Wick ordering of creation and annihilation operators is of fundamental importance for computing averages and correlations in quantum field theory and, by extension, in the Hudson–Parthasarathy theory of quantum stochastic processes, quantum mechanics, stochastic processes, and probability. This book develops the unified combinatorial framework behind these examples, starting with the simplest mathematically, and working up to the Fock space setting for quantum fields. Emphasizing ideas from combinatorics such as the role of lattice of partitions for multiple stochastic integrals by Wallstrom–Rota and combinatorial species by Joyal, it presents insights coming from quantum probability. It also introduces a 'field calculus' which acts as a succinct alternative to standard Feynman diagrams and formulates quantum field theory (cumulant moments, Dyson–Schwinger equation, tree expansions, 1-particle irreducibility) in this language. Featuring many worked examples, the book is aimed at mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students.
Author(s): John Gough, Joachim Kupsch
Series: Cambridge Studies in Advanced Mathematics 171
Publisher: Cambridge University Press
Year: 2018
Language: English
Pages: 341
Contents......Page 8
Preface......Page 11
Notation......Page 14
1.1 Counting: Balls and Urns......Page 18
1.2 Statistical Physics......Page 20
1.3 Combinatorial Coefficients......Page 31
1.4 Sets and Bags......Page 34
1.5 Permutations and Partitions......Page 36
1.6 Occupation Numbers......Page 39
1.7 Hierarchies (= Phylogenetic Trees = Total Partitions)......Page 42
1.8 Partitions......Page 44
1.9 Partition Functions......Page 48
2.1 Random Variables......Page 54
2.2 Key Probability Distributions......Page 56
2.3 Stochastic Processes......Page 59
2.4 Multiple Stochastic Integrals......Page 61
2.5 Iterated Ito¯ Integrals......Page 65
2.6 Stratonovich Integrals......Page 68
2.7 Rota–Wallstrom Theory......Page 71
3 Quantum Probability......Page 73
3.1 The Canonical Anticommutation Relations......Page 74
3.2 The Canonical Commutation Relations......Page 76
3.3 Wick Ordering......Page 86
4.1 Green’s Functions......Page 91
4.2 A First Look at Boson Fock Space......Page 103
5 Combinatorial Species......Page 106
5.1 Operations on Species......Page 108
5.2 Graphs......Page 111
5.3 Weighted Species......Page 112
5.4 Differentiation of Species......Page 114
6.1 Basic Concepts......Page 116
6.2 Functional Integrals......Page 120
6.3 Tree Expansions......Page 130
6.4 One-Particle Irreducibility......Page 132
7.1 Entropy and Information......Page 139
7.2 Law of Large Numbers and Large Deviations......Page 143
7.3 Large Deviations and Stochastic Processes......Page 150
8.1 Hilbert Spaces......Page 155
8.2 Tensor Spaces......Page 157
8.3 Symmetric Tensors......Page 161
8.4 Antisymmetric Tensors......Page 168
9.1 Operators on Fock Spaces......Page 174
9.2 Exponential Vectors and Weyl Operators......Page 189
9.3 Distributions of Boson Fields......Page 194
9.4 Thermal Fields......Page 200
9.5 q-deformed Commutation Relations......Page 201
10.1 The Bargmann–Fock Representation......Page 206
10.2 Wiener Product and Wiener–Segal Representation......Page 208
10.3 Ito–Fock Isomorphism......Page 210
11 Local Fields on the Boson Fock Space: Free Fields......Page 215
11.1 The Free Scalar Field......Page 216
11.2 Canonical Operators for the Free Field......Page 248
12.1 Interacting Neutral Scalar Fields......Page 254
12.2 Interaction with a Classical Current......Page 263
13.1 Operators on Guichardet Fock Space......Page 269
13.2 Wick Integrals......Page 279
13.3 Chronological Ordering......Page 281
13.4 Quantum Stochastic Processes on Fock Space......Page 284
13.5 Quantum Stochastic Calculus......Page 286
13.6 Quantum Stratonovich Integrals......Page 293
13.7 The Quantum White Noise Formulation......Page 294
13.8 Quantum Stochastic Exponentials......Page 296
13.9 The Belavkin–Holevo Representation......Page 299
14.1 A Quantum Wong Zakai Theorem......Page 309
14.2 A Microscopic Model......Page 329
References......Page 333
Index......Page 339
