Supervariable and BRST Approaches to a
Reparameterization Invariant Non-Relativistic System

A. K. Rao(a), A. Tripathi(a), R. P. Malik(a,b)
(a) Physics Department, Institute of Science,
Banaras Hindu University, Varanasi - 221 005, (U.P.), India

(b) DST Centre for Interdisciplinary Mathematical Sciences,
Institute of Science, Banaras Hindu University, Varanasi - 221 005, India
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: We exploit the theoretical strength of the supervariable and Becchi-
Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e. off-shell
nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations
for the reparameterization invariant model of a non-relativistic (NR) free
particle whose space $(x)$ and time $(t)$ variables are function of an
evolution parameter $(\tau)$. The infinitesimal reparameterization (i.e. 1D
diffeomorphism) symmetry transformation of our theory is defined w.r.t. this
evolution parameter $(\tau)$. We apply the modified Bonora-Tonin (BT)
supervariable approach (MBTSA) as well as the (anti-)chiral supervariable
approach (ACSA) to BRST formalism to discuss various aspects of our present
system. For this purpose, our 1D ordinary theory (parameterized by $\tau$) is
generalized onto a $(1,2)$-dimensional supermanifold which is characterized by
the superspace coordinates $Z^{M}=(\tau,\theta,\bar{\theta})$ where a pair of
Grassmannian variables satisfy the fermionic relationships:
$\theta^{2}={\bar{\theta}}^{2}=0,\,\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$
and $\tau$ is the bosonic evolution parameter. In the context of ACSA, we take
into account only the (1, 1)-dimensional (anti-)chiral super sub-manifolds of
the general (1, 2)-dimensional supermanifold. The derivation of the universal
Curci-Ferrari (CF)-type restriction, from various underlying theoretical
methods, is a novel observation in our present endeavor. Furthermore, we note
that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and
non-SUSY system is exactly same as that of the reparameterization invariant
SUSY (i.e. spinning) and non-SUSY (i.e. scalar) relativistic particles. This
is a novel observation, too.

PACS numbers: 11.15.-q; 12.20.-m; 11.30.Pb.; 02.20.+b

Keywords: Reparameterization invariant non-SUSY and non-relativistic system;
modified BT-supervariable approach; horizontality condition; (anti-)chiral
supervariable approach; symmetry invariant restrictions; (anti-)BRST
symmetries and (anti-)BRST charges; off-shell nilpotency and absolute
anticommutativity properties; CF-type restriction

##  1 Introduction

During the last few years, there has been an upsurge of interest in the study
of diffeomorphism invariant theories because one of the key and decisive
features of the gravitational and (super)string theories is the observation
that they respect the classical diffeomorphism symmetry transformations. The
latter symmetry transformations can be exploited within the framework of
Becchi-Rouet-Stora-Tyutin (BRST) formalism [1-4] where the classical
diffeomorphism symmetry transformation is elevated to the quantum (anti-)BRST
symmetry transformations. In fact, it is the key feature of the BRST formalism
that the classical diffeomorphism transformation parameter is traded with the
fermionic (anti-)ghost fields/variables at the quantum level. In other words,
the (anti-)BRST transformations are of the supersymmetric (SUSY) kind under
which the bosonic type of fields/variables transform to the fermionic type
fields/variables and vice-versa. Two of the key properties of the (anti-)BRST
transformations are the on-shell/off-shell nilpotency and absolute
anticommutativity. These key properties encompass in their folds the fermionic
as well as independent natures of the quantum BRST and anti-BRST symmetries at
the level of physical interpretation. The nilpotency property (i.e. fermionic
nature) of the (anti-)BRST symmetries (and their corresponding charges) is
also connected with some aspects of the cohomological properties of
differential geometry and a few decisive features of supersymmetry.

The BRST formalism has been exploited in the covariant canonical quantization
of the gauge and diffeomorphism invariant theories in the past. At the
classical level, the gauge theories are characterized by the existence of the
first-class constraints [5, 6] on them. This fundamental feature is
translated, at the quantum level, into the language of the existence of the
Curci-Ferrari (CF)-type restriction(s) when the classical theory is quantized
by exploiting the theoretical richness of BRST formalism. Hence, the existence
of the CF-type restriction(s) is the key signature of a BRST-quantized version
of the gauge and/or diffeomorphism invariant theory. The CF-type restrictions
are (i) deeply connected with the geometrical objects called gerbes [7, 8],
(ii) responsible for the absolute anticommutativity of the quantum (anti-)BRST
transformations, and (iii) the root-cause behind the existence of the coupled
(but equivalent) Lagrangians/Lagrangian densities for the (anti-)BRST
invariant quantum theories (corresponding to the classically
gauge/diffeomorphism invariant theories). The Abelian 1-form gauge theory is
an exception where we obtain a unique (anti-) BRST invariant Lagrangian
density because the CF-type restriction is trivial in this case. However, this
restriction turns out to be the limiting case of the non-Abelian 1-form gauge
theory where the non-trivial CF-condition exists [9].

It is the supervariable/superfield approaches [10-21] to BRST formalism that
provide the geometrical basis for the off-shell nilpotency and absolute
anticommutativity of the (anti-) BRST symmetries as well as the existence of
the CF-type restrictions for a BRST-quantized gauge/diffeomorphism invariant
theory. In the usual superfield approaches (USFA), it is the horizontality
condition (HC) that plays a decisive role as it leads to (i) the derivation of
the (anti-)BRST symmetry transformations for only the gauge and anti-)ghost
fields, as well as (ii) the derivation of the CF-type restriction(s). The
augmented version of the superfield approach (AVSA) is an extension of USFA
where, in addition to the HC, the gauge [i.e. (anti-)BRST] invariant
restrictions are exploited together which lead to the derivation of the
(anti-)BRST symmetry transformations for the gauge, (anti-)ghost and matter
fields together in an interacting gauge theory. It has been a challenging
problem to incorporate the diffeomorphism transformation within the ambit of
superfield approach to gauge theories (see, e.g. [14-16]) so that one can
discuss the gravitational and (super)string theories within the framework of
USFA/AVSA. In this direction, a breakthrough has recently been made by Bonora
[22] where the superfield approach has been applied to derive the proper
(anti-)BRST transformations as well as the CF-type restriction for the
D-dimensional diffeomorphism invariant theory. This approach has been
christened by us as the modified Bonora-Tonin (BT) superfield approach (MBTSA)
to BRST formalism. In a recent couple of papers [23, 24], we have applied the
theoretical beauty of the MBTSA as well as ACSA [i.e. (anti-)chiral
superfield/supervariable approach] to BRST formalism [25-29] in the context of
the 1D diffeomorphism (i.e. reparameterization) invariant theories of the non-
SUSY (i.e. scalar) as well as SUSY (i.e. spinning) relativistic free
particles.

The central theme of our present investigation is to concentrate on the
reparameterization (i.e. 1D diffeomorphism) invariant theory of a massive non-
supersymmetric (NSUSY) and non-relativistic (NR) free particle where the
standard NR Lagrangian $L_{0}^{(t)}(x,\dot{x})=\frac{1}{2}m\,{\dot{x}}^{2}$
(with $\dot{x}=dx/dt)$ is rendered reparameterization invariant by treating
the “time” variable on a par with $x$ variable [30] that are parameterized by
an evolution parameter $\tau$ such that the new Lagrangian
$L_{0}^{(\tau)}(x,\,\dot{x},\,t,\,\dot{t})=\frac{m\,{\dot{x}}^{2}}{2\,\dot{t}}$
where $\dot{x}=dx/d\tau$ and $\dot{t}=dt/d\tau$. The latter Lagrangian
respects the reparameterization symmetry [31, 32] and it has been discussed in
different theoretical settings where the noncommutativity of the spacetime
appears by the symmetry considerations, constraints analysis, redefinitions of
variables, etc. This reparameterization invariant model of the free particle
($\dot{p}_{x}=\dot{p}_{t}=0$) has been discussed by us within the frameworks
of BRST formalism as well as quantum groups [32]. However, in the BRST
analysis, we have exploited the gauge symmetry of this NSUSY and NR system
[32] without discussing anything about the reparameterization transformations.
In our present investigation, we have applied the beautiful blend of
theoretical ideas from MBTSA and ACSA to derive the proper (anti-)BRST
symmetries and CF-type restriction for this NR system. This model is
interesting in its own right as it is a NR system (unlike our earlier
discussions [23, 24] on the relativistic systems) and “time” itself has been
treated as a physical observable that depends on the evolution parameter
$\tau$. The latter property of our present NR system is important as “time”
has also been treated as an observable in quantum mechanics instead of an
evolution parameter (see, e.g. [30]).

The following motivating factors have been at the heart of our curiosity in
pursuing our present endeavor. First, so far, we have been able to apply the
beautiful blend of theoretical ideas behind MBTSA and ACSA to BRST formalism
in the cases of reparameterization invariant systems of (i) relativistic non-
supersymmetric (NSUSY) scalar free particle, and (ii) supersymmetric (SUSY)
(i.e. spinning) relativistic free particle. Thus, it has been a challenging
problem for us to apply the same theoretical ideas to discuss the NSUSY and
non-relativistic (NR) system of a reparameterization invariant free particle.
We have accomplished this goal in our present investigation. Second, we have
shown the universality of the CF-type restriction in the cases of
reparameterization (i.e. 1D diffeomorphism) invariant NSUSY as well as SUSY
systems of the free relativistic particles. Thus, we have been motivated to
see the existence of the same CF-type of restriction in our present case of
reparameterization invariant system of NSUSY and NR free particle. We have
been able to demonstrate that it is the same CF-type of restriction that
exists in the BRST approach to our present NSUSY and NR system. Third, we have
found out that the gauge-fixing and Faddeev-Popov (FP) ghost terms for the
systems of non-SUSY (i.e. scalar) and SUSY (i.e. spinning) relativistic
particles are same within the ambit of BRST formalism. Thus, we have been
curious to find out the gauge-fixing and FP-ghost terms for our present non-
SUSY and NR system. It is surprising that the above terms are same for our
present system, too. Finally, our present work is our another modest initial
step towards our main goal of applying the theoretical potential of MBTSA and
ACSA to the physical four (3+1)-dimensional (4D) diffeomorphism invariant
gravitational and (super)sting theories in the higher dimensions (i.e. $D>4$)
of spacetime.

The contents of our present endeavor are organized as follows. In Sec. 2, we
recapitulate the bare essentials of the Lagrangian formulation of our
reparameterization invariant non-relativistic system and discuss the BRST
quantization of this model by exploiting its infinitesimal gauge symmetry
transformations. Our Sec. 3 is devoted to the application of MBTSA for the
derivation of (i) the quantum (anti-)BRST symmetry transformations for the
phase space variables, and (ii) the underlying (anti-)BRST invariant Curci-
Ferrari (CF)-type restriction (corresponding to the classical infinitesimal
reparameterization symmetry transformations). The theoretical content of Sec.
4 is related to the derivation of the full set of (anti-)BRST symmetry
transformations by requiring the off-shell nilpotency and absolute
anticommutativity properties. We also show the existence of the CF-type
restriction and deduce the coupled (anti-)BRST invariant Lagrangians for our
theory. In Sec. 5, we derive the (anti-)BRST symmetry transformations for the
other variables (i.e. different from the phase variables) within the purview
of ACSA. Our Sec. 6 deals with the proof of the equivalence of the coupled
Lagrangians within the framework of ACSA to BRST formalism. In Sec. 7, we
prove the off-shell nilpotency and absolute anticommutativity of the conserved
(anti-)BRST charges in the ordinary space and superspace (by exploiting the
theoretical richness of ACSA to BRST formalism). Finally, in Sec. 8, we
discuss our key results and point out a few future directions for further
investigation(s).

In our Appendices A, B and C, we perform some explicit computations that
supplement as well as corroborate the key claims that have been made in the
main body of our text.

Convention and Notations: We adopt the convention of the left-derivative
w.r.t. all the fermionic variables of our theory. In the whole body of the
text, we denote the fermionic (anti-)BRST transformations by the symbols
$s_{(a)b}$ and corresponding conserved and nilpotent charges carry the
notations $Q_{(a)b}$ and $Q_{(\bar{B})B}$ in different contexts. The general
$(1,2)$-dimensional supermanifold is parameterized by
$Z^{M}=(\tau,\theta,\bar{\theta})$ and its chiral and anti-chiral super sub-
manifolds are characterized by $(\tau,\theta)$ and $(\tau,\bar{\theta})$,
respectively, where the bosonic coordinate is represented by the evolution
parameter $(\tau)$ and the Grassmannian variables $(\theta,\bar{\theta})$ obey
the fermionic relationships:
$\theta^{2}=0,\,{\bar{\theta}}^{2}=0,\,\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$.

##  2 Preliminaries: Lagrangian Formulation

Our present section is divided into two parts. In subsection 2.1, we discuss
the classical infinitesimal reparameterization and gauge symmetry
transformations. Our subsection 2.2 deals with the BRST quantization of our
system by exploiting the classical gauge symmetry transformations (which are
also infinitesimal and continuous).

###  2.1 Classical Infinitesimal Symmetries

We begin with the three equivalent reparameterization invariant Lagrangians
for the free non-relativistic and non-SUSY particle as (see e.g. [32] for
details)

$\displaystyle
L_{0}(x,\,\dot{x},\,t,\dot{t})=\frac{m\,{\dot{x}}^{2}}{2\,\dot{t}},$
$\displaystyle
L_{f}(x,\,\dot{x},\,t,\dot{t},\,p_{x},\,p_{t})=p_{x}\,\dot{x}+p_{t}\,\dot{t}-\frac{1}{2}\,E\,(p_{x}^{2}+2\,m\,p_{t}),$
$\displaystyle
L_{s}(x,\,\dot{x},\,t,\dot{t})=\frac{{\dot{x}}^{2}}{2\,E}+\frac{m\,{\dot{x}}^{2}}{2\,\dot{t}}\Big{[}\frac{E\,m}{\dot{t}}-1\Big{]},$
(1)

where the trajectory of the free non-relativistic particle is embedded in a 2D
configuration space characterized by the coordinates $[x(\tau),\,t(\tau)]$ and
the parameter $\tau$ specifies the trajectory of the particle as an evolution
parameter. The momenta variables $(p_{x},\,p_{t})$ are defined by:
$p_{x}=({\partial\,L}/{\partial\,\dot{x}})$,
$p_{t}=({\partial\,L}/{\partial\,\dot{t}})$ where
$\dot{x}=({d\,x}/{d\,\tau})$, $\dot{t}=({d\,t}/{d\,\tau})$ are the generalized
“velocities” w.r.t. the coordinates $[x(\tau),t(\tau)]$ and $L$ stands for any
of the three Lagrangians of Eq. (1). It is self-evident that the 4D phase
space, corresponding to the 2D configuration space, is characterized by
$[x(\tau),t(\tau),p_{x}(\tau),p_{t}(\tau)]$. In the above equation (1), the
mass of the non-relativistic particle is denoted by $m$ and $E(\tau)$ is the
Lagrange multiplier variable that incorporates the constraint
$p_{x}^{2}+2\,m\,p_{t}\approx 0$ in the Lagrangians $L_{f}$ and $L_{s}$. It is
straightforward to note that $L_{0}$ and $L_{s}$ contain variables (and their
first-order derivative) in the denominator but $L_{f}$ (i.e. the first-order
Lagrangian) does not incorporate any variable (and/or its derivative) in its
denominator. Furthermore, the starting Lagrangian***For a free massive NR
particle, the standard action integral is:
$S_{t}=\int_{-\infty}^{+\infty}d\,t\,(\frac{1}{2}\,m\,{\dot{x}}^{2})$ where
$\dot{x}=(dx/dt)$ and time “t” is the evolution parameter. This action has no
reparameterization invariance. If the evolution parameter is $\tau$, then, the
action integral is:
$S_{\tau}=({m}/{2})\,\int_{-\infty}^{+\infty}d\,\tau\,(dt/d\tau)\,(dx/d\tau)\,(d\tau/dt)\,(dx/d\tau)\,(d\tau/dt)$
which leads [32, 31] to the final action integral as:
$S_{\tau}=(m/2)\,\int_{-\infty}^{+\infty}d\,\tau[(dx/d\tau)^{2}\,(d\tau/dt)]\,\equiv\,\int_{-\infty}^{+\infty}d\,\tau\,L_{0}$.
Hence, the starting Lagrangian becomes
$L_{0}(x,\,\dot{x},\,t,\dot{t})=\frac{m\,{\dot{x}}^{2}}{2\,\dot{t}}$ where
$\dot{x}=(dx/d\tau)$ and $\dot{t}=(dt/d\tau)$. $L_{0}$ does not permit the
massless $(m=0)$ limit but the massless $(m=0)$ limits are well-defined for
$L_{f}$ and the second-order Lagrangian $L_{s}$. We would like to stress that
the Lagrange multiplier variable behaves like the “gauge” variable due to its
transformation property.

For our further discussions, we shall concentrate on the first-order
Lagrangian $L_{f}$ because it has maximum number of variables (i.e.
$x,\,p_{x},\,t,\,p_{t},\,E)$, allows massless limit and there are no variables
(and/or their first-order derivative w.r.t. $\tau$) in its denominator. It is
straightforward to check that under the following infinitesimal and continuous
1D diffeomorphism (i.e. reparameterization) symmetry transformations
$(\delta_{r})$, namely;

$\displaystyle\delta_{r}\,x=\epsilon\,\dot{x},\quad\qquad\delta_{r}\,p_{x}=\epsilon\,{\dot{p}}_{x},\quad\qquad\delta_{r}\,t=\epsilon\,\dot{t},\quad\qquad\delta_{r}\,p_{t}=\epsilon\,{\dot{p}}_{t},$
$\displaystyle\delta_{r}\,E=\frac{d}{d\,\tau}\,(\epsilon\,E),\quad\qquad\delta_{r}\,L_{f}=\frac{d}{d\,\tau}\,(\epsilon\,L_{f}),$
(2)

the action integral $S=\int_{-\infty}^{+\infty}d\,\tau\,L_{f}$ remains
invariant (i.e. $\delta_{r}\,S=0$) for the physically well-defined variables
in $L_{f}$ and the infinitesimal diffeomorphism transformation parameter
$\epsilon(\tau)$ in:
$\tau\longrightarrow\tau^{{}^{\prime}}=g(\tau)=\tau-\epsilon(\tau)$ where
$g(\tau)$ is a physically well-defined function of $\tau$ such that it is
finite at $\tau=0$ and vanishes off at $\tau=\pm\infty$ . In fact, the
infinitesimal and continuous reparameterization symmetry transformation
$(\delta_{r})$ is defined as:
$\delta_{r}\,\phi(\tau)=\phi^{{}^{\prime}}(\tau)-\phi(\tau)$ for the generic
variable $\phi(\tau)=x(\tau),\,p_{x}(\tau),\,t(\tau),\,p_{t}(\tau),\,E(\tau)$
of our present theory.

The above infinitesimal and continuous reparameterization symmetry
transformations (2) encompass in their folds the gauge symmetry
transformations $(\delta_{g})$ which are generated (see, e.g. [32] for
details) by the first-class constraints: $\Pi_{E}\approx
0,\,(p_{x}^{2}+2\,m\,p_{t})\approx 0$ where $\Pi_{E}$ is the canonical
conjugate momentum corresponding to the variable $E(\tau)$. Using the
following Euler-Lagrange equations of motion (EL-EOMs) from $L_{f}$, namely;

$\displaystyle{\dot{p}}_{x}=0,\qquad{\dot{p}}_{t}=0,\qquad\dot{x}=E\,p_{x},\qquad\dot{t}=E\,m,$
(3)

and identifying the transformation parameters
$\epsilon(\tau)\,E(\tau)=\xi(\tau)$, we obtain the infinitesimal and
continuous gauge symmetry transformations $(\delta_{g})$, from the
infinitesimal and continuous reparameterization symmetry transformations (2),
as follows:

$\displaystyle\delta_{g}\,x=\xi\,p_{x},\qquad\delta_{g}\,t=\xi\,m,\qquad\delta_{g}\,p_{x}=0,\qquad\delta_{g}\,p_{t}=0,\qquad\delta_{g}\,E=\dot{\xi}.$
(4)

It is elementary now to check that the first-order Lagrangian $(L_{f})$
transforms to a total derivative under the infinitesimal and continuous gauge
transformations $(\delta_{g})$, namely;

$\displaystyle\delta_{g}\,L_{f}=\frac{d}{d\,\tau}\,\Big{[}\frac{\xi}{2}\,p_{x}^{2}\Big{]},$
(5)

thereby rendering the action integral
$S=\int^{+\infty}_{-\infty}d\,\tau\,L_{f}$ invariant (i.e $\delta_{g}\,S=0$)
under the infinitesimal and continuous gauge symmetry transformations
$(\delta_{g})$.

We end this subsection with the following decisive comments. First, the 1D
diffeomorphism (i.e. reparameterization) transformations (2) are more general
than the infinitesimal and continuous gauge symmetry transformations (4).
Second, the Lagrange multiplier variable $E(\tau)$ behaves like a “gauge”
variable due to its transformation: $\delta_{g}\,E=\dot{\xi}$ in (4). Third,
all the three Lagrangians in (1) are equivalent and all of them respect the
infinitesimal and continuous gauge and reparameterization symmetry
transformations [32]. Fourth, all the Lagrangians describe the free motion
$({\dot{p}}_{x}=0,\,{\dot{p}}_{t}=0)$ of the NR particle. Hence, our system is
a non-relativistic free $(\dot{p}_{x}=\dot{p}_{t}=0)$ particle. Fifth, the
first-order Lagrangian $L_{f}$ is theoretically more interesting to handle
because, as pointed out earlier, it incorporates the maximum number of
variables. Finally, we can exploit the reparameterization and gauge symmetry
transformations (2) and (4) for the BRST quantization. Following the usual
BRST prescription, we note that the (anti-)BRST symmetry transformations,
corresponding to the classical reparameterization symmetry transformation (2),
are

$\displaystyle s_{ab}\,x$ $\displaystyle=$
$\displaystyle\bar{C}\,\dot{x},\quad s_{ab}\,p_{x}=\bar{C}\,\dot{p_{x}},\quad
s_{ab}\,t=\bar{C}\,\dot{t},\quad s_{ab}\,p_{t}=\bar{C}\;\dot{p}_{t},\quad
s_{ab}\,E=\frac{d}{d\,\tau}\,(\bar{C}\,E),$ $\displaystyle s_{b}\,x$
$\displaystyle=$ $\displaystyle
C\,\dot{x},\quad~{}s_{b}\,p_{x}=C\,\dot{p_{x}},\quad~{}s_{b}\,t=C\,\dot{t},\quad~{}~{}s_{b}\,p_{t}=C\;\dot{p}_{t},\quad
s_{b}\,E=\frac{d}{d\,\tau}\,(C\,E),$ (6)

where $(\bar{C})C$ are the fermionic
$(C^{2}=\bar{C}^{2}=0,\,C\,\bar{C}+\bar{C}\,C=0)$ (anti-)ghost variables
corresponding to the classical infinitesimal diffeomorphism transformation
parameter $\epsilon(\tau)$ [cf. Eq. (2)]. In exactly similar fashion, the
(anti-)BRST symmetry transformations, corresponding to the classical gauge
symmetry transformations (4), are

$\displaystyle s_{ab}\,x$ $\displaystyle=$ $\displaystyle\bar{c}\,p_{x},\qquad
s_{ab}\,p_{x}=0,\qquad s_{ab}\,t=\bar{c}\,m,\qquad s_{ab}\,p_{t}=0,\qquad
s_{ab}\,E=\dot{\bar{c}},$ $\displaystyle s_{b}\,x$ $\displaystyle=$
$\displaystyle c\,p_{x},\qquad
s_{ab}\,p_{x}=0,\qquad~{}s_{b}\,t=c\,m,\qquad~{}s_{b}\,p_{t}=0,\qquad~{}s_{b}\,E=\dot{c},$
(7)

where the fermionic $(c^{2}=\bar{c}^{2}=0,\,c\,\bar{c}+\bar{c}\,c=0)$
variables $(\bar{c})c$ are the (anti-)ghost variables corresponding to the
classical gauge symmetry transformation parameter $\xi(\tau)$ of Eq. (4). In
addition to the (anti-)BRST symmetry transformations in (6) and (7), we have
the following standard (anti-)BRST transformations

$\displaystyle s_{ab}\,C$ $\displaystyle=$ $\displaystyle i\,\bar{B},\qquad
s_{ab}\,\bar{B}=0,\qquad s_{b}\,\bar{C}=i\,B,\qquad s_{b}\,B=0,$
$\displaystyle s_{ab}\,c$ $\displaystyle=$ $\displaystyle
i\,\bar{b},\qquad~{}~{}s_{ab}\,\bar{b}=0,\qquad~{}~{}s_{b}\,\bar{c}=i\,b,\qquad~{}~{}s_{b}\,b=0,$
(8)

where the pairs $(B,\bar{B})$ and $(b,\bar{b})$ are the Nakanishi-Lautrup
auxiliary variables in the contexts of the BRST quantization of our
reparameterization and gauge invariant system by exploiting the classical
reparameterization and gauge transformations, respectively.

###  2.2 Quantum Nilpotent (Anti-)BRST Symmetries Corresponding to the
Classical Gauge Symmetry Transformations

We have listed the quantum (anti-)BRST symmetries corresponding to the
classical gauge symmetry transformations (4) in our Eqs. (7) and (8). It is
elementary to check that these quantum symmetries are off-shell nilpotent
$(s_{(a)b}^{2}=0)$ of order two. The requirement of the absolute
anticommutativity $(s_{b}\,s_{ab}+s_{ab}\,s_{b}=0)$ leads to the restriction:
$b+\bar{b}=0\Longrightarrow\bar{b}=-b$. As a consequence, we have the full set
of (anti-)BRST symmetry transformations [corresponding to the classical gauge
symmetry transformations (4)] as follows:

$\displaystyle s_{ab}\,x=\bar{c}\,p_{x},\qquad s_{ab}\,p_{x}=0,\qquad
s_{ab}\,t=\bar{c}\,m,\qquad s_{ab}\,p_{t}=0,$ $\displaystyle
s_{ab}\,E=\dot{\bar{c}},\qquad s_{ab}\,\bar{c}=0,\qquad s_{ab}\,c=-i\,b,\qquad
s_{ab}\,b=0,$ $\displaystyle s_{b}\,x=c\,p_{x},\qquad s_{b}\,p_{x}=0,\qquad
s_{b}\,t=c\,m,\qquad s_{b}\,p_{t}=0,$ $\displaystyle s_{b}\,E=\dot{c},\qquad
s_{b}\,c=0,\qquad s_{b}\,\bar{c}=i\,b,\qquad s_{b}\,b=0.$ (9)

It is straightforward to check that the above (anti-)BRST symmetry
transformations are off-shell nilpotent $(s_{(a)b}^{2}=0)$ and absolutely
anticommuting $(s_{b}\,s_{ab}+s_{ab}\,s_{b}=0)$ in nature. The (anti-)BRST
invariant Lagrangian $L_{b}$ (which is the generalization of the classical
$L_{f}$ to its quantum level) can be written as†††The structure of gauge-
fixing and FP-ghost terms is exactly like the Abelian 1-form
$(A^{(1)}=dx^{\mu}\,A_{\mu})$ gauge theory where we have the BRST-invariant
Lagrangian density: ${\cal
L}_{b}=-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}+s_{b}\,[-\,i\,\bar{c}\,(\partial_{\mu}\,A^{\mu}+\frac{b}{2})]\,\equiv\,-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}+s_{ab}\,[-\,i\,c\,(\partial_{\mu}\,A^{\mu}+\frac{b}{2})]\,\equiv\,-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}+s_{b}\,s_{ab}\,[\frac{i}{2}\,A^{\mu}A_{\mu}-\frac{1}{2}{\bar{c}\,c}]$.
Here $A_{\mu}$ is the vector potential,
$F_{\mu\,\nu}=\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu}$ is the field
strength tensor and rest of the symbols are same as in Eqs. (10) and (11).
Note that the $2$-form $F^{(2)}=d\,A^{(1)}=\frac{1}{2}\,(d\,x^{\mu}\wedge
d\,x^{\nu})\,F_{\mu\nu}$ defines the field strength tensor $F_{\mu\nu}$ (where
$d=d\,x^{\mu}\,\partial_{\mu}$ in $F^{(2)}=d\,A^{(1)}$ stands for the exterior
derivative of the differential geometry).:

$\displaystyle L_{b}$ $\displaystyle=$ $\displaystyle
L_{f}+s_{b}\,\Big{[}-\,i\,\bar{c}\,\Big{(}\dot{E}+\frac{b}{2}\Big{)}\Big{]}\equiv
L_{f}+s_{ab}\,\Big{[}i\,c\,\Big{(}\dot{E}+\frac{b}{2}\Big{)}\Big{]},$ (10)
$\displaystyle\equiv$ $\displaystyle
L_{f}+s_{b}\,s_{ab}\,\Big{[}\frac{i\,E^{2}}{2}-\frac{\bar{c}\,c}{2}\Big{]}\equiv
L_{f}-s_{ab}\,s_{b}\,\Big{[}\frac{i\,E^{2}}{2}-\frac{\bar{c}\,c}{2}\Big{]}.$

In other words, we have expressed the gauge-fixing and Faddeev-Popov (FP)
ghost terms in three different ways which, ultimately, lead to the following
expression for $L_{b}$, namely;

$\displaystyle L_{b}$ $\displaystyle=$ $\displaystyle
L_{f}+b\,\dot{E}+\frac{b^{2}}{2}-i\,\dot{\bar{c}}\,\dot{c},$ (11)
$\displaystyle\equiv$ $\displaystyle
p_{x}\,\dot{x}+p_{t}\,\dot{t}-\frac{1}{2}\,E\,(p_{x}^{2}+2\,m\,p_{t})+b\,\dot{E}+\frac{b^{2}}{2}-i\,\dot{\bar{c}}\,\dot{c}.$

It should be noted that we have dropped the total derivative terms in
obtaining $L_{b}$ from (10). The above equation demonstrates that we have
obtained a unique (anti-)BRST invariant Lagrangian. This has happened because
the CF-type restriction is trivial (i.e. $b+\bar{b}=0$) in our simple case of
NR system. We can explicitly check that:

$\displaystyle
s_{b}\,L_{b}=\frac{d}{d\,\tau}\,\Big{[}\frac{c}{2}\,p_{x}^{2}+b\,\dot{c}\,\Big{]},\qquad
s_{ab}\,L_{b}=\frac{d}{d\,\tau}\,\Big{[}\frac{\bar{c}}{2}\,p_{x}^{2}+b\,\dot{\bar{c}}\,\Big{]},$
(12)

which lead to the derivation of the conserved (anti-)BRST charges $[Q_{(a)b}]$
as follows:

$\displaystyle
Q_{ab}=b\,\dot{\bar{c}}+\frac{1}{2}\,\bar{c}\,(p_{x}^{2}+2\,m\,p_{t})\equiv
b\,\dot{\bar{c}}-\dot{b}\,\bar{c},$ $\displaystyle
Q_{b}=b\,\dot{c}+\frac{1}{2}\,c\,(p_{x}^{2}+2\,m\,p_{t})\equiv
b\,\dot{c}-\dot{b}\,c.$ (13)

In the last step, we have used $\dot{b}=-\,(1/2)\,(p_{x}^{2}+2\,m\,p_{t})$
which emerges out as the EL-EOM from $L_{b}$ w.r.t. the Lagrange multiplier
variable $E(\tau)$ .

We close this sub-section with a few crucial and decisive remarks. First, we
can check that the (anti-)BRST charges are conserved $[{\dot{Q}}_{(a)b}=0]$ by
using the EL-EOMs. Second, the (anti-)BRST charges $[Q_{(a)b}]$ are off-shell
nilpotent $[Q_{(a)b}^{2}=0]$ of order two due to the direct observations that:
$s_{b}\,Q_{b}=s_{b}\,[b\,\dot{c}-\dot{b}\,c]=0$ and
$s_{ab}\,Q_{ab}=s_{ab}\,[b\,\dot{\bar{c}}-\dot{b}\,\bar{c}]=0$ which encode in
their folds $s_{b}\,Q_{b}=-\,i\,\\{Q_{b},Q_{b}\\}=0\Longrightarrow
Q_{b}^{2}=0$ and $s_{ab}\,Q_{ab}=-\,i\,\\{Q_{ab},Q_{ab}\\}=0\Longrightarrow
Q_{ab}^{2}=0$. Third, the above nilpotency is also encoded in:
$Q_{b}=s_{b}\,[b\,E+i\,\dot{\bar{c}}\,c]$ implying that $s_{b}\,Q_{b}=0$ due
to $s_{b}^{2}=0$ and we also point out that $s_{ab}\,Q_{ab}=0$ due to the
nilpotency $(s_{ab}^{2}=0)$ of $s_{ab}$ because
$Q_{ab}=s_{ab}\,[b\,E+i\,\bar{c}\,\dot{c}]$. Fourth, we observe that
$s_{ab}\,Q_{b}=i\,\\{Q_{b},Q_{ab}\\}\equiv-\,i\,b\,\dot{b}+i\,\dot{b}\,b=0$
and $s_{b}\,Q_{ab}=-\,i\,\\{Q_{ab},Q_{b}\\}=i\,b\,\dot{b}-i\,b\,\dot{b}=0$
which explicitly lead to the conclusion that the off-shell nilpotent charges
$Q_{(a)b}$ are also absolutely anticommuting $(Q_{b}\,Q_{ab}+Q_{ab}\,Q_{b}=0)$
in nature. Fifth, the above observation of the absolute anticommutativity can
be also expressed in terms of the nilpotency property because we observe that
$Q_{b}=s_{ab}\,(-\,i\,\dot{c}\,c)$ and
$Q_{ab}=s_{b}\,(i\,\dot{\bar{c}}\,\bar{c})$ which imply that
$s_{ab}\,Q_{b}=-\,i\,\\{Q_{b},Q_{ab}\\}=0$ and
$s_{b}\,Q_{ab}=-\,i\,\\{Q_{ab},Q_{b}\\}=0$ [due to the off-shell nilpotency
$(s_{ab}^{2}=0)$ of the anti-BRST as well as the off-shell nilpotency
$(s_{b}^{2}=0)$ of the BRST symmetry transformations]. Sixth, it can be seen
that the physical space (i.e. $\mid phys>$) in the total Hilbert space of
states is defined by $Q_{b}\mid phys>=0$ which implies that $b\mid
phys>\equiv\Pi_{E}\,\mid phys>=0$ and $\dot{b}\mid
phys>\equiv(p_{x}^{2}+2\,m\,p_{t})\,\mid phys>=0$. In other words, the Dirac
quantization conditions (with the first-class constraints $\Pi_{E}\approx
0,\,p_{x}^{2}+2\,m\,p_{t}\approx 0$) are beautifully satisfied. Finally,
physicality criterion $Q_{b}\mid phys>=0$ implies that the two physical states
$\mid phys^{{}^{\prime}}>$ and $\mid phys>$ belong to the same cohomological
class w.r.t. the nilpotent BRST charge $Q_{b}$ if they differ by a BRST exact
state (i.e. $\mid phys^{{}^{\prime}}>=\mid phys>+Q_{b}\,\mid\chi>$ for non-
null $|\chi>$).

##  3 Nilpotent and Anticommuting (Anti-)BRST Symmetries for the Phase
Variables: MBTSA

This section is devoted to the derivation of the transformations:
$s_{b}\,x=C\,\dot{x},\,s_{b}\,p_{x}=C\,{\dot{p}}_{x},\,s_{b}\,t=C\,\dot{t},\,s_{b}\,p_{t}=C\,{\dot{p}}_{t},\,s_{ab}\,x=\bar{C}\,\dot{x},\,s_{ab}\,p_{x}=\bar{C}\,{\dot{p}}_{x},\,s_{ab}\,t=\bar{C}\,\dot{t},\,s_{ab}\,p_{t}=\bar{C}\,{\dot{p}}_{t}$
by exploiting the theoretical tricks of MBTSA. Before we set out to perform
this exercise, it is essential to pinpoint the off-shell nilpotency and
absolute anticommutativity properties of the (anti-)BRST symmetry
transformations on the phase variables [cf. Eq. (6)]. It can be easily checked
that the off-shell nilpotency requirement (i.e.
$s_{(a)b}^{2}\,S=0,\,S=x,\,p_{x},\,t,\,p_{t}$) leads to the (anti-)BRST
symmetry transformations for the (anti-)ghost variables as:

$\displaystyle s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}},\qquad\qquad
s_{b}\,C=C\,\dot{C}.$ (14)

Furthermore, the absolute anticommutativity requirement:
$\\{s_{b},\,s_{ab}\\}S=0$ for the generic phase variable
$S=x,\,p_{x},\,t,\,p_{t}$ leads to the following

$\displaystyle\\{s_{b},\,s_{ab}\\}S=i\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,\dot{S}\,\quad\Longrightarrow\quad\,B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0.$
(15)

In other words, the absolute anticommutativity property
($s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$) is satisfied if and only if we invoke the
sanctity of the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$. It goes without saying
that the above cited requirements of the off-shell nilpotency and absolute
anticommutativity properties are very sacrosanct within the framework of BRST
approach to gauge and/or reparameterization invariant theories.

Against the backdrop of the above discussions, we set out to deduce the
(anti-)BRST symmetry transformations:
$s_{ab}\,S=\bar{C}\,\dot{S},\,s_{b}\,S=C\,\dot{S}$ (with
$S=x,\,p_{x},\,t,\,p_{t}$) and the CF-type restrictions:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ within the framework of
MBTSA. Towards this end in our mind, first of all, we generalize the classical
function $g(\tau)$ [in
$\tau\longrightarrow\tau^{{}^{\prime}}=g(\tau)\equiv\tau-\epsilon(\tau)$] onto
a $(1,2)$-dimensional supermanifold as

$\displaystyle
g(\tau)\quad\longrightarrow\quad\tilde{g}(\tau,\theta,\bar{\theta})=\tau-\theta\,\bar{C}(\tau)-\bar{\theta}\,C(\tau)+\theta\,\bar{\theta}\,k(\tau),$
(16)

where $(\bar{C})C$ variables are the (anti-)ghost variables of Eq. (6) and
$k(\tau)$ is a secondary variable that has to be determined from the
consistency conditions [that include the off-shell nilpotency as well as
absolute anticommutativity requirements]. It will be noted that, due to the
mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}}\mid_{\theta=0},\,s_{ab}\leftrightarrow\partial_{\theta}\mid_{\bar{\theta}=0}$
[14-16], we have taken the coefficients of $\theta$ and $\bar{\theta}$ in Eq.
(16) as the (anti-)ghost variables $(\bar{C})C$. This has been done due to our
observation in the infinitesimal reparameterization symmetry transformation
$(\delta_{r})$ [where $\delta_{r}\,\tau=-\,\epsilon(\tau)$] at the classical
level. Following the basic tenet of BRST formalism, the infinitesimal
parameter $\epsilon(\tau)$ has been replaced (in the BRST-quantized theory) by
the (anti-)ghost variables thereby leading to the (anti-)BRST symmetry
transformations: $s_{ab}\,\tau=-\,\bar{C},\,s_{b}\,\tau=-\,C$.

For our present 1D diffeomorphism (i.e. reparameterization) invariant theory,
the generic variable $S(\tau)$ can be generalized to a supervariable
[$\tilde{S}(\tilde{g}(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})$] on the
(1, 2)-dimensional supermanifold [22] with the following super expansion along
all the Grassmannian directions of the (1, 2)-dimensional supermanifold,
namely;

$\displaystyle\tilde{S}\big{[}\tilde{g}(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})\big{]}={\cal
S}\big{[}\tilde{g}(\tau,\theta,\bar{\theta})\big{]}+\theta\,\bar{R}\big{[}\tilde{g}(\tau,\theta,\bar{\theta})\big{]}+\bar{\theta}\,R\big{[}\tilde{g}(\tau,\theta,\bar{\theta})\big{]}+\theta\,\bar{\theta}\,Q\big{[}\tilde{g}(\tau,\theta,\bar{\theta})\big{]},$
(17)

where the expression for $\tilde{g}(\tau,\theta,\bar{\theta})$ is given in Eq.
(16). It should be noted that all the primary as well as the secondary
supervariables on the r.h.s. of (17) are function of the $(1,2)$-dimensional
super infinitesimal diffeomorphism transformation (16). At this stage, we can
perform the Taylor expansions for all the supervariables as:

$\displaystyle\theta\,\bar{\theta}\,Q(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,k)$
$\displaystyle=$ $\displaystyle\theta\,\bar{\theta}\,Q(\tau),$
$\displaystyle\bar{\theta}\,R(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,k)$
$\displaystyle=$
$\displaystyle\bar{\theta}\,R(\tau)+\theta\,\bar{\theta}\,\bar{C}(\tau)\,\dot{R}(\tau),$
$\displaystyle\theta\,\bar{R}(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,k)$
$\displaystyle=$
$\displaystyle\theta\,\bar{R}(\tau)-\theta\,\bar{\theta}\,C(\tau)\,\dot{\bar{R}}(\tau),$
$\displaystyle{\cal
S}(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,k)$
$\displaystyle=$ $\displaystyle
S(\tau)-\theta\,\bar{C}(\tau)\,\dot{S}(\tau)-\bar{\theta}\,C(\tau)\,\dot{S}(\tau)$
(18) $\displaystyle+$
$\displaystyle\theta\,\bar{\theta}\,\big{[}k(\tau)\,\dot{S}(\tau)-\bar{C}(\tau)\,C(\tau)\,\ddot{S}(\tau)\big{]}.$

Collecting all these terms and substituting them into (17), we obtain the
following super expansion for the supervariable on the (1, 2)-dimensional
supermanifold, namely;

$\displaystyle\tilde{S}\big{[}\tilde{g}(\tau,\theta,\bar{\theta}),\theta,\bar{\theta}\big{]}$
$\displaystyle=$ $\displaystyle
S(\tau)+\theta\,(\bar{R}-\bar{C}\,\dot{S})+\bar{\theta}\,(R-C\,\dot{S})$ (19)
$\displaystyle+$
$\displaystyle\theta\,\bar{\theta}\,\big{[}Q+\bar{C}\,\dot{R}-C\,\dot{\bar{R}}+k\,\dot{S}+C\,\bar{C}\,\ddot{S}\big{]}.$

We now exploit the horizontality condition (HC) which physically implies that
all the scalar variables should not transform at all under any kind of
spacetime, internal, supersymmetric, etc., transformations. With respect to
the 1D space of trajectory of the particle, all the supervariables on the
l.h.s. and r.h.s. of Eq. (18) are scalars. The HC, in our case, is:

$\displaystyle\tilde{S}\big{[}\tilde{g}(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})\big{]}=S(\tau).$
(20)

The above equality implies that all the coefficients of $\theta$,
$\bar{\theta}$ and $\theta\,\bar{\theta}$ of Eq. (19) should be set equal to
zero. In other words, we have the following:

$\displaystyle R=C\,\dot{S},\qquad\bar{R}=\bar{C}\,\dot{S},\qquad
Q=C\,\dot{\bar{R}}-\bar{C}\,\dot{R}-k\,\dot{S}+\bar{C}\,C\,\ddot{S}.$ (21)

Substitutions of the values of $R$ and $\bar{R}$ into the expression for $Q$
leads to the following:

$\displaystyle
Q=-\,(\bar{C}\,\dot{C}+\dot{\bar{C}}\,C)\,\dot{S}-k\,\dot{S}-\bar{C}\,C\,\ddot{S}.$
(22)

As explained before Eq. (20) (i.e. exploiting the key properties of scalars),
it is evident that (17) can be finally written (with
$\tilde{S}\big{[}\tilde{g}(\tau,\theta,\bar{\theta}),\theta,\bar{\theta}\big{]}=\tilde{S}(\tau,\theta,\bar{\theta})$
as

$\displaystyle\tilde{S}\big{[}\tau,\theta,\bar{\theta})\big{]}$
$\displaystyle=$
$\displaystyle{S}(\tau)+\theta\,\bar{R}(\tau)+\bar{\theta}\,R(\tau)+\theta\,\bar{\theta}\,Q(\tau)$
(23) $\displaystyle\equiv$
$\displaystyle{S}(\tau)+\theta\,(s_{ab}\,S)+\bar{\theta}\,(s_{b}\,S)+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,S),$

where, due to the well known mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}}\mid_{\theta=0},\,s_{ab}\leftrightarrow\partial_{\theta}\mid_{\bar{\theta}=0}$
[14-16], the coefficients of $\theta$ and $\bar{\theta}$ are the anti-BRST and
BRST symmetry transformations [cf. Eq. (6)]. We point out that the key
properties of scalars on the r.h.s. of Eq. (17) implies that we have: ${\cal
S}[\tilde{g}(\tau,\theta,\bar{\theta})]=S(\tau),\,R[\tilde{g}(\tau,\theta,\bar{\theta})]=R(\tau),\,\bar{R}[\tilde{g}(\tau,\theta,\bar{\theta})]=\bar{R}(\tau)$
and $Q[\tilde{g}(\tau,\theta,\bar{\theta})]=Q(\tau)$.

A comparison between (21) and (23) implies that we have already derived the
nilpotent (anti-)BRST symmetry transformations: $R=s_{b}\,S=C\,\dot{S}$ and
$\bar{R}=s_{ab}\,S=\bar{C}\,\dot{S}$. In other words, we have obtained:
$s_{b}\,x=C\,\dot{x},\,s_{b}\,p_{x}=C\,{\dot{p}}_{x},\,s_{b}\,t=C\,\dot{t},\,s_{b}\,p_{t}=C\,{\dot{p}}_{t}$
and
$s_{ab}\,x=\bar{C}\,\dot{x},\,s_{ab}\,p_{x}=\bar{C}\,{\dot{p}}_{x},\,s_{ab}\,t=\bar{C}\,\dot{t},\,s_{ab}\,p_{t}=\bar{C}\,{\dot{p}}_{t}$.
Furthermore, it is evident that:

$\displaystyle
s_{b}\,s_{ab}\,S=Q=-\,(\bar{C}\,\dot{C}+\dot{\bar{C}}\,C)\,\dot{S}-k\,\dot{S}-\bar{C}\,C\,\ddot{S}.$
(24)

The requirement of the absolute anticommutativity: $\\{s_{b},s_{ab}\\}\,S=0$
implies that $s_{b}\,s_{ab}\,S=-\,s_{ab}\,s_{b}\,S$ which, in turn, leads to
the following relationships:

$\displaystyle s_{b}\,s_{ab}\,S$ $\displaystyle=$ $\displaystyle
s_{b}\,\bar{R}=Q\equiv-\,(\bar{C}\,\dot{C}+\dot{\bar{C}}\,C)\,\dot{S}-k\,\dot{S}-\bar{C}\,C\,\ddot{S},$
$\displaystyle-s_{ab}\,s_{b}\,S$ $\displaystyle=$
$\displaystyle-\,s_{ab}\,R=Q\equiv-\,(\bar{C}\,\dot{C}+\dot{\bar{C}}\,C)\,\dot{S}-k\,\dot{S}-\bar{C}\,C\,\ddot{S}.$
(25)

The explicit computation of the following, using the (anti-)BRST symmetry
transformations of the phase variables in Eqs. (6) and (14), are:

$\displaystyle s_{b}\,\bar{R}$ $\displaystyle=$ $\displaystyle
i\,B\,\dot{S}-\bar{C}\,\dot{C}\,\dot{S}-\bar{C}\,C\,\ddot{S}\equiv Q,$
$\displaystyle-\,s_{ab}\,R$ $\displaystyle=$
$\displaystyle-\,i\,\bar{B}\,\dot{S}-\dot{\bar{C}}\,C\,\dot{S}-\bar{C}\,C\,\ddot{S}\equiv
Q.$ (26)

Equating (25) and (26), we obtain the following interesting relationship:

$\displaystyle k=-\,\dot{\bar{C}}\,C-i\,B\equiv
i\,\bar{B}-\bar{C}\,\dot{C}\quad\Longrightarrow\quad
B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0.$ (27)

In other words, it is the consistency conditions of the BRST formalism that
lead to the determination of $k(\tau)$ in Eq. (16) within the ambit of MBTSA.
A close look at Eqs. (25), (26) and (27) establishes that a precise
determination of $Q(\tau)$ in (23) leads to (i) the validity of the absolute
anticommutativity (i.e. $\\{s_{b},s_{ab}\\}\,S=0$) of the off-shell nilpotent
(anti-)BRST symmetries, and (ii) the deduction of the (anti-)BRST
invariant‡‡‡This statement is true only when the whole theory is considered on
a submanifold of the Hilbert space of the quantum variables where the CF-type
restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is
satisfied. In other words, we explicitly compute
$s_{b}\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]=(\frac{d}{d\tau})\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,C-[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,\dot{C}$
and
$s_{ab}\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]=(\frac{d}{d\tau})\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,\bar{C}-[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,\dot{\bar{C}}$
which imply that
$s_{(a)b}\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]=0$ is true only
on the above mentioned submanifold. CF-type restriction
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ on our theory.

We conclude this section with the following useful and crucial remarks. First,
we set out to derive the (anti-)BRST symmetry transformations (corresponding
to the classical reparameterization symmetry transformations) for the phase
variables [cf. Eq. (6)]. We have accomplished this goal in Eq. (21). Second,
we have derived the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ within the purview of
MBTSA [cf. Eq. (27)] which is actually hidden in the determination of
$Q(\tau)$ in Eq. (23). Third, for the application of the theoretical potential
of MBTSA, we have taken the full super expansion of the generic supervariable
[cf. Eq. (17)] along all the possible Grassmannian directions of the (1,
2)-dimensional supermanifold. Fourth, unlike the application of BT-
superfield/supervariable approach to the gauge theories [14-16] where
spacetime does not change, in the case of MBTSA, the super diffeomorphism
transformation (16) has been taken into account in all the basic as well as
secondary supervariables. Fifth, taking into account the inputs from Eqs. (21)
and (26), we obtain the following super expansion of the generic variable
$S(\tau)$, namely;

$\displaystyle\tilde{S}^{(h)}(\tau,\theta,\bar{\theta})$ $\displaystyle=$
$\displaystyle
S(\tau)+\theta\,(\bar{C}\,\dot{S})+\bar{\theta}\,(C\,\dot{S})+\theta\,\bar{\theta}\,[i\,B\,\dot{S}-\bar{C}\,\dot{C}\,\dot{S}-\bar{C}\,C\,\ddot{S}]$
(28) $\displaystyle\equiv$ $\displaystyle
S(\tau)+\theta\,(s_{ab}\,S)+\bar{\theta}\,(s_{b}\,S)+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,S),$

where $S=x,p_{x},t,p_{t}$ and the superscript $(h)$ on the supervariable
$\tilde{S}(\tau,\theta,\bar{\theta})$ denotes that this supervariable has been
obtained after the application of HC. Finally, the standard nilpotent
(anti-)BRST symmetry transformations (8) dictate that we can have the
following (anti-)chiral super expansions for the supervariables corresponding
to $(\bar{C})C$, namely;

$\displaystyle
C(\tau)\quad\longrightarrow\quad{F}^{(c)}(\tau,\theta)=C(\tau)+\theta\,(i\,\bar{B})\equiv
C(\tau)+\theta\,(s_{ab}\,C),$
$\displaystyle\bar{C}(\tau)\quad\longrightarrow\quad{{\bar{F}}^{(ac)}}(\tau,\bar{\theta})=\bar{C}(\tau)+\bar{\theta}\,(i\,B)\equiv\bar{C}(\tau)+\bar{\theta}\,(s_{b}\,\bar{C}),$
(29)

where the superscripts $(c)$ and $(ac)$ denote the chiral and anti-chiral
supervariables. The above observation gives us a clue that we should exploit
the theoretical strength of ACSA to BRST formalism for our further
discussions.

##  4 Coupled Lagrangians and Quantum (Anti-)BRST Symmetries Corresponding to
the Classical Reparameterization Symmetry Transformations

In addition to the quantum (anti-)BRST symmetries in (6), (8) and (14), we
derive all the other off-shell nilpotent and absolutely anticommuting
(anti-)BRST symmetries corresponding to the classical infinitesimal and
continuous reparameterization symmetry transformations (2). We exploit the
strength of the sacrosanct requirements of off-shell nilpotency and absolute
anticommutativity properties. In this context, we point out that we have
already derived $s_{b}\,C=C\,\dot{C},\,s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}}$
by invoking the sanctity of the off-shell nilpotency $(s_{(a)b}^{2}=0)$
property for the phase variables (i.e. $s_{(a)b}^{2}\,S=0,S=x,p_{x},t,p_{t}$).
It is interesting to note the following absolute anticommutativity
requirements, namely;

$\displaystyle\\{s_{b},s_{ab}\\}\,C=0\quad\Longrightarrow\quad
s_{b}\,\bar{B}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C},$
$\displaystyle\\{s_{b},s_{ab}\\}\,\bar{C}=0\quad\Longrightarrow\quad
s_{ab}\,B=\dot{B}\,\bar{C}-B\,\dot{\bar{C}},$ (30)

leads to the derivation of the $s_{b}\,{\bar{B}}$ and $s_{ab}\,B$. We can
readily check that $s_{b}^{2}\,\bar{B}=0,\,s_{ab}^{2}\,B=0$ are satisfied due
to our knowledge of the BRST and anti-BRST symmetry transformations:
$s_{b}\,C=C\,\dot{C},\,s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}}$ and the
fermionic $(C^{2}={\bar{C}}^{2}=0,\,C\,\bar{C}+\bar{C}\,C=0)$ nature of the
(anti-)ghost variables $(\bar{C})\,C$. We further note that
$\\{s_{b},s_{ab}\\}\,B=0$ and $\\{s_{b},s_{ab}\\}\,\bar{B}=0$. The requirement
of the absolute anticommutativity on the $E(\tau)$ variable leads to:

$\displaystyle\\{s_{b},s_{ab}\\}\,E(\tau)=\frac{d}{d\,\tau}\,\Big{[}i\,\big{\\{}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{\\}}\,E(\tau)\Big{]}.$
(31)

Thus, we emphasize that the absolute anticommutativity property
$(s_{b}\,s_{ab}+s_{ab}\,s_{b}=0)$ on the phase variables [cf. Eq. (15)] as
well as on the Lagrange multiplier variable [cf. Eq. (31)] are satisfied if
and only if the CF-type restriction is invoked. In the full blaze of glory,
the quantum (anti-)BRST symmetry transformations [corresponding to the
infinitesimal reparameterization symmetry transformations (2)] are as follows:

$\displaystyle s_{ab}\,x$ $\displaystyle=$
$\displaystyle\bar{C}\,\dot{x},\quad s_{ab}\,p_{x}=\bar{C}\,\dot{p_{x}},\quad
s_{ab}\,t=\bar{C}\,\dot{t},\quad s_{ab}\,p_{t}=\bar{C}\;\dot{p}_{t},\quad
s_{ab}\,E=\frac{d}{d\,\tau}\,(\bar{C}\,E),$ $\displaystyle s_{ab}\,C$
$\displaystyle=$ $\displaystyle i\,\bar{B},\quad
s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}},\quad s_{ab}\,\bar{B}=0,\quad
s_{ab}\,B=\dot{B}\,\bar{C}-B\,\dot{\bar{C}},$ (32) $\displaystyle s_{b}\,x$
$\displaystyle=$ $\displaystyle
C\,\dot{x},\quad~{}s_{b}\,p_{x}=C\,\dot{p_{x}},\quad~{}s_{b}\,t=C\,\dot{t},\quad~{}~{}s_{b}\,p_{t}=C\;\dot{p}_{t},\quad
s_{b}\,E=\frac{d}{d\,\tau}\,(C\,E),$ $\displaystyle s_{b}\,\bar{C}$
$\displaystyle=$ $\displaystyle i\,B,\quad s_{b}\,C=C\,\dot{C},\quad
s_{b}\,B=0,\quad s_{b}\,\bar{B}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}.$ (33)

The above fermionic symmetry transformations are off-shell nilpotent and
absolutely anticommuting provided the whole theory is considered on a
submanifold of the space of quantum variables where the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is satisfied.

The existence of the above CF-type restriction leads to the derivation of the
coupled (but equivalent) Lagrangians (i.e. $L_{B}$ and $L_{\bar{B}}$) as
follows:

$\displaystyle
L_{B}=L_{f}+s_{b}\,s_{ab}\Big{[}\frac{i\,E^{2}}{2}-\frac{\bar{C}\,C}{2}\Big{]},$
$\displaystyle
L_{\bar{B}}=L_{f}-s_{ab}\,s_{b}\Big{[}\frac{i\,E^{2}}{2}-\frac{\bar{C}\,C}{2}\Big{]}.$
(34)

We point out that the terms inside the square brackets are same as in Eq. (10)
for the BRST analysis of the classical gauge symmetry transformations (4).
Furthermore, in contrast to the unique (anti-)BRST invariant Lagrangian [cf.
Eq. (11)] (corresponding to the classical gauge symmetry transformations), we
have obtained here a set of coupled (but equivalent) (anti-)BRST invariant
Lagrangians in Eq. (34). This has happened because of the fact that the CF-
type restriction ($b+\bar{b}=0$) is trivial in the case of the former while it
is a non-trivial restriction
[$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$] in the context of the
latter.

One can readily compute the operation of $s_{(a)b}$ on the quantities in the
square brackets of Eq. (34). In the full blaze of their glory, the coupled
(but equivalent) Lagrangians $L_{B}$ and $L_{\bar{B}}$ are as follows§§§ It
will be worthwhile to mention here that the form of the gauge-fixing and
Faddeev-Popov ghost terms is same as in the cases of NSUSY (i.e. scalar) and
SUSY (i.e. spinning) relativistic particles [23, 24].

$\displaystyle L_{B}$ $\displaystyle=$ $\displaystyle
L_{f}+B\,\Big{[}E\,\dot{E}-i\,(2\,\dot{\bar{C}}\,C+{\bar{C}}\,\dot{C})\Big{]}+\frac{B^{2}}{2}$
$\displaystyle-i\,E\,\dot{E}\,\dot{\bar{C}}\,C-\,i\,E^{2}\,\dot{\bar{C}}\,\dot{C}-\dot{\bar{C}}\,{\bar{C}}\,\dot{C}\,C,$
$\displaystyle L_{\bar{B}}$ $\displaystyle=$ $\displaystyle
L_{f}-\bar{B}\,\Big{[}E\,\dot{E}-i\,(2\,{\bar{C}}\,\dot{C}+\dot{\bar{C}}\,C)\Big{]}+\frac{{\bar{B}}^{2}}{2}$
(35)
$\displaystyle-i\,E\,\dot{E}\,{\bar{C}}\,\dot{C}-\,i\,E^{2}\,\dot{\bar{C}}\,\dot{C}-\dot{\bar{C}}\,{\bar{C}}\,\dot{C}\,C,$

where the subscripts $B$ and $\bar{B}$ on the Lagrangians are appropriate
because $L_{B}$ depends uniquely on the Nakanishi-Lautrup auxiliary variable
$B$ (where ${\bar{B}}$ is not present at all). Similarly, the Lagrangian
$L_{\bar{B}}$ is uniquely dependent on $\bar{B}$. They are coupled because the
EL-EOMs with respect to $B$ and $\bar{B}$ from $L_{B}$ and $L_{\bar{B}}$,
respectively, yield

$\displaystyle
B=-\,E\,\dot{E}+2\,i\,\dot{\bar{C}}\,C+i\,\bar{C}\,\dot{C},\qquad\bar{B}=E\,\dot{E}-2\,i\,{\bar{C}}\,\dot{C}-i\,\dot{\bar{C}}\,C,$
(36)

which lead to the deduction of the CF-type restrictions:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$. Furthermore, the
condition $L_{B}\equiv L_{\bar{B}}$ also demonstrates the existence of the CF-
type restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ on our
theory (cf. Appendix A below).

At this stage, we are in the position to study the (anti-)BRST symmetries of
the Lagrangians $L_{B}$ and $L_{\bar{B}}$. It is straightforward to note that
we have the following:

$\displaystyle
s_{b}\,L_{B}=\frac{d}{d\,\tau}\Big{[}C\,L_{f}+B^{2}\,C-i\,B\,\bar{C}\,\dot{C}\,C+E\,\dot{E}\,B\,C+E^{2}\,B\,\dot{C}\Big{]},$
(37) $\displaystyle
s_{ab}\,L_{\bar{B}}=\frac{d}{d\,\tau}\Big{[}\bar{C}\,L_{f}+{\bar{B}}^{2}\,\bar{C}-i\,\bar{B}\,\dot{\bar{C}}\,\bar{C}\,C-E\,\dot{E}\,\bar{B}\,\bar{C}-E^{2}\,\bar{B}\,\dot{\bar{C}}\Big{]}.$
(38)

The above observations demonstrate that the action integrals
$S_{1}=\int_{-\infty}^{\infty}d\,\tau\,L_{B}$ and
$S_{2}=\int_{-\infty}^{\infty}d\,\tau\,L_{\bar{B}}$ remain invariant under the
SUSY-type (i.e. fermionic) off-shell nilpotent, continuous and infinitesimal
(anti-)BRST symmetry transformations for the physical variables that vanish
off at $\tau=\pm\infty$. At this crucial juncture, we establish the
equivalence of the coupled Lagrangian $L_{B}$ and $L_{\bar{B}}$ w.r.t the
(anti-)BRST symmetry transformations $[s_{(a)b}]$. In this context, we apply
$s_{ab}$ on $L_{B}$ and $s_{b}$ on $L_{\bar{B}}$ to obtain the following

$\displaystyle s_{ab}\,L_{B}$ $\displaystyle=$
$\displaystyle\frac{d}{d\,\tau}\,\Big{[}\bar{C}\,L_{f}+E\,\dot{E}\,(i\,\dot{\bar{C}}\,\bar{C}\,C+B\,\bar{C})+E^{2}\,(i\,\dot{\bar{C}}\,\bar{C}\,\dot{C}+B\,\dot{\bar{C}})$
(39) $\displaystyle+$
$\displaystyle{B}^{2}\,\bar{C}+i\,(2\,B-\bar{B})\,\dot{\bar{C}}\,\bar{C}\,C\Big{]}$
$\displaystyle+$
$\displaystyle\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,(2\,i\,\dot{\bar{C}}\,\bar{C}\,\dot{C}-2\,B\,\dot{\bar{C}}-E\,\dot{E}\,\dot{\bar{C}}+i\,\ddot{\bar{C}}\,\bar{C}\,C)$
$\displaystyle-$
$\displaystyle\frac{d}{d\,\tau}\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,\big{[}B\,\bar{C}+E^{2}\,\dot{\bar{C}}\big{]},$
$\displaystyle s_{b}\,L_{\bar{B}}$ $\displaystyle=$
$\displaystyle\frac{d}{d\,\tau}\,\Big{[}C\,L_{f}+E\,\dot{E}\,(i\,\bar{C}\,C\,\dot{C}-\bar{B}\,C)+E^{2}\,(i\,\dot{\bar{C}}\,C\,\dot{C}-\bar{B}\,\dot{C})$
(40) $\displaystyle+$
$\displaystyle{\bar{B}}^{2}\,C-i\,(2\,\bar{B}-B)\,\bar{C}\,C\,\dot{C}\Big{]}$
$\displaystyle+$
$\displaystyle\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,(-2\,i\,\dot{\bar{C}}\,C\,\dot{C}-2\,\bar{B}\,\dot{C}+E\,\dot{E}\,\dot{C}+i\,\bar{C}\,\ddot{C}\,C)$
$\displaystyle+$
$\displaystyle\frac{d}{d\,\tau}\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,\big{[}+E^{2}\,\dot{C}-\bar{B}\,C\big{]},$

which demonstrate that the coupled Lagrangians $L_{B}$ and $L_{\bar{B}}$ (and
corresponding action integrals) respect both (i.e. BRST and anti-BRST)
symmetry transformations together provided the whole theory is considered on a
supermanifold in the Hilbert space of quantum variables where the CF-type
restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is
satisfied. It should be recalled that, under the latter restriction, we also
have the absolute anticommutativity property (i.e. $\\{s_{b},s_{ab}\\}=0$ of
the (anti-)BRST symmetry transformations.

We end this section with the following key comments. First, the properties of
the off-shell nilpotency and absolute anticommutativity are sacrosanct in the
realm of BRST approach to gauge and/or diffeomorphism invariant theories.
Second, physically, the first property (i.e. off-shell nilpotency) implies
that these fermionic symmetry transformations are supersymmetric-type as they
transform bosonic variables to fermionic variables and vice-versa. Third, the
property of the absolute anticommutativity encodes the linear independence of
the BRST and anti-BRST symmetry transformations. Fourth, the absolute
anticommutativity property owes its origin to the existence of the CF-type
restrictions which are connected with the concepts of gerbes [7, 8]. Fifth, as
the classical gauge theory is characterized by the first-class constraints, in
exactly similar fashion, the quantum gauge and/or diffeomorphism [i.e.
(anti-)BRST] invariant theories are characterized by the existence of the CF-
type restrictions within the ambit of BRST formalism. Sixth, the coupled
Lagrangians $L_{B}$ and $L_{\bar{B}}$ are equivalent because both of them
respect BRST and anti-BRST symmetry transformations as is clear from Eqs.
(37)-(40) provided the whole theory is considered on the submanifold of the
total Hilbert space of the quantum variables where the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is satisfied.

##  5 Quantum Off-Shell Nilpotent (Anti-)BRST Symmetries of the Other
Variables: ACSA

In this section, we derive the nilpotent (anti-)BRST symmetry transformations
$[s_{(a)b}]$ for all the other variables [cf. Eqs. (32), (33)] besides the
phase space variables $(x,p_{x},t,p_{t})$ whose (anti-)BRST symmetries have
already been derived in Sec. 3 by exploiting the theoretical potential of
MBTSA. To achieve the above goal, we exploit the ideas behind ACSA to BRST
formalism [25-29]. In this context, first of all, we focus on the derivation
of the BRST symmetry transformations:
$s_{b}\,B=0,\,s_{b}\,\bar{B}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C},\,s_{b}\,C=C\,\dot{C},\,s_{b}\,E=\dot{E}\,C+E\,\dot{C}$
[cf. Eq. (33)]. For this purpose, we generalize the ordinary variables
$[B(\tau),\,\bar{B}(\tau),\,C(\tau),\,E(\tau)]$ onto a $(1,1)$-dimensional
anti-chiral super sub-manifold as follows

$\displaystyle B(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad{\cal
B}(\tau,\bar{\theta})=B(\tau)+\bar{\theta}\,f_{1}(\tau),$
$\displaystyle\bar{B}(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad{\cal\bar{B}}(\tau,\bar{\theta})=\bar{B}(\tau)+\bar{\theta}\,f_{2}(\tau),$
$\displaystyle C(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad F(\tau,\bar{\theta})=C(\tau)+\bar{\theta}\,b_{1}(\tau),$
$\displaystyle E(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad\Sigma(\tau,\bar{\theta})=E(\tau)+\bar{\theta}\,f_{3}(\tau),$
(41)

where we note that $(f_{1},f_{2},f_{3})$ are the fermionic secondary variables
and $b_{1}(\tau)$ is the bosonic secondary variable because of the fermionic
$({\bar{\theta}}^{2}=0)$ nature of the Grassmannian variable $\bar{\theta}$
which characterizes the anti-chiral super sub-manifold (along with the bosonic
evolution parameter $\tau$). It is elementary to note that the observation
$s_{b}\,B=0$ implies the following super expansion (in view of the fact that
$\partial_{\bar{\theta}}\,\leftrightarrow\,s_{b}$), namely;

$\displaystyle{\cal
B}^{(b)}\,(\tau,\bar{\theta})=B(\tau)+\bar{\theta}\,(0)\equiv
B(\tau)+\bar{\theta}\,(s_{b}\,B),$ (42)

where the superscript $(b)$ on the anti-chiral supervariable ${\cal
B}\,(\tau,\bar{\theta})$ denotes that the coefficient of $\bar{\theta}$ yields
the BRST symmetry transformation: $s_{b}\,B=0$ due to the trivial equality:
${\cal B}\,(\tau,\bar{\theta})=B\,(\tau)$ which emerges from the observation
that the Nakanishi-Lautrup auxiliary variable $B(\tau)$ is a BRST invariant
quantity [cf. Eq. (33)]. In other words, we have found out that the secondary
variable $f_{1}\,(\tau)=0$ in the super expansions (41).

At this stage, we find other non-trivial BRST invariant quantities for the
derivation of the secondary variables: $b_{1},\,f_{2},\,f_{3}$ of Eq. (41). We
observe that¶¶¶We have specifically taken here $s_{b}\,(C\,\dot{x})=0$ for our
purpose. However, one can take the general expression:
$s_{b}\,S=C\,\dot{S}\,(S=x,\,p_{x},\,t,\,p_{t})$ for the derivation of
$b_{1}(\tau)=C\,\dot{C}$.:
$s_{b}\,(C\,\dot{x})=0,\,s_{b}\,[\dot{\bar{B}}\,C-\bar{B}\,\dot{C}]=0,\,s_{b}\,[E\,\dot{C}+\dot{E}\,C]=0$.
The basic tenets of the ACSA to BRST formalism requires that the quantities in
the square brackets have to be independent of the Grassmannian variables
$\bar{\theta}$ when they are generalized onto a (1, 1)-dimensional super sub-
manifold, namely;

$\displaystyle
F(\tau,\bar{\theta})\,\dot{X}^{(h,\,ac)}\,(\tau,\bar{\theta})=C\,(\tau)\,\dot{x}(\tau)$
$\displaystyle{\dot{\bar{\cal
B}}}(\tau,\bar{\theta})\,F(\tau,\bar{\theta})-{\bar{\cal
B}}(\tau,\bar{\theta})\,\dot{F}(\tau,\bar{\theta})=\dot{\bar{B}}(\tau)\,C(\tau)-{\bar{B}}(\tau)\,\dot{C}(\tau)$
$\displaystyle{\Sigma}(\tau,\bar{\theta})\,\dot{F}(\tau,\bar{\theta})+\dot{\Sigma}(\tau,\bar{\theta})\,{F}(\tau,\bar{\theta})=E\,(\tau)\,\dot{C}(\tau)+\dot{E}(\tau)\,C(\tau),$
(43)

where ${X}^{(h,\,ac)}$ is the anti-chiral limit of the full expansion of
${X}^{(h)}\,(\tau,\theta,\bar{\theta})$ obtained after the application of HC
[cf. Eq. (28)], namely;

$\displaystyle{X}^{(h)}\,(\tau,\theta,\bar{\theta})=x(\tau)+\theta\,(\bar{C}\,\dot{x})+\bar{\theta}\,(C\,\dot{x})+\theta\,\bar{\theta}\,[i\,B\,\dot{x}-\bar{C}\,\dot{C}\,\dot{x}-\bar{C}\,C\,\ddot{x}],$
(44)

which has been obtained [cf. Eq. (28)] in Sec. 3 using the theoretical
strength of MBTSA. In other words, from the top entry of Eq. (43), we have the
following restrictions:

$\displaystyle
F(\tau,\bar{\theta})\,\dot{X}^{(h,\,ac)}\,(\tau,\bar{\theta})=C\,(\tau)\,\dot{x}(\tau)$
$\displaystyle\Longrightarrow\big{[}C\,(\tau)+\bar{\theta}\,b_{1}\,(\tau)\big{]}\,\big{[}\dot{x}+\bar{\theta}\,(\dot{C}\,\dot{x}+C\,\ddot{x})\big{]}=C\,(\tau)\,\dot{x}(\tau).$
(45)

From the above relationship we obtain $b_{1}(\tau)=C\,\dot{C}$. Thus, we have
the following

$\displaystyle
F^{(b)}(\tau,\bar{\theta})=C\,(\tau)+\bar{\theta}\,(C\,\dot{C})\equiv
C\,(\tau)+\bar{\theta}\,(s_{b}\,C),$ (46)

where the superscript $(b)$ on the l.h.s. of the supervariable denotes that
the coefficient of $\bar{\theta}$ is nothing but the BRST symmetry
transformation $s_{b}\,C$. We have to use the above super expansion in the
second entry from the top in (43) to obtain the following:

$\displaystyle{\dot{\bar{\cal
B}}}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta})-\bar{\cal
B}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})=\dot{\bar{B}}(\tau)\,C(\tau)-{\bar{B}}(\tau)\,{\dot{C}}(\tau).$
(47)

In other words, we have the following equality

$\displaystyle[\dot{\bar{B}}+\bar{\theta}\,{\dot{f}}_{2}(\tau)]\,[C(\tau)+\bar{\theta}\,(C\,\dot{C})]-[\bar{B}(\tau)+\bar{\theta}\,f_{2}(\tau)]\,[\dot{C}(\tau)+\bar{\theta}\,(C\,\ddot{C})]$
$\displaystyle=\dot{\bar{B}}(\tau)\,C(\tau)-\bar{B}(\tau)\,\dot{C}(\tau),$
(48)

which yields the following condition on the secondary variable $f_{2}$,
namely;

$\displaystyle{\dot{f}}_{2}\,C-f_{2}\,\dot{C}-\dot{\bar{B}}\,\dot{C}\,C+\bar{B}\,\ddot{C}\,C=0.$
(49)

It is straightforward to note that $f_{2}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}$
satisfies the above condition in a precise manner. We point out that the last
entry (from the top) of Eq. (43) can be re-written, in view of our the super
expansion in Eq. (46), as follows:

$\displaystyle\Sigma(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})+\dot{\Sigma}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta})=E(\tau)\,\dot{C}(\tau)+\dot{E}(\tau)\,C(\tau).$
(50)

The substitutions of expansions from (41) and (46) lead to the following
condition on the secondary variable $f_{3}(\tau)$ [present in the expansion of
$\Sigma(\tau,\bar{\theta})$], namely;

$\displaystyle
f_{3}\,\dot{C}+{\dot{f}}_{3}\,C-E\,\ddot{C}\,C-\dot{E}\,\dot{C}\,C=0,$ (51)

which is satisfied by the choice $f_{3}=E\,\dot{C}+\dot{E}\,C$. Hence, we have
the following super expansions [with the BRST symmetry transformations (33) as
input], namely;

$\displaystyle{\bar{\cal B}}^{(b)}(\tau,\bar{\theta})$ $\displaystyle=$
$\displaystyle\bar{B}(\tau)+\bar{\theta}\,(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})\equiv\bar{B}(\tau)+\bar{\theta}\,(s_{b}\,\bar{B}),$
$\displaystyle\Sigma^{(b)}(\tau,\bar{\theta})$ $\displaystyle=$ $\displaystyle
E(\tau)+\bar{\theta}\,(E\,\dot{C}+\dot{E}\,C)\equiv
E(\tau)+\bar{\theta}\,(s_{b}\,E),$ (52)

where the coefficients of $\bar{\theta}$ (in view of
$\partial_{\bar{\theta}}\leftrightarrow s_{b}$) are the BRST symmetry
transformations (33). For the convenience of the readers, we have performed
the explicit computations of $f_{3}=E\,\dot{C}+\dot{E}\,C$ and
$f_{2}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}$ in our Appendix B. It is clear that
we have already computed the BRST transformations
$s_{b}\,B=0,\,s_{b}\,C=C\,\dot{C},\,s_{b}\,\bar{B}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C},\,s_{b}\,E=E\,\dot{C}+\dot{E}\,C$
by exploiting the virtues of ACSA in Eqs. (42), (46) and (52).

We concentrate now on the derivation of the anti-BRST symmetry transformations
(32) by exploiting the theoretical strength of ACSA to BRST formalism. It is
obvious that, in Sec. 3, we have already computed
$s_{ab}\,S=\bar{C}\,\dot{S}\;(S=x,p_{x},t,p_{t})$ and $s_{ab}\,C=i\,\bar{B}$
by exploiting MBTSA to BRST formalism. Our objective in the present part of
our section is to derive:
$s_{ab}\,\bar{B}=0,\,s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}},\,s_{ab}\,B=\dot{B}\,\bar{C}-B\,\dot{\bar{C}},\,s_{ab}\,E=E\,\dot{\bar{C}}+\dot{E}\,\bar{C}$
by exploiting ACSA to BRST formalism. In this context, first of all, we
generalize the ordinary variables onto a $(1,1)$-dimensional chiral super sub-
manifold as

$\displaystyle\bar{B}(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad\bar{\cal
B}(\tau,\theta)=\bar{B}(\tau)+\theta\,{\bar{f}}_{1}(\tau),$ $\displaystyle
B(\tau)\qquad$ $\displaystyle\longrightarrow$ $\displaystyle\qquad{\cal
B}(\tau,\theta)=B(\tau)+\theta\,{\bar{f}}_{2}(\tau),$
$\displaystyle\bar{C}(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad{\bar{F}}(\tau,\theta)=\bar{C}(\tau)+\theta\,{\bar{b}}_{1}(\tau),$
$\displaystyle E(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad\Sigma(\tau,\theta)=E(\tau)+\theta\,{\bar{f}}_{3}(\tau),$
(53)

where $({\bar{f}}_{1},\,{\bar{f}}_{2},{\bar{f}}_{3})$ are the fermionic
secondary variables, ${\bar{b}}_{1}(\tau)$ is a bosonic secondary variable and
the above $(1,1)$-dimensional chiral super sub-manifold is parameterized by
$(\tau,\theta)$. It is straightforward to note that $s_{ab}\,\bar{B}=0$
implies that: ${\bar{\cal B}}(\tau,\theta)=B(\tau)$ and, as a consequence, we
have ${\bar{f}}_{1}(\tau)=0$ which leads to

$\displaystyle{\bar{\cal
B}}^{(ab)}(\tau,\theta)=\bar{B}(\tau)+\theta\,(0)\equiv\bar{B}(\tau)+\theta\,(s_{ab}\,\bar{B}),$
(54)

where the superscript $(ab)$ on the chiral supervariable denotes that we have
obtained $s_{ab}\,\bar{B}=0$ as the coefficient of $\theta$. The other useful
and interesting anti-BRST invariant quantities of our interest [cf. Eq. (32)]
are:

$\displaystyle s_{ab}\,[\dot{B}\,\bar{C}-B\,\dot{\bar{C}}]=0,\qquad
s_{ab}\,[E\,\dot{\bar{C}}+\dot{E}\,\bar{C}]=0,\qquad
s_{ab}\,[\bar{C}\,\dot{x}]=0.$ (55)

The quantities in the square brackets can be generalized onto the
$(1,1)$-dimensional chiral super sub-manifold. Following the fundamental
requirement(s) of ACSA to BRST formalism, these quantities must be independent
of the Grassmannian variable $\theta$. In other words, we have the following
restrictions on the chiral supervariables

$\displaystyle\dot{\cal B}(\tau,\theta)\,\bar{F}(\tau,\theta)-{\cal
B}(\tau,\theta)\,\dot{\bar{F}}(\tau,\theta)$ $\displaystyle=$
$\displaystyle\dot{B}(\tau)\,\bar{C}(\tau)-B(\tau)\,\dot{\bar{C}}(\tau),$
$\displaystyle\Sigma(\tau,\theta)\,\dot{\bar{F}}(\tau,\theta)+\dot{\Sigma}(\tau,\theta)\,\bar{F}(\tau,\theta)$
$\displaystyle=$ $\displaystyle
E(\tau)\,\dot{\bar{C}}(\tau)+\dot{E}(\tau)\,\bar{C}(\tau),$
$\displaystyle\bar{F}(\tau,\theta)\,{\dot{X}}^{(h,c)}(\tau,\theta)$
$\displaystyle=$ $\displaystyle\bar{C}(\tau)\,\dot{x}(\tau),$ (56)

where $X^{(h,c)}(\tau,\theta)$ is the chiral limit of the super expansion in
Eq. (44). In other words, we have the following explicit expression for the
supervariable $X^{(h,c)}(\tau,\theta)$, namely;

$\displaystyle X^{(h,c)}(\tau,\theta)=x(\tau)+\theta\,(\bar{C}\,\dot{x}).$
(57)

Taking the expansions from (53) and (57), we find that the last entry of Eq.
(56) yields: ${\bar{b}}_{1}(\tau)=\bar{C}\,\dot{\bar{C}}$. Hence, we have
obtained the following super expansion

$\displaystyle{\bar{F}}^{(ab)}(\tau,\theta)=\bar{C}(\tau)+\theta\,(\bar{C}\,\dot{\bar{C}})\equiv\bar{C}(\tau)+\theta\,(s_{ab}\,\bar{C}),$
(58)

where the superscript $(ab)$ on the chiral supervariable on the l.h.s. denotes
that it has been derived after the application of the anti-BRST invariant
restriction in (56). The coefficient $\theta$ is nothing but the anti-BRST
symmetry transformation: $s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}}$. This
equation also shows that $\partial_{\theta}\leftrightarrow s_{ab}$ and it
leads to the anti-BRST symmetry for $\bar{C}$.

We utilize now the two top entries of (56) where we use the explicit expansion
for ${\bar{F}}^{(ab)}(\tau,\theta)$ of (58) in the following restrictions on
the supervariables, namely;

$\displaystyle\dot{\cal B}(\tau,\theta)\,{\bar{F}}^{(ab)}(\tau,\theta)-{\cal
B}(\tau,\theta)\,\dot{\bar{F}}^{(ab)}(\tau,\theta)$ $\displaystyle=$
$\displaystyle{\dot{B}}(\tau)\,\bar{C}(\tau)-B(\tau)\,\dot{\bar{C}}(\tau),$
$\displaystyle\Sigma(\tau,\theta)\,\dot{\bar{F}}^{(ab)}(\tau,\theta)+\dot{\Sigma}(\tau,\theta)\,{\bar{F}}^{(ab)}(\tau,\theta)$
$\displaystyle=$ $\displaystyle
E(\tau)\,\dot{\bar{C}}(\tau)+\dot{E}(\tau)\,\bar{C}(\tau).$ (59)

The substitutions from (53) and (58) lead to:

$\displaystyle\dot{\bar{f}}_{2}\,\bar{C}-\bar{f}_{2}\,\dot{\bar{C}}-\dot{B}\,\dot{\bar{C}}\,\bar{C}+B\,\ddot{\bar{C}}\,\bar{C}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\bar{f}}_{3}\,\dot{\bar{C}}+\dot{\bar{f}}_{3}\,{\bar{C}}-E\,\ddot{\bar{C}}\,\bar{C}-E\,\dot{\bar{C}}\,\bar{C}$
$\displaystyle=$ $\displaystyle 0.$ (60)

It is straightforward, following the theoretical tricks of Appendix B, to find
out the solutions for the secondary variables ${\bar{f}}_{2}(\tau)$ and
${\bar{f}}_{3}(\tau)$ which are as follows:

$\displaystyle{\bar{f}}_{2}(\tau)=\dot{B}\,\bar{C}-B\,\dot{\bar{C}},\qquad\qquad{\bar{f}}_{3}=E\,\dot{\bar{C}}+\dot{E}\,\bar{C}.$
(61)

Substitutions of these secondary variables into the super expansions (53)
leads to the determination of the anti-BRST symmetry transformations for the
variables $B(\tau)$ and $E(\tau)$ as the coefficients of $\theta$ in the
following

$\displaystyle\Sigma^{(ab)}\,(\tau,\theta)=E(\tau)+\theta\,(E\,\dot{\bar{C}}+\dot{E}\,\bar{C})\,\equiv\,E(\tau)+\theta\,[s_{ab}\,E(\tau)],$
$\displaystyle{\cal
B}^{(ab)}\,(\tau,\theta)=B(\tau)+\theta\,(\dot{B}\,\bar{C}-B\dot{\bar{C}})\,\equiv\,B(\tau)+\theta\,[s_{ab}\,B(\tau)],$
(62)

where the superscript $(ab)$ on the chiral supervariable denotes that these
supervariables have been obtained after the applications of the anti-BRST
invariant restrictions (56). Moreover, the above observation establishes that:
$s_{ab}\Leftrightarrow\partial_{\theta}$ which implies that the nilpotency
$(s_{ab}^{2}=0,\,\partial_{\theta}^{2}=0)$ properties of $s_{ab}$ and
$\partial_{\theta}$ are connected with each-other. Thus, we have obtained all
the anti-BRST symmetry transformations (besides the phase variables) in our
Eqs. (54), (58) and (62). This completes our discussion on the derivation of
the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry
transformations (32) and (33) within the ambit of ACSA to BRST formalism.

##  6 Symmetry Invariance of the Lagrangians: ACSA

In this section, we establish the equivalence of the coupled Lagrangian
$L_{B}$ and $L_{\bar{B}}$ as far as the (anti-)BRST symmetry invariance
(within the purview of ACSA to BRST formalism) is concerned. We accomplish
this objective by generalizing the ordinary Lagrangians to their counterpart
super Lagrangians as

$\displaystyle L_{\bar{B}}\rightarrow{\tilde{L}}_{\bar{B}}^{(c)}(\tau,\theta)$
$\displaystyle=$ $\displaystyle{\tilde{L}}_{f}^{(c)}(\tau,\theta)-{\bar{\cal
B}}^{(ab)}(\tau,\theta)\Big{[}{\Sigma}^{(ab)}(\tau,\theta)\,{\dot{\Sigma}}^{(ab)}(\tau,\theta)-i\,\big{\\{}2\,{{\bar{F}}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)$
(63) $\displaystyle+$
$\displaystyle{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,{F}^{(ab)}(\tau,\theta)\big{\\}}\Big{]}+\frac{1}{2}\,{\bar{\cal
B}}^{(ab)}(\tau,\theta)\,{\bar{\cal B}}^{(ab)}(\tau,\theta)$ $\displaystyle-$
$\displaystyle
i\,{\Sigma}^{(ab)}(\tau,\theta)\,{\Sigma}^{(ab)}(\tau,\theta)\,{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)$
$\displaystyle-$ $\displaystyle
i\,{\Sigma}^{(ab)}(\tau,\theta)\,{\dot{\Sigma}}^{(ab)}(\tau,\theta)\,{{\bar{F}}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)$
$\displaystyle-$
$\displaystyle{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,{\bar{F}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta),$
$\displaystyle L_{B}\rightarrow{\tilde{L}}_{B}^{(ac)}(\tau,\bar{\theta})$
$\displaystyle=$ $\displaystyle{\tilde{L}}_{f}^{(ac)}(\tau,\bar{\theta})+{\cal
B}^{(b)}(\tau,\bar{\theta})\Big{[}{\Sigma}^{(b)}(\tau,\bar{\theta})\,{\dot{\Sigma}}^{(b)}(\tau,\bar{\theta})-i\,\big{\\{}2\,{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta})$
(64) $\displaystyle+$
$\displaystyle{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})\big{\\}}\Big{]}+\frac{1}{2}\,{\cal
B}^{(b)}(\tau,\bar{\theta})\,{\cal B}^{(b)}(\tau,\bar{\theta})$
$\displaystyle-$ $\displaystyle
i\,{\Sigma}^{(b)}(\tau,\bar{\theta})\,{\Sigma}^{(b)}(\tau,\bar{\theta})\,{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})$
$\displaystyle-$ $\displaystyle
i\,{\Sigma}^{(b)}(\tau,\bar{\theta})\,{\dot{\Sigma}}^{(b)}(\tau,\bar{\theta})\,{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{F}^{(b)}(\tau,\bar{\theta})$
$\displaystyle-$
$\displaystyle{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta}),$

where ${\tilde{L}}_{f}^{(c)}$ and ${\tilde{L}}_{f}^{(ac)}$ are the
generalizations of the first-order Lagrangian $(L_{f})$ to its counterpart
chiral and anti-chiral super Lagrangians as

$\displaystyle{\tilde{L}}_{f}^{(c)}(\tau,\theta)$ $\displaystyle=$
$\displaystyle
P_{x}^{(h,c)}(\tau,\theta)\,{\dot{X}}^{(h,c)}(\tau,\theta)+P_{t}^{(h,c)}(\tau,\theta)\,{\dot{T}}^{(h,c)}(\tau,\theta)$
$\displaystyle-$
$\displaystyle\frac{\Sigma^{(ab)}(\tau,\theta)}{2}\,\Big{[}P_{x}^{(h,c)}(\tau,\theta)\,P_{x}^{(h,c)}(\tau,\theta)+2\,m\,P_{t}^{(h,c)}(\tau,\theta)\Big{]},$
$\displaystyle{\tilde{L}}_{f}^{(ac)}(\tau,\bar{\theta})$ $\displaystyle=$
$\displaystyle
P_{x}^{(h,ac)}(\tau,\bar{\theta})\,{\dot{X}}^{(h,ac)}(\tau,\bar{\theta})+P_{t}^{(h,ac)}(\tau,\bar{\theta})\,{\dot{T}}^{(h,ac)}(\tau,\bar{\theta})$
(65) $\displaystyle-$
$\displaystyle\frac{\Sigma^{(b)}(\tau,\bar{\theta})}{2}\,\Big{[}P_{x}^{(h,ac)}(\tau,\bar{\theta})\,P_{x}^{(h,ac)}(\tau,\bar{\theta})+2\,m\,P_{t}^{(h,ac)}(\tau,\bar{\theta})\Big{]},$

where the superscripts $(c)$ and $(ac)$ denote the chiral and anti-chiral
generalizations and the rest of the supervariables with superscripts $(b)$ and
$(ab)$ have already been explained earlier in Sec. 5. The supervariables with
superscripts $(h,c)$ and $(h,ac)$ are the chiral and anti-chiral limits of the
super phase variables $(X^{(h)},P_{x}^{(h)},T^{(h)},P_{t}^{(h)})$ that have
been obtained after the application of HC. Thus, these are the counterparts of
the ordinary phase variables $(x,\,p_{x},\,t,\,p_{t})$ and they have been
explained in Sec. 3. In the above equation (65), the super phase variables
with superscript $(h,c)$ and $(h,ac)$ can be expressed in terms of the generic
supervariable as follows

$\displaystyle S(\tau)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad{\cal
S}^{(h,c)}(\tau,\theta)=S(\tau)+\theta\,[\bar{C}\,\dot{S}(\tau)],$
$\displaystyle S(\tau)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad{\cal
S}^{(h,ac)}(\tau,\bar{\theta})=S(\tau)+\bar{\theta}\,[C\,\dot{S}(\tau)],$ (66)

where the (anti-)chiral supervariables on the l.h.s. stand for the super phase
variables $(X,P_{x},T,P_{t})$ with the proper chiral and anti-chiral
superspace coordinates $(\tau,\theta)$ and $(\tau,\bar{\theta})$ as their
arguments. The set of supervariables $(X,P_{x},T,P_{t})$ are the
generalizations of the ordinary phase variables $(x,\,p_{x},\,t,\,p_{t})$ to
their (anti-)chiral counterparts onto the $(1,1)$-dimensional (anti-)chiral
super submanifolds of the general $(1,2)$-dimensional supermanifold. It is
straightforward to check that the following is true, namely;

$\displaystyle\frac{\partial}{\partial\,\theta}\,{\tilde{L}}_{f}^{(c)}(\tau,\theta)=\frac{d}{d\,\tau}[\bar{C}\,L_{f}]\quad$
$\displaystyle\Longleftrightarrow$ $\displaystyle\quad
s_{ab}\,L_{f}=\frac{d}{d\,\tau}\,[\bar{C}\,L_{f}],$
$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,{\tilde{L}}_{f}^{(ac)}(\tau,\bar{\theta})=\frac{d}{d\,\tau}[C\,L_{f}]\quad$
$\displaystyle\Longleftrightarrow$ $\displaystyle\quad
s_{b}\,L_{f}=\frac{d}{d\,\tau}\,[C\,L_{f}].$ (67)

In other words, we have captured the (anti-)BRST invariance of the first-order
Lagrangian $(L_{f})$ in view of the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$.
Since in the ordinary space, the (anti-) BRST symmetry transformations acting
on $L_{f}$ produce the total derivatives [cf. Eq. (67)], the action integral
$S=\int^{+\infty}_{-\infty}\,d\,\tau\,L_{f}$ remains invariant under the
transformations $s_{(a)b}$.

At this stage, we focus on the (anti-)BRST invariance of the coupled
Lagrangians $L_{B}$ and $L_{\bar{B}}$ [cf. Eqs. (38), (37)]. We can express
these invariances within the ambit of ACSA (in view of the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$),
namely;

$\displaystyle\frac{\partial}{\partial\,\theta}\,{\tilde{L}}_{\bar{B}}^{(c)}(\tau,\theta)=\frac{d}{d\,\tau}\,\Big{[}\bar{C}\,L_{f}-e\,\dot{e}\,\bar{B}\,\bar{C}-e^{2}\,\bar{B}\,\dot{\bar{C}}+{\bar{B}}^{2}\,\bar{C}-i\,\bar{B}\,\dot{\bar{C}}\,\bar{C}\,C\Big{]}=s_{ab}\,L_{\bar{B}},$
(68)
$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,{\tilde{L}}_{B}^{(ac)}(\tau,\bar{\theta})=\frac{d}{d\,\tau}\,\Big{[}C\,L_{f}+e\,\dot{e}\,B\,C+e^{2}\,B\,\dot{C}+B^{2}\,C-i\,B\,\bar{C}\,\dot{C}\,C\Big{]}=s_{b}\,L_{B},$
(69)

where the super Lagrangians ${\tilde{L}}_{\bar{B}}^{(c)}(\tau,\theta)$ and
${\tilde{L}}_{B}^{(ac)}(\tau,\bar{\theta})$ have been already quoted in Eqs.
(63) and (64). It is interesting to note that the r.h.s. of (68) and (69) are
same as we have found in the ordinary space [cf. Eq. (38), (37)]. To prove the
equivalence of the Lagrangians $L_{B}$ and $L_{\bar{B}}$ w.r.t. the
(anti-)BRST symmetry transformations [cf. Eqs. (32), (33)] within the purview
of ACSA, we generalize the ordinary Lagrangians $L_{B}$ and $L_{\bar{B}}$ as
follows

$\displaystyle L_{B}\rightarrow{\tilde{L}}_{B}^{(c)}(\tau,\theta)$
$\displaystyle=$ $\displaystyle{\tilde{L}}_{f}^{(c)}(\tau,\theta)+{\cal
B}^{(ab)}(\tau,\theta)\Big{[}\Sigma^{(ab)}(\tau,\theta)\,{\dot{\Sigma}}^{(ab)}(\tau,\theta)-i\,\\{2\,{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta)$
(70) $\displaystyle+$
$\displaystyle{\bar{F}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)\\}\Big{]}+\frac{1}{2}\,{\cal
B}^{(ab)}(\tau,\theta)\,{\cal B}^{(ab)}(\tau,\theta)$ $\displaystyle-$
$\displaystyle
i\,\Sigma^{(ab)}(\tau,\theta)\,\Sigma^{(ab)}(\tau,\theta)\,{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)$
$\displaystyle-$ $\displaystyle
i\,\Sigma^{(ab)}(\tau,\theta)\,{\dot{\Sigma}}^{(ab)}(\tau,\theta)\,{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,{F}^{(ab)}(\tau,\theta)$
$\displaystyle-$
$\displaystyle{\dot{\bar{F}}}^{(ab)}(\tau,\theta)\,{\bar{F}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta),$
$\displaystyle
L_{\bar{B}}\rightarrow{\tilde{L}}_{\bar{B}}^{(ac)}(\tau,\bar{\theta})$
$\displaystyle=$
$\displaystyle{\tilde{L}}_{f}^{(ac)}(\tau,\bar{\theta})-{\bar{\cal
B}}^{(b)}(\tau,\bar{\theta})\Big{[}\Sigma^{(b)}(\tau,\bar{\theta})\,{\dot{\Sigma}}^{(b)}(\tau,\bar{\theta})-i\,\big{\\{}2\,{{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})$
(71) $\displaystyle+$
$\displaystyle{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{F}^{(b)}(\tau,\bar{\theta})\big{\\}}\Big{]}+\frac{1}{2}\,{\bar{\cal
B}}^{(b)}(\tau,\bar{\theta})\,{\bar{\cal B}}^{(b)}(\tau,\bar{\theta})$
$\displaystyle-$ $\displaystyle
i\,\Sigma^{(b)}(\tau,\bar{\theta})\,\Sigma^{(b)}(\tau,\bar{\theta})\,{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})$
$\displaystyle-$ $\displaystyle
i\,\Sigma^{(b)}(\tau,\bar{\theta})\,{\dot{\Sigma}}^{(b)}(\tau,\bar{\theta})\,{{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})$
$\displaystyle-$
$\displaystyle{\dot{\bar{F}}}^{(b)}(\tau,\bar{\theta})\,{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta}),$

where all the notations and symbols have already been explained earlier. To
find out the result of the operations of $s_{b}$ on $L_{\bar{B}}$ and $s_{ab}$
on $L_{B}$, we observe the following (in view of the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$),
namely;

$\displaystyle\frac{\partial}{\partial\,\theta}\,{\tilde{L}}_{B}^{(c)}(\tau,\theta)$
$\displaystyle=$
$\displaystyle\frac{d}{d\,\tau}\,\Big{[}\bar{C}\,L_{f}+E\,\dot{E}\,(i\,\dot{\bar{C}}\,\bar{C}\,C+B\,\bar{C})+E^{2}\,(i\,\dot{\bar{C}}\,\bar{C}\,\dot{C}+B\,\dot{\bar{C}})$
(72) $\displaystyle+$
$\displaystyle{B}^{2}\,\bar{C}+i\,(2\,B-\bar{B})\,\dot{\bar{C}}\,\bar{C}\,C\Big{]}$
$\displaystyle+$
$\displaystyle\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,(2\,i\,\dot{\bar{C}}\,\bar{C}\,\dot{C}-E\,\dot{E}\,\dot{\bar{C}}-2\,B\,\dot{\bar{C}}+i\,\ddot{\bar{C}}\,\bar{C}\,C)$
$\displaystyle-$
$\displaystyle\frac{d}{d\,\tau}\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,(B\,\bar{C}+E^{2}\,\dot{\bar{C}})\equiv
s_{ab}\,L_{B},$
$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,{\tilde{L}}_{\bar{B}}^{(ac)}(\tau,\bar{\theta})$
$\displaystyle=$
$\displaystyle\frac{d}{d\,\tau}\,\Big{[}C\,L_{f}-E\,\dot{E}\,(i\,\bar{C}\,\dot{C}\,C+\bar{B}\,C)-E^{2}\,(i\,\dot{\bar{C}}\,\dot{C}\,C+\bar{B}\,\dot{C})$
(73) $\displaystyle+$
$\displaystyle{\bar{B}}^{2}\,C+i\,(2\,\bar{B}-B)\,\bar{C}\,\dot{C}\,C\Big{]}$
$\displaystyle+$
$\displaystyle\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,\big{[}i\,\bar{C}\,\ddot{C}\,C+2\,i\,\dot{\bar{C}}\,\dot{C}\,C-2\,\bar{B}\,\dot{C}+E\,\dot{E}\,\dot{C}\big{]}$
$\displaystyle+$
$\displaystyle\frac{d}{d\,\tau}\big{[}i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)+B+\bar{B}\big{]}\,(E^{2}\,\dot{C}-\bar{B}\,C)\equiv
s_{b}\,L_{\bar{B}},$

within the framework of ACSA. It is self-evident, from the r.h.s. of (72) and
(73), that we have the BRST invariance of $L_{\bar{B}}$ and anti-BRST
invariance of $L_{B}$ if and only if our whole theory is considered on the
submanifold of the Hilbert space of quantum variables where the CF-type
restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is
satisfied.

We end this section with the following crucial remarks. First of all, we have
captured the BRST and anti-BRST invariance of $L_{B}$ and $L_{\bar{B}}$,
respectively, in the terminology of ACSA on the (anti-)chiral super
submanifolds [cf. Eqs. (68), (69)]. Second, we have also demonstrated the
anti-BRST invariance of $L_{B}$ and BRST invariance of $L_{\bar{B}}$ in the
superspace formalism [cf. Eqs. (72), (73)] where the theoretical techniques of
ACSA have played very important roles. Third, we have also expressed the
(anti-)BRST invariance of the first-order Lagrangian $L_{f}$ in Eq. (67).
Finally, we have proven the equivalence  of $L_{B}$ and $L_{\bar{B}}$ within
the framework of ACSA in the Eqs. (68), (69), (72) and (73).

##  7 Nilpotency and Absolute Anticommutativity Properties of the (Anti-)BRST
Charges: ACSA

Our present section is divided into two subsections. In subsection 7.1, we
discuss the off-shell nilpotency and absolute anticommutativity of the
conserved (anti-)BRST charges in the ordinary space. Our subsection 7.2 deals
with the above properties within the realm of ACSA to BRST formalism. In other
words, we capture the off-shell nilpotency and absolute anticommutativity of
the conserved fermionic (anti-)BRST charges in the superspace by taking the
theoretical inputs from ACSA.

###  7.1 Nilpotency and Anticommutativity: Ordinary Space

The perfect symmetry invariance of $L_{\bar{B}}$ under the anti-BRST symmetry
transformations [cf. Eq. (38)] and $L_{B}$ under the BRST symmetry
transformations [cf. Eq. (37)] allow us to compute the Noether conserved
charges by using the standard techniques of Noether’s theorem (applied to the
action integrals corresponding to the Lagrangians $L_{\bar{B}}$ and $L_{B}$)
as

$\displaystyle Q_{\bar{B}}$ $\displaystyle=$
$\displaystyle\frac{\bar{C}\,E}{2}\,(p_{x}^{2}+2\,m\,p_{t})+{\bar{B}}^{2}\,\bar{C}-i\,\bar{B}\,\dot{\bar{C}}\,\bar{C}\,C-\bar{B}\,E\,\dot{E}\,\bar{C}-\bar{B}\,E^{2}\,\dot{\bar{C}},$
$\displaystyle Q_{B}$ $\displaystyle=$
$\displaystyle\frac{C\,E}{2}\,(p_{x}^{2}+2\,m\,p_{t})+B\,E^{2}\,\dot{C}-i\,B\,\bar{C}\,\dot{C}\,C+B\,E\,\dot{E}\,C+B^{2}\,C,$
(74)

where conserved $({\dot{Q}}_{\bar{B}}=0,\,{\dot{Q}}_{B}=0)$ (anti-)BRST
charges are denoted by $Q_{(\bar{B})B}$. The conservation law
$({\dot{Q}}_{\bar{B}}=0,\,{\dot{Q}}_{B}=0)$ can be proven by using the EL-EOMs
derived from the coupled Lagrangians $L_{\bar{B}}$ and $L_{B}$. For readers’
convenience, we prove the conservation $({\dot{Q}}_{B}=0)$ of the BRST charge
by using the EL-EOMs derived from $L_{B}$ in our Appendix C.

First of all, we concentrate on the proof of the off-shell nilpotency
properties of the (anti-)BRST charges $Q_{(\bar{B})B}$. In this context, we
note that the following EL-EOMs w.r.t. the variable $E$ from $L_{\bar{B}}$ and
$L_{B}$, respectively, yield the following:

$\displaystyle\dot{\bar{B}}\,E-i\,E\,\dot{\bar{C}}\,\dot{C}+i\,E\,\bar{C}\,\ddot{C}-\frac{1}{2}\,(p_{x}^{2}+2\,m\,p_{t})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\dot{B}\,E+i\,E\,\dot{\bar{C}}\,\dot{C}-i\,E\,\ddot{\bar{C}}\,C+\frac{1}{2}\,(p_{x}^{2}+2\,m\,p_{t})$
$\displaystyle=$ $\displaystyle 0.$ (75)

The above equations can be used to recast $Q_{\bar{B}}$ and $Q_{B}$ as
follows:

$\displaystyle Q_{\bar{B}}^{(1)}$ $\displaystyle=$ $\displaystyle
E^{2}\,(\dot{\bar{B}}\,{\bar{C}}-{\bar{B}}\,\dot{\bar{C}}+i\,\dot{\bar{C}}\,\bar{C}\,\dot{C})-i\,\bar{B}\,\dot{\bar{C}}\,{\bar{C}}\,C-\bar{B}\,E\,\dot{E}\,\bar{C}+\bar{B}^{2}\,\bar{C},$
$\displaystyle Q_{B}^{(1)}$ $\displaystyle=$ $\displaystyle
E^{2}\,(B\,\dot{C}-\dot{B}\,C-i\,\dot{\bar{C}}\,\dot{C}\,C)-i\,B\,\bar{C}\,\dot{C}\,C+B\,E\,\dot{E}\,C+B^{2}\,C,$
(76)

Using the following EL-EOMs w.r.t. the variables $C$ and $\bar{B}$,
respectively, from $L_{\bar{B}}$, namely;

$\displaystyle
i\,\bar{B}\,\dot{\bar{C}}+2\,i\,\dot{\bar{B}}\,\bar{C}-3\,i\,E\,\dot{E}\,\dot{\bar{C}}-i\,E^{2}\,\ddot{\bar{C}}-i\,{\dot{E}}^{2}\,\bar{C}-i\,E\,\ddot{E}\,\bar{C}+\ddot{\bar{C}}\,\bar{C}\,C+2\,\dot{\bar{C}}\,\bar{C}\,\dot{C}=0,$
$\displaystyle\bar{B}=E\,\dot{E}-i\,(2\,\bar{C}\,\dot{C}+\dot{\bar{C}}\,C),$
(77)

we obtain the following exact and interesting expression for the anti-BRST
charge:

$\displaystyle Q_{\bar{B}}^{(1)}\quad\longrightarrow\quad Q_{\bar{B}}^{(2)}$
$\displaystyle=$ $\displaystyle
E^{2}\,(\dot{\bar{B}}\,{\bar{C}}-{\bar{B}}\,\dot{\bar{C}}+i\,\dot{\bar{C}}\,\bar{C}\,\dot{C})+i\,E^{2}\,\ddot{\bar{C}}\,\bar{C}\,C+2\,i\,E\,\dot{E}\,\dot{\bar{C}}\,\bar{C}\,C$
(78) $\displaystyle\equiv$ $\displaystyle
s_{ab}\,[i\,E^{2}\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)].$

At this juncture, we apply the basic principle behind the relationship between
the continuous symmetry transformations (e.g. $s_{ab}$) and its generator
$[Q_{\bar{B}}^{(2)}]$ which implies that:

$\displaystyle
s_{ab}\,Q_{\bar{B}}^{(2)}=-\,i\,\\{Q_{\bar{B}}^{(2)},Q_{\bar{B}}^{(2)}\\}=0\quad\Rightarrow\quad[Q_{\bar{B}}^{(2)}]^{2}=0\quad\Leftrightarrow\quad
s_{ab}^{2}=0.$ (79)

Thus, we observe that the off-shell nilpotency $([Q_{\bar{B}}^{(2)}]^{2}=0)$
of the anti-BRST charge $Q_{\bar{B}}^{(2)}$ and the anti-BRST symmetry
transformations $(s_{ab})$ are inter-related. Thus, we have proven the off-
shell nilpotency of the anti-BRST charge $Q_{\bar{B}}^{(2)}$. In exactly
similar fashion, we exploit the following EL-EOMs w.r.t. the variables
$\bar{C}$ and $B$ from the Lagrangian $L_{B}$

$\displaystyle
i\,B\,\dot{C}+2\,i\,\dot{B}\,C+3\,i\,E\,\dot{E}\,\dot{C}+i\,E^{2}\,\ddot{C}+i\,{\dot{E}}^{2}\,C+i\,E\,\ddot{E}\,C+\bar{C}\,\ddot{C}\,C+2\,\dot{\bar{C}}\,\dot{C}\,C=0,$
$\displaystyle B=-E\,\dot{E}+i\,(2\,\dot{\bar{C}}\,C+\bar{C}\,\dot{C}),$ (80)

to recast the BRST charge $Q_{B}^{(1)}$ into another interesting form (i.e.
$Q_{B}^{(2)}$) as

$\displaystyle Q_{B}^{(1)}\quad\rightarrow\quad Q_{B}^{(2)}$ $\displaystyle=$
$\displaystyle
E^{2}\,(B\,\dot{C}-i\,\dot{\bar{C}}\,\dot{C}\,C-\dot{B}\,C)-2\,i\,E\,\dot{E}\,\bar{C}\,\dot{C}\,C-i\,E^{2}\,\bar{C}\,\ddot{C}\,C$
(81) $\displaystyle\equiv$ $\displaystyle
s_{b}\,[i\,E^{2}\,(\dot{\bar{C}}\,C-\bar{C}\,\dot{C})]$

which turns out to be an exact quantity w.r.t. $s_{b}$. Thus, we find that we
have the following:

$\displaystyle
s_{b}\,Q_{B}^{(2)}=-\,i\,\\{Q_{B}^{(2)},Q_{B}^{(2)}\\}=0\quad\Rightarrow\quad[Q_{B}^{(2)}]^{2}=0\quad\Leftrightarrow\quad
s_{b}^{2}=0.$ (82)

In other words, we have proven the off-shell nilpotency
$([Q_{B}^{(2)}]^{2}=0)$ of the BRST charge $Q_{B}^{(2)}$. Once again, we find
that off-shell nilpotency $(s_{b}^{2}=0)$ of the BRST symmetry transformations
and the off-shell nilpotency $([Q_{B}^{(2)}]^{2}=0)$ are intertwined an
intimate manner.

We now focus on the proof of the absolute anticommutativity property of the
BRST charge with the anti-BRST charge and vice-versa. First of all, let us
focus on the BRST charge $Q_{B}^{(2)}$ [cf. Eq. (81)]. Using the CF-type
restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$, we can
easily check the following transformation:

$\displaystyle Q_{B}^{(2)}\quad\longrightarrow\quad Q_{B}^{(3)}$
$\displaystyle=$ $\displaystyle
E^{2}\,(\dot{\bar{B}}\,C-2\,i\,\dot{\bar{C}}\,\dot{C}\,C-\bar{B}\,\dot{C})-2\,i\,E\,\dot{E}\,\bar{C}\,\dot{C}\,C$
(83) $\displaystyle\equiv$ $\displaystyle s_{ab}\,[i\,E^{2}\,\dot{C}\,C].$

In other words we have been able to express the above BRST charge as an exact
form w.r.t. the anti-BRST symmetry transformations $(s_{ab})$. This is an
interesting observation because using the relationship between the continuous
symmetry transformations and their generators, we can obtain the following
from (83), namely;

$\displaystyle
s_{ab}\,Q_{B}^{(3)}=-i\,\\{Q_{B}^{(3)},Q_{\bar{B}}^{(3)}\\}=0\quad\Leftrightarrow\quad
s_{ab}^{2}=0.$ (84)

Thus we have been able to demonstrate that the absolute anticommutativity of
the BRST charge with the anti-BRST charge is connected with the off-shell
nilpotency $(s_{ab}^{2}=0)$ of the anti-BRST symmetry transformation
$(s_{ab})$. In exactly similar fashion, we can have a different form of the
anti-BRST charge $Q_{\bar{B}}^{(2)}$ [cf. Eq. (78)] by using the CF-type
restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$. In other
words, we have the following interesting transformation:

$\displaystyle Q_{\bar{B}}^{(3)}\quad\longrightarrow\quad Q_{\bar{B}}^{(3)}$
$\displaystyle=$ $\displaystyle
E^{2}\,(B\,\dot{\bar{C}}+2\,i\,\dot{\bar{C}}\,\bar{C}\,\dot{C}-\dot{B}\,\bar{C})+2\,i\,E\,\dot{E}\,\dot{\bar{C}}\,\bar{C}\,C$
(85) $\displaystyle\equiv$ $\displaystyle
s_{b}\,[i\,E^{2}\,\dot{\bar{C}}\,\bar{C}].$

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

$\displaystyle
s_{b}\,Q_{\bar{B}}^{(3)}=-\,i\,\\{Q_{\bar{B}}^{(3)},Q_{B}^{(3)}\\}=0\quad\Leftrightarrow\quad
s_{b}^{2}=0.$ (86)

In other words, we point out that the absolute anticommutativity of the anti-
BRST charge with the BRST charge is intimately connected with the off-shell
nilpotency $(s_{b}^{2}=0)$ of the BRST symmetry transformations $(s_{b})$.
This completes our discussions on the off-shell nilpotency and absolute
anticommutativity of the conserved (anti-)BRST charges in the ordinary space.
In a subtle manner, the observations in (83) and (85) prove the validity of
the CF-type restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$
on our theory.

###  7.2 Nilpotency and Anticommutativity: ACSA

The key observations of subsection $7.1$ can be translated into the superspace
by using the basic terminology of ACSA. Keeping in our mind the mappings
$\partial_{\bar{\theta}}\leftrightarrow
s_{b},\,\partial_{\theta}\leftrightarrow s_{ab}$, we note that the (anti-)BRST
charges $Q_{(\bar{B})B}$ [cf. Eqs. (78), (81)] can be expressed as:

$\displaystyle
Q_{\bar{B}}=\frac{\partial}{\partial\,\theta}\,\Big{[}i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\big{\\{}{\bar{F}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)-\dot{\bar{F}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta)\big{\\}}\Big{]}$
$\displaystyle=\int
d\,\theta\,\Big{[}i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\big{\\{}{\bar{F}}^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)-\dot{\bar{F}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta)\big{\\}}\Big{]}$
(87) $\displaystyle
Q_{B}=\frac{\partial}{\partial\,\bar{\theta}}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\big{\\{}\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{F}^{(b)}(\tau,\bar{\theta})-{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})\big{\\}}\Big{]}$
$\displaystyle=\int
d\,\bar{\theta}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\big{\\{}\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{F}^{(b)}(\tau,\bar{\theta})-{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\dot{F}}^{(b)}(\tau,\bar{\theta})\big{\\}}\Big{]}.$
(88)

It is straightforward to observe that we have the following:

$\displaystyle\frac{\partial}{\partial\,\theta}\,Q_{\bar{B}}=0\quad$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad
s_{ab}\,Q_{\bar{B}}=0\quad\Leftrightarrow\quad
Q_{\bar{B}}^{2}=0\quad\Leftrightarrow\quad\partial_{\theta}^{2}=0,$
$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,Q_{B}=0\quad$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad
s_{b}\,Q_{B}=0\quad\Leftrightarrow\quad
Q_{B}^{2}=0\quad\Leftrightarrow\quad\partial_{\bar{\theta}}^{2}=0.$ (89)

Thus, the off-shell nilpotency of the (anti-)BRST charges is connected with
the nilpotency $(\partial_{\theta}^{2}=0,\,\partial_{\bar{\theta}}^{2}=0)$ of
the translational generators $(\partial_{\theta},\partial_{\bar{\theta}})$
along the Grassmannian directions of the chiral and anti-chiral
$(1,1)$-dimensional super sub-manifolds. This observation is consistent with
our discussion of the nilpotency property in the ordinary space if we remember
the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$
[14-16].

As far as the absolute anticommutativity property is concerned, we note that
the expressions of the (anti-)BRST charges in (85) and (83) can be translated
into the superspace where we can exploit the theoretical tools of ACSA. To
accomplish this goal, we keep in our knowledge the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$
to recast the expressions (85) and (83) as:

$\displaystyle Q_{\bar{B}}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\bar{F}}^{(b)}(\tau,\bar{\theta})\Big{]}$
$\displaystyle\equiv$ $\displaystyle\int
d\,\bar{\theta}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\bar{F}}^{(b)}(\tau,\bar{\theta})\Big{]},$
$\displaystyle Q_{B}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\,\theta}\,\Big{[}-i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,{\dot{F}}^{(ab)}(\tau,\theta)\,{F}^{(ab)}(\tau,\theta)\Big{]}$
(90) $\displaystyle\equiv$ $\displaystyle\int
d\,\theta\,\Big{[}-i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\dot{F}^{(ab)}(\tau,\theta)\,{F}^{(ab)}(\tau,\theta)\Big{]}.$

It is now straightforward to check that the following are true, namely;

$\displaystyle\partial_{\bar{\theta}}\,Q_{\bar{B}}^{(3)}=0\quad$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad
s_{b}\,Q_{\bar{B}}^{(3)}=0\quad\Leftrightarrow\quad\\{Q_{\bar{B}}^{(3)},Q_{B}^{(3)}\\}=0\quad\Leftrightarrow\quad\partial_{\bar{\theta}}^{2}=0,$
$\displaystyle\partial_{\theta}\,Q_{B}^{(3)}=0\quad$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad
s_{ab}\,Q_{B}^{(3)}=0\quad\Leftrightarrow\quad\\{Q_{B}^{(3)},Q_{\bar{B}}^{(3)}\\}=0\quad\Leftrightarrow\quad\partial_{\theta}^{2}=0,$
(91)

which establishes the fact that the ACSA to BRST formalism distinguishes
between the two types of absolute anticommutativity properties. In other
words, we observe that the absolute anticommutativity of the BRST charge with
the anti-BRST charge is connected with the nilpotency
$(\partial_{\theta}^{2}=0)$ of the translational generator
$(\partial_{\theta})$ along the Grassmannian direction of the (1,
1)-dimensional chiral super sub-manifold. On the contrary, the absolute
anticommutativity of the anti-BRST charge with the BRST charge is connected
with the nilpotency $(\partial^{2}_{\bar{\theta}}=0)$ of the translational
generator $(\partial_{\bar{\theta}})$ along the Grassmannian direction of the
(1, 1)-dimensional anti-chiral super sub-manifold.

##  8 Conclusions

In our present endeavor, we have purposely taken a reparameterization
invariant NR and NSUSY system so that we could discuss theoretical aspects
that are different from our earlier works on the NSUSY relativistic scalar and
SUSY relativistic spinning particles [23, 24]. We have demonstrated, however,
in our present investigation that $(i)$ the CF-type restriction, and $(ii)$
the sum of gauge-fixing and Faddeev-Popov ghost terms are same for our present
NR and NSUSY system as have been shown by us for the relativistic particles
(in our earlier works [23, 24]). The above observations are interesting
results of our present investigation which establish the universality of the
(anti-)BRST invariant CF-type restriction for the 1D diffeomorphism invariant
(i.e. reparameterization) theories.

The CF-type restriction(s) are the hallmark of a quantum theory that is BRST-
quantized. In fact, for a D-dimensional diffeomorphism invariant theory, it
has been shown [22, 33] that the universal CF-type restrictions for a BRST-
quantized theory is:
$B_{\mu}+{\bar{B}}_{\mu}+i\,({\bar{C}}^{\rho}\,\partial_{\rho}\,C_{\mu}+{C}^{\rho}\,\partial_{\rho}\,\bar{C}_{\mu})=0$
where $\mu,\rho=0,1,2,...D-1$, $B_{\mu}$ and $\bar{B}_{\mu}$ are the
Nakanishi-Lautrup auxilary fields and the (anti-)ghost fields
$(\bar{C}_{\mu})\,C_{\mu}$ correspond to the D-dimensional diffeomorphism
parameter $\epsilon_{\mu}(x)$ in the infinitesimal transformation:
$x_{\mu}\rightarrow x_{\mu}^{\prime}=x_{\mu}-\epsilon_{\mu}(x)$. The
universality of the above CF-type restriction implies that, for our 1D
diffeomorphism (i.e. reparameterization) invariant theory, the CF-type
restriction is: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$. This is
what we have obtained from various theoretical tricks in our present endeavor.
The existence of the CF-type restriction is very fundamental to a BRST-
quantized theory as it is connected with the geometrical objects called gerbes
[7, 8]. Physically, the existence of the CF-type restriction leads to the
independent nature of the BRST and anti-BRST symmetries (and corresponding
conserved charges) at the quantum level (that are connected with a given
classical local symmetry).

Our present work (and earlier works [23, 24]) can be generalized to the cases
of (super)string and gravitational theories which are also diffeomorphism
invariant. In fact, in our earlier work on a bosonic string theory [34], we
have shown the existence of the CF-type restriction in the context of its BRST
quantization and it has turned out to be the 2D version of the universal CF-
type restriction for the D-dimensional diffeomorphism invariant theory. It is
gratifying to pinpoint the fact that we have derived the CF-type restrictions:
$B^{a}+\bar{B}^{a}+i\;\big{(}\bar{C}^{m}\;\partial_{m}\;C^{a}+C^{m}\;\partial_{m}\;\bar{C}^{a}\big{)}=0$
(with $a,m=0,1$) for a model of bosonic string theory [34]. This has happened
because the bosonic string theory has the 2D diffeomorphism invariance on the
2D world-sheet. We have applied the beautiful blend of MBTSA and ACSA to
derive all the (anti-)BRST symmetries as well as the 2D version of the CF-type
restriction in the case of a bosonic string theory of our interest [35]. In
our present investigation, we have utilized only two and/or one Grassmannian
variables because there are only two nilpotent symmetries in the theory. If a
theory is endowed with the nilpotent (anti-)BRST as well as (anti-)co-BRST
symmetries, then, we have to invoke four number of Grassmannian variables. We
are currently exploring such kinds of possibilities.

Appendix A: The CF-Type Restriction from $L_{B}\equiv L_{\bar{B}}$

In this Appendix, we provide the step-by-step derivation of the CF-type
restriction by requiring the equivalence of the coupled Lagrangian $L_{B}$ and
$L_{\bar{B}}$ [cf. Eq. (35)]. A close look at them demonstrates that if we
demand $L_{B}\equiv L_{\bar{B}}$, the terms that are common would cancel out.
For instance, we have cancellations of terms
$L_{f},\,-i\,E^{2}\,\dot{\bar{C}}\,\dot{C}$ and
$-\,\dot{\bar{C}}\,\bar{C}\,\dot{C}\,C$ that are present both in $L_{B}$ and
$L_{\bar{B}}$. Thus, we are left with the following equality:

$\frac{B^{2}}{2}+B\,[E\,\dot{E}-i\,(2\,\dot{\bar{C}}\,C+\bar{C}\,\dot{C})]-i\,E\,\dot{E}\,\dot{\bar{C}}\,C$
$\equiv\frac{\bar{B}^{2}}{2}+\bar{B}\,[E\,\dot{E}-i\,(2\,{\bar{C}}\,\dot{C}+\dot{\bar{C}}\,C)]-i\,E\,\dot{E}\,{\bar{C}}\,\dot{C}.$
$None$

From the above equation, it is evident that we have

$E\,\dot{E}\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)],$ $None$

on the l.h.s. when we bring all the terms from the r.h.s. to the l.h.s.. At
this stage, excluding (A.2), the left-over terms on the l.h.s. and the r.h.s.
are

$\frac{B^{2}}{2}-\frac{\bar{B}^{2}}{2}-2\,i\,B\,\dot{\bar{C}}\,C-i\,B\,\bar{C}\,\dot{C}-2\,i\,\bar{B}\,\bar{C}\dot{C}-i\,\bar{B}\dot{\bar{C}}\,C=0.$
$None$

The above equation can be expressed as:

$\frac{B^{2}}{2}-\frac{\bar{B}^{2}}{2}-i\,(B+\bar{B})\,\dot{\bar{C}}\,C-i\,(B+\bar{B})\,\bar{C}\,\dot{C}-i\
B\,\dot{\bar{C}}\,C-i\,\bar{B}\,\bar{C}\,\dot{C}=0.$ $None$

The re-arrangements of the terms produce the following

$\frac{B^{2}}{2}-\frac{\bar{B}^{2}}{2}-i\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,\dot{\bar{C}}\,C-i\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]\,\bar{C}\,\dot{C}$
$-i\,B\dot{\bar{C}}\,C-i\,\bar{B}\,\bar{C}\,\dot{C}=0.$ $None$

Taking into account $(A.2)$, we have the following

$\big{[}E\,\dot{E}-i\,\dot{\bar{C}}\,C-i\,\bar{C}\,\dot{C}\big{]}\,\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}+\frac{B^{2}}{2}-\frac{\bar{B}^{2}}{2}$
$-i\ B\,\dot{\bar{C}}\,C-i\,\bar{B}\,\bar{C}\,\dot{C}=0.$ $None$

Substituting for
$-\,i\,B\,\dot{\bar{C}}\,C=-\,\frac{i}{2}\,B\,\dot{\bar{C}}\,C-\,\frac{i}{2}\,B\,\dot{\bar{C}}\,C$
and
$-\,i\,\bar{B}\bar{C}\dot{C}=-\,\frac{i}{2}\,\bar{B}\,\bar{C}\,\dot{C}-\frac{i}{2}\,\bar{B}\,\bar{C}\,\dot{C}$
and re-arranging the terms, we end up with the following final result:

$\big{[}B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)\big{]}\,\big{[}E\,\dot{E}+\frac{1}{2}\,\\{B-\bar{B}-3\,i\,(\dot{\bar{C}}\,C+\bar{C}\,\dot{C})\,\\}\big{]}=0.$
$None$

The above equation establishes the existence of the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ on our theory due to the
equivalence of the coupled (but equivalent) Lagrangians (i.e. $L_{B}\equiv
L_{\bar{B}}$). This is due to the fact that, in no way, we can state that the
other combination:
$E\,\dot{E}+\frac{1}{2}\,\\{B-\bar{B}-3\,i\,(\dot{\bar{C}}\,C+\bar{C}\,\dot{C})\\}=0$.
On the contrary, the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ has been proven from
various angles [cf. Eq. (36)].

We end this Appendix with the concluding remark that we have derived the CF-
type restriction on our theory from theoretical requirements related with the
symmetries of the coupled (but equivalent) Lagrangians and the absolute
anticommutativity properties. However, our present derivation of the CF-type
restriction is more direct as well as transparent.

Appendix B: On the derivation of $f_{3}=E\,\dot{C}+\dot{E}\,C$ and
$f_{2}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}$

The theoretical content of this Appendix is, first of all, devoted to the
explicit derivation of $f_{3}(\tau)$ in the expansion of
$\Sigma(\tau,\bar{\theta})$ in the anti-chiral super expansions (41). Towards
this objective in mind, we focus on Eq. (51) where the first-order
differential equation w.r.t. the evolution parameter $\tau$ for $f_{3}$ has
been expressed. We can re-write it as

$f_{3}\,\dot{C}+{\dot{f}}_{3}\,C-\dot{E}\,\dot{C}\,C-E\,\ddot{C}\,C=0\Rightarrow$
$\frac{d}{d\,\tau}\,[f_{3}\,C]-\frac{d}{d\,\tau}\,[(\dot{E}\,C+E\,\dot{C})\,C]=0,$
$None$

where we have used the fermionic property $(C^{2}=0)$ of the ghost variable
$(C)$. The above equation can be re-expressed in a different but useful form
as the total derivative w.r.t. $\tau$:

$\frac{d}{d\,\tau}\,[\\{f_{3}-(E\,\dot{C}+\dot{E}\,C)\\}\,C]=0.$ $None$

Integrating the above equation from $\tau=-\infty$ to $\tau=+\infty$ (which
are the limiting cases for $\tau$ in our theory), we obtain the following
relationship:

$[f_{3}-(E\,\dot{C}+\dot{E}\,C)]\,C=0.$ $None$

We would like to point out that, while deriving (B.3) from (B.2), we have
assumed that all the physical variables of the Lagrangian $L_{B}$ and the
secondary variable $f_{3}(\tau)$ vanish off at $\tau=\pm\,\infty$. For $C\neq
0$, we obtain the desired result: $f_{3}=E\,\dot{C}+\dot{E}\,C$. We have taken
$C\neq 0$ because the whole set of BRST symmetry transformations in Eq. (33)
is true only when the ghost variable $C(\tau)$ has the non-trivial and non-
zero value.

We now concentrate on the precise determination of $f_{2}(\tau)$ of the super-
expansion (41). In other words, we wish to show that
$f_{2}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}$. For this purpose, we note that we
have a first-order differential equation w.r.t. the evolution parameter $\tau$
for the secondary variable $f_{2}(\tau)$ in Eq. (49). This can be re-expressed
as follows

$f_{2}\,\dot{C}+{\dot{f}}_{2}\,C-2\,f_{2}\,\dot{C}-2\,\dot{\bar{B}}\,\dot{C}\,C+\dot{\bar{B}}\,\dot{C}\,C+\bar{B}\,\ddot{C}\,C=0$
$None$

where we have added and subtracted $f_{2}\,\dot{C}$ and
$\dot{\bar{B}}\,\dot{C}\,C$. The above equation implies that we have now its
modified form (with total derivatives) as:

$\frac{d}{d\,\tau}\,[f_{2}\,C]-2\,(f_{2}\,\dot{C}+\dot{\bar{B}}\,\dot{C}\,C)+\frac{d}{d\,\tau}\,[\bar{B}\,\dot{C}\,C]=0$
$\Rightarrow\quad\frac{d}{d\,\tau}\,\Big{[}f_{2}\,C+\bar{B}\,\dot{C}\,C\Big{]}-2\,\Big{[}(f_{2}-\dot{\bar{B}}\,C)\,\dot{C}\Big{]}=0.$
$None$

Using the fermionic $(C^{2}=\dot{C}^{2}=0)$ property of the ghost variables
$C$ and $\dot{C}$, we can recast the above equation in the following
interesting form where $[f_{2}-(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})]$ appears
very nicely in the individual terms of the following difference, namely;

$\frac{d}{d\,\tau}\,\Big{[}\\{f_{2}-(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})\\}\,C\Big{]}-2\,\Big{[}\\{f_{2}-(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})\\}\,\dot{C}\Big{]}=0.$
$None$

We can expand the total derivative in the first term to obtain:

$\frac{d}{d\,\tau}\,\Big{[}\\{f_{2}-(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})\\}\Big{]}\,C-\Big{[}\\{f_{2}-(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})\\}\,\dot{C}\Big{]}=0.$
$None$

Defining $f_{2}-(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})=\chi$ leads us to the
following

$\Big{(}\frac{d}{d\,\tau}\,\chi\Big{)}\,C-\chi\,\dot{C}=0\qquad\Rightarrow\qquad\dot{\chi}\,C=\chi\,\dot{C}.$
$None$

Multiplying from the right by $C$ and taking into account the fermionic (i.e.
$C^{2}=0$) nature of the ghost variable $C$, we obtain the following

$0=\chi\,\dot{C}\,C\qquad\Rightarrow\qquad\chi=0\;\;\;\qquad\big{[}\text{for}\quad
C\,\dot{C}\neq 0\big{]}.$ $None$

It should be noted that we have the off-shell nilpotent BRST symmetry
transformation: $s_{b}\,C=C\,\dot{C}$ [cf. Eq. (33)]. As a consequence, the
combination of the variables $C\,\dot{C}\neq 0$. If the symmetry of a theory
is the guiding principle behind its beauty, it is physically correct to assume
that $s_{b}\,C=C\,\dot{C}\neq 0$. In fact, if we take $C\dot{C}=0$ the whole
beauty and sacrosanct properties (i.e. off-shell nilpotency and absolute
anticommutativity) of the (anti-)BRST symmetry transformations of our present
theory will be spoiled. As a consequence, the CF-type restriction will no
longer remain (anti-)BRST invariant. It should be recalled, however, that we
have invoked the CF-type restriction in proving the equivalence of the
Lagrangians [cf. Eq. (35)] w.r.t. the (anti-)BRST symmetry transformations.
Hence, our conclusion in (B.9) is correct which leads to the derivation of
$f_{2}=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}$ from $\chi=0$.

Appendix C: On the Proof of the Conservation Law

We take up here the expression for the BRST charge $Q_{B}$ [cf. Eq. (74)] that
has been derived using the Noether theorem [cf. Sec. 7]. We exploit the EL-EOM
derived from the Lagrangian $L_{B}$ to recast the expression for
${\dot{Q}}_{B}$, namely;

${\dot{Q}}_{B}=\frac{\dot{C}\,E}{2}\,(p_{x}^{2}+2\,m\,p_{t})+\frac{C\,\dot{E}}{2}\,(p_{x}^{2}+2\,m\,p_{t})+C\,E\,(p_{x}\,{\dot{p}}_{x}+m\,p_{t})$
$+2\,E\,\dot{E}\,B\,\dot{C}+E^{2}\,B\,\dot{C}+E^{2}\,B\,\ddot{C}+{\dot{E}}^{2}\,B\,C+E\,\ddot{E}\,B\,C+E\,\dot{E}\,\dot{B}\,C$
$+E\,\dot{E}\,B\,\dot{C}+2\,B\,\dot{B}\,C-i\,\dot{B}\,\bar{C}\,\dot{C}\,C-i\,B\,\dot{\bar{C}}\,\dot{C}\,C-i\,B\,\bar{C}\,\ddot{C}\,C,$
$None$

into the following form

${\dot{Q}}_{B}=-i\,E^{2}\,\ddot{\bar{C}}\,\dot{C}\,C-i\,E\,\dot{E}\,\dot{\bar{C}}\,\dot{C}\,C+3\,E\,\dot{E}\,B\,\dot{C}+E^{2}\,B\,\ddot{C}+{\dot{E}}^{2}\,B\,C$
$+E\,\ddot{E}\,B\,C+2\,B\,\dot{B}\,C+B^{2}\,\dot{C}-i\,\dot{B}\,\bar{C}\,\dot{C}\,C-i\,B\,\dot{\bar{C}}\,\dot{C}\,C-i\,B\,\bar{C}\,\ddot{C}\,C,$
$None$

where we have used the EL-EOM from $L_{B}$ as:

${\dot{p}}_{x}=0,\qquad{\dot{p}}_{t}=0,\qquad\frac{1}{2}\,(p_{x}^{2}+2\,m\,p_{t})=-\,\dot{B}\,E+i\,E\,\ddot{\bar{C}}\,C-i\,E\,\dot{\bar{C}}\,\dot{C}.$
$None$

The expression (C.2) can be further changed to a reduced form as follows

${\dot{Q}}_{B}=i\,B\,\dot{\bar{C}}\,\dot{C}\,C-i\,\dot{B}\,\bar{C}\,\dot{C}\,C-i\,E\,\dot{E}\,\dot{\bar{C}}\,\dot{C}\,C-i\,E^{2}\,\ddot{\bar{C}}\,\dot{C}\,C,$
$None$

if we use the EL-EOM from $L_{B}$ w.r.t. the variable $\bar{C}$ as

$2\,\dot{B}\,C+B\,\dot{C}+3\,E\,\dot{E}\,\dot{C}+E^{2}\,\ddot{C}+{\dot{E}}^{2}\,C+E\,\ddot{E}\,C-i\,\bar{C}\,\ddot{C}\,C-2\,i\,\dot{\bar{C}}\,\dot{C}\,C=0.$
$None$

The expression in (C.4) can be proven to be equal to zero by using the
following EL-EOM from $L_{B}$ w.r.t. the variable $C$, namely;

$i\,\dot{B}\,\bar{C}-i\,B\,\dot{\bar{C}}+i\,E\,\dot{E}\,\dot{\bar{C}}+i\,E^{2}\,\ddot{\bar{C}}-\ddot{\bar{C}}\,\bar{C}\,C-2\,\dot{\bar{C}}\,\bar{C}\,\dot{C}=0.$
$None$

We end this Appendix with the remark that, in exactly similar fashion, we can
prove the conservation law $({\dot{Q}}_{\bar{B}}=0)$ of the anti-BRST charge
[cf. Eq. (74)] which has been derived by exploiting the theoretical tricks of
the Noether theorem.

Acknowledgments

Two of us (AKR and AT) gratefully acknowledge the financial support from the
BHU-fellowship program of the Banaras Hindu University (BHU), Varanasi (U.
P.), under which, the present investigation has been carried out. Fruitful and
enlightening comments by our esteemed Reviewer are thankfully acknowledged,
too.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest

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