Quantum Fields and Processes: A Combinatorial Approach

✍ Scribed by John Gough, Joachim Kupsch

• Publisher: Cambridge University Press
• Year: 2018
• Tongue: English
• Leaves: 341
• Series: Cambridge Studies in Advanced Mathematics (Book 171)
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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