The Path Integral Quantization And The Construction Of The S-matrix In The Abelian And Non-Abelian Chern-Simons Theories

TL;DR

This paper develops a path integral quantization approach for abelian and non-Abelian Chern-Simons theories, addressing mathematical issues, constructing the S-matrix, and deriving topological unitarity identities in 2+1 dimensions.

Contribution

It introduces a regularization method using Maxwell terms, constructs the S-matrix, and derives novel topological unitarity identities for Chern-Simons theories.

Findings

Regularized the theory with Maxwell terms for superrenormalizability.

Constructed the S-matrix operator for Chern-Simons theories.

Derived topological unitarity identities demanding vanishing sums of certain diagram parts.

Abstract

The cvariant path integral quantization of the theory of the scalar and spinor particles interacting through the abelian and non-Abelian Chern-Simons gauge fields is carried out and is shown to be mathematically ill defined due to the absence of the transverse components of these gauge fields. This is remedied by the introduction of the Maxwell or the Maxwell-type (in the non-Abelian case)term which makes the theory superrenormalizable and guarantees its gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be formally the same as those of QED (or QCD) in 2+1 dimensions with the substitution of the Chern-Simons propagator for the photon (gluon) propagator. By constructing the propagator in the general case, the existence of two limits; pure Chern-Simons and QED (QCD) after renormalization is demonstrated. By carrying out carefully the path integral quantization of the non-Abelian Chern-Simons theories using the De Witt-Fadeev-Popov and the Batalin-Fradkin- Vilkovisky methods it is demonstrated that there is no need to quantize the dimensionless charge of the theory. The main reason is that the action in the exponent of the path integral is BRST-invariant which acquires a zero winding number and guarantees the BRST renormalizability of the model. The S-matrix operator is constructed, and starting from this S-matrix operator novel topological unitarity identities are derived that demand the vanishing of the gauge-invariant sum of the imaginary parts of the Feynman diagrams with a given number of intermediate on-shell topological photon lines in each order of perturbation theory. These identities are illustrated by an explicit example.