Quantal Canonical Symmetry In Constrained System And Its Applications To Supersymmetry Chern-Simons System

We review the history of constrained Hamilton path integral methods and the progress in canonical symmetry of constrained system, and also comment on supersymmetric Chern-Simons model and NJL model. Besides that, we present the path integral quantization for constrained Hamilton system in Faddeev-Senjanovic (FS) scheme and the symmetric character of constrained Hamilton system.We found problems in the identification of quantal global canonical Noether theorem. The problem is that the global transformation is extended to local transformation, which means the global symmetry is not conserved strictly. Without the extention from global transformation to local transformation, we deduce the global canonical Norther theorem with two-order-tensor superscripts of a transformation group at quantum level using different methods from the past.Using the framework of path integral quantization for constrained Hamilton system in FS scheme,we quantize SU(n) N=2 supersymmetric gauge field with non-abelian Chern-Simons term in 2+1 dimensions, and utilize consistency of a gauge condition naturally to deduce another gauge condition. Further, we get the generating functional of Green function in phase space. Based on the quantum global Noether theorem we achieve the total angular momentum, and obtain the fractional spin of this supersymmetric system. We further find that this anomalous fractional spin has the contribution from the group superscript components.Using Faddeev-Senjanovic path integral quantization for constrained Hamilton system, SU(n) N=2 supersymmetric gauge field system with non-abelian Chern-Simons topological term is quantized in 2+1 dimensions, and the generating functional of Green function in phase space is obtained. Further, we deduced the total angular momentum based on the global canonical Noether theorem at quantum level, which includes the orbital angular momenta, spin angular momenta and anomalous fractional spin angular momenta related to the non-abelian gauge fields. Finally, we discover that the fractional spin has the contributions both from the group superscript components and the zeroth component charge of the non-abelian gauge fields.In terms of theory of constrained Hamilton system, the Faddeev-Senjanovic path integral quantizations of extended NJL model and its bosonization model are given. The generating functional of Green function in phase space is obtained. The Chiral transformation of fields, composite fields and their conjugate momenta were deduced. Utilizing the invariance of generating functional, we obtain Ward identities for both extended model and bosonized extended model.