Quantization of $gl(1,{\bf R})$ Generalized Chern-Simons Theory in 1+1 Dimensions

TL;DR

This paper develops a quantization method for a simple 1+1 dimensional generalized Chern-Simons theory with $gl(1,{f R})$ algebra, revealing a new quantum degree of freedom absent classically.

Contribution

It applies Batalin-Vilkovisky formalisms to quantize a generalized CS theory with on-shell reducibility and regularity violations, showing their equivalence and the emergence of a quantum degree of freedom.

Findings

Quantization via BV and BFV formalisms yields equivalent results.

A new physical degree of freedom appears at the quantum level.

The model exemplifies challenges in quantizing theories with reducibility and irregularity.

Abstract

We present a quantization of previously proposed generalized Chern-Simons theory with $gl(1,{\bf R})$ algebra in 1+1 dimensions. This simplest model shares the common features of generalized CS theories: on-shell reducibility and violations of regularity. On-shell reducibility of the theory requires us to use the Lagrangian Batalin-Vilkovisky and/or Hamiltonian Batalin-Fradkin-Vilkovisky formulation. Since the regularity condition is violated, their quantization is not straightforward. In the present case we can show that both formulations give an equivalent result. It leads to an interpretation that a physical degree of freedom which does not exist at the classical level appears at the quantum level.