Gauge fixing and abelianization in simple BRST quantization

TL;DR

This paper explores advanced gauge fixing and abelianization techniques in BRST quantization, extending previous decompositions using gauge fixing operators and providing solutions related to Dirac quantization.

Contribution

It introduces new decompositions of the BRST charge via gauge fixing operators, including abelianization, and connects solutions to Dirac quantization methods.

Findings

Derived new decompositions of the BRST charge using gauge fixing operators.

Established solutions involving abelian algebra structures within BRST framework.

Linked BRST solutions to Dirac quantization for inner product space analysis.

Abstract

In a previous paper \cite{Simple} it was 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 $Q$. In this paper further decompositions are derived but now by means of gauge fixing operators. As in \cite{Simple} it is shown that $\del=c^{\dag a}\phi_a$ where $c^a$ are new ghosts and $\phi_a$ are nonhermitian variables satisfying the gauge algebra. However, in distinction to \cite{Simple} also solutions of the form $\del=c^{\dag a}A_a$ where $A_a$ satisfy an abelian algebra is derived (abelianization). By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del^{\dag}|ph\hb=0$ on inner product spaces whose general solutions are expressed in terms of the solutions to a proper Dirac quantization. Thus, the procedure provides for inner products for the solutions of a Dirac quantization.