

Energy Eigenfunctions

954

(2)

Quantum Mechancis on a Klein Bottle

956

(1)

Periodicity Conditions and Lagrangian

956

(1)

Energy Eigenfunctions

956

(1)

Homotopy

957

(2)

Ghosts for Ghosts

959

(1)

Linearly Dependent Flow Vectors

959

(1)

Nonsingular Operators

960

(1)

Mode Functions

961

(3)

Green's Function Relations

964

(2)

Nonstationary Background

966

(3)

Massless Antisymmetric Tensor Field

969

(1)

Flow Vectors

969

(1)

Matris and Operators

970

(1)

Four-dimensional Minkowski Spacetime

971

(1)

Heat Kernels and Vacuum Persistence Amplitude

972

(4)

Appendix A: Superanalysis

976

(1)

Supernumbers

976

(1)

Superanalytic Functions

977

(1)

Functions of Real Variables

978

(1)

Integration

979

(2)

Supervector Spaces

981

(2)

Bases

983

(1)

Change of Basis

984

(1)

Shifting Indices. The Supertranspose

985

(2)

Dual Supervector Spaces

987

(1)

Dual Bases

988

(1)

The Supertrace and Superdeterminant

989

(1)

Integration over Rcm x Ran

990

(1)

Transformation of Variables

991

(3)

Gaussian Integrals

994

(2)

Supermanifolds

996

(2)

Structures on Supermanifolds

998

(1)

Super Lie Groups

999

(1)

Super Hilbert Spaces

1000

(1)

Linear Operators. Superalgebras

1001

(1)

Physical Observables

1002

(1)

Basic Inequalities for Ordinary Hilbert Space. Averages and Variances

1003

(1)

Supercommutator and Antisupercommutator

1004

(2)

Appendix B

1006

(1)

The Structure Functions to All Orders

1006

(4)

Appendix C

1010

(1)

The Case in Which is not Supersymmetric

1010

(4)

Appendix D: Properties of Vilkovisky's Connection

1014

(1)

Horizontal Geodesics

1014

(1)

Derivation of Eq. (24.40)

1014

(1)

Vertical Geodesics

1015

(1)

Derivation of Eq. (24.41)

1016

(2)

Appendix E: Analytic Continuation in Dimension

1018

(1)

Compilation of Familiar Equations

1018

(1)

Axioms for Analytic Continuation in (Euclidean) Dimensions

1019

(1)

Illustrative Examples

1020

(5)

Integrals over Euclidean Spaces of Even Negative Dimension

1025

(1)

External Parameters

1026

(2)

Integrals of Tensors

1028

(2)

Change of Integration Order

1030

(1)

Integration Over Minkowski Spacetime

1031

(4)

Analytic Continuation in the Presence of Dirac Matrices

1035

(1)

Analytic Continuation of Traces

1036

(2)

Appendix F: Conformally Flat Spacetimes

1038

(1)

Conformal Transformations

1038

(1)

Conformal Flatness in Two Dimensions

1039

(1)

Conformal Flatness in Three Dimensions

1040

(1)

Conformal Flatness in Dimensions Greater Than Three

1040

(2)

Spacetimes of Isotropic Curvature

1042

Index

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