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. Contents Preface Notation 1 Introduction to Combinatorics 1.1 Counting: Balls and Urns 1.2 Statistical Physics 1.3 Combinatorial Coefficients 1.4 Sets and Bags 1.5 Permutations and Partitions 1.6 Occupation Numbers 1.7 Hierarchies (= Phylogenetic Trees = Total Partitions) 1.8 Partitions 1.9 Partition Functions 2 Probabilistic Moments and Cumulants 2.1 Random Variables 2.2 Key Probability Distributions 2.3 Stochastic Processes 2.4 Multiple Stochastic Integrals 2.5 Iterated Ito ̄ Integrals 2.6 Stratonovich Integrals 2.7 Rota–Wallstrom Theory 3 Quantum Probability 3.1 The Canonical Anticommutation Relations 3.2 The Canonical Commutation Relations 3.3 Wick Ordering 4 Quantum Fields 4.1 Green’s Functions 4.2 A First Look at Boson Fock Space 5 Combinatorial Species 5.1 Operations on Species 5.2 Graphs 5.3 Weighted Species 5.4 Differentiation of Species 6 Combinatorial Aspects of Quantum Fields: Feynman Diagrams 6.1 Basic Concepts 6.2 Functional Integrals 6.3 Tree Expansions 6.4 One-Particle Irreducibility 7 Entropy, Large Deviations, and Legendre Transforms 7.1 Entropy and Information 7.2 Law of Large Numbers and Large Deviations 7.3 Large Deviations and Stochastic Processes 8 Introduction to Fock Spaces 8.1 Hilbert Spaces 8.2 Tensor Spaces 8.3 Symmetric Tensors 8.4 Antisymmetric Tensors 9 Operators and Fields on the Boson Fock Space 9.1 Operators on Fock Spaces 9.2 Exponential Vectors and Weyl Operators 9.3 Distributions of Boson Fields 9.4 Thermal Fields 9.5 q-deformed Commutation Relations 10 L2-Representations of the Boson Fock Space 10.1 The Bargmann–Fock Representation 10.2 Wiener Product and Wiener–Segal Representation 10.3 Ito–Fock Isomorphism 11 Local Fields on the Boson Fock Space: Free Fields 11.1 The Free Scalar Field 11.2 Canonical Operators for the Free Field 12 Local Interacting Boson Fields 12.1 Interacting Neutral Scalar Fields 12.2 Interaction with a Classical Current 13 Quantum Stochastic Calculus 13.1 Operators on Guichardet Fock Space 13.2 Wick Integrals 13.3 Chronological Ordering 13.4 Quantum Stochastic Processes on Fock Space 13.5 Quantum Stochastic Calculus 13.6 Quantum Stratonovich Integrals 13.7 The Quantum White Noise Formulation 13.8 Quantum Stochastic Exponentials 13.9 The Belavkin–Holevo Representation 14 Quantum Stochastic Limits 14.1 A Quantum Wong Zakai Theorem 14.2 A Microscopic Model References Index This book gets to the heart of the combinatorics that binds together quantum field theory and probability with a unified framework for Wick (normal) ordering and its applications. Featuring many worked examples, it is for mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students. John Gough, Aberystwyth University, Joachim Kupsch, University Of Kaiserslautern. Includes Bibliographical References (pages 316-321) And Index.
