Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review

TL;DR

This paper reviews the limitations of Noether's theorem in deriving off-shell nilpotent (anti-)BRST charges in gauge theories and proposes a systematic method to obtain these charges, supported by multiple physical examples.

Contribution

It introduces a systematic approach to derive off-shell nilpotent (anti-)BRST charges from non-nilpotent Noether charges, enhancing understanding of gauge symmetry quantization.

Findings

Proposed a method to obtain nilpotent (anti-)BRST charges systematically.

Validated the approach with examples from various gauge theories.

Clarified the relationship between Noether charges and BRST symmetry properties.

Abstract

In some of the physically interesting gauge systems, we show that the application of the Noether theorem does not lead to the deduction of the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST charges that obey precisely the off-shell nilpotency property despite the fact that these charges are $(i)$ derived by using the off-shell nilpotent (anti-)BRST symmetry transformations, $(ii)$ the generators of the above continuous symmetry transformations, and $(iii)$ conserved w.r.t. the time-evolution due to the Euler-Lagrange equations of motion derived from the Lagrangians/Lagrangian densities (that describe the dynamics of the suitably chosen physical systems). We propose a systematic method for the derivation of the off-shell nilpotent (anti-)BRST charges from the corresponding {non-nilpotent Noether conserved (anti-)BRST charges. To corroborate the sanctity and preciseness of our proposal, we take into account the examples of $(i)$ the one ($0 + 1$)-dimensional (1D) system of a massive spinning (i.e. SUSY) relativistic particle, $(ii)$ the D-dimensional non-Abelian 1-form gauge theory, and $(iii)$ the Abelian 2-form and the St${\ddot u}$ckelberg-modified version of the massive Abelian 3-form gauge theories in any arbitrary D-dimension of spacetime. Our present endeavor is a brief review where some decisive proposals have been made and a few novel results have been obtained as far as the nilpotency property is concerned.