Quantum Fields and Processes: A Combinatorial Approach

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