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

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 (unless we exploit the theoretical strength of the Gauss divergence theorem and use the appropriate equations of motion at suitable places). This happens because of the presence of the non-trivial Curci-Ferrari (CF) condition on our non-Abelian 1-form gauge theory (whose limiting case is the Abelian 1-form 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 quantities together). We exploit the theoretical strength of the basic canonical approach to prove (i) the nilpotency of the Noether (anti-)BRST charges by using the Gauss divergence theorem, and (ii) the (anti-)BRST invariance of the consistently modified versions of the Noether (anti-)BRST charges. We very briefly comment on the nilpotency property of the consistently modified versions of the Noether (anti-)BRST charges and demonstrate that the (anti-)BRST invariant versions of the conserved (anti-)BRST charges are useful in the discussion on the physicality criteria (and their consistency with the Dirac quantization conditions for the systems that are endowed with constraints).