SYMMETRIES IN CONSTRAINED CANONICAL SYSTEMS(精)

A dynamical system described by a singular Lagrangian is subject to some in- herent phase space constraints and is called a constrained canonical system (or con- strained Hamiltonian system). Gauge field theories (for example, QED, QFD, QCD, Supergravity, Superstring) belong to this category. These theories now dominate particle physics. The theory of a constrained canonical system can also be applied to the theories of condensed matter and some other research area (for ex- ample, the quantum field theories of anyons). The theories of a constrained canon- ical system play a fundamental role in modern field theories. The quantization of a constrained canonical system with a singular Lagrangian remains one of the key problems of quantum field theory and is being intensively discussed in the litera- ture. Symmetry is now a fundamental concept in modern physics. As it is well known, the discussion of symmetry properties of a system is usually based on the examination of the Lagranigan in configuration space. In quantum theory, the phase-space path integral is more fundamental than the configuration-space path integral, the latterprovides for a Hamiltonian quadratic in canonical momenta, whereas the former can be applied to an arbitrary Hamiltonian. Thus, the phase- space form of the path integral is a necessary precursor to the configuration-space form of the path integral. Therefore, the investigation of symmetry properties of a system in phase space has a basic sense in quantum theories, which can be applied to more general cases.

Chapter I Constrained Canonical Systems and Their Quantization
1.1 The Canonical Formalism for a System with a Singular Lagrangian
1.2 First-Class Constraints and Second-Class Constraints
1 3 Dirac Brackets
1 4 The First-Class Constraints and Gauge Transformation
I 5 Gauge Fixing
1 6 Field Theory with Canonical Constraints
1 7 Dirac Canonical Quantization
1 8 The Path Integral Quantization
1 9 Path Integral Quantization of Constrained canonical Systems
1 10 Quantization of the Yang-Mills Fields
1 11 The BFV Path Integral Quantization
1 12 BFV Quantization of Scalar QED with Chern-Simons Terms
Chapter II Classical Canonical Symmetries
2.1 Global Canonical Symmetry for a System with a Singular Lagrangian
2.2 Canonical Noether Identities in Phase Space
2.3 Nonrelativistic Charge Particles in an Electromagnetic Field
2.4 The Generators of Gauge Transformation
2.5 Poincare-Cartan Integral Invariant for a Constrained Canonical System

A dynamical system described by a singular Lagrangian is subject to some in- herent phase space constraints and is called a constrained canonical system (or con- strained Hamiltonian system). Gauge field theories (for example, QED, QFD, QCD, Supergravity, Superstring) belong to this category. These theories now dominate particle physics. The theory of a constrained canonical system can also be applied to the theories of condensed matter and some other research area (for ex- ample, the quantum field theories of anyons). The theories of ..