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

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