Full Description
Gauge field theories underlie all models now used in elementary particle physics. These theories refer to the class of singular theories which are also theories with constraints. The quantization of singular theories remains one of the key problems of quantum field theory and is being intensively discussed in the literature. This book is an attempt to fill the need for a comprehensive analysis of this problem, which has not heretofore been met by the available monographs and reviews. The main topics are canonical quantization and the path integral method. In addition, the Lagrangian BRST quantization is completely described, for the first time in a monograph. The book also presents a number of original results obtained by the authors, in particular, a complete description of the physical sector of an arbitrary gauge theory, quantization of singular theories with higher theories with time-dependent constraints, and correct derivatives, quantization of canonical quantization of theories of a relativistic point-like particle. As a general illustration we present quantization of field theories such as electrodynamics, Yang-Mills theory, and gravity. It should be noted that this monograph is aimed not only at giving the reader the rules of quantization according to the principle "if you do it this way, it will be good", but also at presenting strong arguments based on the modem interpretation of the classical and quantum theories which show that these methods· are the natural, if not the only possible ones.
Contents
1. Introduction.- 1.1 General.- 1.2 Singular Theories.- 2. The Structure of the Classical Singular Theory.- 2.1 Hamiltonian Formalism.- 2.2 Constraints.- 2.3 Theories with Second-Class Constraints.- 2.4 Theories with First-Class Constraints.- 2.5 Physical Interpretation of Degenerate Theories.- 2.6 Physical Quantities and Gauges in the Hamiltonian Formalism.- 2.7 Lagrangian Formalism, First-Class Constraints and Gauge Invariance.- 2.8 Formulation of a Singular Boson- and Fermion Theory in the Berezin Algebra.- 3. Canonical Quantization of Singular Theories.- 3.1 Dirac Quantization.- 3.2 The Generating Functional of Green Functions.- 3.3 Functional Integrals.- 3.4 Representation of the Generating Functional of Green Functions in the Form of a Functional Integral.- 4. Canonical Quantization of Physical Field Theories.- 4.1 Quantization of Electrodynamics in Canonical Gauges.- 4.2 Quantization of Electrodynamics in Covariant Gauges.- 4.3 Models with Yang-Mills Fields.- 4.4 Gravity.- 5. Lagrangian Quantization of Gauge Theories.- 5.1 Effective Action.- 5.2 Operator Formulation.- 6. Gauge Theories of Special Form.- 6.1 Structure of the Theories of Special Form.- 6.2 Alternative Methods of Canonical Quantization of the Theories of Special Form.- 6.3 Equivalence of the Definitions of Physical Functions in the Lagrangian and Hamiltonian Formalisms.- 7. Quantization of Singular Theories with Higher Derivatives and Theories with Time-Dependent Constraints.- 7.1 Theories with Higher Derivatives.- 7.2 Theories with Explicitly Time-Dependent Lagrangians and Constraints.- 7.3 Quantization of a Relativistic Particle.- A. Some Properties of Constraints.- B. Canonical Transformation Theorems.- C. Systems of Ordinary Differential Equations Unsolvable with Respect to the Highest Derivatives.- D. Berezin Algebra.- E. Some Properties of the Dirac Bracket.- F. Some Singular Functions.- References.- Notation.
