THE PATH-INTEGRAL QUANTIZATION AND THE CONSTRUCTION OF THE S-MATRIX OPERATOR IN THE ABELIAN AND NON-ABELIAN CHERN-SIMONS THEORIES

The covariant path integral quantization of the theory of the scalar a nd spinor fields interacting through the Abelian and non-Abelian Chern -Simons gauge fields in 2 + 1 dimensions is carried out using the De W itt-Fadeev-Popov method. The mathematical ill-definiteness of the path integral of theories with pure Chern-Simons' fields is remedied by th e introduction of the Maxwell or Maxwell-type (in the non-Abelian case ) terms, which make the resulting theories super-renormalizable and gu arantees their gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be the same as th ose of quantum electrodynamics (quantum chromodynamics) in 2 + 1 dimen sions with the substitution of the Chern-Simons propagator for the pho ton (gluon) propagator. By constructing the propagator in the general case, the existence of two limits; pure Chern-Simons and quantum elect rodynamics (quantum chromodynamics) after renormalization is demonstra ted. The Batalin-Fradkin-Vilkovisky method is invoked to quantize the theory of spinor non-Abelian fields interacting via the pure Chern-Sim ons gauge held and the equivalence of the resulting generating functio nal to the one given by the De Witt-Fadeev-Popov method is demonstrate d. The S-matrix operator is constructed, and starting from this S-matr ix operator novel topological unitarity identities are derived that de mand the vanishing of the gauge-invariant sum of the imaginary parts o f the Feynman diagrams with a given number of intermediate on-shell to pological photon lines in each order of perturbation theory. These ide ntities are illustrated by explicit examples.