General Solution of Quantum Master Equation in Finite-Dimensional Case

TL;DR

This paper explicitly constructs the general solution to the quantum master equation in finite-dimensional systems, revealing its triviality and emphasizing the importance of locality for meaningful physical results in gauge field quantization.

Contribution

It provides an explicit finite-dimensional solution to the quantum master equation and highlights the necessity of locality for nontrivial physical outcomes.

Findings

Finite-dimensional solutions are physically trivial.

Locality is essential for nontrivial gauge field quantization.

The solution cannot be directly extended to local field theories.

Abstract

The general solution to the quantum master equation (and its $Sp(2)$ symmetric counterpart) is constructed explicitly in case of finite number of variables. It is shown that the finite-dimensional solution is physically trivial and thus can not be extended directly to cover the case of a local field theory. In this way we conclude that the locality condition plays an important role by making it possible to obtain nontrivial physical results when quantizing gauge field theories on the basis of field-antifield formalism.