The Quantal Symmetries Of The Constrained Hamiltonian Systems And Their Applications

Several quantization formalisms for constrained Hamiltonian system are reviewed in this thesis. The Faddeev-Senjanovic(FS) path-integral quantization formalism is a mainly one. Based on the phase-space generating functional of Green function for a regular/singular Lagrangian with finite degrees of freedom, the canonical Noether theorem at the quantum level is derived. Appling to the Emden's equation, the results show that the conserved quantity at the classical level may not hold true at the quantum level. Appling to a system of interacting electron-phonon, the conserved quantity at the classical level still hold true at the quantum level. Quantal canonical Noether identities under the local transformation for a system with finite degree freedom is derived. The equations of transformation properties for global transformation in gauge field theory at the quantum level are also derived. Appling to non-Abelian Chern-Simons(CS) field, the quantal BRST charge is derived and the properties of conformal symmetries are also discussed in this system. The Poincaré-Cartan integral invariant (PCII) plays an important role in classical mechanics and field theories. It can be treated as a fundamental principle of dynamic in classical theories. In this dissertation, based on the invariance of phase space generating function of Green function, considering the transformation property in the extended phase space, along the quantal motion trajectories, the quantal PCII in field theory for a system with a regular/singular Lagrangian are derived and the results are also generalized to higher-order system. For these cases in which the Jacobian of the transformation does not equal to unity, the quantal PCII can be still derived. These cases are different from the quantal first Noether theorem. It is proved that the PCII is equivalent to the quantal canonical equations, thus the classical PCII is extended to the quantum level. The quantal PCII connected with canonical equations and canonical transformation and Hamilton-Jacobi equations are also discussed. Anyons have attracted much attention due to their possible relevance to...