Masujima Michio. Path Integral Quantization and Stochastic Quantization

2 ed. — Springer-Verlag, Berlin, Heidelberg, 2009. — 286 p.In this book, we discuss the path integral quantization and the stochastic quantization of classical mechanics and classical field theory.Path Integral Representation of Quantum Mechanics
Quantum Mechanics in the Lagrangian Formalism
Contact Transformation in Analytical Mechanics and Quantum Mechanics
The Lagrangian and the Action Principle
The Feynman Path Integral Formula
The Time-Dependent SchrSdinger Equation
The Principle of Superposition and the Composition Law
Path Integral Representation of Quantum Mechanics in the Hamiltonian Formalism
Review of Quantum Mechanics in the Hamiltonian Formalism
Phase Space Path Integral Representation of the Transformation Function
Matrix Element of a Time-Ordered Product
Wave Function of the Vacuum
Generating Functional of the Green's Function
Configuration Space Path Integral Representation
Weyl Correspondence
Weyl Correspondence
Path Integral Formula in a Cartesian Coordinate System
Path Integral Formula in a Curvilinear Coordinate SystemPath Integral Representation of Quantum Field Theory
Path Integral Quantization of Field Theory
Review of Classical Field Theory
Phase Space Path Integral Quantization of Field Theory
Configuration Space Path Integral Quantization of Field Theory
Covariant Perturbation Theory
Generating Functional of Green's Functions of the Free Field
Generating Functional of Full Green's Functions of an Interacting System
Feynman-Dyson Expansion and Wick's Theorem
Symanzik Construction
Equation of Motion of the Generating Functional
Method of the Functional Fourier Transform
External Field Problem
Schwinger Theory of the Green's Function
Definition of the Green's Function and the Equation of Motion
Proper Self-Energy Parts and the Vertex Operator
Equivalence of Path Integral Quantization and Canonical Quantization
Feynman's Action Principle
The Operator, Equation of Motion and Time-Ordered Product
Canonically Conjugate Momentum and Equal-Time Canonical (Anti-)CommutatorsPath Integral Quantization of Gauge Field
Review of Lie Groups
Group Theory
Lie Groups
Non-Abelian Gauge Field Theory
Gauge Principle: "U(1) => SU(2) Isospin"
Non-Abelian Gauge Field Theory
Abelian Gauge Fields vs. Non-Abelian Gauge Fields
Examples
Path Integral Quantization of Gauge Fields
The First Faddeev-Popov Formula
The Second Faddeev-Popov Formula
Choice of Gauge-Fixing Condition
The Ward-Takahashi-Slavnov-Taylor Identity and Gauge Independence of the Physical S-Matrix
Spontaneous Symmetry Breaking and the Gauge Field
Goldstone's Theorem
Higgs-Kibble Mechanism
Path Integral Quantization of the Gauge Field in the Rksi-Gauge
Ward-Takahashi-Slavnov-Taylor Identity and the ksi-Independence of the Physical S-MatrixPath Integral Representation of Quantum Statistical Mechanics
Partition Function of the Canonical Ensemble and the Grand Canonical Ensemble
The Canonical Ensemble and the Bloch Equation
Extension to the Grand Canonical Ensemble
Fradkin Construction
Density Matrix of Relativistic Quantum Field Theory at Finite Temperature
Functional Differential Equation of the Partition Function of the Grand Canonical Ensemble
Schwinger-Dyson Equation
The Schwinger-Dyson Equation
Nonrelativistic Limit
Methods of the Auxiliary Field
Method of the Auxiliary Field in the Lagrangian Formalism
Stratonovich-Hubbard Transformation: Gaussian MethodStochastic Quantization
Review of the Theory of Probability and Stochastic Processes
Random Variables and the Notion of Convergence
Stochastic Processes
Evolution Equation of Quantum Mechanics and Quantum Field Theory
Stochastic Quantization of Non-Abelian Gauge Field
Parisi-Wu Equation and Fokker-Planck Equation
Stochastic Quantization of Abelian and Non-Abelian Gauge Fields
Covariant Nonholonomic Gauge-Fixing Condition and Stochastic Quantization of the Non-Abelian Gauge FieldAppendices
Gaussian Integration
Fermion Number Integration
Functional Integration
Gauge Invariance of "D[Amunu]DF[Amunu]
Minkowskian and Euclidean Spinors
Multivariate Normal Analysis