Simple BRST quantization of general gauge models

TL;DR

This paper presents a simplified BRST quantization method for gauge models with Lie algebra symmetry, decomposing the BRST charge into nilpotent parts using only gauge generators and Lagrange multipliers.

Contribution

It introduces a decomposition of the BRST charge into two nilpotent operators without adding extra matter variables, simplifying the quantization process for gauge theories.

Findings

Decomposition of BRST charge into el and el with nilpotency and commutation properties.

Explicit form of el in terms of ghosts and gauge generators.

Simplified solutions to the BRST condition using bigrading.

Abstract

It is shown that the BRST charge $Q$ for any gauge model with a Lie algebra symmetry may be decomposed as $$Q=\del+\del^{\dag}, \del^2=\del^{\dag 2}=0, [\del, \del^{\dag}]_+=0$$ provided dynamical Lagrange multipliers are used but without introducing other matter variables in $\del$ than the gauge generators in $Q$. Furthermore, $\del$ is shown to have the form $\del=c^{\dag a}\phi_a$ (or $\phi'_ac^{\dag a}$) where $c^a$ are anticommuting expressions in the ghosts and Lagrange multipliers, and where the non-hermitian operators $\phi_a$ satisfy the same Lie algebra as the original gauge generators. By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del^{\dag}|ph\hb=0$ which is naturally solved by $c^a|ph\hb=\phi_a|ph\hb=0$ (or $c^{\dag a}|ph\hb={\phi'_a}^{\dag}|ph\hb=0$). The general solutions are shown to have a very simple form.