Gespeichert in:
| Titel: | Functional methods in quantum field theory and statistical physics |
|---|---|
| Von: |
A. N. Vasiliev
|
| Person: |
Vasilʹev, Aleksandr N.
1940-2006 Verfasser aut |
| Hauptverfasser: | |
| Format: | Buch |
| Sprache: | Englisch Russisch |
| Veröffentlicht: |
Amsterdam [u.a.]
Gordon & Breach
1998
|
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008076234&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Aus dem Russ. übers. |
| Beschreibung: | XIV, 312 S. |
| ISBN: | 9056990357 |
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| 245 | 1 | 0 | |a Functional methods in quantum field theory and statistical physics |c A. N. Vasiliev |
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|---|---|
| DE-BY-UBR_katkey | 2343781 |
| DE-BY-UBR_location | UB Lesesaal Physik UB Handapparat Physik Prof. Schäfer UB Handapparat Physik Prof. V. Braun |
| DE-BY-UBR_media_number | 069022634069 TEMP3037001 TEMP3037002 |
| _version_ | 1835113849795117056 |
| adam_text | IMAGE 1
FUNCTIONAL METHODS IN QUANTUM FIELD
THEORY AND STATISTICAL PHYSICS
A.N. VASILIEV DEPARTMENT OF THEORETICAL PHYSICS ST PETERSBURG STATE
UNIVERSITY, RUSSIA
TRANSLATED FROM THE RUSSIAN BY PATRICIA A. MILLARD
GORDON AND BREACH SCIENCE PUBLISHERS
AUSTRALIA * CANADA * CHINA * FRANCE * GERMANY * INDIA JAPAN * LUXEMBOURG
* MALAYSIA * THE NETHERLANDS * RUSSIA SINGAPORE * SWITZERLAND * THAILAND
IMAGE 2
CONTENTS
FOREWORD XI
PREFACE XIII
CHAPTER 1 THE BASIC FORMALISM OF FIELD THEORY 1 1.1 FIELDS AND PRODUCTS
1
1.1.1 CANONICAL QUANTIZATION 1
1.1.2 THE CLASSICAL FREE THEORY 3
1.1.3 ANTICOMMUTING FIELDS 4
1.1.4 THE NORMAL-ORDERED PRODUCT OF FREE-FIELD OPERATORS 7 1.2
FUNCTIONAL FORMULATIONS OF WICK S THEOREM 9
1.2.1 WICK S THEOREM FOR A SIMPLE PRODUCT 9
1.2.2 THE SYM-PRODUCT AND THE *-PRODUCT 12
1.2.3 WICK S THEOREM FOR SYMMETRIC PRODUCTS 15
1.2.4 REDUCTION FORMULAS FOR OPERATOR FUNCTIONALS 18
1.2.5 THE WICK AND DYSON *-PRODUCTS 21
1.3 THE S-MATRIX AND GREEN FUNCTIONS 23
1.3.1 DEFINITIONS 23
1.3.2 TRANSFORMATION TO THE INTERACTION PICTURE IN THE EVOLUTION
OPERATOR 25
1.3.3 TRANSFORMATION TO THE INTERACTION PICTURE FOR THE GREEN FUNCTIONS
27
1.3.4 INTERACTIONS CONTAINING TIME DERIVATIVES OF THE FIELD 31 1.3.5
GENERATING FUNCTIONALS FOR THE S-MATRIX AND GREEN FUNCTIONS 34
1.4 GRAPHS 37
1.4.1 PERTURBATION THEORY 37
1.4.2 SOME CONCEPTS FROM GRAPH THEORY 39
1.4.3 SYMMETRY COEFFICIENTS 39
1.4.4 RECURSION RELATION FOR THE SYMMETRY COEFFICIENTS 41
1.4.5 TRANSFORMATION TO MAYER GRAPHS FOR THE EXPONENTIAL INTERACTION 42
1.4.6 GRAPHS FOR THE YUKAWA-TYPE INTERACTION 43
1.4.7 GRAPHS FOR A PAIR INTERACTION 48
1.4.8 CONNECTEDNESS OF THE LOGARITHM OF /?( P) 49
1.4.9 GRAPHS FOR THE GREEN FUNCTIONS 50
1.5 UNITARITY OF THE S MATRIX 53
1.5.1 THE CONJUGATION OPERATION 53
1.5.2 FORMAL UNITARITY OF THE OFF-SHELL 5 MATRIX 55
IMAGE 3
VI
CONTENTS
1.6 FUNCTIONAL INTEGRALS 58
1.6.1 GAUSSIAN INTEGRALS 58
1.6.2 INTEGRALS ON A GRASSMANN ALGEBRA 61
1.6.3 GAUSSIAN INTEGRALS ON A GRASSMANN ALGEBRA 63
1.6.4 GAUSSIAN INTEGRALS IN FIELD THEORY 65
1.6.5 REPRESENTATIONS OF GENERATING FUNCTIONALS FOR THE S-MATRIX AND
GREEN FUNCTIONS BY FUNCTIONAL INTEGRALS 68
1.6.6 THE STATIONARY-PHASE METHOD 70
1.6.7 THE DOMINICIS-ENGLERT THEOREM 73
1.7 EQUATIONS IN VARIATIONAL DERIVATIVES 75
1.7.1 THE SCHWINGER EQUATIONS 75
1.7.2 LINEAR EQUATIONS FOR CONNECTED GREEN FUNCTIONS 77
1.7.3 GENERAL METHOD OF DERIVING THE EQUATIONS 78
1.7.4 ITERATION SOLUTION OF THE EQUATIONS 82
1.8 ONE-IRREDUCIBLE GREEN FUNCTIONS 84
1.8.1 DEFINITIONS 84
1.8.2 THE EQUATIONS OF MOTION FOR * 87
1.8.3 ITERATION SOLUTION OF THE EQUATIONS AND PROOF OF 1-IRREDUCIBILITY
88
1.9 RENORMALIZATION TRANSFORMATIONS 91
1.10 ANOMALOUS GREEN FUNCTIONS AND SPONTANEOUS SYMMETRY BREAKING 94
CHAPTER 2 SPECIFIC SYSTEMS 97
2.1 QUANTUM MECHANICS 97
2.1.1 THE OSCILLATOR 97
2.1.2 THE FREE PARTICLE 99
2.2 NONRELATIVISTIC FIELD THEORY 104
2.2.1 THE QUANTUM BOSE AND FERMI GASES 108
2.2.2 THE ATOM 109
2.2.3 ELECTRONS IN SOLIDS AND PHONONS 110
2.3 RELATIVISTIC FIELD THEORY 111
2.4 INTEGRAL REPRESENTATIONS OF THE TRANSITION AMPLITUDE 114 2.5 THE
SPACE E(A) FOR VARIOUS SYSTEMS 120
2.6 FUNCTIONAL INTEGRALS OVER PHASE SPACE 122
CHAPTER 3 THE MASSLESS YANG-MILLS FIELD 127
3.1 QUANTIZATION OF THE YANG-MILLS FIELD 127
3.1.1 THE CLASSICAL THEORY 127
IMAGE 4
CONTENTS
VLL
3.1.2 A GENERAL RECIPE FOR QUANTIZATION 128
3.1.3 PERTURBATION THEORY FOR GAUGES NB + * = 0 131
3.1.4 THE GENERALIZED FEYNMAN GAUGE 134
3.1.5 THE S-MATRIX GENERATING FUNCTIONAL 134
3.2 GAUGE INVARIANCE 136
3.2.1 THE WARD-SLAVNOV IDENTITIES 136
3.2.2 TRANSVERSALITY AND GAUGE INVARIANCE OF THE S-MATRIX IN
ELECTRODYNAMICS 137
3.2.3 TRANSVERSALITY AND GAUGE INVARIANCE OF THE ON-SHELL S-MATRIX FOR
THE YANG-MILLS FIELD 139
CHAPTER 4 EUCLIDEAN FIELD THEORY 141
4.1 THE EUCLIDEAN ROTATION 141
4.1.1 DEFINITIONS 141
4.1.2 THE FORMAL EUCLIDEAN ROTATION OF THE ACTION FUNCTIONAL 142
4.1.3 EUCLIDEAN ROTATION OF THE GREEN FUNCTIONS 143
4.1.4 PROPERTIES OF THE FIELD * AND THE ACTION S E (*) 145
4.1.5 ROTATION OF THE LORENTZ GROUP INTO 0 4 146
4.1.6 EXAMPLES 148
4.2 FUNCTIONAL INTEGRAL REPRESENTATIONS 150
4.3 CONVEXITY PROPERTIES 154
4.3.1 QUASIPROBABILISTIC THEORIES 154
4.3.2 CONVEXITY AND SPECTRAL REPRESENTATIONS 156
CHAPTER 5 STATISTICAL PHYSICS 159
5.1 THE QUANTUM STATISTICS OF FIELD SYSTEMS 159
5.1.1 DEFINITIONS 159
5.1.2 THE FREE THEORY 161
5.1.3 THE AVERAGE OF AN OPERATOR IN NORMAL-ORDERED FORM 162 5.1.4
DIAGRAMMATIC REPRESENTATION OF THE PARTITION FUNCTION AND THE GREEN
FUNCTIONS 165
5.1.5 PERIODIC EXTENSIONS OF THE GREEN FUNCTIONS 166
5.1.6 REPRESENTATIONS BY FUNCTIONAL INTEGRALS 169
5.1.7 THE ZERO-TEMPERATURE LIMIT 170
5.1.8 THE FEYNMAN-KAC FORMULA 171
5.1.9 CONVEXITY PROPERTIES 172
5.1.10 CONVEXITY OF THE LOGARITHM OF THE PARTITION FUNCTION 174 5.1.11
REPRESENTATION OF THE PARTITION FUNCTION OF THE FREE THEORY BY A
FUNCTIONAL INTEGRAL 176
IMAGE 5
VLLL
CONTENTS
5.2 LATTICE SPIN SYSTEMS 180
5.2.1 THE ISING MODEL 180
5.2.2 THE HEISENBERG QUANTUM FERROMAGNET 182
5.3 THE NONIDEAL CLASSICAL GAS 185
5.3.1 A GAS WITH TWO-BODY FORCES 185
5.3.2 A GAS WITH MANY-BODY FORCES 188
CHAPTER 6 VARIATIONAL METHODS AND FUNCTIONAL LEGENDRE TRANSFORMS 191
6.1 PHASE TRANSITIONS 191
6.1.1 INTRODUCTION 191
6.1.2 TRANSFORMATION TO THE VARIATIONAL PROBLEM IN THERMODYNAMICS 192
6.1.3 THE INFINITE-VOLUME LIMIT 197
6.1.4 SINGULAR AND CRITICAL POINTS 199
6.1.5 DESCRIPTION OF PHASE TRANSITIONS 201
6.1.6 CRITICAL AND GOLDSTONE FLUCTUATIONS 204
6.2 LEGENDRE TRANSFORMATIONS OF THE GENERATING FUNCTIONAL OF CONNECTED
GREEN FUNCTIONS 207
6.2.1 FUNCTIONAL FORMULATIONS OF THE VARIATIONAL PRINCIPLE 207
6.2.2 THE EQUATIONS OF MOTION IN CONNECTED VARIABLES 213
6.2.3 THE EQUATIONS OF MOTION IN 1-IRREDUCIBLE VARIABLES 220
6.2.4 LINEAR EQUATIONS AND THEIR GENERAL SOLUTIONS 224
6.2.5 ITERATION SOLUTION OF THE EQUATIONS 229
6.2.6. THE SECOND LEGENDRE TRANSFORM 232
6.2.7 THE SELF-CONSISTENT FIELD APPROXIMATION 235
6.2.8 THE THIRD LEGENDRE TRANSFORM 238
6.2.9 THE FOURTH TRANSFORM 243
6.2.10 STATIONARITY EQUATIONS, RENORMALIZATION, AND PARQUET GRAPHS 249
6.2.11 SYMMETRY PROPERTIES OF THE COMPLETE LEGENDRE TRANSFORM AND THE
SPONTANEOUS INTERACTION 252
6.2.12 THE GROUND-STATE ENERGY 256
6.2.13 STABILITY AND CONVEXITY PROPERTIES OF FUNCTIONAL LEGENDRE
TRANSFORMS 258
6.3 LEGENDRE TRANSFORMS OF THE LOGARITHM OF THE 5-MATRIX GENERATING
FUNCTIONAL 261
IMAGE 6
CONTENTS IX
6.3.1 DEFINITIONS AND GENERAL PROPERTIES 261
6.3.2 THE CLASSICAL NONIDEAL GAS AND THE VIRIAL EXPANSION 266 6.3.3 THE
ISING MODEL 269
6.3.4 ANALYSIS OF NONSTAR GRAPHS FOR THE ISING MODEL 272
6.3.5 THE SECOND LEGENDRE TRANSFORM FOR THE CLASSICAL GAS 278 6.3.6 THE
STATIONARITY EQUATIONS AND THE SELF-CONSISTENT FIELD APPROXIMATION 284
APPENDIX 1 NONSTATIONARY PERTURBATION
THEORY FOR A DISCRETE LEVEL 289
APPENDIX 2 GRAPHS AND SYMMETRY COEFFICIENTS 301
REFERENCES 305
SUBJECT INDEX 309
|
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| author | Vasilʹev, Aleksandr N. 1940-2006 |
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| id | DE-604.BV011946284 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T10:21:30Z |
| institution | BVB |
| isbn | 9056990357 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008076234 |
| oclc_num | 39632571 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-188 DE-20 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-188 DE-20 |
| physical | XIV, 312 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Gordon & Breach |
| record_format | marc |
| spellingShingle | Vasilʹev, Aleksandr N. 1940-2006 Functional methods in quantum field theory and statistical physics Champs, Théorie quantique des ram Kwantumveldentheorie gtt Statistische mechanica gtt Teoria quântica de campo larpcal Functional analysis Quantum field theory Quantum statistics Statistische Physik (DE-588)4057000-9 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Funktional (DE-588)4155667-7 gnd |
| subject_GND | (DE-588)4057000-9 (DE-588)4047984-5 (DE-588)4155667-7 |
| title | Functional methods in quantum field theory and statistical physics |
| title_alt | Funkcional'nye metody kvantovoj teorii polja i statistike |
| title_auth | Functional methods in quantum field theory and statistical physics |
| title_exact_search | Functional methods in quantum field theory and statistical physics |
| title_full | Functional methods in quantum field theory and statistical physics A. N. Vasiliev |
| title_fullStr | Functional methods in quantum field theory and statistical physics A. N. Vasiliev |
| title_full_unstemmed | Functional methods in quantum field theory and statistical physics A. N. Vasiliev |
| title_short | Functional methods in quantum field theory and statistical physics |
| title_sort | functional methods in quantum field theory and statistical physics |
| topic | Champs, Théorie quantique des ram Kwantumveldentheorie gtt Statistische mechanica gtt Teoria quântica de campo larpcal Functional analysis Quantum field theory Quantum statistics Statistische Physik (DE-588)4057000-9 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Funktional (DE-588)4155667-7 gnd |
| topic_facet | Champs, Théorie quantique des Kwantumveldentheorie Statistische mechanica Teoria quântica de campo Functional analysis Quantum field theory Quantum statistics Statistische Physik Quantenfeldtheorie Funktional |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008076234&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT vasilʹevaleksandrn funkcionalnyemetodykvantovojteoriipoljaistatistike AT vasilʹevaleksandrn functionalmethodsinquantumfieldtheoryandstatisticalphysics |
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