Holdings: The global approach to quantum field theory

The global approach to quantum field theory: 1

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Bibliographic Details
Main Author:	De Witt, Bryce S. (Author)
Format:	Book
Language:	English
Published:	Oxford [u.a.] Oxford Univ. Press 2005
Edition:	repr. 2004 (with corr.), 2005
Series:	... 114 The international series of monographs on physics ... Oxford science publications
Online Access:	Inhaltsverzeichnis
Item Description:	Hier auch später erschienene, unveränderte Nachdrucke
Physical Description:	XXXI, 528, 11 S. graph. Darst.
ISBN:	019852790x 9780198527909 9780198712862

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adam_text	CONTENTS VOLUME 1 I CLASSICAL DYNAMICAL THEORY 1 Fundamentals 3 Classical Dynamical Systems 3 Space of Histories. Functional Differentiation 4 Examples of Simple Functionals. Notational Abbreviations 6 Condensed and Supercondensed Notations 9 Changing Topology 11 Comments on Chapter 1 12 Reference 12 2 Dynamics and Invariance Transformations 13 Action Functional and Dynamical Equations 13 Invariance Transformations 14 Commutator of Invariance Transformations 17 Gauge Algebras. Gauge Groups. Orbits 18 Functional Differential Equation for the Structure Functions 22 Further Simplification of Notation 24 Physical Observables 25 Gauge Groups and Manifest Covariance 26 Comments on Chapter 2 28 References 29 3 Small Disturbances and Green s Functions 30 The Equations of Small Disturbances. Jacobi Fields 30 The Structure of ,·. S ; 31 Other Identities Satisfied by the Coefficient Matrices 32 Supplementary Conditions 33 Retarded and Advanced Green s Functions of F 35 Equality of Left and Right Green s Functions 36 Reciprocity Relations 37 Explicit Structure of F 38 Relation between the Greeďs Functions of 5 and F 40 x CONTENTS The Green s Functions Off Shell. Coherent Green s Functions 42 Landau Green s Functions 44 Transformation Laws for the Green s Functions 44 Comments on Chapter 3 45 4 The Peierls Bracket 46 Disturbances in Physical Observables 46 The Reciprocity Relation for Physical Observables 47 Invariance Properties of D J B 47 The Peierls Bracket and the Supercommutator Function 49 The Bracket Identities 50 Standard Canonical Systems. Equivalence of Peierls and Poisson Brackets 51 The Wronskian Operator 54 The Cauchy Problem for Jacobi Fields 56 Comments on Chapter 4 58 Reference 59 5 Finite Disturbances. Tree Theorems. Asymptotic Fields 60 Differential and Integro-differential Equations for Finite Disturbances 60 Iterative Solutions. Tree Functions 63 Background Field. Transformation Laws 64 Changes in P and к 66 Gauge Transformations 67 Tree Amplitudes and Tree Theorems 67 Tree Theorems When the Configuration Space is not a Vector Space 69 Change of Chart 70 Asymptotic In and Out Fields 73 Comments on Chapter 5 77 Reference 77 6 Conservation Laws 78 Alternative Action Functional 78 Internal Sources 78 Killing Fields. Currents, and Charges 79 The Charge Group 81 Flow Invariance of the Charges 82 Fundamental Lemma 83 A Special Transformation 85 Pages 1-528 appear in Volume 1, pages 529-1042 appear in Volume 2 CONTENTS xi The Peierls Bracket (qA, qB). The View from the Remote Past or Future 87 The View from Spatial Infinity 90 Compact Universes and Linearization Constraints 93 Pseudocovariants 94 Externally Impressed Fields 95 Charges in the Presence of External Fields . 96 Peierls Bracket for the Charges 98 Differences from the Previous Theory 98 External Gravitational Field. Diffeomorphism Group 101 Stress-energy Density 103 Killing Vector Fields. The Poincaré Group 107 Energy-momentum Vector and Angular Momentum Tensor 108 Local Conformai Group 109 Conformai Killing Vector Fields. The Global Conformai Group 110 Conformai Charges and their Peierls Brackets 111 Comments on Chapter 6 112 References 113 II THE HEURISTIC ROAD TO QUANTIZATION. THE QUANTUM FORMALISM AND ITS INTERPRETATION 7 Classical Theory of Measurement 117 System. Apparatus. Coupling 117 Apparatus Inertia 118 Disturbance in the Apparatus 120 Design of the Coupling. Accuracy of the Measurement 121 Compensation Term 122 Stem-Gerlach Experiment 123 Electric Field Measurement 125 Comments on Chapter 7 130 Reference 130 8 Quantum Theory of Measurement 131 Measurement of Two Observables 131 Mutual Uncertainties 132 A Universal Uncertainty Principle 133 The Heuristic Quantization Rule 134 A Preferred Basis 136 Pages 1-528 appear in Volume 1, pages 529-1042 appear in Volume 2 xii CONTENTS Relative States. Good Measurements 138 Many Worlds 140 Unobservability of the Splits 141 Comments on Chapter 8 144 References 144 9 Interpretation of the Quantum Formalism I 145 Reality 145 Signalling by Permutations 146 Equal Likelihood 147 Unequal Probabilities 149 Irrational Probabilities 150 Expectation Value. Single System vs. an Ensemble 152 Relative Frequency Function. The Ensemble Interpretation of Quantum-mechanical Probabilities 153 Density Operator. Decoherence 156 Imperfect Measurements 159 Comments on Chapter 9 161 Reference 161 10 The Schwinger Variational Principle and the Feynman Functional Integral 162 Transition Amplitudes 162 The Schwinger Variational Principle 162 External Sources and Chronological Products 165 The Operator Dynamical Equations and the Measure Functional 168 Functional Fourier Analysis. The Feynman Functional Integral 171 Comments on the Feynman Integral 173 The Schwinger Variational Principle Revisited 175 Expressions Involving 5,- 176 Comments on Chapter 10 177 References 177 11 The Quantum Mechanics of Standard Canonical Systems 178 Problems with the Naïve Quantization Rule 178 The Position Mapping 179 Vector Operators 180 The Momentum Operator 183 Restriction to a Local Chart 184 Overlapping Charts. Transformation of Coordinates 186 Pages 1-528 appear in Volume 1. pages 529-1042 appear in Volume 2 CONTENTS xiii The Position Representation 188 The Projection m-form 1 89 The Momentum Operator in the Position Representation 190 The Schrödinger Equation ] 92 Remarks on Global Consistency 194 Comments on Chapter 11 194 Reference 194 12 Interpretation of the Quantum Formalism II 195 Tracing Out 195 The Model System 196 Density Operator 198 Localization. Sharp Decoherence 199 Discussion 201 Coarse Graining. Decoherence Function 202 Interpretation of the Diagonal Elements 203 Emergence of Classicality 205 Many Worlds Again. Probability as an Emergent Concept 207 Comments on Chapter 12 209 References 209 III EVALUATION AND APPROXIMATION OF FEYNMAN FUNCTIONAL INTEGRALS 13 The Functional Integral for Standard Canonical Systems 213 The Path Integral 213 The Point-to-point Amplitude. The Action Function 213 Formal Computation of detG+[x] in the Lagrangian Formalism 216 Formal Computation of det G+[x. p] in the Hamiltonian Formalism 217 Ambiguity in the Path Integral 219 Homotopy 220 The Universal Covering Space. Covering Translations 221 The Relation of Homotopy to Homology and Cohomology 223 The Path Integral in Č 225 Fundamental Domains 227 Partial Amplitudes in C. The Total Amplitude 228 Combination Law 229 The Hamiltonian Form of the Path Integral and the Schrödinger Equation 230 Pages 3—528 appear in Volume 1. pases 529-1042 appear in Volume 2 xiv CONTENTS Homotopy in Quantum Field Theory 232 Comments on Chapter 13 233 References 233 14 Approximation and Evaluation of the Path Integral 234 The Van Vleck-Morette Determinant 234 Jacobi Fields and Green s Functions for the Trajectory xc 236 Determinantal Relations 239 The Loop Expansion 241 The WKB Approximation 243 Normalization 245 Other Boundary Conditions 246 WKB Approximation 247 Comments on Chapter 14 250 References 250 15 The Nonrelativistic Particle in a Curved Space 251 From the Hamiltonian Path Integral to the Lagrangian Path Integral 251 The Nonrelativistic Particle in a Curved Space 252 Covariant Variation 253 Covariant Differentiation with Respect to t 254 The Dynamical Equations 255 Covariant Functional Differentiation 255 The Measure for the Lagrangian Path Integral 257 Computation of HQ 259 Normalization of the Path Integral 262 A Two-loop Calculation 263 Analysis of the Jacobi Field Operator 265 The Morse Index 267 Morse Index Theorem 268 Application to Path Integration 271 The Generalized Morse Index 272 Comments on Chapter 15 274 References 274 16 The Heat Kernel 275 History 275 Geodesic Normal Coordinates 275 Coïncidence Limits 277 Caustics 279 Riemanman Connection. The World Function 281 Pages 1-528 appear in Volume 1, pages 529-1042 appear in Volume 2 CONTENTS xv Auxiliary Geometrical Quantities. Modified Covariant Derivative 283 Further Coincidence Limits 284 Heat-kernel Expansion. Recursion Relations 285 Comments on Chapter 16 286 References 287 IV LINEAR SYSTEMS 17 Linear Boson Fields in Stationary Backgrounds 291 The Scalar Field 291 Stationary Backgrounds 292 The Field Equations 293 The Energy 295 Energy Bounds 296 Mode Functions 297 Alternative Representation 298 Multiple Roots 300 Wronskian Relations 301 Zero Roots 302 Wronskian Relations Again 304 General Solution. Supercommutator Function. Energy 307 Matrix Identities 307 Mode Functions for the Scalar Field 309 The Massless Scalar Field in a Compact Universe 310 The Vector Field 312 Rescaling of Time and Energy. Canonical Form for the Energy 315 Conformally Invariant Scalar and Vector Fields 318 Comments on Chapter 17 319 18 Quantization of Linear Boson Fields 320 Green s Functions 320 Quantization 321 Super Hubert and Fock Spaces 322 Nonstationary Backgrounds and Inequivalent Vacua 327 Nonuniqueness of д/дх0 328 Vacua Defined by Symmetry Properties 330 The Feynman Propagator 330 Feynman Propagators for the Scalar and Vector Fields 332 When the Energy is Unbounded from Below 335 Pages 1-528 appear in Volume 1. pages 529-1042 appear in Volume 2 xvi CONTENTS When K~l Μ is Fully Diagonalizable and has Pure Imaginary Eigenvalues 337 Comments on Chapter 18 339 References 340 19 Linear Fermion Fields. Stationary Backgrounds 341 Local Lorentz Frames 341 The Dirac Matrices 342 Spin Structures 344 Space Inversions and Pin Structures 345 Spinor Fields 347 The Spin Connection 349 Generalized Spin or Pin Structures 350 Real Representations 351 Lagrange Function. Conformai Invariance. Majorana Representation 351 Stress-energy Density 353 Leibniz Rule. Matrix Identities. Canonical Form for the Energy 355 Energy-momentum Conservation and the Poincaré Group 357 The Dirac Operator 359 Stationary Backgrounds. A Special Field of Local Lorentz Frames 360 A Model System 362 Mode Functions 362 When —iB is not Positive Definite 365 Comments on Chapter 19 365 References 366 20 Quantization of Linear Fermion Fields 367 Greeks Functions 367 Quantization 367 Fock Space 368 Hole Theory t 370 The Feynman Propagator 372 Comments on Chapter 20 376 21 Linear Fields in Nonstationary Backgrounds 377 In and Out Regions 377 Bogoliubov Relations 379 in and Out Fock Spaces and the S-matrix 381 Particle Production and Annihilation Amplitudes 382 One-particle Scattering Amplitudes 385 Pages 1-528 appear in Voïume I. pages 529-1042 appear in Volume 2 CONTENTS xvii Nonsingularity of a 386 Unitarity. The Vacuum Persistence Amplitude 387 Unitarily Inequivalent Fock Spaces 389 Green s Function Representation of e w. The Feynman Propagator 391 Functional Integral Representation of e w 393 Comments on Chapter 21 394 References 395 22 Linear (or Linearized) Fields Possessing Invariant Flows 396 The Electromagnetic Field in a Stationary Curved Background 396 Other Linear or Linearized Fields on Stationary Backgrounds 397 Mode Functions for F and S 399 Mode-function Decompositions of G and Ò 401 Quantization 402 Fields on Nonstationary Backgrounds 402 Green s Function Representation of e w 404 Comments on Chapter 22 406 V NONLINEAR FIELDS 23 The Effective Action, the 5-matrix, and Slavnov-Taylor Identities 409 Anticipation of Counter Terms 409 Correlation Functions 4 і () The Effective Action 411 The Relation of Γ to W. The Legendre Transform 413 The Correlation Functions as Tree Functions 413 Structure of the Effective Action 414 The Loop Expansion 4 j 6 Reai-Yaluedness of the Counter Terms. Perturbative Renormali zability 419 Asymptotic Fields 421 Asymptotic States 422 Wave-packet States 424 The S-rnatrix and the Scattering Operator 426 The Lehmann-Symanzik-Zimmermann Theorem 427 Mode Functions for the Effective Action 430 The Effective Action as the Generator of Quantum Dynamics 432 Coherent States as Relative Vacua 433 Pages 1-528 appear in Volume I. pages 529-1042 appear in Volume 2 xviii CONTENTS Evaluation of (+, rel vac]-, rel vac) 436 Expression of the S-matrix in Terms of Quantum Tree Functions 438 The Cluster Decomposition Principle 439 Use of a (Super)Classical Background 440 Construction of the Relative S-matrix 442 Structure of the Relative S-matrix 444 When the Configuration Space is Not a Supervector Space 446 Relation to the Classical Tree Theorems 449 Slavnov-Taylor Identities 450 Current Algebra 452 Relation to the Effective Action. Finiteness of the Current Operator 453 Comments on Chapter 23 454 References 455 24 Gauge Theories I. General Formalism 456 Structure of the Space of Field Histories 456 Fibre-Adapted Coordinate Patches 457 A Metric for Φ/α 459 Vilkovisky s Connection 460 Properties of Vilkovisky s Connection 462 The Functional Integral for In-Out Amplitudes 463 Properties of the Jacobian J [ψ] 465 A Special Choice for Ω[φ] and a New Measure Functional 467 Explicit Form for the New Measure 468 Consistency of Eq. (24.80) 470 Ghosts 471 Relation to the Functional 6. The Batalin-Vilkovisky Equation 473 Counter Terms, the Measure Functional, and the Quantum BV Equation 474 The Slavnov Operator and BRST Transformations 476 Cohomology of the Slavnov Operator 477 Loop Decomposition of the Measure 478 The Role of the Measure Functional 480 The Effective Action 482 The Zinn-Justin Equation. Proof of Eq. (24.125) 484 The Yang-Mills Field in Four Dimensions 485 Rigidity of the Gauge Group 488 Renormalization Constants 489 Comments on Chapter 24 490 References 491 Pages 1-528 appear in Volume 1. pages 529-1042 appear in Volume 2 CONTENTS xix 25 Gauge Theories II. Background Field Methods. Scattering Theory 492 Invariants. The Quantum Slavnov Operator 492 Integrating Out the Ghosts 493 Reduced Effective Action 494 Introduction of a Background Field 495 Gauge Fixing 496 Loop Expansion of the Reduced Effective Action 496 Structure of the Reduced Effective Action 498 The Full Quantum Shell 500 Alternative Loop Expansions 500 TheS-matrix 501 Mode Functions for the Effective Action 503 ^-matrix Relative to an Arbitrary Background 505 Use of the (Supe^Classical Background 507 A Special Phenomenon 509 Use of Г[^] in Constructing the S-matrix 510 Comments on Chapter 25 511 Reference 512 26 Case-I Gauge Theory without Ghosts. Description of Cases II and III 513 Geodesic Normal Fields 513 A New Effective Action 514 The Illusory Ghost 516 The Loop Series 518 Rules for Differentiating the Measure 519 Invariance of the Loop Graphs 520 Renormalization 522 Difficulties in Applying the LSZ Theorem 523 Case-II Systems 524 Uncertain Cohomology of the Slavnov Operator 525 Decomposition of the Effective Action 525 Scattering Theory 525 Case-Ill Systems 526 Comments on Chapter 26 528 Reference 528 Index [1] Pages 1-528 appear in Volume i. pages 529-1042 appear in Volume 2
adam_txt	CONTENTS VOLUME 1 I CLASSICAL DYNAMICAL THEORY 1 Fundamentals 3 Classical Dynamical Systems 3 Space of Histories. Functional Differentiation 4 Examples of Simple Functionals. Notational Abbreviations 6 Condensed and Supercondensed Notations 9 Changing Topology 11 Comments on Chapter 1 12 Reference 12 2 Dynamics and Invariance Transformations 13 Action Functional and Dynamical Equations 13 Invariance Transformations 14 Commutator of Invariance Transformations 17 Gauge Algebras. Gauge Groups. Orbits 18 Functional Differential Equation for the Structure Functions 22 Further Simplification of Notation 24 Physical Observables 25 Gauge Groups and Manifest Covariance 26 Comments on Chapter 2 28 References 29 3 Small Disturbances and Green's Functions 30 The Equations of Small Disturbances. Jacobi Fields 30 The Structure of ,·. S ; 31 Other Identities Satisfied by the Coefficient Matrices 32 Supplementary Conditions 33 Retarded and Advanced Green's Functions of F 35 Equality of Left and Right Green's Functions 36 Reciprocity Relations 37 Explicit Structure of F 38 Relation between the Greeďs Functions of 5 and F 40 x CONTENTS The Green's Functions Off Shell. Coherent Green's Functions 42 Landau Green's Functions 44 Transformation Laws for the Green's Functions 44 Comments on Chapter 3 45 4 The Peierls Bracket 46 Disturbances in Physical Observables 46 The Reciprocity Relation for Physical Observables 47 Invariance Properties of D J B 47 The Peierls Bracket and the Supercommutator Function 49 The Bracket Identities 50 Standard Canonical Systems. Equivalence of Peierls and Poisson Brackets 51 The Wronskian Operator 54 The Cauchy Problem for Jacobi Fields 56 Comments on Chapter 4 58 Reference 59 5 Finite Disturbances. Tree Theorems. Asymptotic Fields 60 Differential and Integro-differential Equations for Finite Disturbances 60 Iterative Solutions. Tree Functions 63 Background Field. Transformation Laws 64 Changes in P and к 66 Gauge Transformations 67 Tree Amplitudes and Tree Theorems 67 Tree Theorems When the Configuration Space is not a Vector Space 69 Change of Chart 70 Asymptotic "In" and "Out" Fields 73 Comments on Chapter 5 77 Reference 77 6 Conservation Laws 78 Alternative Action Functional 78 Internal Sources 78 Killing Fields. Currents, and Charges 79 The Charge Group 81 Flow Invariance of the Charges 82 Fundamental Lemma 83 A Special Transformation 85 Pages 1-528 appear in Volume 1, pages 529-1042 appear in Volume 2 CONTENTS xi The Peierls Bracket (qA, qB). The View from the Remote Past or Future 87 The View from Spatial Infinity 90 Compact Universes and Linearization Constraints 93 Pseudocovariants 94 Externally Impressed Fields 95 Charges in the Presence of External Fields . 96 Peierls Bracket for the Charges 98 Differences from the Previous Theory 98 External Gravitational Field. Diffeomorphism Group 101 Stress-energy Density 103 Killing Vector Fields. The Poincaré Group 107 Energy-momentum Vector and Angular Momentum Tensor 108 Local Conformai Group 109 Conformai Killing Vector Fields. The Global Conformai Group 110 Conformai Charges and their Peierls Brackets 111 Comments on Chapter 6 112 References 113 II THE HEURISTIC ROAD TO QUANTIZATION. THE QUANTUM FORMALISM AND ITS INTERPRETATION 7 Classical Theory of Measurement 117 System. Apparatus. Coupling 117 Apparatus Inertia 118 Disturbance in the Apparatus 120 Design of the Coupling. Accuracy of the Measurement 121 Compensation Term 122 Stem-Gerlach Experiment 123 Electric Field Measurement 125 Comments on Chapter 7 130 Reference 130 8 Quantum Theory of Measurement 131 Measurement of Two Observables 131 Mutual Uncertainties 132 A Universal Uncertainty Principle 133 The Heuristic Quantization Rule 134 A Preferred Basis 136 Pages 1-528 appear in Volume 1, pages 529-1042 appear in Volume 2 xii CONTENTS Relative States. Good Measurements 138 Many Worlds 140 Unobservability of the Splits 141 Comments on Chapter 8 144 References 144 9 Interpretation of the Quantum Formalism I 145 Reality 145 Signalling by Permutations 146 Equal Likelihood 147 Unequal Probabilities 149 Irrational Probabilities 150 Expectation Value. Single System vs. an Ensemble 152 Relative Frequency Function. The Ensemble Interpretation of Quantum-mechanical Probabilities 153 Density Operator. Decoherence 156 Imperfect Measurements 159 Comments on Chapter 9 161 Reference 161 10 The Schwinger Variational Principle and the Feynman Functional Integral 162 Transition Amplitudes 162 The Schwinger Variational Principle 162 External Sources and Chronological Products 165 The Operator Dynamical Equations and the Measure Functional 168 Functional Fourier Analysis. The Feynman Functional Integral 171 Comments on the Feynman Integral 173 The Schwinger Variational Principle Revisited 175 Expressions Involving 5,- 176 Comments on Chapter 10 177 References 177 11 The Quantum Mechanics of Standard Canonical Systems 178 Problems with the Naïve Quantization Rule 178 The Position Mapping 179 Vector Operators 180 The Momentum Operator 183 Restriction to a Local Chart 184 Overlapping Charts. Transformation of Coordinates 186 Pages 1-528 appear in Volume 1. pages 529-1042 appear in Volume 2 CONTENTS xiii The Position Representation 188 The Projection m-form 1 89 The Momentum Operator in the Position Representation 190 The Schrödinger Equation ] 92 Remarks on Global Consistency 194 Comments on Chapter 11 194 Reference 194 12 Interpretation of the Quantum Formalism II 195 Tracing Out 195 The Model System 196 Density Operator 198 Localization. Sharp Decoherence 199 Discussion 201 Coarse Graining. Decoherence Function 202 Interpretation of the Diagonal Elements 203 Emergence of Classicality 205 Many Worlds Again. Probability as an Emergent Concept 207 Comments on Chapter 12 209 References 209 III EVALUATION AND APPROXIMATION OF FEYNMAN FUNCTIONAL INTEGRALS 13 The Functional Integral for Standard Canonical Systems 213 The Path Integral 213 The Point-to-point Amplitude. The Action Function 213 Formal Computation of detG+[x] in the Lagrangian Formalism 216 Formal Computation of det G+[x. p] in the Hamiltonian Formalism 217 Ambiguity in the Path Integral 219 Homotopy 220 The Universal Covering Space. Covering Translations 221 The Relation of Homotopy to Homology and Cohomology 223 The Path Integral in Č 225 Fundamental Domains 227 Partial Amplitudes in C. The Total Amplitude 228 Combination Law 229 The Hamiltonian Form of the Path Integral and the Schrödinger Equation 230 Pages 3—528 appear in Volume 1. pases 529-1042 appear in Volume 2 xiv CONTENTS Homotopy in Quantum Field Theory 232 Comments on Chapter 13 233 References 233 14 Approximation and Evaluation of the Path Integral 234 The Van Vleck-Morette Determinant 234 Jacobi Fields and Green's Functions for the Trajectory xc 236 Determinantal Relations 239 The Loop Expansion 241 The WKB Approximation 243 Normalization 245 Other Boundary Conditions 246 WKB Approximation 247 Comments on Chapter 14 250 References 250 15 The Nonrelativistic Particle in a Curved Space 251 From the Hamiltonian Path Integral to the Lagrangian Path Integral 251 The Nonrelativistic Particle in a Curved Space 252 Covariant Variation 253 Covariant Differentiation with Respect to t 254 The Dynamical Equations 255 Covariant Functional Differentiation 255 The Measure for the Lagrangian Path Integral 257 Computation of HQ 259 Normalization of the Path Integral 262 A Two-loop Calculation 263 Analysis of the Jacobi Field Operator 265 The Morse Index 267 Morse Index Theorem 268 Application to Path Integration 271 The Generalized Morse Index 272 Comments on Chapter 15 274 References 274 16 The Heat Kernel 275 History 275 Geodesic Normal Coordinates 275 Coïncidence Limits 277 Caustics 279 Riemanman Connection. The World Function 281 Pages 1-528 appear in Volume 1, pages 529-1042 appear in Volume 2 CONTENTS xv Auxiliary Geometrical Quantities. Modified Covariant Derivative 283 Further Coincidence Limits 284 Heat-kernel Expansion. Recursion Relations 285 Comments on Chapter 16 286 References 287 IV LINEAR SYSTEMS 17 Linear Boson Fields in Stationary Backgrounds 291 The Scalar Field 291 Stationary Backgrounds 292 The Field Equations 293 The Energy 295 Energy Bounds 296 Mode Functions 297 Alternative Representation 298 Multiple Roots 300 Wronskian Relations 301 Zero Roots 302 Wronskian Relations Again 304 General Solution. Supercommutator Function. Energy 307 Matrix Identities 307 Mode Functions for the Scalar Field 309 The Massless Scalar Field in a Compact Universe 310 The Vector Field 312 Rescaling of Time and Energy. Canonical Form for the Energy 315 Conformally Invariant Scalar and Vector Fields 318 Comments on Chapter 17 319 18 Quantization of Linear Boson Fields 320 Green's Functions 320 Quantization 321 Super Hubert and Fock Spaces 322 Nonstationary Backgrounds and Inequivalent Vacua 327 Nonuniqueness of д/дх0 328 Vacua Defined by Symmetry Properties 330 The Feynman Propagator 330 Feynman Propagators for the Scalar and Vector Fields 332 When the Energy is Unbounded from Below 335 Pages 1-528 appear in Volume 1. pages 529-1042 appear in Volume 2 xvi CONTENTS When K~l Μ is Fully Diagonalizable and has Pure Imaginary Eigenvalues 337 Comments on Chapter 18 339 References 340 19 Linear Fermion Fields. Stationary Backgrounds 341 Local Lorentz Frames 341 The Dirac Matrices 342 Spin Structures 344 Space Inversions and Pin Structures 345 Spinor Fields 347 The Spin Connection 349 Generalized Spin or Pin Structures 350 Real Representations 351 Lagrange Function. Conformai Invariance. Majorana Representation 351 Stress-energy Density 353 Leibniz' Rule. Matrix Identities. Canonical Form for the Energy 355 Energy-momentum Conservation and the Poincaré Group 357 The Dirac Operator 359 Stationary Backgrounds. A Special Field of Local Lorentz Frames 360 A Model System 362 Mode Functions 362 When —iB is not Positive Definite 365 Comments on Chapter 19 365 References 366 20 Quantization of Linear Fermion Fields 367 Greeks Functions 367 Quantization 367 Fock Space 368 Hole Theory t 370 The Feynman Propagator 372 Comments on Chapter 20 376 21 Linear Fields in Nonstationary Backgrounds 377 "In" and "Out" Regions 377 Bogoliubov Relations 379 "in" and "Out" Fock Spaces and the S-matrix 381 Particle Production and Annihilation Amplitudes 382 One-particle Scattering Amplitudes 385 Pages 1-528 appear in Voïume I. pages 529-1042 appear in Volume 2 CONTENTS xvii Nonsingularity of a 386 Unitarity. The Vacuum Persistence Amplitude 387 Unitarily Inequivalent Fock Spaces 389 Green's Function Representation of e'w. The Feynman Propagator 391 Functional Integral Representation of e'w 393 Comments on Chapter 21 394 References 395 22 Linear (or Linearized) Fields Possessing Invariant Flows 396 The Electromagnetic Field in a Stationary Curved Background 396 Other Linear or Linearized Fields on Stationary Backgrounds 397 Mode Functions for F and S 399 Mode-function Decompositions of G and Ò 401 Quantization 402 Fields on Nonstationary Backgrounds 402 Green's Function Representation of e'w 404 Comments on Chapter 22 406 V NONLINEAR FIELDS 23 The Effective Action, the 5-matrix, and Slavnov-Taylor Identities 409 Anticipation of Counter Terms 409 Correlation Functions 4 і () The Effective Action 411 The Relation of Γ to W. The Legendre Transform 413 The Correlation Functions as Tree Functions 413 Structure of the Effective Action 414 The Loop Expansion 4 j 6 Reai-Yaluedness of the Counter Terms. Perturbative Renormali zability 419 Asymptotic Fields 421 Asymptotic States 422 Wave-packet States 424 The S-rnatrix and the Scattering Operator 426 The Lehmann-Symanzik-Zimmermann Theorem 427 Mode Functions for the Effective Action 430 The Effective Action as the Generator of Quantum Dynamics 432 Coherent States as Relative Vacua 433 Pages 1-528 appear in Volume I. pages 529-1042 appear in Volume 2 xviii CONTENTS Evaluation of (+, rel vac]-, rel vac) 436 Expression of the S-matrix in Terms of Quantum Tree Functions 438 The Cluster Decomposition Principle 439 Use of a (Super)Classical Background 440 Construction of the Relative S-matrix 442 Structure of the Relative S-matrix 444 When the Configuration Space is Not a Supervector Space 446 Relation to the Classical Tree Theorems 449 Slavnov-Taylor Identities 450 Current Algebra 452 Relation to the Effective Action. Finiteness of the Current Operator 453 Comments on Chapter 23 454 References 455 24 Gauge Theories I. General Formalism 456 Structure of the Space of Field Histories 456 Fibre-Adapted Coordinate Patches 457 A Metric for Φ/α 459 Vilkovisky's Connection 460 Properties of Vilkovisky's Connection 462 The Functional Integral for "In-Out" Amplitudes 463 Properties of the Jacobian J [ψ] 465 A Special Choice for Ω[φ] and a New Measure Functional 467 Explicit Form for the New Measure 468 Consistency of Eq. (24.80) 470 Ghosts 471 Relation to the Functional 6. The Batalin-Vilkovisky Equation 473 Counter Terms, the Measure Functional, and the Quantum BV Equation 474 The Slavnov Operator and BRST Transformations 476 Cohomology of the Slavnov Operator 477 Loop Decomposition of the Measure 478 The Role of the Measure Functional 480 The Effective Action 482 The Zinn-Justin Equation. Proof of Eq. (24.125) 484 The Yang-Mills Field in Four Dimensions 485 Rigidity of the Gauge Group 488 Renormalization Constants 489 Comments on Chapter 24 490 References 491 Pages 1-528 appear in Volume 1. pages 529-1042 appear in Volume 2 CONTENTS xix 25 Gauge Theories II. Background Field Methods. Scattering Theory 492 Invariants. The Quantum Slavnov Operator 492 Integrating Out the Ghosts 493 Reduced Effective Action 494 Introduction of a Background Field 495 Gauge Fixing 496 Loop Expansion of the Reduced Effective Action 496 Structure of the Reduced Effective Action 498 The Full Quantum Shell 500 Alternative Loop Expansions 500 TheS-matrix 501 Mode Functions for the Effective Action 503 ^-matrix Relative to an Arbitrary Background 505 Use of the (Supe^Classical Background 507 A Special Phenomenon 509 Use of Г[^] in Constructing the S-matrix 510 Comments on Chapter 25 511 Reference 512 26 Case-I Gauge Theory without Ghosts. Description of Cases II and III 513 Geodesic Normal Fields 513 A New Effective Action 514 The Illusory Ghost 516 The Loop Series 518 Rules for Differentiating the Measure 519 Invariance of the Loop Graphs 520 Renormalization 522 Difficulties in Applying the LSZ Theorem 523 Case-II Systems 524 Uncertain Cohomology of the Slavnov Operator 525 Decomposition of the Effective Action 525 Scattering Theory 525 Case-Ill Systems 526 Comments on Chapter 26 528 Reference 528 Index [1] Pages 1-528 appear in Volume i. pages 529-1042 appear in Volume 2
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id	DE-604.BV021520440
illustrated	Illustrated
index_date	2024-07-02T14:22:25Z
indexdate	2024-07-09T20:37:42Z
institution	BVB
isbn	019852790x 9780198527909 9780198712862
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-014736947
oclc_num	315194993
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physical	XXXI, 528, 11 S. graph. Darst.
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publisher	Oxford Univ. Press
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series2	... The international series of monographs on physics Oxford science publications
spelling	De Witt, Bryce S. Verfasser aut The global approach to quantum field theory 1 Bryce DeWitt repr. 2004 (with corr.), 2005 Oxford [u.a.] Oxford Univ. Press 2005 XXXI, 528, 11 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier ... 114 The international series of monographs on physics ... Oxford science publications Hier auch später erschienene, unveränderte Nachdrucke (DE-604)BV014651048 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014736947&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	De Witt, Bryce S. The global approach to quantum field theory
title	The global approach to quantum field theory
title_auth	The global approach to quantum field theory
title_exact_search	The global approach to quantum field theory
title_exact_search_txtP	The global approach to quantum field theory
title_full	The global approach to quantum field theory 1 Bryce DeWitt
title_fullStr	The global approach to quantum field theory 1 Bryce DeWitt
title_full_unstemmed	The global approach to quantum field theory 1 Bryce DeWitt
title_short	The global approach to quantum field theory
title_sort	the global approach to quantum field theory
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014736947&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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