Functional Methods in Quantum Field Theory and Statistical Physics 1st Edition by Vasiliev 0367841738 9780367841737

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Product details:

ISBN 10: 0367841738

ISBN 13: 9780367841737

Author: A.N Vasiliev

Providing a systematic introduction to the techniques which are fundamental to quantum field theory, this book pays special attention to the use of these techniques in a wide variety of areas, including ordinary quantum mechanics, quantum mechanics in the second-quantized formulation, relativistic quantum field theory, Euclidean field theory, quant

Table of contents:

Chapter 1 THE BASIC FORMALISM OF FIELD THEORY

1.1 Fields and Products

1.1.1 Canonical quantization

1.1.2 The classical free theory

1.1.3 Anticommuting fields

1.1.4 The normal-ordered product of free-field operators

1.2 Functional Formulations of Wick’s Theorem

1.2.1 Wick’s theorem for a simple product

1.2.2 The Sym-product and the T-product

1.2.3 Wick’s theorem for symmetric products

1.2.4 Reduction formulas for operator functionals

1.2.5 The Wick and Dyson T-products

1.3 The S-Matrix and Green Functions

1.3.1 Definitions

1.3.2 Transformation to the interaction picture in the evolution operator

1.3.3 Transformation to the interaction picture for the Green functions

1.3.4 Interactions containing time derivatives of the field

1.3.5 Generating functionals for the S-matrix and Green functions

1.4 Graphs

1.4.1 Perturbation theory

1.4.2 Some concepts from graph theory

1.4.3 Symmetry coefficients

1.4.4 Recursion relation for the symmetry coefficients

1.4.5 Transformation to Mayer graphs for the exponential interaction

1.4.6 Graphs for the Yukawa-type interaction

1.4.7 Graphs for a pair interaction

1.4.8 Connectedness of the logarithm of R(φ)

1.4.9 Graphs for the Green functions

1.5 Unitarity of the S Matrix

1.5.1 The conjugation operation

1.5.2 Formal unitarity of the off-shell S matrix

1.6 Functional Integrals

1.6.1 Gaussian integrals

1.6.2 Integrals on a Grassmann algebra

1.6.3 Gaussian integrals on a Grassmann algebra

1.6.4 Gaussian integrals in field theory

1.6.5 Representations of generating functionals for the S-matrix and Green functions by functional integrals

1.6.6 The stationary-phase method

1.6.7 The Dominicis–Englert theorem

1.7 Equations in Variational Derivatives

1.7.1 The Schwinger equations

1.7.2 Linear equations for connected Green functions

1.7.3 General method of deriving the equations

1.7.4 Iteration solution of the equations

1.8 One-Irreducible Green Functions

1.8.1 Definitions

1.8.2 The equations of motion for Γ

1.8.3 Iteration solution of the equations and proof of 1-irreducibility

1.9 Renormalization Transformations

1.10 Anomalous Green Functions and Spontaneous Symmetry Breaking

Chapter 2 SPECIFIC SYSTEMS

2.1 Quantum Mechanics

2.1.1 The oscillator

2.1.2 The free particle

2.2 Nonrelativistic Field Theory

2.2.1 The quantum Base and Fermi gases

2.2.2 The atom

2.2.3 Electrons in solids and phonons

2.3 Relativistic Field Theory

2.4 Integral Representations of the Transition Amplitude

2.5 The Space E(Δ) for Various Systems

2.6 Functional Integrals Over Phase Space

Chapter 3 THE MASSLESS YANG-MILLS FIELD

3.1 Quantization of the Yang–Mills Field

3.1.1 The classical theory

3.1.2 A general recipe for quantization

3.1.3 Perturbation theory for gauges nB+c = 0

3.1.4 The generalized Feynman gauge

3.1.5 The S-matrix generating functional

3.2 Gauge Invariance

3.2.1 The Ward–Slavnov identities

3.2.2 Transversality and gauge invariance of the S-matrix in electrodynamics

3.2.3 Transversality and gauge invariance of the on-shell S-matrix for the Yang–Mills field

Chapter 4 EUCLIDEAN FIELD THEORY

4.1 The Euclidean Rotation

4.1.1 Definitions

4.1.2 The formal Euclidean rotation of the action functional

4.1.3 Euclidean rotation of the Green functions

4.1.4 Properties of the field φ¯ and the action Se(φ¯)

4.1.5 Rotation of the Lorentz group into O4

4.1.6 Examples

4.2 Functional Integral Representations

4.3 Convexity Properties

4.3.1 Quasiprobabilistic theories

4.3.2 Convexity and spectral representations

Chapter 5 STATISTICAL PHYSICS

5.1 The Quantum Statistics of Field Systems

5.1.1 Definitions

5.1.2 The free theory

5.1.3 The average of an operator in normal-ordered form

5.1.4 Diagrammatic representation of the partition function and the Green functions

5.1.5 Periodic extensions of the Green functions

5.1.6 Representations by functional integrals

5.1.7 The zero-temperature limit

5.1.8 The Feynman-Kac formula

5.1.9 Convexity properties

5.1.10 Convexity of the logarithm of the partition function

5.1.11 Representation of the partition function of the free theory by a functional integral

5.2 Lattice Spin Systems

5.2.1 The Ising model

5.2.2 The Heisenberg quantum ferromagnet

5.3 The Nonideal Classical Gas

5.3.1 A gas with two-body forces

5.3.2 A gas with many-body forces

Chapter 6 VARIATIONAL METHODS AND FUNCTIONAL LEGENDRE TRANSFORMS

6.1 Phase Transitions

6.1.1 Introduction

6.1.2 Transformation to the variational problem in thermodynamics

6.1.3 The infinite-volume limit

6.1.4 Singular and critical points

6.1.5 Description of phase transitions

6.1.6 Critical and Goldstone fluctuations

6.2 Legendre Transformations of the Generating Functional of Connected Green Functions

6.2.1 Functional formulations of the variational principle

6.2.2 The equations of motion in connected variables

6.2.3 The equations of motion in 1-irreducible variables

6.2.4 Linear equations and their general solutions

6.2.5 Iteration solution of the equations

6.2.6 The second Legendre transform

6.2.7 The self-consistent field approximation

6.2.8 The third Legendre transform

6.2.9 The fourth transform

6.2.10 Stationarity equations, renormalization, and parquet graphs

6.2.11 Symmetry properties of the complete Legendre transform and the “spontaneous interaction”

6.2.12 The ground-state energy

6.2.13 Stability and convexity properties of functional Legendre transforms

6.3 Legendre Transforms of the Logarithm of the S-Matrix Generating Functional

6.3.1 Definitions and general properties

6.3.2 The classical nonideal gas and the virial expansion

6.3.3 The Ising model

6.3.4 Analysis of nonstar graphs for the Ising model

6.3.5 The second Legendre transform for the classical gas

6.3.6 The stationarity equations and the self-consistent field approximation

Appendix 1 NONSTATIONARY PERTURBATION THEORY FOR A DISCRETE LEVEL

Appendix 2 GRAPHS AND SYMMETRY COEFFICIENTS

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