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

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 $\left(i\right)$ 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.

1. Introduction

It is an undeniable truth that the symmetries of all kinds (e.g., global, local, discrete, continuous, spacetime, internal, etc.) have played a decisive role in the realm of theoretical physics as they have provided a set of deep insights into various working aspects of the physical systems of interest. The basic principles behind the local gauge symmetries and diffeomorphism symmetries provide the precise theoretical descriptions of the standard model of particle physics and theory of gravity (i.e., general theory of relativity and (super)string theories). The Becchi–Rouet–Stora–Tyutin (BRST) formalism [1,2,3,4] is applied fruitfully to the gauge theories as well as the diffeomorphism invariant theories. Some of the salient features of BRST formalism are (i) it covariantly quantizes the gauge theories which are characterized by the existence of the first-class constraints on them in the terminology of Dirac’s prescription for the classifications of constraints (see, e.g., [5,6] for details), ($ii$) it is consistent with the Dirac quantization scheme because the physicality criteria with the nilpotent and conserved (anti-)BRST charges imply that the physical states, in the total quantum Hilbert space, are those that are annihilated by the operator form of the first-class constraints of the gauge theories (see, e.g., [7,8,9] for details), ($iii$) it maintains the unitarity and quantum gauge (i.e., BRST) invariance at any arbitrary order of perturbative computations of a given physical process that is allowed by an interacting gauge theory, and ($iv$) it has deep connections with some of the key ideas behind differential geometry and its (anti-)BRST transformations resemble the $\mathcal{N}=2$ supersymmetry transformations. There is a decisive difference, however, between the key properties associated with the (anti-)BRST symmetries and the $\mathcal{N}=2$ supersymmetries in the sense that the nilpotent BRST and anti-BRST symmetries are absolutely anticommuting in nature but the nilpotent $\mathcal{N}=2$ supersymmetries are not. To sum up, we note that the application of the BRST formalism is physically very useful and mathematically, its horizon is quite wide.

The purpose of our present endeavor is related with the derivations of the conserved (anti-)BRST charges by exploiting the theoretical potential of Noether’s theorem and a thorough study of their nilpotency property. We take into account diverse examples of physically interesting models of gauge theories and demonstrate that the Noether theorem does not always lead to the derivation of conserved and nilpotent (anti-)BRST charges1. One has to apply specific set of theoretical tricks and techniques to obtain the nilpotent versions of the (anti-)BRST charges from the Noether conserved (anti-)BRST charges which are found to be non-nilpotent. In simple examples, we show that one equation of motion is good enough to convert the non-nilpotent Noether conserved (anti-)BRST charges into the conserved and nilpotent (anti-)BRST charges. For instance, in the case of a 1D massive spinning (i.e., SUSY) relativistic particle (see, e.g., [10,11,12,13] and references therein), the Euler–Lagrange (EL) equations of motion (EoM) with respect to the “gauge” and “supergauge” variables are sufficient to convert a non-nilpotent set of (anti-)BRST charges into nilpotent of order two (cf. Section 2). However, in the case of a D-dimensional non-Abelian 1-form theory, the Gauss divergence theorem and EL-EoM with respect to the gauge field are needed to obtain the nilpotent (anti-)BRST charges from the conserved and non-nilpotent Noether (anti-)BRST charges. We have purposely chosen these simple examples so that it becomes clear that the celebrated Noether theorem does not lead to the derivation of nilpotent (anti-)BRST charges where $\left(i\right)$ the non-trivial (anti-)BRST invariant Curci–Ferrari (CF) type restrictions exist, and ($ii$) a set of coupled (but equivalent) Lagrangians/Lagrangian densities respects the off-shell nilpotent (anti-)BRST symmetry transformations.

It is worthwhile to mention here that the limiting cases of the above two examples are the free scalar relativistic particle and D-dimensional Abelian 1-form gauge theory where there is the existence of a single Lagrangian/Lagrangian density. In these cases, the Noether conserved (anti-)BRST charges are off-shell nilpotent automatically because the CF-type restriction is trivial. In fact, it is observed that the trivial CF-type restriction of the scalar relativistic particle is the limiting case of the non-trivial CF-type restriction of the spinning relativistic particle and the trivial CF-type restriction of the Abelian 1-form gauge theory is the limiting case of the non-trivial CF-condition of non-Abelian 1-form gauge theory.

One of the central issue we address in our present investigation is the cases of the BRST approach to higher p-form ($p=2,3,\dots$) gauge theories2 where there is always the existence of (i) a set of coupled (but equivalent) Lagrangian densities and ($ii$) a set of (anti-)BRST invariant CF-type restrictions. In such cases, the Noether theorem always leads to the derivation of conserved (anti-)BRST charges which are found to be non-nilpotent. In fact, the expressions for these charges are quite complicated, and the EL-EoMs are too many because of the presence of too many fields in the theory (cf. Section 5 below for details). We make a systematic proposal which enables us to obtain the off-shell nilpotent (anti-) BRST charges from the conserved Noether (anti-)BRST charges which are found to be non-nilpotent. One of the key ingredients of our proposal is the observation that one has to start with the EL-EoM with respect to the gauge field of the p-form gauge theories where, most of the time, the Gauss divergence theorem is required to be applied (before we exploit the potential and power of the EL-EoM with respect to the gauge fields). To be precise, all the D-dimensional ($D\ge 2$) higher p-form gauge theories require the application of the Gauss divergence theorem before we exploit the potential of EL-EoM with respect to the gauge field. After this, it is the interplay amongst (i) the application of the EL-EoMs, ($ii$) use of the (anti-) BRST transformations, and ($iii$) the requirement of Gauss’s divergence theorem that lead to the derivation of the nilpotent versions of the (anti-)BRST charges from the conserved Noether (anti-)BRST charges which are found to be non-nilpotent (cf. Section 4, Section 5 and Section 6).

The theoretical contents of our present endeavor are organized as follows. In Section 2, we exploit the theoretical potential and power of Noether’s theorem in the context of a gauge system of 1D spinning relativistic particles to deduce the explicit expressions for the conserved (anti-)BRST charges $Q_{\left(a\right)b}$ and establish that they are not off-shell nilpotent. We focus, in Section 3, on the D-dimensional non-Abelian 1-form gauge theory (without any interaction with matter fields) and demonstrate that the Noether conserved (anti-)BRST charges, once again, are non-nilpotent to begin with. We pinpoint the specific EL-EoMs that have to be used to make these (anti-)BRST charges off-shell nilpotent. Section 4 is devoted to the discussion of Noether’s theorem in the context of D-dimensional (anti-) BRST invariant Abelian 2-form theory and discuss the nitty-gritty details of the nilpotency property. The theoretical content of Section 5 is concerned with the (anti-)BRST invariant coupled (but equivalent) Lagrangian densities of the (anti-)BRST invariant modified massive Abelian three-form gauge theory, and our discussion is centered around the property of the off-shell nilpotency of the (anti-)BRST charges. Finally, in Section 6, we make some concluding remarks and comment on the future prospects of our present investigation.

In our Appendix A, we discuss the St$\ddot{u}$ckelberg-modified D-dimensional massive Abelian two-form theory and deduce the off-shell nilpotent versions of the (anti-)BRST charges [$Q_{\left(a\right)b}^{\left(1\right)}$)] from the non-nilpotent Noether conserved (anti-)BRST charges [$Q_{\left(a\right)b}$].

Convention and Notations for the 1D Massive Spinning (SUSY) Relativistic Particle and D-dimensional Abelian Two-Form as well as Three-Form Theories: We follow the convention of the left derivatives with respect to all the fermionic variables/fields of our theory in the computations of the canonical conjugate momenta and the Noether conserved currents. The flat metric tensor ${\eta }_{\mu \nu }=$ diag $(+1,-1,-1,\dots )$ is chosen for the D-dimensional flat Minkowskian space so that the dot product between two non-null vectors $P_{\mu }$ and $Q_{\mu }$ is denoted by $P·Q={\eta }_{\mu \nu }{P}^{\mu }{Q}^{\nu }=P_0{Q}_0-P_i{Q}_i$ where the Greek indices $\mu ,\nu ,\lambda ,\dots =0,1,2,\dots D-1$ correspond to the time and space directions and the Latin indices $i,j,k\dots =1,2,\dots D-1$ stand for the space directions only. Throughout the whole body of our text, the nilpotent (anti-)BRST symmetry transformations carry the symbol $s_{\left(a\right)b}$, and the corresponding conserved (anti-) BRST charges are denoted by $Q_{\left(a\right)b}$ for all kinds of the Abelian systems that have been chosen for our present discussion. For our discussions on the D-dimensional non-Abelian gauge theory, we shall adopt a different convention in Section 3.

2. Preliminary: (Anti-)BRST Charges and Nilpotency for a Massive Spinning Relativistic Particle

We begin with the following coupled (but equivalent) (anti-)BRST invariant (see, e.g., [12,13]) Lagrangians that describe the dynamics of a one (0 + 1)-dimensional massive spinning (i.e., supersymmetric) relativistic particle in the D-dimensional target space, namely:

$$\begin{array}{ccc}\hfill L_b& =& L_f+b^2+b(\dot{e}+2\overline{\beta }\beta )-i\dot{\overline{c}}\dot{c}+{\overline{\beta }}^2{\beta }^2+2i\chi (\beta \dot{\overline{C}}-\overline{\beta }\dot{c})-2e(\overline{\beta }\dot{\beta }+\gamma \chi )\hfill \\ & +& 2\gamma (\beta \overline{c}-\overline{\beta }c)+m(\overline{\beta }\dot{\beta }-\dot{\overline{\beta }}\beta +\gamma \chi )-\dot{\gamma }{\psi }_5,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill L_{\overline{b}}& =& L_f+{\overline{b}}^2-\overline{b}(\dot{e}-2\overline{\beta }\beta )-i\dot{\overline{c}}\dot{c}+{\overline{\beta }}^2{\beta }^2+2i\chi (\beta \dot{\overline{C}}-\overline{\beta }\dot{c})+2e(\dot{\overline{\beta }}\beta -\gamma \chi )\hfill \\ & +& 2\gamma (\beta \overline{c}-\overline{\beta }c)+m(\overline{\beta }\dot{\beta }-\dot{\overline{\beta }}\beta +\gamma \chi )-\dot{\gamma }{\psi }_5,\hfill \end{array}$$

(2)

where $L_f$ is the first-order Lagrangian for our system [10]

$$\begin{array}{c}\hfill L_f=p_{\mu }{\dot{x}}^{\mu }+\frac{i}{2}({\psi }_{\mu }{\dot{\psi }}^{\mu }-{\psi }_5{\dot{\psi }}_5)-\frac{e}{2}(p^2-m^2)+i\chi (p_{\mu }{\psi }^{\mu }-m{\psi }_5).\end{array}$$

(3)

In the above, the target space canonical conjugate quantities $\left(x_{\mu }\left(\tau \right),p^{\mu }\left(\tau \right)\right)$ are the bosonic coordinates $\left(x_{\mu }\right)$ and canonical momenta $\left(p^{\mu }\right)$, respectively, with the Greek indices $\mu =0,1,2,\dots ,D-1$ corresponding to the D-dimensional flat Minkowskian target space. The trajectory of the spinning particle is parameterized by $\tau$ and the generalized velocities: ${\dot{x}}_{\mu }=(d{x}^{\mu }/d\tau ),{\dot{\psi }}_{\mu }=(d{\psi }^{\mu }/d\tau )$ is defined with respect to it. The pair of fermionic (${\psi }_{\mu }^2=0;$${\psi }_{\mu }{\psi }_{\nu }+{\psi }_{\nu }{\psi }_{\mu }=0,{\psi }_5^2=0,{\psi }_5{\psi }_{\mu }+{\psi }_{\mu }{\psi }_5=0,$ etc.) variables $({\psi }_{\mu },{\psi }_5)$ are introduced in the theory to (i) maintain the SUSY gauge symmetry transformations and (ii) incorporate the mass-shell condition $p^2-m^2=0$ where m is the rest mass of the particle. The variable $e\left(\tau \right)$ and $\chi \left(\tau \right)$ are the superpartners of each other where $e\left(\tau \right)$ is the einbein variable and the fermionic $({\chi }^2=0)$ variable $\chi \left(\tau \right)$ is its superpartner and both of them behave as the “gauge” and “supergauge” variables. We have incorporated a pair of variables $(b,\overline{B})$ as the bosonic Nakanishi–Lautrup type auxiliary variables which participate in defining the CF-type restriction3: $b+\overline{B}+2\overline{\beta }\beta =0$ where the bosonic (anti-)ghost variables $\left(\overline{\beta }\right)\beta$ are the counterparts of the fermionic$(c^2=0,{\overline{c}}^2=0,c\overline{C}+\overline{C}c=0)$ (anti-)ghost variables $\left(\overline{C}\right)c$ in our supersymmetric (i.e., spinning) system of a 1D diffeomorphism invariant theory [10,12,13]. We require an additional auxiliary variable $\left(\gamma \right)$ that is fermionic$({\gamma }^2=0)$ in nature as it anticommutes with all the other fermionic variables of our theory (i.e., $\gamma \chi +\chi \gamma =0,$$\gamma {\psi }_5+{\psi }_5\gamma =0,\gamma {\psi }_{\mu }+{\psi }_{\mu }\gamma =0,\gamma c+c\gamma =0,\gamma \overline{C}+\overline{C}\gamma =0$, etc.).

The above Lagrangians $L_b$ and $L_{\overline{B}}$ are coupled (but equivalent) on the submanifold of the quantum variables where the CF-type restriction: $b+\overline{B}+2\overline{\beta }\beta =0$ is satisfied [12,13]. Furthermore, it is straightforward to check that under the following off-shell nilpotent $[s_{\left(a\right)b}^2=0]$ (anti-)BRST symmetry transformations $[s_{\left(a\right)b}]$, namely;

$$\begin{array}{ccc}& & s_{ab}{x}_{\mu }=\overline{c}{p}_{\mu }+\overline{\beta }{\psi }_{\mu },s_{ab}e=\dot{\overline{c}}+2\overline{\beta }\chi ,s_{ab}{\psi }_{\mu }=i\overline{\beta }{p}_{\mu },\hfill \\ & & s_{ab}\overline{c}=-i{\overline{\beta }}^2,s_{ab}c=i\overline{b},s_{ab}\overline{\beta }=0,s_{ab}\beta =-i\gamma ,s_{ab}{p}_{\mu }=0,\hfill \\ & & s_{ab}\gamma =0,s_{ab}\overline{b}=0,s_{ab}\chi =i\dot{\overline{\beta }},s_{ab}b=2i\overline{\beta }\gamma ,s_{ab}{\psi }_5=i\overline{\beta }m,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & s_b{x}_{\mu }=c{p}_{\mu }+\beta {\psi }_{\mu },s_{b}e=\dot{c}+2\beta \chi ,s_b{\psi }_{\mu }=i\beta p_{\mu },\hfill \\ & & s_{b}c=-i{\beta }^2,s_b\overline{c}=ib,s_b\beta =0,s_b\overline{\beta }=i\gamma ,s_b{p}_{\mu }=0,\hfill \\ & & s_b\gamma =0,s_{b}b=0,s_b\chi =i\dot{\beta },s_b\overline{b}=-2i\beta \gamma ,s_b{\psi }_5=i\beta m,\hfill \end{array}$$

(5)

the Lagrangians $L_b$ and $L_{\overline{B}}$ transform4 to the total derivatives as:

$$\begin{array}{c}\hfill s_{ab}{L}_{\overline{b}}=\frac{d}{d\tau }\left[\frac{\overline{C}}{2}(p^2+m^2)+\frac{\overline{\beta }}{2}(p_{\mu }{\psi }^{\mu }+m{\psi }_5)-\overline{b}(\dot{\overline{c}}+2\overline{\beta }\chi )\right],\end{array}$$

(6)

$$\begin{array}{c}\hfill s_b{L}_b=\frac{d}{d\tau }\left[\frac{c}{2}(p^2+m^2)+\frac{\beta }{2}(p_{\mu }{\psi }^{\mu }+m{\psi }_5)+b(\dot{c}+2\beta \chi )\right].\end{array}$$

(7)

As a consequence, the action integrals $S_1={\int }_{-\infty }^{\infty }d\tau L_b$ and $S_2={\int }_{-\infty }^{\infty }d\tau L_{\overline{b}}$ remain invariant for the physically well-defined variables that vanish off as $\tau \to \pm \infty$. The application of Noether’s theorem yields the following explicit expressions for the conserved (anti-)BRST charges [$Q_{\left(a\right)b}$] for our 1D SUSY system of a relativistic spinning particle, namely:

$$\begin{array}{ccc}\hfill Q_{ab}& =& \frac{\overline{c}}{2}(p^2-m^2)+\overline{\beta }(p_{\mu }{\psi }^{\mu }-m{\psi }_5)-\overline{b}\dot{\overline{c}}-2\overline{b}\overline{\beta }\chi -im\overline{\beta }\gamma -\overline{\beta }{}^2\dot{c}-2\beta \overline{\beta }{}^2\chi ,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill Q_b& =& \frac{c}{2}(p^2-m^2)+\beta (p_{\mu }{\psi }^{\mu }-m{\psi }_5)+b\dot{c}+2b\beta \chi -im\beta \gamma +\beta {}^2\dot{\overline{c}}+2\overline{\beta }\beta {}^2\chi ,\hfill \end{array}$$

(9)

where we have used the following explicit relationships

$$\begin{array}{c}{Q}_{ab}=\left(s_{ab}{x}_{\mu }\right)\frac{\partial L_{\overline{b}}}{\partial {\dot{x}}_{\mu }}+\left(s_{ab}{\psi }_{\mu }\right)\frac{\partial L_{\overline{b}}}{\partial {\dot{\psi }}_{\mu }}+\left(s_{ab}e\right)\frac{\partial L_{\overline{b}}}{\partial \dot{e}}+\left(s_{ab}c\right)\frac{\partial L_{\overline{b}}}{\partial \dot{c}}+\left(s_{ab}\overline{c}\right)\frac{\partial L_{\overline{b}}}{\partial \dot{\overline{c}}}\hfill \\ \hspace{1em} +\left(s_{ab}\beta \right)\frac{\partial L_{\overline{b}}}{\partial \dot{\beta }}+\left(s_{ab}{\psi }_5\right)\frac{\partial L_{\overline{b}}}{\partial \dot{{\psi }_5}}-X,\hfill \end{array}$$

(10)

$$\begin{array}{c}{Q}_b=\left(s_b{x}_{\mu }\right)\frac{\partial L_b}{\partial {\dot{x}}_{\mu }}+\left(s_b{\psi }_{\mu }\right)\frac{\partial L_b}{\partial {\dot{\psi }}_{\mu }}+\left(s_{b}e\right)\frac{\partial L_b}{\partial \dot{e}}+\left(s_{b}c\right)\frac{\partial L_b}{\partial \dot{c}}+\left(s_b\overline{c}\right)\frac{\partial L_b}{\partial \dot{\overline{c}}}\hfill \\ \hspace{1em} +\left(s_b\overline{\beta }\right)\frac{\partial L_b}{\partial \dot{\overline{\beta }}}+\left(s_b{\psi }_5\right)\frac{\partial L_b}{\partial \dot{{\psi }_5}}-Y,\hfill \end{array}$$

(11)

for the derivations of conserved (anti-)BRST charges $Q_{\left(a\right)b}$.

The following points are pertinent as far as Equations (10) and (11) are concerned. First, even though some of the auxiliary variables (e.g., $b,\overline{B},\chi$) transform (cf. Equations (4) and (5)) under the (anti-)BRST symmetry transformations, we have not used their contributions because their “time” derivative is not present in the Lagrangians $L_b$ and $L_{\overline{B}}$. Second, we have not taken into account the contributions of $\beta$ and $\overline{\beta }$ in (11) and (10), respectively, because $s_{ab}\overline{\beta }=0$ and $s_b\overline{\beta }=0$. Third, the expressions X and Y are the quantities that are present in the square brackets of (6) and (7). Fourth, the direct applications of the EL-EoMs ensure that $Q_{\left(a\right)b}$ are conserved quantities [13]. Finally, the conserved charges $Q_{\left(a\right)b}$ are the generators for the continuous (anti-)BRST symmetry transformations (4) and (5) because it can be checked that the following is true, namely:

$$\begin{array}{c}\hfill s_r\mathrm{\Phi }=-i{[\mathrm{\Phi },Q_r]}_{\pm },r=b,ab,\end{array}$$

(12)

where the $(\pm )$ signs (as the subscript) on the square bracket, on the r.h.s., denote that the square bracket is an (anti)commutator for the given variable $\mathrm{\Phi }$ being fermionic/bosonic in nature. Here, $\mathrm{\Phi }$ denotes the generic variable of $L_b$ and $L_{\overline{B}}$. In other words, we have: $\mathrm{\Phi }=x_{\mu },{\varphi }_{\mu },{\psi }_{\mu },{\psi }_5,b,\overline{B},e,\chi ,c,\overline{C},\beta ,\overline{\beta },\gamma$. Using the principle behind the continuous symmetries and their corresponding generators (cf. Equation (12)), we have the following

$$\begin{array}{c}\hfill s_b{Q}_b=-i\{Q_b,Q_b\},s_{ab}{Q}_{ab}=-i\{Q_{ab},Q_{ab}\},\end{array}$$

(13)

where the l.h.s. of both the entries can be directly computed by using the (anti-)BRST symmetry transformations (cf. Equations (4) and (5)) and the explicit expressions for the Noether conserved (anti-)BRST charges (8) and (9). It is interesting to point out that the explicit computations of $s_b{Q}_b$ and $s_{ab}{Q}_{ab}$ are as follows:

$$\begin{array}{ccc}& & s_b{Q}_b=i{\beta }^2\left[\dot{b}+\frac{1}{2}(p^2-m^2)+2\gamma \chi +2\overline{\beta }\dot{\beta }\right]\equiv -i\{Q_b,Q_b\},\hfill \\ & & s_{ab}{Q}_{ab}=i{\overline{\beta }}^2\left[\frac{1}{2}(p^2-m^2)-\dot{\overline{b}}+2\gamma \chi -2\beta \dot{\overline{\beta }}\right]\equiv -i\{Q_{ab},Q_{ab}\}.\hfill \end{array}$$

(14)

It is obvious, from the above expressions, that the (anti-)BRST charges $Q_{\left(a\right)b}$ are not off-shell nilpotent. However, these non-nilpotent conserved charges can be made off-shell nilpotent if $\left(i\right)$ we use the EL-EoMs with respect to the “gauge” and “supergauge” variables $e\left(\tau \right)$ and $\chi \left(\tau \right)$, respectively, which are, in some sense, superpartners of each other, $\left(ii\right)$ we apply the principle that the off-shell nilpotent continuous (anti-)BRST transformations are generated by the conserved (anti-)BRST charges, and $\left(iii\right)$ we apply the (anti-)BRST symmetry transformations at appropriate places.

We propose here a systematic method to obtain the off-shell nilpotent version $Q_b^{\left(1\right)}$ of the BRST charge $Q_b$. Our aim would be to obtain $Q_b^{\left(1\right)}$ from the non-nilpotent Noether BRST charge from $Q_b$ such that $s_b{Q}_b^{\left(1\right)}=-i\{Q_b^{\left(1\right)},Q_b^{\left(1\right)}\}=0$. In other words, the l.h.s. (i.e., $s_b{Q}_b^{\left(1\right)}$) should be precisely equal to zero. Toward this goal in mind, first of all, we focus on the EL-EoMs that emerge out from $L_b$ with respect to $e\left(\tau \right)$ and $\chi \left(\tau \right)$ which are the “gauge” and “supergauge” variables. These, in their useful form, are as follows:

$$\begin{array}{ccc}& & \frac{1}{2}(p^2-m^2)=-\dot{b}-2(\overline{\beta }\dot{\beta }+\gamma \chi ),\hfill \\ & & (p_{\mu }{\psi }^{\mu }-m{\psi }_5)=-im\gamma +2ie\gamma -2(\beta \dot{\overline{C}}-\overline{\beta }\dot{c}).\hfill \end{array}$$

(15)

In the second step, the substitutions of EL-EoMs with respect to “gauge” and “supergauge” variables in the appropriate terms of Noether conserved BRST charge $Q_b$. For instance, the substitutions of (15) lead to the modifications of the following terms:

$$\begin{array}{ccc}& & \frac{c}{2}(p^2-m^2)=-\dot{b}c-2c(\overline{\beta }\dot{\beta }+\gamma \chi ),\hfill \\ & & \beta (p_{\mu }{\psi }^{\mu }-m{\psi }_5)=-im\beta \gamma +2ie\beta \gamma -2\beta (\beta \dot{\overline{C}}-\overline{\beta }\dot{c}).\hfill \end{array}$$

(16)

In the third step, we observe whether the above “modified” terms add, subtract and/or cancel out with some of the terms of $Q_b$. For instance, we note that in our present case, only the term which is added in (16) is “$-im\beta \gamma$” from $Q_b$. The total sum of the expressions in (16) and this term is the following explicit expression:

$$\begin{array}{c}\hfill -\dot{b}c-2c(\overline{\beta }\dot{\beta }+\gamma \chi )-2im\beta \gamma +2ie\beta \gamma -2\beta (\beta \dot{\overline{C}}-\overline{\beta }\dot{c}).\end{array}$$

(17)

In the fourth step, we apply the BRST transformations on (17) which yields:

$$\begin{array}{c}\hfill -i\dot{b}{\beta }^2+2i{\beta }^2\chi \gamma -2i{\beta }^2\dot{\beta }\overline{\beta }.\end{array}$$

(18)

In our fifth step, we keenly observe whether some of terms of the Noether conserved charge $Q_b$ should be modified so that the terms of (18) cancel out precisely when we apply the BRST symmetry transformations on them. In our present case, we observe luckily that

$$\begin{array}{c}\hfill s_b[{\beta }^2\dot{\overline{C}}+2{\beta }^2\overline{\beta }\chi ]=i\dot{b}{\beta }^2+2i{\beta }^2\gamma \chi +2i{\beta }^2\dot{\beta }\overline{\beta },\end{array}$$

(19)

which cancels out whatever we have obtained in (18). In the final step, we apply the BRST symmetry transformations on the leftover terms of the Noether conserved charge $Q_b$. It turns out that we have the following

$$\begin{array}{c}\hfill s_b[b\dot{c}+2b\beta \chi ]=-2ib\beta \dot{\beta }+2ib\beta \dot{\beta }=0.\end{array}$$

(20)

It is pertinent to point out that all the terms that cancel out due to the application of the BRST symmetry transformations should be present in the off-shell nilpotent version of the (anti-)BRST charges $Q_{\left(a\right)b}^{\left(1\right)}$. For instance, all the terms of (17) and the terms, on the l.h.s. of (19) and (20) in the square bracket, will be present5 in $Q_b^{\left(1\right)}$. Ultimately, in our present case, we obtain the following off-shell nilpotent version of the BRST charge:

$$\begin{array}{c}\hfill Q_b⟶Q_b^{\left(1\right)}=b\dot{c}-\dot{b}c+2\beta [ie\gamma +\overline{\beta }\dot{c}-im\gamma +b\chi +\beta \overline{\beta }\chi ]-{\beta }^2\dot{\overline{c}}-2c[\overline{\beta }\dot{\beta }+\gamma \chi ].\end{array}$$

(21)

At this stage, it is straightforward to note that the following observation is true, namely:

$$\begin{array}{c}\hfill s_b{Q}_b^{\left(1\right)}=-i\{Q_b^{\left(1\right)},Q_b^{\left(1\right)}\},⟹{[Q_b^{\left(1\right)}]}^2=0.\end{array}$$

(22)

In other words, we point out that the off-shell nilpotent version of the conserved BRST charge [$Q_b^{\left(1\right)}$] is obtained from the non-nilpotent Noether conserved charge by using the EL-EoMs with respect to the “gauge” variable $e\left(\tau \right)$ and “supergauge” variable $\chi \left(\tau \right)$ and the application of the (anti-)BRST symmetry transformations at appropriate places.

Against the backdrop of the above paragraph, we note that to obtain the off-shell nilpotent version of the conserved anti-BRST charge, we use the following EL-EoMs

$$\begin{array}{ccc}& & \frac{1}{2}(p^2-m^2)=\dot{\overline{b}}+2(\dot{\overline{\beta }}\beta -\gamma \chi ),\hfill \\ & & (p_{\mu }{\psi }^{\mu }-m{\psi }_5)=2ie\gamma -im\gamma -2(\beta \dot{\overline{C}}-\overline{\beta }\dot{c}).\hfill \end{array}$$

(23)

that are derived from the perfectly anti-BRST invariant $L_{\overline{B}}$. We follow exactly the same steps as in the case of BRST charge $Q_b$ to obtain the off-shell nilpotent version of the anti-BRST $Q_{ab}^{\left(1\right)}$. In fact, the substitution of (23) into the expression for $Q_{ab}$ (cf. Equation (8)) at appropriate places leads to the following expression for $Q_{ab}^{\left(1\right)}$, namely:

$$\begin{array}{ccc}\hfill Q_{ab}⟶Q_{ab}^{\left(1\right)}& =& \dot{\overline{b}}\overline{c}-\overline{b}\dot{\overline{c}}+2\overline{\beta }[ie\gamma -\beta \dot{\overline{C}}-\overline{b}\chi -im\gamma -\beta \overline{\beta }\chi ]\hfill \\ & +& 2\overline{c}(\overline{\beta }\dot{\beta }+\gamma \chi )+{\overline{\beta }}^2\dot{c}.\hfill \end{array}$$

(24)

It is now straightforward to note that we have the following:

$$\begin{array}{c}\hfill s_{ab}{Q}_{ab}^{\left(1\right)}=-i\{Q_{ab}^{\left(1\right)},Q_{ab}^{\left(1\right)}\},⟹{[Q_{ab}^{\left(1\right)}]}^2=0.\end{array}$$

(25)

The above observation is nothing but the proof of the off-shell nilpotency of the anti-BRST charge $Q_{ab}^{\left(1\right)}$ where the l.h.s. is computed explicitly by using (4) and (24).

We end this section with the following remarks. First of all, we note that it is the EL-EoMs with respect to the “gauge” and “supergauge” variables that have been used and these have been singled out from the rest of the EL-EoM. This observation is one of the key ingredients of our proposal followed by the steps that have been discussed from Equation (15) to Equation (20). Second, in our present simple case of a 1D spinning relativistic particle, only a single step is good enough to enable us to obtain an off-shell nilpotent set of (anti-)BRST conserved charges. However, we shall see that in the context of Abelian two-form and three-form gauge theories defined in any arbitrary dimension of spacetime, more steps will be required to obtain the off-shell nilpotent set of conserved (anti-)BRST charges from the non-nilpotent forms of the (anti)BRST charges (that are derived directly from the applications of Noether’s theorem). Third, it is interesting to mention that in the limiting case of the spinning relativistic particle when $\beta =\overline{\beta }=\gamma =0$, we obtain the (anti-)BRST charges for the scalar relativistic particle from (8) and (9) as

$$\begin{array}{c}\hfill Q_{ab}⟶Q_{ab}^{\left(sr\right)}=\frac{\overline{c}}{2}(p^2-m^2)-\overline{b}\dot{\overline{c}},Q_b⟶Q_b^{\left(sr\right)}=\frac{c}{2}(p^2-m^2)+b\dot{c},\end{array}$$

(26)

which are off-shell nilpotent of order two because $s_b{Q}_b^{\left(sr\right)}=0$ and $s_{ab}{Q}_{ab}^{\left(sr\right)}=0$ due to: $s_{b}c=0,s_{b}b=0,s_b{p}_{\mu }=0$ and $s_{ab}\overline{C}=0,s_{ab}\overline{B}=0,s_{ab}{p}_{\mu }=0$ which are true for the scalar relativistic particle. In the above equation, the superscript $\left(sr\right)$ on the charges stand for the conserved and off-shell nilpotent (anti-)BRST charges for the scalar relativistic particle. Finally, it can be explicitly checked that the modified versions of the (anti-)BRST charges $Q_{\left(a\right)b}^{\left(1\right)}$ are also conserved quantities if we use the proper EL-EoMs that are derived from the coupled (but equivalent) Lagrangians $L_b$ and $L_{\overline{B}}$ of our 1D massive SUSY gauge theory.

3. (Anti-)BRST Charges and Nilpotency: Arbitrary Dimensional Non-Abelian 1-Form Gauge Theory

In this section, we show that the Noether conserved (anti-)BRST charges for the D-dimensional non-Abelian one-form gauge theory are non-nilpotent. However, following our proposal, we can obtain the appropriate forms of the conserved and off-shell nilpotent expressions for the (anti-)BRST charges for our non-Abelian theory (without any interactions with the matter fields). We begin with the following coupled (but equivalent) Lagrangian densities (see, e.g., [9]) in the Curci–Ferrari gauge (see, e.g., [15,16])

$$\begin{array}{c}\hfill {\mathcal{L}}_B=-\frac{1}{4}{F}^{\mu \nu }·F_{\mu \nu }+B·\left({\partial }_{\mu }{A}^{\mu }\right)+\frac{1}{2}(B·B+\overline{B}·\overline{B})-i{\partial }_{\mu }\overline{C}·D^{\mu }C,\\ \hfill {\mathcal{L}}_{\overline{B}}=-\frac{1}{4}{F}^{\mu \nu }·F_{\mu \nu }-\overline{B}·\left({\partial }_{\mu }{A}^{\mu }\right)+\frac{1}{2}(B·B+\overline{B}·\overline{B})-i{D}^{\mu }\overline{C}·{\partial }_{\mu }C,\end{array}$$

(27)

where the field strength tensor $F_{\mu \nu }\equiv F_{\mu \nu }^a{T}^a(a=1,2,\dots ,N^2-1)$ has been derived from the non-Abelian two-form: $F^{\left(2\right)}=d{A}^{\left(1\right)}+i{A}^{\left(1\right)}\wedge A^{\left(1\right)}$ where the one-form $A^{\left(1\right)}=d{x}^{\mu }{A}_{\mu }\equiv d{x}^{\mu }{A}_{\mu }^q{T}^a$ defines the non-Abelian gauge field ($A_{\mu }^a$) so that we have: $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+i{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$. For the $SU\left(N\right)$ Lie algebraic space, we have the Lie algebra: $[T^a,T^b]=f^{abc}{T}^c$ that is satisfied by the $SU\left(N\right)$ generators $T^a(a=1,2,\dots ,N^2-1)$ where $f^{abc}$ are the structure constants that can be chosen [17] to be totally antisymmetric in all the indices for the semi-simple Lie group $SU\left(N\right)$. In this section, we adopt the dot and cross products only in the $SU\left(N\right)$ Lie algebraic space where we have: $S·T=S^a{T}^a,{(S\times T)}^a=f^{abc}{S}^b{T}^c$ for the two non-null vectors $S^a$ and $T^a$ (in this space and $a,b,c,\dots =1,2,\dots N^2-1$). We have also taken into account the summation convention where the repeated indices are summed over and $D_{\mu }C={\partial }_{\mu }C+i(A_{\mu }\times C)$ and $D_{\mu }\overline{C}={\partial }_{\mu }\overline{C}+i(A_{\mu }\times \overline{C})$ are the covariant derivatives in the adjoint representation of the $SU\left(N\right)$ Lie algebra.

The above coupled (but equivalent) Lagrangian densities (27) respect the following off-shell nilpotent [$s_{\left(a\right)b}^2=0$] (anti-)BRST symmetry transformations ($s_{\left(a\right)b}$)

$$\begin{array}{ccc}& & s_{ab}{A}_{\mu }=D_{\mu }\overline{C},s_{ab}\overline{C}=-\frac{i}{2}(\overline{C}\times \overline{C}),s_{ab}C=i\overline{B},s_{ab}\overline{B}=0,\hfill \\ & & s_{ab}{F}_{\mu \nu }=i(F_{\mu \nu }\times \overline{C}),s_{ab}\left({\partial }_{\mu }{A}^{\mu }\right)={\partial }_{\mu }{D}^{\mu }\overline{C},s_{ab}B=i(B\times \overline{C}),\hfill \\ & & s_b{A}_{\mu }=D_{\mu }C,s_{b}C=-\frac{i}{2}(C\times C),s_b\overline{C}=iB,s_{b}B=0,\hfill \\ & & s_b\overline{B}=i(\overline{B}\times C),s_b\left({\partial }_{\mu }{A}^{\mu }\right)={\partial }_{\mu }{D}^{\mu }C,s_b{F}_{\mu \nu }=i(F_{\mu \nu }\times C),\hfill \end{array}$$

(28)

because of the following observations

$$\begin{array}{ccc}\hfill s_b{\mathcal{L}}_B& =& {\partial }_{\mu }(B·D^{\mu }C),s_{ab}{\mathcal{L}}_{\overline{B}}=-{\partial }_{\mu }(\overline{B}·D^{\mu }\overline{C}),\hfill \\ \hfill s_b{\mathcal{L}}_{\overline{B}}& =& {\partial }_{\mu }[\{B+(C\times \overline{C})\}·{\partial }^{\mu }C]-\{B+\overline{B}+(C\times \overline{C})\}·D_{\mu }{\partial }^{\mu }C,\hfill \\ \hfill s_{ab}{\mathcal{L}}_B& =& -{\partial }_{\mu }\left[\{\overline{B}+(C\times \overline{C})\}·{\partial }^{\mu }\overline{C}\right]+\{(B+\overline{B}+(C\times \overline{C})\}·D_{\mu }{\partial }^{\mu }\overline{C},\hfill \end{array}$$

(29)

which establish that the action integrals $S_1=\int d^{D}x{\mathcal{L}}_B,S_2=\int d^{D}x{\mathcal{L}}_{\overline{B}}$ remain invariant ($s_b{S}_1=0,s_{ab}{S}_2=0$) under the BRST and anti-BRST symmetry transformations, respectively, because the physical fields vanish off as $x⟶\pm \infty$ due to Gauss’s divergence theorem. If we confine our whole discussion on the submanifold of the total quantum Hilbert space of fields where the CF-condition: $B+\overline{B}+(C\times \overline{C})=0$ is respected [18], we note that both the Lagrangian densities respect both the nilpotent symmetries. In other words, we have $s_b{\mathcal{L}}_{\overline{B}}=-{\partial }_{\mu }[\overline{B}·{\partial }^{\mu }C],s_{ab}{\mathcal{L}}_B={\partial }_{\mu }[B·{\partial }^{\mu }\overline{C}]$ on the above submanifold of Hilbert space of quantum fields. We christen the transformations: $s_b{\mathcal{L}}_B={\partial }_{\mu }[B·D^{\mu }C],s_{ab}{\mathcal{L}}_{\overline{B}}=-{\partial }_{\mu }[\overline{B}·D^{\mu }\overline{C}]$ as perfect symmetry transformations because we do not use any EL-EoMs and/or CF-type condition for their proof.

In addition to the equivalence of the coupled Lagrangian densities (27) from the point of view of the (anti-)BRST symmetry considerations, we note that the absolute anticommutativity (i.e., $\{s_b,s_{ab}\}=0$) property of the (anti-)BRST symmetry transformations is satisfied if and only if we invoke the sanctity of the CF-condition: $B+\overline{B}+(C\times \overline{C})=0$. This becomes obvious when we observe that the following are true, namely:

$$\begin{array}{ccc}\hfill \{s_b,s_{ab}\}{A}_{\mu }& =& i{D}_{\mu }\left[B+\overline{B}+(C\times \overline{C})\right],\hfill \\ \hfill \{s_b,s_{ab}\}{F}_{\mu \nu }& =& -F_{\mu \nu }\times \left[B+\overline{B}+(C\times \overline{C})\right].\hfill \end{array}$$

(30)

Thus, it is clear that the absolute anticommutativity properties: $\{s_b,s_{ab}\}{A}_{\mu }=0$ and $\{s_b,s_{ab}\}{F}_{\mu \nu }=0$ are true if and only if $B+\overline{B}+(C\times \overline{C})=0$. We further note that $\{s_b,s_{ab}\}\mathrm{\Phi }=0$ where $\mathrm{\Phi }=C,\overline{C},B,\overline{B}$ is the generic field of the theory (besides $A_{\mu }$ and $F_{\mu \nu }$). Thus, the absolute anticommutativity property (i.e., $\{s_b,s_{ab}\}\mathrm{\Phi }=0$) is automatically satisfied for the fields: $B,\overline{B},C,\overline{C}$ due to the off-shell nilpotent (anti-)BRST symmetry transformations (28). It is very interesting to point out that the straightforward equivalence $({\mathcal{L}}_B={\mathcal{L}}_{\overline{B}})$ of both the Lagrangian densities (27) of our theory leads to

$$\begin{array}{c}\hfill \left({\partial }_{\mu }{A}^{\mu }\right)·\left[B+\overline{B}+(C\times \overline{C})\right]=0,\end{array}$$

(31)

modulo a total spacetime derivative. The above observation establishes the fact that both the Lagrangian densities ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$ of Equation (27) are coupled in the sense that the Nakanishi–Lautrup auxiliary fields B and $\overline{B}$ are not free but these specific fields are restricted to obey $B+\overline{B}+(C\times \overline{C})=0$ (which is nothing but the CF condition [18]). This condition, for the $SU\left(N\right)$ non-Abelian gauge theory, is physically sacrosanct because it is an (anti-)BRST invariant (i.e., $s_{\left(a\right)b}[B+\overline{B}+(C\times \overline{C})]=0$) quantity, which can be verified by using the (anti-)BRST symmetry transformations (28).

The perfect symmetry invariance of the Lagrangian densities ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$ under the infinitesimal, continuous and off-shell nilpotent $[s_{\left(a\right)b}^2=0]$ BRST and anti-BRST symmetry transformations, respectively, leads to the derivation of the following expressions for the conserved Noether currents

$$\begin{array}{ccc}& & J_{\left(B\right)}^{\mu }=\left(s_b{A}_{\nu }\right)\frac{\partial {\mathcal{L}}_B}{\partial \left({\partial }_{\mu }{A}_{\nu }\right)}+\left(s_b\overline{C}\right)\frac{\partial {\mathcal{L}}_B}{\partial \left({\partial }_{\mu }\overline{C}\right)}+\left(s_{b}C\right)\frac{\partial {\mathcal{L}}_B}{\partial \left({\partial }_{\mu }C\right)}-B·D^{\mu }C,\hfill \\ & & J_{\left(\overline{B}\right)}^{\mu }=\left(s_{ab}{A}_{\nu }\right)\frac{\partial {\mathcal{L}}_{\overline{B}}}{\partial \left({\partial }_{\mu }{A}_{\nu }\right)}+\left(s_{ab}C\right)\frac{\partial {\mathcal{L}}_{\overline{B}}}{\partial \left({\partial }_{\mu }C\right)}+\left(s_{ab}\overline{C}\right)\frac{\partial {\mathcal{L}}_{\overline{B}}}{\partial \left({\partial }_{\mu }\overline{C}\right)}+\overline{B}·D^{\mu }\overline{C},\hfill \end{array}$$

(32)

where we have followed the convention of the left derivative with respect to all the fermionic fields. The explicit form of the (anti-)BRST Noether currents are:

$$\begin{array}{ccc}& & J_{\left(\overline{B}\right)}^{\mu }=-F^{\mu \nu }·D_{\nu }\overline{C}-\overline{B}·D^{\mu }\overline{C}-\frac{1}{2}(\overline{C}\times \overline{C})·{\partial }^{\mu }C,\hfill \\ & & J_{\left(B\right)}^{\mu }=B·D^{\mu }C-F^{\mu \nu }·D_{\nu }C+\frac{1}{2}{\partial }^{\mu }\overline{C}·(C\times C).\hfill \end{array}$$

(33)

The conservation law ${\partial }_{\mu }{J}_{\left(r\right)}^{\mu }=0$ (with $r=\overline{B},B$) can be proven by exploiting the power and potential of the EL-EoMs. For the proof of ${\partial }_{\mu }{J}_{\left(\overline{B}\right)}^{\mu }=0$, we have to use the following EL-EOMs with respect to the gauge field $A_{\mu }$ and (anti-)ghost fields $\left(\overline{C}\right)C$, namely:

$$\begin{array}{c}\hfill D_{\mu }{F}^{\mu \nu }+{\partial }^{\nu }\overline{B}+(\overline{C}\times {\partial }^{\nu }C)=0,{\partial }_{\mu }\left(D^{\mu }\overline{C}\right)=0,D_{\mu }\left({\partial }^{\mu }C\right)=0,\end{array}$$

(34)

that are derived from ${\mathcal{L}}_{\overline{B}}$. In exactly similar fashion, for the proof of ${\partial }_{\mu }{J}_{\left(B\right)}^{\mu }=0$, we utilize the following EL-EoMs with respect to $A_{\mu },C$ and $\overline{C}$, namely:

$$\begin{array}{c}\hfill D_{\mu }{F}^{\mu \nu }-{\partial }^{\nu }B-({\partial }^{\nu }\overline{C}\times C)=0,{\partial }_{\mu }\left(D^{\mu }C\right)=0,D_{\mu }\left({\partial }^{\mu }\overline{C}\right)=0,\end{array}$$

(35)

which are derived from the Lagrangian density ${\mathcal{L}}_B$. Ultimately, we claim that the Noether currents (33) are conserved, and they lead to the derivation of conserved (anti-)BRST charges for our D-dimensional non-Abelian one-form gauge theory. Following the sacrosanct prescription of Noether theorem, we derive the expressions for the conserved (anti-)BRST charges ($Q_{\left(\overline{B}\right)B}=\int d^{D-1}x{J}_{\left(\overline{B}\right)B}^0)$ as follows:

$$\begin{array}{ccc}& & Q_{\overline{B}}=-\int d^{D-1}x[F^{0i}·D_i\overline{C}+\overline{B}·D_0\overline{C}+\frac{1}{2}(\overline{C}\times \overline{C})·\dot{C}],\hfill \\ & & Q_B=\int d^{D-1}x[B·D_{0}C-F^{0i}·D_{i}C+\frac{1}{2}\dot{\overline{C}}·(C\times C)].\hfill \end{array}$$

(36)

A few comments, at this juncture, are in order. First of all, it can be checked that ${\partial }_0{Q}_{\overline{B}}=0$ and ${\partial }_0{Q}_B=0$ where we have to use the EL-EoMs from ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$, and (ii) the above conserved (anti-)BRST charges are the generators of all the symmetry transformations (28) provided we use the canonical (anti-)commutators by deriving the explicit expressions for the canonical conjugate momenta from ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$. Using the principle behind the continuous symmetry transformations and their generators as the Noether conserved charges, we note that the expressions (36) lead to the following

$$\begin{array}{ccc}& & s_{ab}{Q}_{\overline{B}}=-\int d^{D-1}x\left[i(F^{0i}\times \overline{C})·D_i\overline{C}-\frac{i}{2}(\overline{C}\times \overline{C})·\dot{\overline{B}}\right]\ne 0,\hfill \\ & & s_b{Q}_B=\int d^{D-1}x\left[-i(F^{0i}\times C)·D_{i}C+\frac{i}{2}\dot{B}·(C\times C)\right]\ne 0,\hfill \end{array}$$

(37)

which are not equal to zero. In other words, we note that: $s_{ab}{Q}_{\overline{B}}=-i\{Q_{\overline{B}},Q_{\overline{B}}\}\ne 0$ and $s_b{Q}_B=-i\{Q_B,Q_B\}\ne 0$. Hence, the expressions for the Noether (anti-)BRST charges (36) are not off-shell nilpotent (i.e., $Q_{\overline{B}}^2\ne 0,Q_B^2\ne 0$) of order two.

Following the proposal mentioned in the context of 1D massive spinning relativistic particle, in the first step, we have to find out the EL-EOMs with respect to the non-Abelian gauge field and substitute it in the non-nilpotent Noether conserved charges $Q_B$ and $Q_{\overline{B}}$. In our present case, it can be done only after the application of the Gauss divergence theorem so that we have the following for the first term in $Q_{\overline{B}}$ and the second term $Q_B$, namely:

$$\begin{array}{ccc}\hfill \int d^{D-1}x\left[-F^{0i}·({\partial }_i\overline{C}+i{A}_i\times \overline{C})\right]& \equiv & +\int d^{D-1}x\left({\partial }_i{F}^{0i}\right)·\overline{C}\hfill \\ & -& i\int d^{D-1}x\left[F^{0i}·(A_i\times \overline{C})\right],\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill \int d^{D-1}x\left[-F^{0i}·({\partial }_{i}C+i{A}_i\times C)\right]& =& +\int d^{D-1}x\left({\partial }_i{F}^{0i}\right)·C\hfill \\ & -& i\int d^{D-1}x\left[F^{0i}·(A_i\times C)\right],\hfill \end{array}$$

(39)

where we can use the following EL-EoM with respect to the gauge field, namely:

$$\begin{array}{ccc}\hfill \left({\partial }_i{F}^{0i}\right)·\overline{C}& =& \dot{\overline{B}}·\overline{C}+(\overline{C}\times \dot{C})·\overline{C}-i(A_i\times F^{0i})·\overline{C},\hfill \\ \hfill \left({\partial }_i{F}^{0i}\right)·C& =& -\dot{B}·C+(\dot{\overline{C}}\times C)·C-i(A_i\times F^{0i})·C.\hfill \end{array}$$

(40)

In the second step, we have to find out if there are addition, subtraction and/or cancellations with the rest of the terms of the conserved Noether charges $Q_{\overline{B}}$ and $Q_B$ in (36). At this stage, taking the help of: $+(\overline{C}\times \dot{C})·\overline{C}=+(\overline{C}\times \overline{C})·\dot{C}$ and $-(\dot{\overline{C}}\times C)·C=-\dot{\overline{C}}·(C\times C)$, we have the following expressions for $Q_{\left(\overline{B}\right)B}^{\left(1\right)}$ from the expressions for $Q_{\overline{B}}$ and $Q_B$, namely:

$$\begin{array}{ccc}\hfill Q_{\overline{B}}\to Q_{\overline{B}}^{\left(1\right)}& =& \int d^{D-1}x\left[\dot{\overline{B}}·\overline{C}-\overline{B}·D_0\overline{C}+\frac{1}{2}(\overline{C}\times \overline{C})·\dot{C}\right],\hfill \\ \hfill Q_B\to Q_B^{\left(1\right)}& =& \int d^{D-1}x\left[B·D_{0}C-\dot{B}·C-\frac{1}{2}\dot{\overline{C}}·(C\times C)\right],\hfill \end{array}$$

(41)

where (i) a cancellation has taken place between $\left[-i(A_i\times F^{0i})·\overline{C}\right]$ and $\left[-i{F}^{0i}·(A_i\times \overline{C})\right]$ in the expression for $Q_{\overline{B}}$, ($ii$) in the expression for $Q_B$, there has been a cancellation between $\left[-i(A_i\times F^{0i})·C\right]$ and $\left[-i{F}^{0i}·(A_i\times C)\right]$, ($iii$) the pure ghost terms have been added to yield $\left[+\frac{1}{2}(\overline{C}\times \overline{C})·\dot{C}\right]$ in $Q_{\overline{B}}$ and $-\frac{1}{2}\dot{\overline{C}}·(C\times C)$ in the expression for $Q_B$, and ($iv$) the final contributions, after the applications of the Gauss divergence theorem (cf. Equations (38) and (39)) and the equation of motion (40), are as follows

$$\begin{array}{c}\hfill \dot{\overline{B}}·\overline{C}+\frac{1}{2}(\overline{C}\times \overline{C})·\dot{C},-\dot{B}·C-\frac{1}{2}\dot{\overline{C}}·(C\times C),\end{array}$$

(42)

in $Q_{\overline{B}}$ and $Q_B$, respectively. In the third step, we have to apply the anti-BRST and BRST symmetry transformations on (42), respectively. It turns out that we have the following:

$$\begin{array}{c}\hfill s_{ab}\left[\dot{\overline{B}}·\overline{C}+\frac{1}{2}(\overline{C}\times \overline{C})·\dot{C}\right]=0,s_b\left[-\dot{B}·C-\frac{1}{2}\dot{\overline{C}}·(C\times C)\right]=0.\end{array}$$

(43)

Thus, our all the relevant steps terminate here, and we have the final expression for the (anti-)BRST charges as quoted in (41) because it is elementary to check that the leftover terms of (36) are (anti-)BRST invariant: $s_{ab}(\overline{B}·D_0\overline{C})=0,s_b(B·D_{0}C)=0$. It is straightforward now to observe that the above expressions for the conserved (anti-)BRST charges are off-shell nilpotent [${(Q_{(\overline{B})B}^{\left(1\right)})}^2=0$] of order two. To corroborate this statement, we have the following observations related with the modified version of $Q_{(\overline{B})B}^{\left(1\right)}$

$$\begin{array}{ccc}& & s_{ab}{Q}_{\overline{B}}^{\left(1\right)}=-i\{Q_{\overline{B}}^{\left(1\right)},Q_{\overline{B}}^{\left(1\right)}\}=0⟹{\left[Q_{\overline{B}}^{\left(1\right)}\right]}^2=0,\hfill \\ & & s_b{Q}_B^{\left(1\right)}=-i\{Q_B^{\left(1\right)},Q_B^{\left(1\right)}\}=0⟹{\left[Q_B^{\left(1\right)}\right]}^2=0,\hfill \end{array}$$

(44)

where the l.h.s. of the above equation can be computed explicitly by taking the help of (28) and (34). Thus, the (anti-)BRST charges $Q_{(\overline{B})B}^{\left(1\right)}$ are off-shell nilpotent.

We end this section with the following useful remarks. First, we observe that the Noether conserved (anti-)BRST charges $Q_{\overline{B}}$ and $Q_B$ are not off-shell nilpotent of order two as was the case with the massive spinning (i.e., SUSY) relativistic particle. Second, to obtain the conserved and off-shell nilpotent expressions for the (anti-)BRST charges $Q_{\overline{B}}^{\left(1\right)}$ and $Q_B^{\left(1\right)}$, we have, first of all, taken the help of Gauss’s divergence theorem and, then, used only the equation of motion for the gauge field from the coupled Lagrangian densities ${\mathcal{L}}_{\overline{B}}$ and ${\mathcal{L}}_B$. This observation is similar to our observation in the context of the 1D massive spinning relativistic particle (modulo Gauss’s divergence theorem). Finally, we observe that the Abelian limit (i.e., $f^{abc}=0$ plus no dot and/or cross products plus no covariant derivative, etc.) of the Noether conserved charges $Q_{\overline{B}}$ and $Q_B$ from (31) are:

$$\begin{array}{ccc}& & Q_{\overline{B}}⟶Q_{ab}=-\int d^{D-1}x\left[F^{0i}{\partial }_i\overline{C}-B\dot{\overline{C}}\right],\hfill \\ & & Q_B⟶Q_b=\int d^{D-1}x\left[F^{0i}{\partial }_{i}C-B\dot{C}\right],\hfill \end{array}$$

(45)

where $Q_{\left(a\right)b}$ are the (anti-)BRST charges for the free Abelian 1-form gauge theory. It is an elementary exercise to note that the above expressions for the charges are off-shell nilpotent ($s_b{Q}_b=-i\{Q_b,Q_b\}=0,s_{ab}{Q}_{ab}=-i\{Q_{ab},Q_{ab}\}=0$) where we have to apply the analogues of the (anti-)BRST transformations (28) on $Q_{\left(a\right)b}$ for the Abelian one-form theory theory, which are: $s_b{F}_{0i}=0,s_{b}C=0,s_{b}B=0$ and $s_{ab}{F}_{0i}=0,s_{ab}\overline{C}=0,s_{ab}\overline{B}=0$.

4. (Anti-)BRST Charges and Nilpotency: Arbitrary Dimensional Abelian 2-Form Gauge Theory

We begin with the (anti-)BRST invariant coupled (but equivalent) Lagrangian densities for the free D-dimensional Abelian two-form gauge theory as follows (see, e.g., [8,19] for details)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_B& =& \frac{1}{12}{H}^{\mu \nu \kappa }{H}_{\mu \nu \kappa }+B^{\mu }\left({\partial }^{\nu }{B}_{\nu \mu }-{\partial }_{\mu }\varphi \right)+B·B+{\partial }_{\mu }\overline{\beta }{\partial }^{\mu }\beta \hfill \\ & +& ({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu })\left({\partial }^{\mu }{C}^{\nu }\right)+(\partial ·C-\lambda )\rho +(\partial ·\overline{C}+\rho )\lambda ,\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\overline{B}}& =& \frac{1}{12}{H}^{\mu \nu \kappa }{H}_{\mu \nu \kappa }+{\overline{B}}^{\mu }\left({\partial }^{\nu }{B}_{\nu \mu }+{\partial }_{\mu }\varphi \right)+\overline{B}·\overline{B}+{\partial }_{\mu }\overline{\beta }{\partial }^{\mu }\beta \hfill \\ & +& ({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu })\left({\partial }^{\mu }{C}^{\nu }\right)+(\partial ·C-\lambda )\rho +(\partial ·\overline{C}+\rho )\lambda ,\hfill \end{array}$$

(47)

where the totally antisymmetric tensor $H_{\mu \nu \lambda }={\partial }_{\mu }{B}_{\nu \lambda }+{\partial }_{\nu }{B}_{\lambda \mu }+{\partial }_{\lambda }{B}_{\mu \nu }$ is derived from the Abelian three-form $H^{\left(3\right)}=d{B}^{\left(2\right)}\equiv \left[(d{x}^{\mu }\wedge d{x}^{\nu }\wedge d{x}^{\lambda })/3!\right]H_{\mu \nu \lambda }$. Here, the Abelian two-form $B^{\left(2\right)}=\left[(d{x}^{\mu }\wedge d{x}^{\nu })/2!\right]B_{\mu \nu }$ is an antisymmetric $(B_{\mu \nu }=-B_{\nu \mu })$ tensor gauge field and $d=d{x}^{\mu }{\partial }_{\mu }$ (with $d^2=0$) is the exterior derivative. The gauge-fixing term for the gauge field has its origin in the co-exterior derivative of differential geometry (see, e.g., [20,21,22,23]), as it is straightforward to check that $\delta B^{\left(2\right)}=-\ast d\ast B^{\left(2\right)}\equiv \left({\partial }^{\nu }{B}_{\nu \mu }\right)d{x}^{\mu }$ where * is the Hodge duality operator on the flat D-dimesnional spectime manifold. A derivative on a scalar field $\left(\varphi \right)$ has been incorporated into the gauge-fixing term on dimensional ground. The Nakanishi–Lautrup-type auxiliary vector fields $B_{\mu }$ and ${\overline{B}}_{\mu }$ are restricted to obey the CF-type restriction: $B_{\mu }-{\overline{B}}_{\mu }-{\partial }_{\mu }\varphi =0$ (see, e.g., [24]). Here, the fermionic ($C_{\mu }^2=0,{\overline{C}}_{\mu }^2=0,C_{\mu }{C}_{\nu }+C_{\nu }{C}_{\mu }=0,{\overline{C}}_{\mu }{\overline{C}}_{\nu }+{\overline{C}}_{\nu }{\overline{C}}_{\mu }=0,C_{\mu }{\overline{C}}_{\nu }+{\overline{C}}_{\nu }{C}_{\mu }=0$, etc.) vector (anti-)ghost fields $\left({\overline{C}}_{\mu }\right)C_{\mu }$ carry the ghost numbers $(-1)+1$, respectively, and the bosonic (anti-)ghost fields $\left(\overline{\beta }\right)\beta$ are endowed with the ghost numbers $(-2)+2$, respectively. The auxiliary (anti-)ghost fields $\left(\rho \right)\lambda$ are fermionic (${\rho }^2={\lambda }^2=0,\rho \lambda +\lambda \rho =0$, etc.) in nature, and they also carry the ghost numbers $(-1)+1$, respectively, due to the fact that $\lambda =\frac{1}{2}(\partial ·C)$ and $\rho =-\frac{1}{2}(\partial ·\overline{C})$. The (anti-)ghost fields are invoked to maintain the unitarity in the theory.

The above coupled Lagrangian densities ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$ respect the following perfect off-shell nilpotent [$s_{\left(a\right)b}^2=0$] (anti-)BRST symmetry transformations [$s_{\left(a\right)b}$], namely:

$$\begin{array}{ccc}& & s_{ab}{B}_{\mu \nu }=-({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu }),s_{ab}{\overline{C}}_{\mu }=-{\partial }_{\mu }\overline{\beta },s_{ab}{C}_{\mu }={\overline{B}}_{\mu },\hfill \\ & & s_{ab}\varphi =\rho ,s_{ab}\beta =-\lambda ,s_{ab}{B}_{\mu }={\partial }_{\mu }\rho ,s_{ab}\left[\rho ,\lambda ,\overline{\beta },{\overline{B}}_{\mu },H_{\mu \nu \kappa }\right]=0,\hfill \\ & & s_b{B}_{\mu \nu }=-({\partial }_{\mu }{C}_{\nu }-{\partial }_{\nu }{C}_{\mu }),s_b{C}_{\mu }=-{\partial }_{\mu }\beta ,s_b{\overline{C}}_{\mu }=-B_{\mu },\hfill \\ & & s_b\varphi =\lambda ,s_b\overline{\beta }=-\rho ,s_b{\overline{B}}_{\mu }=-{\partial }_{\mu }\lambda ,s_b\left[\rho ,\lambda ,\beta ,B_{\mu },H_{\mu \nu \kappa }\right]=0,\hfill \end{array}$$

(48)

due to our observations that:

$$\begin{array}{c}{s}_{ab}{\mathcal{L}}_{\overline{B}}=-{\partial }_{\mu }[({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }){\overline{B}}_{\nu }-\rho {\overline{B}}^{\mu }+\lambda {\partial }^{\mu }\overline{\beta }],\hfill \\ s_b{\mathcal{L}}_B=-{\partial }_{\mu }[({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu })B_{\nu }+\rho {\partial }^{\mu }\beta +\lambda B^{\mu }].\hfill \end{array}$$

(49)

Thus, it is crystal clear that the action integrals $S_1=\int d^{D}x{\mathcal{L}}_B$ and $S_2=\int d^{D}x{\mathcal{L}}_{\overline{B}}$ remain invariant (i.e., $s_b{S}_1=0,s_{ab}{S}_2=0$) for the physical fields that vanish-off as $x⟶\pm \infty$ due to Gauss’s divergence theorem. It should be noted that the above (anti-)BRST symmetry transformations are absolutely anticommuting only on the submanifold of the Hilbert space of quantum fields where the CF-type restriction: $B_{\mu }-{\overline{B}}_{\mu }-{\partial }_{\mu }\varphi =0$ is satisfied. This statement can be corroborated by the following observation

$$\begin{array}{c}\hfill \{s_b,s_{ab}\}{B}_{\mu \nu }={\partial }_{\mu }(B_{\nu }-{\overline{B}}_{\nu })-{\partial }_{\nu }(B_{\mu }-{\overline{B}}_{\mu }),\end{array}$$

(50)

which establishes that $\{s_b,s_{ab}\}{B}_{\mu \nu }=0$ if and only if we invoke the sanctity of CF-type restriction $B_{\mu }-{\overline{B}}_{\mu }-{\partial }_{\mu }\varphi =0$. It can be checked that the absolute anticommutativity property $\{s_b,s_{ab}\}\Psi =0$ is satisfied automatically if we use (48) for the generic field $\mathrm{\Psi }=C_{\mu },{\overline{C}}_{\mu },\varphi ,\lambda ,\rho ,\beta ,\overline{\beta },B_{\mu },{\overline{B}}_{\mu }$ of ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$.

The above infinitesimal, continuous and off-shell nilpotent (anti-)BRST symmetry transformations lead to the derivations of the Noether conserved currents

$$\begin{array}{c}{J}_{\left(ab\right)}^{\mu }=\rho {\overline{B}}^{\mu }-({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }){\overline{B}}_{\nu }-({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu }){\partial }_{\nu }\overline{\beta }-\lambda {\partial }^{\mu }\overline{\beta }-H^{\mu \nu \kappa }\left({\partial }_{\nu }{\overline{C}}_{\kappa }\right),\hfill \\ J_{\left(b\right)}^{\mu }=({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }){\partial }_{\nu }\beta -({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu })B_{\nu }-\lambda B^{\mu }-\rho {\partial }^{\mu }\beta -H^{\mu \nu \kappa }\left({\partial }_{\nu }{C}_{\kappa }\right),\hfill \end{array}$$

(51)

where it is quite straightforward to check (see, e.g., [19] for details) that the conservation law (${\partial }_{\mu }{J}_{\left(r\right)}^{\mu },r=ab,b$) is true provided we use the EL-EoMs from the coupled (but equivalent) (anti-)BRST invariant Lagrangian densities ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$, respectively. The conserved Noether (anti-)BRST charges $Q_r=\int d^{D-1}x{J}_{\left(r\right)}^0(r=ab,b)$ are as follows

$$\begin{array}{c}\hfill Q_{ab}={\displaystyle \int }{d}^{D-1}x[\rho {\overline{B}}^0-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\overline{B}}_i-({\partial }^0{C}^i-{\partial }^i{C}^0){\partial }_i\overline{\beta }-\lambda {\partial }^0\overline{\beta }-H^{0ij}\left({\partial }_i{\overline{C}}_j\right)],\\ \hfill Q_b={\displaystyle \int }{d}^{D-1}x[({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\partial }_i\beta -({\partial }^0{C}^i-{\partial }^i{C}^0)B_i-\lambda B^0-\rho {\partial }^0\beta -H^{0ij}\left({\partial }_i{C}_j\right)],\end{array}$$

(52)

which are the generators for the infinitesimal, continuous and off-shell nilpotent [$s_{\left(a\right)b}^2=0$] (anti-)BRST symmetry transformations (48). At this stage, it is worthwhile to point out that the following observations are true, namely:

$$\begin{array}{c}\hfill s_{ab}{Q}_{ab}=-i\{Q_{ab},Q_{ab}\}={\displaystyle \int }{d}^{D-1}x[-({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0){\partial }_i\overline{\beta }]\ne 0,\\ \hfill s_b{Q}_b=-i\{Q_b,Q_b\}={\displaystyle \int }{d}^{D-1}x[-({\partial }^0{B}^i-{\partial }^i{B}^0){\partial }_i\beta ]\ne 0,\end{array}$$

(53)

when we apply the principle behind the continuous symmetry transformations and their generators as the conserved Noether charges. The above observations establish that the Noether conserved charges $Q_b$ and $Q_{ab}$ are not off-shell nilpotent (i.e., $Q_b^2\ne 0,Q_{ab}^2\ne 0$).

At this juncture, we follow our proposal to obtain the off-shell nilpotent versions [$Q_{\left(a\right)b}^{\left(1\right)}$] of the (anti-)BRST conserved charges from the conserved Noether (anti-)BRST charge $Q_{\left(a\right)b}$ [cf. Equation (52)] which are found to be non-nilpotent [cf. Equation (53)]. Our objective is to prove the validity of $s_{\left(a\right)b}{Q}_{\left(a\right)b}^{\left(1\right)}=-i\{Q_{\left(a\right)b}^{\left(1\right)},Q_{\left(a\right)b}^{\left(1\right)}\}=0$ by computing precisely the l.h.s. (i.e., $s_{\left(a\right)b}{Q}_{\left(a\right)b}^{\left(1\right)}$). In the first step, we have to use the equation of motion with respect to the gauge field. Toward this goal in mind, first of all, we note that the last terms of $Q_b$ and $Q_{ab}$ can be re-expressed as follows

$$\begin{array}{c}\hfill -\int d^{D-1}x{H}^{0ij}{\partial }_i{C}_j=-\int d^{D-1}x{\partial }_i\left[H^{0ij}{C}_j\right]+\int d^{D-1}x\left({\partial }_i{H}^{0ij}\right)C_j,\\ \hfill -\int d^{D-1}x{H}^{0ij}{\partial }_i{\overline{C}}_j=-\int d^{D-1}x{\partial }_i\left[H^{0ij}{\overline{C}}_j\right]+\int d^{D-1}x\left({\partial }_i{H}^{0ij}\right){\overline{C}}_j,\end{array}$$

(54)

where the first-term will be zero due to Gauss’s divergence theorem for the physical fields (which vanish-off as $x⟶\mp \infty$), and we can apply the first step of our proposal where the equations of motion with respect to the gauge field from ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$ are as follows:

$$\begin{array}{c}\hfill {\partial }_i{H}^{0ij}=({\partial }^0{B}^j-{\partial }^j{B}^0),{\partial }_i{H}^{0ij}=({\partial }^0{\overline{B}}^j-{\partial }^j{\overline{B}}^0).\end{array}$$

(55)

Thus, the last terms of $Q_{ab}$ and $Q_b$ are as follows:

$$\begin{array}{c}\hfill \int d^{D-1}x\left({\partial }_i{H}^{0ij}\right){\overline{C}}_j=\int d^{D-1}x[({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0){\overline{C}}_i],\\ \hfill \int d^{D-1}x\left({\partial }_i{H}^{0ij}\right)C_j=\int d^{D-1}x\left[({\partial }^0{B}^i-{\partial }^i{B}^0)C_i\right].\end{array}$$

(56)

We remark here that the terms in (56) here emerged out from the EL-EoMs with respect to the gauge field, and they are sacrosanct. As a consequence, they will be present in the off-shell nilpotent version of $Q_{\left(a\right)b}^{\left(1\right)}$. In the second step, we apply the (anti-)BRST symmetry transformations on the above terms (modulo integration) which lead to the following:

$$\begin{array}{c}\hfill s_{ab}[({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0){\overline{C}}_i]=-({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0){\partial }_i\overline{\beta },\\ \hfill s_b\left[({\partial }^0{B}^i-{\partial }^i{B}^0)C_i\right]=-({\partial }^0{B}^i-{\partial }^i{B}^0){\partial }_i\beta .\end{array}$$

(57)

In the third step, we have to modify6 appropriate terms of $Q_{ab}$ and $Q_b$ so that (57) cancels out when we apply the (anti-)BRST symmetry transformations on them. With this goal in mind, first of all, we focus on the BRST charge $Q_b$ and explain clearly the third step of our proposal. It can be seen that the first term of $Q_b$ can be re-written as

$$\begin{array}{c}\hfill ({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\partial }_i\beta =2({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\partial }_i\beta -({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\partial }_i\beta .\end{array}$$

(58)

It is clear that if we apply the BRST symmetry transformations on the second term of the above equation, it will serve our purpose and cancel out the second entry in (57). At this stage, we comment that the second term of (58) and (56) will be present in the off-shell nilpotent version ($Q_b^{\left(1\right)}$) of the BRST charge. This is due to the fact that our central aim is to show that $s_b{Q}_b^{\left(1\right)}=0$. Now, we focus on the first term of (57), which can be re-written in an appropriate form:

$$\begin{array}{c}\hfill 2({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\partial }_i\beta ={\partial }_i[2({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\beta ]-2{\partial }_i[{\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0]\beta .\end{array}$$

(59)

The above terms are inside the integration with respect to $d^{D-1}x$. Hence, the first term of the above equation will vanish due to the Gauss divergence theorem. The second term can be written, using the following equations of motion, derived from ${\mathcal{L}}_B$, as:

$$\begin{array}{ccc}& & \square {\overline{C}}_{\mu }-{\partial }_{\mu }(\partial ·\overline{C})-{\partial }_{\mu }\rho =0,\hfill \\ & & ⟹-2{\partial }_i[{\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0]\beta =2\beta {\partial }^0\rho \equiv 2\beta \dot{\rho }.\hfill \end{array}$$

(60)

In the fourth step, we have to apply the BRST symmetry transformations on $\left(2\beta \dot{\rho }\right)$ which turns out to be zero. We comment that this term will be present in the off-shell nilpotent version $(Q_b^{\left(1\right)})$ of the BRST charge due to our objective to show that $s_b{Q}_b^{\left(1\right)}=0$. Hence, all the steps of our proposal terminate at this stage. As a consequence, we have the following off-shell nilpotent version of the BRST charge:

$$\begin{array}{ccc}\hfill Q_b⟶Q_b^{\left(1\right)}& =& {\displaystyle \int }{d}^{D-1}x[({\partial }^0{B}^i-{\partial }^i{B}^0)C_i-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\partial }_i\beta \hfill \\ & +& 2\beta \dot{\rho }-({\partial }^0{C}^i-{\partial }^i{C}^0)B_i-\lambda B^0-\rho \dot{\beta }],\hfill \end{array}$$

(61)

where we have taken into account the appropriate term from (56), the second term from (58) and the r.h.s. of (60). It is straightforward now to check that

$$\begin{array}{c}\hfill s_b{Q}_b^{\left(1\right)}=-i\{Q_b^{\left(1\right)},Q_b^{\left(1\right)}\}=0⟹{[Q_b^{\left(1\right)}]}^2=0,\end{array}$$

(62)

which proves the off-shell nilpotency of the modified version (i.e., $Q_b^{\left(1\right)}$) of the non-nilpotency Noether conserved charges $Q_b$. It is worthwhile to mention that $s_b[\lambda B^0+\rho \dot{\beta }]=0$. Hence, these terms (i.e., $-(\lambda B^0+\rho \dot{\beta })$) remain intact and they are present in the expression for $Q_b^{\left(1\right)}$. We follow exactly the above prescription of our proposal in the case of the non-nilpotent Noether anti-BRST charge $Q_{ab}$ to obtain its modified off-shell nilpotent version as

$$\begin{array}{ccc}\hfill Q_{ab}⟶Q_{ab}^{\left(1\right)}& =& {\displaystyle \int }{d}^{D-1}x[({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0){\overline{C}}_i+({\partial }^0{C}^i-{\partial }^i{C}^0){\partial }_i\overline{\beta }\hfill \\ & -& ({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0){\overline{B}}_i+\rho {\overline{B}}^0-2\overline{\beta }\dot{\lambda }-\lambda \dot{\overline{\beta }}],\hfill \end{array}$$

(63)

which satisfies the following relationship:

$$\begin{array}{c}\hfill s_{ab}{Q}_{ab}^{\left(1\right)}=-i\{Q_{ab}^{\left(1\right)},Q_{ab}^{\left(1\right)}\}=0⟹{[Q_{ab}^{\left(1\right)}]}^2=0.\end{array}$$

(64)

The above observations establish the sanctity of our proposal that enables us to obtain precisely the off-shell nilpotent versions of the (anti-)BRST charges [$Q_{\left(a\right)b}^{\left(1\right)}$] from the conserved Noether (anti-)BRST charges [$Q_{\left(a\right)b}$] which are non-nilpotent.

We end this section with the following remarks. First, in the case of the 1D massive spinning particle, only the EL-EoMs with respect to the “gauge” and “supergauge” variables were good enough to convert the non-nilpotent Noether (anti-)BRST charges into the off-shell nilpotent versions of the (anti-)BRST conserved charges. Second, to obtain the off-shell nilpotent versions of the (anti-)BRST conserved charges in the context of the D-dimensional non-Abelian one-form theory, we required Gauss’s divergence theorem plus the EL-EoMs with respect to the gauge field. Third, in the context of the D-dimensional Abelian two-form theory, we invoked twice the Gauss divergence theorem and the EL-EoMs with respect to the gauge field and fermionic (anti-)ghost fields $\left({\overline{C}}_{\mu }\right)C_{\mu }$ to obtain the off-shell nilpotent versions of the (anti-)BRST charges from the conserved Noether non-nilpotent (anti-)BRST charges. In all the above examples, we have also used the strength of the (anti-)BRST symmetry transformations, at appropriate places, so that we could obtain the off-shell nilpotent versions of the (anti-)BRST charges from the Noether conserved charges, which are non-nilpotent.

5. (Anti-)BRST Charges and Nilpotency: Arbitrary Dimensional St$\ddot{\mathit{u}}$ckelberg-Modified Massive Abelian Three-Form Gauge Theory

We begin our present section with the (anti-)BRST invariant coupled (but equivalent) Lagrangian densities7 for the St$\ddot{u}$ckelberg-modified massive Abelian three-form gauge theory (see, e.g., [25] for details)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_B& =& {\mathcal{L}}_S+\left({\partial }_{\mu }{A}^{\mu \nu \lambda }\right)B_{\nu \lambda }-\frac{1}{2}{B}_{\mu \nu }{B}^{\mu \nu }+\frac{1}{2}{B}^{\mu \nu }\left[{\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }\mp m{\mathrm{\Phi }}_{\mu \nu }\right]\hfill \\ & -& \left({\partial }_{\mu }{\mathrm{\Phi }}^{\mu \nu }\right)B_{\nu }-\frac{1}{2}{B}^{\mu }{B}_{\mu }+\frac{1}{2}{B}^{\mu }\left[\pm m{\varphi }_{\mu }-{\partial }_{\mu }\varphi \right]+\frac{m^2}{2}{\overline{C}}_{\mu \nu }{C}^{\mu \nu }\hfill \\ & +& ({\partial }_{\mu }{\overline{C}}_{\nu \lambda }+{\partial }_{\nu }{\overline{C}}_{\lambda \mu }+{\partial }_{\lambda }{\overline{C}}_{\mu \nu })\left({\partial }^{\mu }{C}^{\nu \lambda }\right)\pm m\left({\partial }_{\mu }{\overline{C}}^{\mu \nu }\right)C_{\nu }\pm m{\overline{C}}^{\nu }\left({\partial }^{\mu }{C}_{\mu \nu }\right)\hfill \\ & +& ({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu })\left({\partial }^{\mu }{C}^{\nu }\right)-\frac{1}{2}\left[\pm m{\overline{\beta }}^{\mu }-{\partial }^{\mu }\overline{\beta }\right]\left[\pm m{\beta }_{\mu }-{\partial }_{\mu }\beta \right]\hfill \\ & -& ({\partial }_{\mu }{\overline{\beta }}_{\nu }-{\partial }_{\nu }{\overline{\beta }}_{\mu })\left({\partial }^{\mu }{\beta }^{\nu }\right)-{\partial }_{\mu }{\overline{C}}_2{\partial }^{\mu }{C}_2-m^2{\overline{C}}_2{C}_2+[(\partial ·\overline{\beta })\mp m\overline{\beta }]B\hfill \\ & -& [(\partial ·\varphi )\mp m\varphi ]B_1-[(\partial ·\beta )\mp m\beta ]B_2+\left[{\partial }_{\nu }{\overline{C}}^{\nu \mu }+{\partial }^{\mu }{\overline{C}}_1\mp \frac{m}{2}{\overline{C}}^{\mu }\right]f_{\mu }\hfill \\ & -& 2F^{\mu }{f}_{\mu }-2Ff-\left[{\partial }_{\nu }{C}^{\nu \mu }+{\partial }^{\mu }{C}_1\mp \frac{m}{2}{C}^{\mu }\right]F_{\mu }+\left[\frac{1}{2}(\partial ·C)\mp m{C}_1\right]F\hfill \\ & -& \left[\frac{1}{2}(\partial ·\overline{C})\mp m{\overline{C}}_1\right]f-B{B}_2-\frac{1}{2}{B}_1^2,\hfill \end{array}$$

(65)

$$\begin{array}{lll}{\mathcal{L}}_{\overline{B}}& =& {\mathcal{L}}_S-\left({\partial }_{\mu }{A}^{\mu \nu \lambda }\right){\overline{B}}_{\nu \lambda }-\frac{1}{2}{\overline{B}}_{\mu \nu }{\overline{B}}^{\mu \nu }+\frac{1}{2}{\overline{B}}^{\mu \nu }\left[{\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\mu }{\varphi }_{\nu }\pm m{\mathrm{\Phi }}_{\mu \nu }\right]\\ & +& \left({\partial }_{\mu }{\mathrm{\Phi }}^{\mu \nu }\right){\overline{B}}_{\nu }-\frac{1}{2}{\overline{B}}^{\mu }{\overline{B}}_{\mu }+\frac{1}{2}{\overline{B}}^{\mu }\left[\pm m{\varphi }_{\mu }-{\partial }_{\mu }\varphi \right]+\frac{m^2}{2}{\overline{C}}_{\mu \nu }{C}^{\mu \nu }\\ & +& ({\partial }_{\mu }{\overline{C}}_{\nu \lambda }+{\partial }_{\nu }{\overline{C}}_{\lambda \mu }+{\partial }_{\lambda }{\overline{C}}_{\mu \nu })\left({\partial }^{\mu }{C}^{\nu \lambda }\right)\pm m\left({\partial }_{\mu }{\overline{C}}^{\mu \nu }\right)C_{\nu }\pm m{\overline{C}}^{\nu }\left({\partial }^{\mu }{C}_{\mu \nu }\right)\\ & +& ({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu })\left({\partial }^{\mu }{C}^{\nu }\right)-\frac{1}{2}\left[\pm m{\overline{\beta }}^{\mu }-{\partial }^{\mu }\overline{\beta }\right]\left[\pm m{\beta }_{\mu }-{\partial }_{\mu }\beta \right]\\ & -& ({\partial }_{\mu }{\overline{\beta }}_{\nu }-{\partial }_{\nu }{\overline{\beta }}_{\mu })\left({\partial }^{\mu }{\beta }^{\nu }\right)-{\partial }_{\mu }{\overline{C}}_2{\partial }^{\mu }{C}_2-m^2{\overline{C}}_2{C}_2+[(\partial ·\overline{\beta })\mp m\overline{\beta }]B\\ & -& [(\partial ·\varphi )\mp m\varphi ]B_1-[(\partial ·\beta )\mp m\beta ]B_2+\left[{\partial }_{\nu }{C}^{\nu \mu }-{\partial }^{\mu }{C}_1\mp \frac{m}{2}{C}^{\mu }\right]{\overline{f}}_{\mu }\\ & +& 2{\overline{F}}^{\mu }{\overline{f}}_{\mu }+2\overline{F}\overline{f}-\left[{\partial }_{\nu }{\overline{C}}^{\nu \mu }-{\partial }^{\mu }{\overline{C}}_1\mp \frac{m}{2}{\overline{C}}^{\mu }\right]{\overline{F}}_{\mu }+\left[\frac{1}{2}(\partial ·\overline{C})\pm m{\overline{C}}_1\right]\overline{F}\\ & -& \left[\frac{1}{2}(\partial ·C)\pm m{C}_1\right]\overline{f}-B{B}_2-\frac{1}{2}{B}_1^2,\end{array}$$

(66)

where ${\mathcal{L}}_S$ is the St$\ddot{u}$ckelberg-modified classical Lagrangian density that incorporates into it the totally antisymmetric tensor gauge field $\left(A_{\mu \nu \lambda }\right)$ and antisymmetric $({\mathrm{\Phi }}_{\mu \nu }=-{\mathrm{\Phi }}_{\nu \mu })$ St$\ddot{u}$ckelberg field $\left({\mathrm{\Phi }}_{\mu \nu }\right)$ along with their kinetic terms as follows [25]:

$$\begin{array}{c}\hfill {\mathcal{L}}_S=\frac{1}{24}{H}^{\mu \nu \lambda \zeta }{H}_{\mu \nu \lambda \zeta }-\frac{m^2}{6}{A}^{\mu \nu \lambda }{A}_{\mu \nu \lambda }\pm \frac{m}{3}{A}^{\mu \nu \lambda }{\Sigma }_{\mu \nu \lambda }-\frac{1}{6}{\Sigma }^{\mu \nu \lambda }{\Sigma }_{\mu \nu \lambda }.\end{array}$$

(67)

In the above, the kinetic term with the tensor field $H_{\mu \nu \lambda \zeta }$ for the gauge field $A_{\mu \nu \lambda }$ owes its origin to the exterior derivative $d=d{x}^{\mu }{\partial }_{\mu }(\mu =0,1\dots D-1)$ because the Abelian four-form is: $H^{\left(4\right)}=d{A}^{\left(3\right)}=\left[(d{x}^{\mu }\wedge d{x}^{\nu }\wedge d{x}^{\lambda }\wedge d{x}^{\xi })/4!\right]H_{\mu \nu \lambda \xi }$ where the Abelian three-form totally antisymmetric gauge field is defined through: $A^{\left(3\right)}=\left[(d{x}^{\mu }\wedge d{x}^{\nu }\wedge d{x}^{\lambda })/3!\right]A_{\mu \nu \lambda }$. In addition, the Abelian three-form ${\Sigma }^{\left(3\right)}=\left[(d{x}^{\mu }\wedge d{x}^{\nu }\wedge d{x}^{\lambda })/3!\right]{\Sigma }_{\mu \nu \lambda }$ is defined through the Abelian St$\ddot{u}$ckelberg two-form ${\mathrm{\Phi }}^{\left(2\right)}=\left[(d{x}^{\mu }\wedge d{x}^{\nu })/2!\right]{\mathrm{\Phi }}_{\mu \nu }$ as: ${\Sigma }^{\left(3\right)}=d{\mathrm{\Phi }}^{\left(2\right)}$, which implies that ${\Sigma }_{\mu \nu \lambda }={\partial }_{\mu }{\mathrm{\Phi }}_{\nu \lambda }+{\partial }_{\nu }{\mathrm{\Phi }}_{\lambda \mu }+{\partial }_{\lambda }{\mathrm{\Phi }}_{\mu \nu }$. It is also evident that $H_{\mu \nu \lambda \xi }$ is equal to

$$\begin{array}{c}\hfill H_{\mu \nu \lambda \zeta }={\partial }_{\mu }{A}_{\nu \lambda \zeta }-{\partial }_{\nu }{A}_{\lambda \zeta \mu }+{\partial }_{\lambda }{A}_{\zeta \mu \nu }-{\partial }_{\zeta }{A}_{\mu \nu \lambda },\end{array}$$

(68)

which is invoked in the definition of the kinetic term for the gauge field $A_{\mu \nu \lambda }$.

In the (anti-)BRST invariant Lagrangian densities ${\mathcal{L}}_B$ and ${\mathcal{L}}_{\overline{B}}$, we have the bosonic auxiliary fields $(B_{\mu \nu },{\overline{B}}_{\mu \nu },B_{\mu },{\overline{B}}_{\mu }{B}_1,B_2,B)$ and a fermionic set of auxiliary fields is $(F_{\mu },{\overline{F}}_{\mu },f_{\mu },{\overline{f}}_{\mu },F,\overline{F},f,\overline{f})$ out of which the two bosonic auxiliary fields $(B,B_2)$ carry the ghost numbers $(-2,+2)$, respectively, and a set of fermionic auxiliary fields $(F_{\mu },{\overline{f}}_{\mu },\overline{f},F)$ carry the ghost number $(-1)$. On the other hand, a fermionic set of auxiliary fields $({\overline{F}}_{\mu },f_{\mu },f,\overline{F})$ is endowed with ghost number $(+1)$. To maintain the unitarity in the theory, we need the fermionic set of ghost fields $(C_{\mu \nu },{\overline{C}}_{\mu \nu },C_{\mu },{\overline{C}}_{\mu },{\overline{C}}_2,C_2,{\overline{C}}_1,C_1)$ as well as the bosonic set of (anti-)ghost fields $({\overline{\beta }}_{\mu },{\beta }_{\mu },\overline{\beta },\beta )$ where the latter (anti-)ghost fields carry the ghost numbers $(-2,+2)$. To be precise, the set of bosonic anti-ghost fields $({\overline{\beta }}_{\mu },\overline{\beta })$ and the ghost fields $({\beta }_{\mu },\beta )$ are endowed with $(-2)$ and $(+2)$ ghost numbers, respectively. The fermionic set of (anti-)ghost fields $\left({\overline{C}}_2\right)C_2$ has the ghost numbers $(-3)+3$, respectively, and all the rest of the fermionic (anti-)ghost fields: $({\overline{C}}_{\mu },{\overline{C}}_1)$ and $(C_{\mu },C_1)$ carry the ghost numbers $(-1)$ and $(+1)$, respectively. We have the bosonic vector and scalar fields $({\varphi }_{\mu },\varphi )$, too, in our theory, which appear in the gauge-fixing terms.

The following off-shell nilpotent $[s_{\left(a\right)b}^2=0]$ anti-BRST transformations $\left[s_{\left(a\right)b}\right]$

$$\begin{array}{ccc}& & s_{ab}{A}_{\mu \nu \lambda }={\partial }_{\mu }{\overline{C}}_{\nu \lambda }+{\partial }_{\nu }{\overline{C}}_{\lambda \mu }+{\partial }_{\lambda }{\overline{C}}_{\mu \nu },s_{ab}{\overline{C}}_{\mu \nu }={\partial }_{\mu }{\overline{\beta }}_{\nu }-{\partial }_{\nu }{\overline{\beta }}_{\mu },\hfill \\ & & s_{ab}{B}_{\mu \nu }=-({\partial }_{\mu }{F}_{\nu }-{\partial }_{\nu }{F}_{\mu })\equiv ({\partial }_{\mu }{\overline{f}}_{\nu }-{\partial }_{\nu }{\overline{f}}_{\mu }),s_{ab}{C}_{\mu \nu }={\overline{B}}_{\mu \nu },\hfill \\ & & s_{ab}{\mathrm{\Phi }}_{\mu \nu }=\pm m{\overline{C}}_{\mu \nu }-({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu }),s_{ab}{F}_{\mu }=-{\partial }_{\mu }{B}_2,\hfill \\ & & s_{ab}{\overline{C}}_{\mu }=\pm m{\overline{\beta }}_{\mu }-{\partial }_{\mu }\overline{\beta },s_{ab}{\varphi }_{\mu }={\overline{f}}_{\mu },s_{ab}{\overline{\beta }}_{\mu }={\partial }_{\mu }{\overline{C}}_2,\hfill \end{array}$$

$$\begin{array}{ccc}& & s_{ab}{B}_{\mu }=\pm m{\overline{f}}_{\mu }-{\partial }_{\mu }\overline{f},s_{ab}{\beta }_{\mu }={\overline{F}}_{\mu },s_{ab}{f}_{\mu }={\partial }_{\mu }{B}_1,\hfill \\ & & s_{ab}{C}_{\mu }={\overline{B}}_{\mu },s_{ab}f=\pm m{B}_1,s_{ab}\varphi =\overline{f},\hfill \\ & & s_{ab}\overline{\beta }=\pm m{\overline{C}}_2,s_{ab}\beta =\overline{F},s_{ab}F=\mp m{B}_2\hfill \\ & & s_{ab}{\overline{C}}_1=-B_2,s_{ab}{C}_1=B_1,s_{ab}{C}_2=B,\hfill \\ & & s_{ab}[H_{\mu \nu \lambda \zeta },{\overline{B}}_{\mu \nu },{\overline{B}}_{\mu },{\overline{F}}_{\mu },{\overline{f}}_{\mu },\overline{F},\overline{f},B,B_1,B_2,{\overline{C}}_2]=0,\hfill \end{array}$$

(69)

are the symmetry transformations for the action integral $S_1=\int d^{D-1}x{\mathcal{L}}_{\overline{B}}$ because the Lagrangian density ${\mathcal{L}}_{\overline{B}}$ transforms to a total spacetime derivative under the infinitesimal, continuous and off-shell nilpotent $(s_{ab}^2=0)$ anti-BRST symmetry transformations $s_{ab}$:

$$\begin{array}{ccc}\hfill s_{ab}{\mathcal{L}}_{\overline{B}}& =& {\partial }_{\mu }[{\overline{B}}^{\mu \nu }{\overline{f}}_{\nu }-({\partial }^{\mu }{\overline{C}}^{\nu \lambda }+{\partial }^{\nu }{\overline{C}}^{\lambda \mu }+{\partial }^{\lambda }{\overline{C}}^{\mu \nu }){\overline{B}}_{\nu \lambda }+B{\partial }^{\mu }{\overline{C}}_2\hfill \\ & -& B_2{\overline{F}}^{\mu }-B_1{\overline{f}}^{\mu }-({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }){\overline{F}}_{\nu }+\frac{1}{2}(\pm m{\overline{\beta }}^{\mu }-{\partial }^{\mu }\overline{\beta })\overline{F}\hfill \\ & -& ({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }){\overline{B}}_{\nu }\mp m{\overline{B}}^{\mu \nu }{\overline{C}}_{\nu }-\frac{1}{2}{\overline{B}}^{\mu }\overline{f}\hfill \\ & \pm & m{C}^{\mu \nu }(\pm m{\overline{\beta }}_{\nu }-{\partial }_{\nu }\overline{\beta })\pm m({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu })C_{\nu }].\hfill \end{array}$$

(70)

On the other hand, under the infinitesimal, continuous and off-shell nilpotent $(s_b^2=0)$ BRST symmetry transformations $\left(s_b\right)$

$$\begin{array}{ccc}& & s_b{A}_{\mu \nu \lambda }={\partial }_{\mu }{C}_{\nu \lambda }+{\partial }_{\nu }{C}_{\lambda \mu }+{\partial }_{\lambda }{C}_{\mu \nu },s_b{C}_{\mu \nu }={\partial }_{\mu }{\beta }_{\nu }-{\partial }_{\nu }{\beta }_{\mu },\hfill \\ & & s_b{\overline{B}}_{\mu \nu }=-({\partial }_{\mu }{\overline{F}}_{\nu }-{\partial }_{\nu }{\overline{F}}_{\mu })\equiv ({\partial }_{\mu }{f}_{\nu }-{\partial }_{\nu }{f}_{\mu }),s_b{\overline{C}}_{\mu \nu }=B_{\mu \nu },\hfill \\ & & s_b{\mathrm{\Phi }}_{\mu \nu }=\pm m{C}_{\mu \nu }-({\partial }_{\mu }{C}_{\nu }-{\partial }_{\nu }{C}_{\mu }),s_b{\overline{F}}_{\mu }=-{\partial }_{\mu }B,\hfill \\ & & s_b{C}_{\mu }=\pm m{\beta }_{\mu }-{\partial }_{\mu }\beta ,s_b{\varphi }_{\mu }=f_{\mu },s_b{\beta }_{\mu }={\partial }_{\mu }{C}_2,\hfill \\ & & s_b{\overline{B}}_{\mu }=\pm m{f}_{\mu }-{\partial }_{\mu }f,s_b{\overline{f}}_{\mu }=-{\partial }_{\mu }{B}_1,s_b{\overline{\beta }}_{\mu }=F_{\mu },\hfill \\ & & s_b{\overline{C}}_{\mu }=B_{\mu },s_b{\overline{C}}_2=B_2,s_b{\overline{C}}_1=-B_1,\hfill \\ & & s_b\varphi =f,s_b\beta =\pm m{C}_2,s_b\overline{\beta }=F,\hfill \\ & & s_b\overline{F}=\mp mB,s_b\overline{f}=\mp m{B}_1,s_b{C}_1=-B,\hfill \\ & & s_b[H_{\mu \nu \lambda \zeta },B_{\mu \nu },B_{\mu },f_{\mu },F_{\mu },F,f,B,B_1,B_2,C_2]=0,\hfill \end{array}$$

(71)

the Lagrangian density ${\mathcal{L}}_B$ transforms to a total spacetime derivative

$$\begin{array}{ccc}\hfill s_b{\mathcal{L}}_B& =& {\partial }_{\mu }[({\partial }^{\mu }{C}^{\nu \lambda }+{\partial }^{\nu }{C}^{\lambda \mu }+{\partial }^{\lambda }{C}^{\mu \nu })B_{\nu \lambda }+B^{\mu \nu }{f}_{\nu }-B_2{\partial }^{\mu }{C}_2\hfill \\ & -& B_1{f}^{\mu }+B{F}^{\mu }-({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu })F_{\nu }+\frac{1}{2}(\pm m{\beta }^{\mu }-{\partial }^{\mu }\beta )F\hfill \\ & +& ({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu })B_{\nu }\pm m{B}^{\mu \nu }{C}_{\nu }-\frac{1}{2}{B}^{\mu }f\hfill \\ & \mp & m{\overline{C}}^{\mu \nu }\left(\pm m{\beta }_{\nu }-{\partial }_{\nu }\beta \right)\mp m({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }){\overline{C}}_{\nu }],\hfill \end{array}$$

(72)

which establishes the fact that the action integral $S_2=\int d^{D-1}x{\mathcal{L}}_B$ respects the above infinitesimal, continuous and nilpotent BRST symmetry transformations ($s_b$).

According to Noether’s theorem, the above observations imply that the BRST conserved currents can be derived, by exploiting the standard theoretical formula, as [26]:

$$\begin{array}{ccc}\hfill J_{\left(b\right)}^{\mu }& =& H^{\mu \nu \lambda \zeta }\left({\partial }_{\nu }{C}_{\lambda \zeta }\right)\pm \frac{m}{2}[\pm m{\overline{\beta }}^{\mu }-{\partial }^{\mu }\overline{\beta }]C_2\pm m{\overline{C}}^{\mu \nu }(\pm m{\beta }_{\nu }-{\partial }_{\nu }\beta )+({\partial }^{\mu }{C}^{\nu \lambda }\hfill \\ & +& {\partial }^{\nu }{C}^{\lambda \mu }+{\partial }^{\lambda }{C}^{\mu \nu })B_{\nu \lambda }-({\partial }^{\mu }{\overline{C}}^{\nu \lambda }+{\partial }^{\nu }{\overline{C}}^{\lambda \mu }+{\partial }^{\lambda }{\overline{C}}^{\mu \nu })({\partial }_{\nu }{\beta }_{\lambda }-{\partial }_{\lambda }{\beta }_{\nu })\mp m{C}^{\mu \nu }{B}_{\nu }\hfill \\ & -& B_1{f}^{\mu }+[\pm m{A}^{\mu \nu \lambda }-{\Sigma }^{\mu \nu \lambda }][\pm m{C}_{\nu \lambda }-({\partial }_{\nu }{C}_{\lambda }-{\partial }_{\lambda }{C}_{\nu })]-B_2{\partial }^{\mu }{C}_2+B^{\mu \nu }{f}_{\nu }\hfill \\ & -& ({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu })\left({\partial }_{\nu }{C}_2\right)-({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu })F_{\nu }+({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu })B_{\nu }+B{F}^{\mu }\hfill \\ & -& \frac{1}{2}{B}^{\mu }f+\frac{1}{2}\left(\pm m{\beta }^{\mu }-{\partial }^{\mu }\beta \right)F-({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu })(\pm m{\beta }_{\nu }-{\partial }_{\nu }\beta ).\hfill \end{array}$$

(73)

In exactly similar fashion, we obtain the precise expression for the conserved $({\partial }_{\mu }{J}_{\left(ab\right)}^{\mu }=0)$ anti-BRST Noether current $J_{\left(ab\right)}^{\mu }$ [26]:

$$\begin{array}{ccc}\hfill J_{\left(ab\right)}^{\mu }& =& H^{\mu \nu \lambda \zeta }\left({\partial }_{\nu }{\overline{C}}_{\lambda \zeta }\right)\pm \frac{m}{2}[\pm m{\beta }^{\mu }-{\partial }^{\mu }\beta ]{\overline{C}}_2\mp m{C}^{\mu \nu }(\pm m{\overline{\beta }}_{\nu }-{\partial }_{\nu }\overline{\beta })-({\partial }^{\mu }{\overline{C}}^{\nu \lambda }\hfill \\ & +& {\partial }^{\nu }{\overline{C}}^{\lambda \mu }+{\partial }^{\lambda }{\overline{C}}^{\mu \nu }){\overline{B}}_{\nu \lambda }+({\partial }^{\mu }{C}^{\nu \lambda }+{\partial }^{\nu }{C}^{\lambda \mu }+{\partial }^{\lambda }{C}^{\mu \nu })({\partial }_{\nu }{\overline{\beta }}_{\lambda }-{\partial }_{\lambda }{\overline{\beta }}_{\nu })\pm m{\overline{C}}^{\mu \nu }{\overline{B}}_{\nu }\hfill \\ & -& {\overline{f}}^{\mu }{B}_1+[\pm m{A}^{\mu \nu \lambda }-{\Sigma }^{\mu \nu \lambda }][\pm m{\overline{C}}_{\nu \lambda }-({\partial }_{\nu }{\overline{C}}_{\lambda }-{\partial }_{\lambda }{\overline{C}}_{\nu })]+B{\partial }^{\mu }{\overline{C}}_2+{\overline{B}}^{\mu \nu }{\overline{f}}_{\nu }\hfill \\ & -& ({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu })\left({\partial }_{\nu }{\overline{C}}_2\right)-({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }){\overline{F}}_{\nu }-({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }){\overline{B}}_{\nu }-B_2{\overline{F}}^{\mu }\hfill \\ & -& \frac{1}{2}{\overline{B}}^{\mu }\overline{f}+\frac{1}{2}(\pm m{\overline{\beta }}^{\mu }-{\partial }^{\mu }\overline{\beta })\overline{F}+({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu })(\pm m{\overline{\beta }}_{\nu }-{\partial }_{\nu }\overline{\beta }).\hfill \end{array}$$

(74)

The conservation law $({\partial }_{\mu }{J}_{\left(a\right)b}^{\mu }=0)$ can be checked by using the EL-EoMs that have been derived in our earlier work [26]. In these proofs, the algebra is a bit involved, but it is quite straightforward to check that ${\partial }_{\mu }{J}_{\left(r\right)}^{\mu }=0$ with $r=b,ab$.

The conserved Noether charges $Q_{\left(a\right)b}=\int d^{D-1}x{J}_{\left(a\right)b}^0$ (that emerge out from the conserved currents) $J_{\left(a\right)b}^{\mu }$ are as follows [26]:

$$\begin{array}{ccc}\hfill Q_{ab}& =& \int d^{D-1}x{J}_{\left(ab\right)}^0\equiv \int d^{D-1}x[H^{0ijk}\left({\partial }_i{\overline{C}}_{jk}\right)\pm \frac{m}{2}[\pm m{\beta }^0-{\partial }^0\beta ]{\overline{C}}_2\hfill \\ & -& [\pm m{C}^{0i}-({\partial }^0{C}^i-{\partial }^i{C}^0)](\pm m{\overline{\beta }}_i-{\partial }_i\overline{\beta })+[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)]{\overline{B}}_i\hfill \\ & -& ({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}){\overline{B}}_{ij}+({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)\hfill \\ & -& {\overline{f}}^0{B}_1+[\pm m{A}^{0ij}-{\Sigma }^{0ij}][\pm m{\overline{C}}_{ij}-({\partial }_i{\overline{C}}_j-{\partial }_j{\overline{C}}_i)]+B{\partial }^0{\overline{C}}_2-\frac{1}{2}{\overline{B}}^0\overline{f}-B_2{\overline{F}}^0\hfill \\ & +& {\overline{B}}^{0i}{\overline{f}}_i-({\partial }^0{\beta }^i-{\partial }^i{\beta }^0)\left({\partial }_i{\overline{C}}_2\right)-({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\overline{F}}_i+\frac{1}{2}(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })\overline{F}],\hfill \end{array}$$

(75)

$$\begin{array}{ccc}\hfill Q_b& =& \int d^{D-1}x{J}_{\left(b\right)}^0\equiv \int d^{D-1}x[H^{0ijk}\left({\partial }_i{C}_{jk}\right)\pm \frac{m}{2}[\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta }]C_2\hfill \\ & +& [\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)](\pm m{\beta }_i-{\partial }_i\beta )-[\pm m{C}^{0i}-({\partial }^0{C}^i-{\partial }^i{C}^0)]B_i\hfill \\ & +& ({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})B_{ij}-({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & -& B_1{f}^0+[\pm m{A}^{0ij}-{\Sigma }^{0ij}][\pm m{C}_{ij}-({\partial }_i{C}_j-{\partial }_j{C}_i)]-B_2{\partial }^0{C}_2-\frac{1}{2}{B}^{0}f+B^{0i}{f}_i\hfill \\ & -& ({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0)\left({\partial }_i{C}_2\right)-({\partial }^0{\beta }^i-{\partial }^i{\beta }^0)F_i+B{F}^0+\frac{1}{2}\left(\pm m{\beta }^0-{\partial }^0\beta \right)F].\hfill \end{array}$$

(76)

These conserved charges are the generators of all the (anti-)BRST symmetry transformations. However, they are not off-shell nilpotent of order two. Exploiting the principle behind the continuous symmetry transformations and their generators, we obtain the following

$$\begin{array}{ccc}\hfill s_b{Q}_b=-i\{Q_b,Q_b\}& =& \int d^{D-1}x[\pm \frac{m}{2}\left(\pm m{F}^0-{\partial }^{0}F\right)C_2-({\partial }^0{F}^i-{\partial }^i{F}^0)\left({\partial }_i{C}_2\right)\hfill \\ & -& ({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & +& \left[\pm m{B}^{0i}-({\partial }^0{B}^i-{\partial }^i{B}^0)](\pm m{\beta }_i-{\partial }_i\beta )\right]\ne 0,\hfill \\ \hfill s_{ab}{Q}_{ab}=-i\{Q_{ab},Q_{ab}\}& =& \int d^{D-1}x[\pm \frac{m}{2}\left(\pm m{\overline{F}}^0-{\partial }^0\overline{F}\right){\overline{C}}_2-({\partial }^0{\overline{F}}^i-{\partial }^i{\overline{F}}^0)\left({\partial }_i{\overline{C}}_2\right)\hfill \\ & +& ({\partial }^0{\overline{B}}^{ij}+{\partial }^i{\overline{B}}^{j0}+{\partial }^j{\overline{B}}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)\hfill \\ & -& \left[\pm m{\overline{B}}^{0i}-({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0)](\pm m{\overline{\beta }}_i-{\partial }_i\overline{\beta })\right]\ne 0,\hfill \end{array}$$

(77)

where the l.h.s. of the above equations has been explicitly computed by using the (anti-) BRST symmetry transformations (cf. Equations (69) and (71)) and the expressions for the Noether conserved charges $Q_{\left(a\right)b}$ (cf. Equations (75) and (76)). The above observations establish that the conserved Noether (anti-)BRST charges $Q_{\left(a\right)b}$ are not off-shell nilpotent of order two.

We now follow our step-by-step proposal to obtain the off-shell nilpotent expressions for the (anti-)BRST charges $Q_{\left(a\right)b}^{\left(1\right)}$ where the off-shell nilpotency is proven by using the principle behind the continuous symmetry transformations and their generators as: $s_b{Q}_b^{\left(1\right)}=-i\{Q_b^{\left(1\right)},Q_b^{\left(1\right)}\}=0$ and $s_{ab}{Q}_{ab}^{\left(1\right)}=-i\{Q_{ab}^{\left(1\right)},Q_{ab}^{\left(1\right)}\}=0$. First of all, we focus on the BRST charge $Q_b$. The first step is to use the equation of motion with respect to the gauge field. Toward this goal in mind, we note that the first term of $Q_b$ can be re-expressed as:

$$\begin{array}{c}\hfill \int d^{D-1}x\left[H^{0ijk}{\partial }_i{C}_{jk}\right]=\int d^{D-1}x\left[{\partial }_i\left\{H^{0ijk}{C}_{jk}\right\}\right]-\int d^{D-1}x\left[\left({\partial }_i{H}^{0ijk}\right)C_{jk}\right].\end{array}$$

(78)

The first term on the r.h.s. of the above equation vanishes for the physical fields due to celebrated Gauss’s divergence theorem. In the second term, we apply the following equation of motion with respect to the gauge field $A_{\mu \nu \lambda }$, namely:

$$\begin{array}{c}\hfill {\partial }_{\rho }{H}^{\rho \mu \nu \lambda }+m^2{A}^{\mu \nu \lambda }\mp m{\Sigma }^{\mu \nu \lambda }+({\partial }^{\mu }{B}^{\nu \lambda }+{\partial }^{\nu }{B}^{\lambda \mu }+{\partial }^{\lambda }{B}^{\mu \nu })=0\\ \hfill ⟹-\left({\partial }_i{H}^{0ijk}\right)C_{jk}=\mp m\left[\pm m{A}^{0ij}-{\Sigma }^{0ij}\right]C_{ij}-({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{oi})C_{ij}.\end{array}$$

(79)

In the second step, we look for the terms of $Q_b$ that cancel out with some of the terms that emerge out after the first step. In this context, we note that such an appropriate term is

$$\begin{array}{c}\hfill +\left[\pm m{A}^{0ij}-{\Sigma }^{0ij}\right]\left[\pm m{C}_{ij}-({\partial }_i{C}_j-{\partial }_j{C}_i)\right].\end{array}$$

(80)

The sum of (79) and (80) yields the following:

$$\begin{array}{ccc}& -& \left({\partial }_i{H}^{0ijk}\right)C_{jk}+\left[\pm m{A}^{0ij}-{\Sigma }^{0ij}\right]\left[\pm m{C}_{ij}-({\partial }_i{C}_j-{\partial }_j{C}_i)\right]\hfill \\ & \equiv & -({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{oi})C_{ij}-2\left[\pm m{A}^{0ij}-{\Sigma }^{0ij}\right]\left({\partial }_i{C}_j\right).\hfill \end{array}$$

(81)

We comment here that as pointed out earlier, one of the key ingredients of our proposal is the use of EL-EoM with respect to the gauge field. This step opens up a decisive door for (i) our further applications of Gauss’s divergence theorem, (ii) use of the appropriate EL-EoM, and (iii) the application of the BRST symmetry transformations ($s_b$) at appropriate places. In the third step, we focus on the second term of the r.h.s., which can be written as:

$$\begin{array}{c}\hfill -2{\partial }_i\left[(\pm m{A}^{0ij}-{\Sigma }^{0ij})C_j\right]+2{\partial }_i\left[(\pm m{A}^{0ij}-{\Sigma }^{0ij})\right]C_j.\end{array}$$

(82)

The terms in (82) are inside the integration. Thus, the first term of (82) will vanish due to Gauss’s divergence theorem for physical fields. Using the following equation of motion with respect to the St$\ddot{u}$ckelberg field ${\mathrm{\Phi }}_{\mu \nu }$ from the Lagrangian density ${\mathcal{L}}_B$, we obtain:

$$\begin{array}{c}\hfill {\partial }_{\mu }{\Sigma }^{\mu \nu \lambda }\mp m\left({\partial }_{\mu }{A}^{\mu \nu \lambda }\right)+\frac{1}{2}({\partial }^{\nu }{B}^{\lambda }-{\partial }^{\lambda }{B}^{\nu })\mp \frac{m}{2}{B}^{\nu \lambda }=0\\ \hfill ⟹2{\partial }_i\left[\pm m{A}^{0ij}-{\Sigma }^{0ij}\right]C_j=\pm m{B}^{0i}{C}_i-({\partial }^0{B}^i-{\partial }^i{B}^0)C_i,\end{array}$$

(83)

where the l.h.s. is nothing but the second term of (82). Using (83), we can re-write the whole sum of (81) as follows:

$$\begin{array}{c}\hfill +\left[\pm m{B}^{0i}-({\partial }^0{B}^i-{\partial }^i{B}^0)\right]C_i-({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{oi})C_{ij}.\end{array}$$

(84)

Within the framework of our proposal, the terms in (84) will be present in the off-shell nilpotent form of the BRST charge ($Q_b^{\left(1\right)}$) because they have come out from the use of EL-EoMs and, hence, will remain intact. This is also required due to the fact that, as pointed out earlier, we wish to obtain $s_b{Q}_b^{\left(1\right)}=0$ which will automatically imply that ${[Q_b^{\left(1\right)}]}^2=0$. In the fourth step, we apply the BRST symmetry transformation $\left(s_b\right)$ on (84), which leads to the following explicit expression:

$$\begin{array}{ccc}& & +\left[\pm m{B}^{0i}-({\partial }^0{B}^i-{\partial }^i{B}^0)\right](\pm m{\beta }_i-{\partial }_i\beta )\hfill \\ & & -({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{oi})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i).\hfill \end{array}$$

(85)

In the fifth step, we modify some of the terms of $Q_b$ so that when $s_b$ acts on a part of them, there is precise cancellation between the ensuing result and (85). In this context, we modify the following terms from the explicit expression for $Q_b$ (cf. Equation (76)), namely:

$$\begin{array}{ccc}& -& ({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{oi})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)=+({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{oi})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & -& 2({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{oi})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i),\hfill \\ & +& \left[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right](\pm m{\beta }_i-{\partial }_i\beta )=2[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i\hfill \\ & -& {\partial }^i{\overline{C}}^0\left)\right](\pm m{\beta }_i-{\partial }_i\beta )-\left[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right](\pm m{\beta }_i-{\partial }_i\beta ).\hfill \end{array}$$

(86)

In the sixth step, we note that in the above modified expressions, when we apply $s_b$ on a part of (86) as explicitly written below, namely:

$$\begin{array}{ccc}& & s_b[({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & & -\left\{\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right\}(\pm m{\beta }_i-{\partial }_i\beta )],\hfill \end{array}$$

(87)

we obtain the desired result

$$\begin{array}{ccc}& & ({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & & -\left\{\pm m{B}^{0i}-({\partial }^0{B}^i-{\partial }^i{B}^0)\right\}(\pm m{\beta }_i-{\partial }_i\beta )],\hfill \end{array}$$

(88)

which precisely cancels out with whatever we have obtained in (85). At this juncture, we point out that in the off-shell nilpotent version of the BRST charge $Q_b^{\left(1\right)}$, the terms in the square bracket of (87) will be always (along with (84) (as pointed out earlier)) present. We now focus on the leftover terms of (86), which can be re-expressed as follows:

$$\begin{array}{ccc}& & \pm 2m\{\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\}{\beta }_i\hfill \\ & & -2\{\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\}{\partial }_i\beta \hfill \\ & & -4\left[{\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}\right]\left({\partial }_i{\beta }_j\right).\hfill \end{array}$$

(89)

All the above terms are inside the integration. Hence, we can apply the following algebraic tricks on the last two terms of (89) to obtain the following

$$\begin{array}{ccc}\hfill -2\{\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\}{\partial }_i\beta & =& {\partial }_i\left[-2\left\{\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right\}\beta \right]\hfill \\ & +& 2{\partial }_i\left[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right]\beta \hfill \\ \hfill -4\left[{\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}\right]{\partial }_i{\beta }_j& =& {\partial }_i\{-4\left\{{\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}\}{\beta }_j\right\}\hfill \\ & +& 4{\partial }_i\left[{\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}\right]{\beta }_j,\hfill \end{array}$$

(90)

so that the total space derivative terms vanish due to Gauss’s divergence theorem for the physical fields, and we are left with the following from the above Equation (90):

$$\begin{array}{c}\hfill 2{\partial }_i\left[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right]\beta +4{\partial }_i\left[{\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}\right]{\beta }_j.\end{array}$$

(91)

In the above, we apply the following equation of motion:

$$\begin{array}{c}\hfill {\partial }_{\mu }[{\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }]-\frac{1}{2}{\partial }^{\nu }F\mp m({\partial }_{\mu }{\overline{C}}^{\mu \nu })\pm \frac{m}{2}{F}^{\nu }=0,\\ \hfill ⟹2{\partial }_i[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)]\beta =-(\pm m{F}^0-{\partial }^{0}F)\beta ,\end{array}$$

(92)

which provides the concise value of the first term of (91). Exactly, in a similar fashion, the application of the following equation of motion

$$\begin{array}{ccc}& & {\partial }_{\mu }[{\partial }^{\mu }{\overline{C}}^{\nu \lambda }+{\partial }^{\nu }{\overline{C}}^{\lambda \mu }+{\partial }^{\lambda }{\overline{C}}^{\mu \nu }]\pm \frac{m}{2}({\partial }^{\nu }{\overline{C}}^{\lambda }-{\partial }^{\lambda }{\overline{C}}^{\nu })\hfill \\ & & +\frac{1}{2}({\partial }^{\nu }{F}^{\lambda }-{\partial }^{\lambda }{F}^{\nu })-\frac{m^2}{2}{\overline{C}}^{\nu \lambda }=0,\hfill \\ & & ⟹4{\partial }_i[{\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}]{\beta }_j=2({\partial }^0{F}^i-{\partial }^i{F}^0){\beta }_i\hfill \\ & & \mp 2m\left[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)\right]{\beta }_i,\hfill \end{array}$$

(93)

leads to the alternative (but appropriate) form of the second term of (91). The sum of the r.h.s. of (92) and (93) and the leftover term (i.e., the first term) of (89) is equal to:

$$\begin{array}{c}\hfill 2({\partial }^0{F}^i-{\partial }^i{F}^0){\beta }_i-(\pm m{F}^0-{\partial }^{0}F)\beta .\end{array}$$

(94)

At this stage, we take the seventh step and apply the BRST symmetry transformations $\left(s_b\right)$ on (94) which yields the following explicit expression, namely:

$$\begin{array}{c}\hfill -2({\partial }^0{F}^i-{\partial }^i{F}^0)\left({\partial }_i{C}_2\right)+(\pm m{F}^0-{\partial }^{0}F)(\pm m{C}_2).\end{array}$$

(95)

We emphasize that the term in (94) will be present in the off-shell nilpotent version of the BRST charge $(Q_b^{\left(1\right)})$. We take now the eighth step and modify some of the terms of $Q_b$ so that when we apply the BRST transformations $\left(s_b\right)$ on a part of them, the resulting expressions must cancel out (95) in a precise manner. Toward this goal in mind, we modify the following appropriate terms of the original expression for $Q_b$ (cf. Equation (76)):

$$\begin{array}{ccc}& -& ({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2\pm \frac{1}{2}m(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })C_2\hfill \\ & \equiv & 2({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2\mp m(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })C_2\hfill \\ & \pm & \frac{3}{2}(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })C_2-3({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2.\hfill \end{array}$$

(96)

It is evident that if we apply the BRST symmetry transformations $(s_b)$ on the first two terms of (96), the resulting expressions will cancel out the terms that are written in (95). Hence, along with (94), the above first two terms will be present in the off-shell nilpotent version of the BRST charge $(Q_b^{\left(1\right)})$ so that our central objective $s_b{Q}_b^{\left(1\right)}=0$ can be fulfilled. At this juncture, we focus on the last term of (96), which can be re-written as

$$\begin{array}{c}\hfill -3({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2={\partial }_i\left[-3({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0)C_2\right]+3{\partial }_i\left[({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0)\right]C_2.\end{array}$$

(97)

The above terms are inside the integral. Hence, the first term on the r.h.s. will vanish due to Gauss’s divergence theorem and only the second term on the r.h.s. of (97) will survive. Now, we apply the following EL-EoM

$$\begin{array}{ccc}& & \square {\overline{\beta }}_{\mu }-{\partial }_{\mu }(\partial ·\overline{\beta })+{\partial }_{\mu }{B}_2-\frac{m^2}{2}{\overline{\beta }}_{\mu }\pm \frac{m}{2}{\partial }_{\mu }\overline{\beta }=0\hfill \\ \hfill ⟹& & 3{\partial }_i\left[{\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0\right]C_2=3{\dot{B}}_2{C}_2\mp \frac{3}{2}m(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })C_2,\hfill \end{array}$$

(98)

which will replace the last term of (96). The substitution of (98) into (96) yields the following concise and beautiful result, namely:

$$\begin{array}{c}\hfill \pm \frac{3}{2}m(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })C_2+3{\partial }_i\left[{\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0\right]C_2=3{\dot{B}}_2{C}_2.\end{array}$$

(99)

If we apply further the BRST symmetry transformation $\left(s_b\right)$ on (99), it turns out to be zero. Here, all our steps terminate (according to our proposal). It is self-evident that the r.h.s. of (99) will be part of $Q_b^{\left(1\right)}$. Ultimately, the off-shell nilpotent version $Q_b^{\left(1\right)}$, from the non-nilpotent Noether conserved charge $Q_b$, is as follows

$$\begin{array}{ccc}\hfill Q_b⟶Q_b^{\left(1\right)}& =& \int d^{D-1}x[({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})B_{ij}-[\pm m{C}^{0i}-({\partial }^0{C}^i-{\partial }^i{C}^0)]B_i\hfill \\ & -& ({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{0i})C_{ij}+[\pm m{B}^{0i}-({\partial }^0{B}^i-{\partial }^i{B}^0)]C_i\hfill \\ & -& [\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)(\pm m{\beta }_i-{\partial }_i\beta )]+2({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2\hfill \\ & \mp & m(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })C_2+({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & +& 2({\partial }^0{F}^i-{\partial }^i{F}^0){\beta }_i-(\pm m{F}^0-{\partial }^{0}F)\beta +3{\dot{B}}_2{C}_2-B_2{\dot{C}}_2+B{F}^0\hfill \\ & -& B_1{f}^0-\frac{1}{2}{B}^{0}f+B^{0i}{f}_i+\frac{1}{2}(\pm m{\beta }^0-{\partial }^0\beta )F-({\partial }^0{\beta }^i-{\partial }^i{\beta }^0)F_i].\hfill \end{array}$$

(100)

It will be noted that we have not touched several terms of the Noether conserved charge $Q_b$ (cf. Equation (76)) because these terms are BRST invariant. For instance, we have the following explicit observations:

$$\begin{array}{ccc}& & s_b\left[({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})B_{ij}\right]=0,s_b\left[\left\{\pm m{C}^{0i}-({\partial }^0{C}^i-{\partial }^i{C}^0)\right\}{B}_i\right]=0,\hfill \\ & & s_b\left[(\pm m{F}^0-{\partial }^{0}F)\beta \right]=0,s_b\left(B_1{f}^0\right)=0,s_b\left(B^{0}f\right)=0,s_b\left(B^{0i}{f}_i\right)=0,\hfill \\ & & s_b\left[(\pm m{\beta }^0-{\partial }^0\beta )F\right]=0,s_b\left[({\partial }^0{\beta }^i-{\partial }^i{\beta }^0)F_i\right]=0,s_b({\dot{B}}_2{C}_2)=0,\hfill \\ & & s_b(B_2{\dot{C}}_2)=0,s_b\left(B{F}^0\right)=0.\hfill \end{array}$$

(101)

It is straightforward to check that the following is true, namely:

$$\begin{array}{c}\hfill s_b{Q}_b^{\left(1\right)}=-i\{Q_b^{\left(1\right)},Q_b^{\left(1\right)}\}=0⟹{[Q_b^{\left(1\right)}]}^2=0,\end{array}$$

(102)

where the l.h.s. is computed directly by using the BRST symmetry transformations (71) on the expression for the modified version of the BRST charge $Q_b^{\left(1\right)}$.

We follow the prescriptions and proposal outlined above to compute the exact expression for the off-shell nilpotent version of the anti-BRST charge $Q_{ab}^{\left(1\right)}$ from the non-nilpotent conserved Noether anti-BRST charge $Q_{ab}$ as follows:

$$\begin{array}{ccc}\hfill Q_{ab}⟶Q_{ab}^{\left(1\right)}& =& \int d^{D-1}x[({\partial }^0{\overline{B}}^{ij}+{\partial }^i{\overline{B}}^{j0}+{\partial }^j{\overline{B}}^{0i}){\overline{C}}_{ij}-[\pm m{\overline{B}}^{0i}-({\partial }^0{\overline{B}}^i-{\partial }^i{\overline{B}}^0)]{\overline{C}}_i\hfill \\ & -& ({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}){\overline{B}}_{ij}+[\pm m{\overline{C}}^{0i}-({\partial }^0{\overline{C}}^i-{\partial }^i{\overline{C}}^0)]{\overline{B}}_i\hfill \\ & +& [\pm m{C}^{0i}-({\partial }^0{C}^i-{\partial }^i{C}^0)](\pm m{\overline{\beta }}_i-{\partial }_i\overline{\beta })+2({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\partial }_i{\overline{C}}_2\hfill \\ & \mp & m(\pm m{\beta }^0-{\partial }^0\beta ){\overline{C}}_2-({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)\hfill \\ & +& 2({\partial }^0{\overline{F}}^i-{\partial }^i{\overline{F}}^0){\overline{\beta }}_i-(\pm m{\overline{F}}^0-{\partial }^0\overline{F})\overline{\beta }-3\dot{B}{\overline{C}}_2+B{\dot{\overline{C}}}_2-B_2{\overline{F}}^0\hfill \\ & -& B_1{\overline{f}}^0-\frac{1}{2}{\overline{B}}^0\overline{f}+{\overline{B}}^{0i}{\overline{f}}_i+\frac{1}{2}(\pm m{\overline{\beta }}^0-{\partial }^0\overline{\beta })\overline{F}-({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\overline{F}}_i].\hfill \end{array}$$

(103)

It is now straightforward to check that:

$$\begin{array}{c}\hfill s_{ab}{Q}_{ab}^{\left(1\right)}=-i\{Q_{ab}^{\left(1\right)},Q_{ab}^{\left(1\right)}\}=0⟹{\left[Q_{ab}^{\left(1\right)}\right]}^2=0.\end{array}$$

(104)

The above observation proves that we have derived the off-shell nilpotent version $[Q_{ab}^{\left(1\right)}]$ of the non-nilpotent Noether conserved charge $Q_{ab}$ in a precise and logical manner. Our final results are the expressions for $Q_{\left(a\right)b}^{\left(1\right)}$ in (104) and (100).

6. Conclusions

In our present investigation, for a few physically interesting gauge systems, we have shown that wherever there is existence of the coupled (but equivalent) Lagrangians/Lagrangian densities due to the presence of the (anti-)BRST invariant CF-type restriction(s), we observe that the Noether theorem does not lead to the derivation of the off-shell nilpotent versions of the conserved (anti-)BRST charges $\left[Q_{\left(a\right)b}\right]$ within the framework of BRST formalism. These Noether conserved charges are found to be the generators for the infinitesimal, continuous and off-shell nilpotent (anti-)BRST symmetry transformations from which they are derived by exploiting the theoretical strength of Noether’s theorem. However, they are found to be non-nilpotent in the sense that $s_b{Q}_b=-i\{Q_b,Q_b\}\ne 0$ and $s_{ab}{Q}_{ab}=-i\{Q_{ab},Q_{ab}\}\ne 0$, which can be verified by applying the (anti-)BRST transformations directly on the Noether conserved charges $Q_{\left(a\right)b}$. In other words, by directly computing the explicit expressions for $s_b{Q}_b$ and $s_{ab}{Q}_{ab}$, which are the l.h.s. of: $s_b{Q}_b=-i\{Q_b,Q_b\}$ and $s_{ab}{Q}_{ab}=-i\{Q_{ab},Q_{ab}\}$, we show that these charges are not nilpotent (see, e.g., Equation (77)). However, a close and careful look at them demonstrates that $s_b{Q}_b$ and $s_{ab}{Q}_{ab}$ are proportional to the EL-EoM. For instance, we are sure that in the simple case of a BRST-invariant 1D massive spinning relativistic particle, the r.h.s. of (14) is zero due to the EL-EoMs (15) and (23) that are derived from $L_b$ and $L_{\overline{B}}$, respectively.

In the derivations of the off-shell nilpotent versions of the (anti-)BRST charges $\left[Q_{\left(a\right)b}^{\left(1\right)}\right]$, the crucial roles are played by (i) the Gauss divergence theorem, (ii) the appropriate EL-EoMs from the appropriate Lagrangian/Lagrangian density of a set of coupled (but equivalent) Lagrangians/Lagrangian densities, and (iii) the application of the (anti-)BRST symmetry transformations at appropriate places (cf. Section 5 for details). We lay emphasis on the fact that for the D-dimensional (D ≥ 2) higher p-form ($p=1,2,3,\dots$) gauge theories, it is imperative to first exploit the Gauss divergence theorem so that we can use the EL-EoMs with respect to the gauge field. For the 1D case of a spinning (i.e., SUSY) relativistic particle, we have shown that the Gauss divergence theorem, for obvious reasons, is not required at all and we directly use the EL-EoMs, right in the beginning, with respect to the “gauge” and “supergauge” variables. This is not the case with the rest of the examples considered in our present investigation which are connected with the D-dimensional (D ≥ 2) (non-)Abelian p-form $(p=1,2,3,\dots )$ massless and massive gauge theories.

The sequence of our proposal, to obtain the off-shell nilpotent versions of the (anti-) BRST charges from the non-nilpotent Noether (anti-)BRST charges, is as follows. In the first step, we apply the Gauss divergence theorem and take the help of EL-EoM with respect to the gauge field. This is the crucial and key first step of our proposal. In the next (i.e., second) step, we observe carefully whether there are any addition, subtraction and/or cancellation of the resulting terms (from the first step) with any of the terms of the non-nilpotent Noether conserved charges. The existing terms, after these two steps, will always be present in the off-shell nilpotent version of the (anti-)BRST charges $Q_{\left(a\right)b}^{\left(1\right)}$. After the above two steps, we apply the (anti-)BRST symmetry transformations on the existing terms. In the third step, we modify some of the appropriate terms of the non-nilpotent Noether conserved (anti-) BRST charges and see to it that a part of these modified terms cancel precisely with the terms that have appeared after the application of the nilpotent (anti-)BRST transformations on the existing terms (after the first two steps of our proposal). The parts which participate in the above cancellation are also always present in the off-shell nilpotent versions of (anti-)BRST charges $[Q_{\left(a\right)b}^{\left(1\right)}]$. After this step, it is the interplay amongst the Gauss divergence theorem, appropriate8 EL-EoMs and application of the nilpotent (anti-) BRST symmetry transformations at appropriate places that lead to the derivation of the precise forms of the off-shell nilpotent versions9 of the (anti-)BRST charges $[Q_{\left(a\right)b}^{\left(1\right)}]$ from the non-nilpotent Noether conserved (anti-)BRST charges (cf. Section 4 and Section 5 for details).

Before we end this section by pointing out our future directions of investigation in the next paragraph, it is worthwhile to point out the physical significance of the conserved and off-shell nilpotent (anti-)BRST charges in the context of a given gauge theory which is endowed with a set of non-trivial CF-type restrictions(s). The physicality condition ($Q_B^{\left(1\right)}\mid phys>=0$, cf. Section 3) ensures that the operator form of the first-class constraints of the given gauge theory annihilate the physical state, which is consistent with the requirements of the Dirac quantization condition for a theory endowed with constraints (see, e.g., [5,6] for details). As has been shown by Weinberg [19], in the Fock space, the gauge-transformed states differ from their original counterparts by the BRST-exact states. Hence, if we have an off-shell nilpotent BRST charge, their difference becomes trivial when we invoke the physicality condition on the states with respect to the BRST charge. This argument has been extended in our earlier research works (see, e.g., [28,29]) where we have been able to prove that the 2D (non-)Abelian one-form and 4D Abelian two-form gauge theories are the tractable field–theoretic models for the Hodge theory. In the case of the 2D (non-)Abelian one-form theories, we have been able to show that the BRST and co-BRST symmetry transformations “gauge away” both the d.o.f. of the gauge fields and these theories become a new kind of topological field theory (see, e.g., [30] for details).

We end this section with the final comment that our proposal is very general and it can be applied to any physical system where (i) the (anti-)BRST invariant non-trivial CF-type restrictions exist, and (ii) the coupled (but equivalent) Lagrangians/Lagrangian densities describe the dynamics of the above physical systems within the framework of BRST formalism. The systematic application of our proposal sheds light on the appropriate directions that should be followed in order to obtain the off-shell nilpotent versions of the (anti-)BRST charges from the non-nilpotent versions of the Noether conserved (anti-)BRST charges. We plan to extend our present ideas in the context of more challenging problems of physical interest in the future.