Path Integral Quantization and Stochastic Quantization

by Masujima, Michio

• ISBN13: 9783540665427
• ISBN10: 3540665420
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: Springer Verlag

Summary

The book gives an overview of path integral quantization and stochastic quantization of classical mechanics and field theory. The non-Abelian gauge field, the gravitational field, and the path integral representation of quantum statistical mechanics and stochastic quantization are described.

Table of Contents

Path Integral Representation of Quantum Mechanics	1	(54)
Quantum Mechanics in the Lagrangian Formalism	3	(19)
Contact Transformation in Analytical Mechanics and Quantum Mechanics	3	(5)
The Lagrangian and the Action Principle	8	(5)
The Feynman Path Integral Formula	13	(3)
The Time-Dependent Schrodinger Equation	16	(4)
The Principle of Superposition and the Composition Law	20	(2)
Path Integral Representation of Quantum Mechanics in the Hamiltonian Formalism	22	(18)
Review of Quantum Mechanics in the Hamiltonian Formalism	23	(2)
Phase Space Path Integral Representation of the Transformation Function	25	(2)
Matrix Element of a Time-Ordered Product	27	(4)
Wave Function of the Vacuum	31	(3)
Generating Functional of the Green's Function	34	(1)
Configuration Space Path Integral Representation	35	(5)
Weyl Correspondence	40	(11)
Weyl Correspondence	41	(2)
Path Integral Formula in a Cartesian Coordinate System	43	(2)
Path Integral Formula in a Curvilinear Coordinate System	45	(6)
Bibliography	51	(4)
Path Integral Representation of Quantum Field Theory	55	(48)
Path Integral Quantization of Field Theory	56	(11)
Review of Classical Field Theory	57	(2)
Phase Space Path Integral Quantization of Field Theory	59	(3)
Configuration Space Path Integral Quantization of Field Theory	62	(5)
Covariant Perturbation Theory	67	(5)
Generating Functional of Green's Functions of the Free Field	68	(2)
Generating Functional of Full Green's Functions of an Interacting System	70	(1)
Feynman--Dyson Expansion and Wick's Theorem	71	(1)
Symanzik Construction	72	(11)
Equation of Motion of the Generating Functional	73	(5)
Method of the Functional Fourier Transform	78	(2)
External Field Problem	80	(3)
Schwinger Theory of the Green's Function	83	(8)
Definition of the Green's Function and the Equation of Motion	83	(4)
Proper Self-Energy Parts and the Vertex Operator	87	(4)
Equivalence of Path Integral Quantization and Canonical Quantization	91	(9)
Feynman's Action Principle	91	(2)
The Operator, Equation of Motion and Time-Ordered Product	93	(3)
Canonically Conjugate Momentum and Equal-Time Canonical (Anti-)Commutators	96	(4)
Bibliography	100	(3)
Path Integral Quantization of Gauge Field	103	(72)
Review of Lie Groups	107	(6)
Group Theory	107	(1)
Lie Groups	108	(5)
Non-Abelian Gange Field Theory	113	(14)
Gauge Principle: ``U(1) ⟹ SU(2) Isospin''	113	(3)
Non-Abelian Gauge Field Theory	116	(5)
Abelian Gauge Fields vs. Non-Abelian Gauge Fields	121	(2)
Examples	123	(4)
Path Integral Quantization of Gauge Fields	127	(23)
The First Faddeev--Popov Formula	128	(6)
The Second Faddeev--Popov Formula	134	(2)
Choice of Gauge-Fixing Condition	136	(10)
The Ward-Takahashi--Slavnov--Taylor Identity and Gauge Independence of the Physical S-Matrix	146	(4)
Spontaneous Symmetry Breaking and the Gauge Field	150	(18)
Goldstone's Theorem	151	(3)
Higgs--Kibble Mechanism	154	(4)
Path Integral Quantization of the Gauge Field in the Rξ-Gauge	158	(9)
Ward--Takahashi--Slavnov--Taylor Identity and the ξ-Independence of the Physical S-Matrix	167	(1)
Bibliography	168	(7)
Path Integral Representation of Quantum Statistical Mechanics	175	(34)
Partition Function of the Canonical Ensemble and the Grand Canonical Ensemble	176	(8)
The Canonical Ensemble and the Block Equation	177	(2)
Extension to the Grand Canonical Ensemble	179	(5)
Fradkin Construction	184	(10)
Density Matrix of Relativistic Quantum Field Theory at Finite Temperature	185	(3)
Functional Differential Equation of the Partition Function of the Grand Canonical Ensemble	188	(6)
Schwinger--Dyson Equation	194	(4)
The Schwinger--Dyson Equation	195	(2)
Nonrelativistic Limit	197	(1)
Methods of the Auxiliary Field	198	(7)
Method of the Auxiliary Field in the Lagrangian Formalism	199	(2)
Stratonovich--Hubbard Transformation: Gaussian Method	201	(4)
Bibliography	205	(4)
Stochastic Quantization	209	(31)
Review of the Theory of Probability and Stochastic Processes	210	(13)
Random Variables and the Notion of Convergence	211	(5)
Stochastic Processes	216	(3)
Evolution Equation of Quantum Mechanics and Quantum Field Theory	219	(4)
Stochastic Quantization of Non-Abelian Gauge Field	223	(17)
Parisi--Wu Equation and Fokker-Planck Equation	223	(7)
Stochastic Quantization of Abelian and Non-Abelian Gauge Fields	230	(5)
Covariant Nonholonomic Gauge-Fixing Condition and Stochastic Quantization of the Non-Abelian Gauge Field	235	(5)
Bibliography	240