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 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 $\left(i\right)$ derived by using the off-shell nilpotent (anti-)BRST symmetry transformations, $\left(ii\right)$ found to be the generators of the above continuous symmetry transformations, and $\left(iii\right)$ conserved with respect to the time-evolution due to the Euler–Lagrange equations of motion derived from the Lagrangians/Lagrangian densities (that describe the dynamics of these 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 (anti-)BRST charges. To corroborate the sanctity and preciseness of our proposal, we take into account the examples of $\left(i\right)$ the one (0 + 1)-dimensional (1D) system of a massive spinning (i.e., SUSY) relativistic particle, $\left(ii\right)$ the D-dimensional non-Abelian one-form gauge theory, and $\left(iii\right)$ the Abelian two-form and the St$\ddot{u}$ckelberg-modified version of the massive Abelian three-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.