Basic Canonical Brackets and Nilpotency Property of Noether (anti-)BRST Charges: Non-Abeian 1-Form Gauge Theory

TL;DR

This paper investigates the non-nilpotency of Noether-derived BRST charges in non-Abelian 1-form gauge theories due to the Curci-Ferrari condition, contrasting with the Abelian case where charges are nilpotent.

Contribution

It demonstrates that the standard Noether procedure does not produce nilpotent BRST charges in non-Abelian theories and proposes modified charges that preserve BRST invariance.

Findings

Noether charges are non-nilpotent in non-Abelian case due to CF condition.

Modified BRST charges can be BRST invariant despite non-nilpotency.

In Abelian limit, charges are both nilpotent and invariant.

Abstract

In the case of a D-dimensional non-Abelian 1-form gauge theory (without any interaction with the matter fields), we show that the application of the Noether theorem does not lead to the derivations of the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST charges that obey (i) the (anti-)BRST invariance, and (ii) the nilpotency property (despite the fact that these charges are derived from the infinitesimal, continuous and nilpotent (anti-)BRST symmetry transformations). This happens because of the presence of the {\it non-trivial} Curci-Ferrari (CF) condition on our non-Abelian theory (whose limiting case is the Abelian gauge theory where the CF-type restriction is trivial and the corresponding Noether (anti-)BRST charges turn out to be nilpotent as well as (anti-)BRST invariant together). We exploit the theoretical strength of the basic canonical approach to prove (i) the non-nilpotency of the conserved Noether (anti-)BRST charges, and (ii) the (anti-)BRST invariance of the consistently modified versions of the Noether conserved (anti-)BRST charges. We very briefly comment on the nilpotency property of the consistently modified versions of the conserved (anti-)BRST charges, too. Our present observations are very sacrosanct because they have been proven by (i) using the beauty of the symmetry properties, and (ii) exploiting the theoretical strength of the basic canonical (anti)commutators.