Reparameterization Invariant Model of a Supersymmetric System: BRST and
Supervariable Approaches

A. Tripathi(a), B. Chauhan (a), A. K. Rao (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><EMAIL_ADDRESS>

Abstract: We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of
the one (0 + 1)-dimensional (1D) model of a free massive spinning relativistic
particle (i.e. a supersymmetric system) by exploiting its classical
infinitesimal and continuous reparameterization symmetry transformations. We
use the modified Bonora-Tonin (BT) supervariable approach (MBTSA) to BRST
formalism to obtain the nilpotent (anti-)BRST symmetry transformations of the
target space variables and the (anti-)BRST invariant Curci-Ferrari (CF)-type
restriction for the 1D model of our supersymmetric (SUSY) system. The
nilpotent (anti-)BRST symmetry transformations for other variables of our
model are derived by using the (anti-)chiral supervariable approach (ACSA) to
BRST formalism where the CF-type restriction appears in the proof of (i) the
invariance of the coupled (but equivalent) Lagrangians, and (ii) the absolute
anticommutativity of the conserved and off-shell nilpotent (anti-)BRST
charges. The application of the MBTSA to a physical SUSY system (i.e. 1D model
of a massive spinning particle) is a novel result in our present endeavor. The
proof of the absolute anticommutativity of the conserved (anti-)BRST charges
(within the framework of ACSA) is another very interesting observation in view
of the fact that only the (anti-)chiral super expansions of the supervariables
have been taken into account.The CF-type restriction is universal in nature as
it turns out to be the same for the SUSY and non-SUSY reparameterizaion (i.e.
1D diffeomorphism) invariant theories.

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

Keywords: A massive spinning (i.e. SUSY) relativistic particle;
reparameterization symmetry; (anti-)BRST symmetries; (anti-)BRST charges;
modified BT-supervariable approach; (anti-)chiral supervariable approach;
nilpotency and absolute anticommutativity; CF-type restriction; symmetry
invariant restrictions, invariance of the Lagrangians

##  1 Introduction

The Becchi-Rouet-Stora-Tyutin (BRST) quantization scheme is one of the most
elegant approaches to quantize the locally gauge and diffeomorphism invariant
theories where the local classical transformation parameters are traded with
the (anti-)ghost fields at the quantum level [1-4]. For the quantization of
the classical supersymmetric gauge theories (with the bosonic and fermionic
transformation parameters), the BRST quantization scheme requires the
fermionic as well as the bosonic (anti-)ghost fields/variables (see, e.g. [5,
6]). Some of the key characteristic features of the BRST quantization scheme
are (i) for a given local gauge and/or diffeomorphism symmetry, there exist
two nilpotent symmetries which are christened as the BRST and anti-BRST
symmetries, (ii) the (anti-)BRST symmetries $(s_{(a)b})$ are fermionic (i.e.
nilpotent) and absolutely anticommuting (i.e. $s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$)
in nature, (iii) the quantum gauge (i.e. BRST) invariance and unitarity are
respected together in the perturbative calculations at any arbitrary order,
and (iv) there is appearence of the Curci-Ferrari (CF)-type restriction(s)
which are responsible for the absolute anticommutativity of the (anti-)BRST
transformations and they lead to the existence of the coupled (but equivalent)
Lagrangian (densities) for the BRST quantized theory which respect both (i.e.
BRST and anti-BRST) nilpotent quantum symmetries.

Physically, the nilpotency property of the (anti-)BRST symmetry
transformations corresponds to the fermionic nature of these quantum
symmetries and the absolute anticommutativity property encodes the linear
independence of the BRST and anti-BRST symmetry transformations. As pointed
out earlier, the absolute anticommutativity property owes its dependence on
the existence of the CF-type restriction(s). The BRST approach to Abelian
1-form gauge theory is an exception  where the CF-type restriction is trivial
(but it turns out to be the limiting case of the non-Abelian 1-form theory
which is endowed with the CF-condition [7]). It is the usual superfield
approach (USFA) to BRST formalism [8-15] which provides the interpretation and
origin for the abstract mathematical properties (i.e. nilpotency and
anticommutativity) that are associated with the (anti-)BRST symmetries.
Furthermore, the USFA leads to the derivation of the CF-condition [7] in the
case of a non-Abelian 1-form theory (see, e.g. [10-12]) which is found to be
an (anti-)BRST invariant quantity. Hence, it is a physical restriction on the
BRST quantized theory.

The USFA to BRST formalism [8-15] leads to the derivation of only the off-
shell nilpotent (anti-)BRST symmetry transformations for the gauge and
associated (anti-)ghost fields/variables. It does not shed any light on the
(anti-)BRST transformations that are associated with the matter fields in an
interacting gauge theory. There have been consistent extensions of the USFA
(see, e.g. [16-20]) where additional quantum gauge invariant restrictions on
the superfields have been imposed to derive the (anti-)BRST symmetry
transformations for the gauge, (anti-)ghost and matter fields together. This
extended version of the superfield approach to BRST formalism has been called
as the augmented version of superfield approach (AVSA). In our recent works
[21-25], we have been able to develop a simpler off-shoot of AVSA where only
the (anti-)chiral superfields/supervariables have been taken into account. The
quantum gauge [i.e. (anti-)BRST] invariant restrictions on these (anti-)chiral
superfields/supervariables have led to the deduction of (anti-)BRST symmetry
transformations for all the fields/variables. This approach to BRST formalism
has been named as the (anti-)chiral superfield/supervariable approach (ACSA)
to BRST formalism where the existence of the CF-type restriction(s) has been
shown by proving (i) the absolute anticommutativity of the (anti-)BRST
charges, and (ii) the invariance of the coupled (but equivalent) Lagrangian
(densities) of the (anti-)BRST invariant theories.

It has been a challenging problem to apply the superfield approach to BRST
formalism in the context of (super)string and gravitational theories which are
diffeomorphism invariant. In a recent paper [26], it has been proposed that
the diffeomorphism symmetry can be taken into consideration within the
framework of superfield approach to BRST formalism. This approach to BRST
formalism has been called as the modified Bonora-Tonin
superfield/supervariable approach (MBTSA) to BRST formalism which has been
recently applied to the physical system of a 1D scalar relativistic particle
[27]. To be precise, judicious combination of MBTSA and ACSA has been very
fruitful in our recent work [27] where we have been able to derive the proper
(anti-)BRST symmetries for all the quantum variables along with the CF-type
restriction in a systematic fashion. In the proposal by Bonora [26], the
diffeomorphism symmetry transformations have been incorporated into the
supefields which are defined on a (D, 2)-dimensional supermanifold on which a
D-dimensional ordinary diffeomorphism invariant theory is generalized. We have
exploited the mathematical rigor and beauty of the MBTSA in our present
endeavor for the BRST analysis as well as quantization of a 1D diffeomorphism
invariant SUSY system.

To be precise, in our present investigaion, we have applied the theoretical
beauty and strength of MBTSA to derive the off-shell nilpotent (anti-)BRST
transformations for the target space canonically conjugate bosonic variables
($x_{\mu}$ and $p^{\mu}$) and fermionic variables $(\psi_{\mu},\psi_{5})$
along with the (anti-)BRST invariant CF-type restrictions:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ which is responsible for
(i) the validity [cf. Eq. (22) below] of the absolute anticommutativity (i.e.
$\\{s_{b},s_{ab}\\}=s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$) of the off-shell nilpotent
($s_{(a)b}^{2}=0$) (anti-)BRST symmetry transformations $s_{(a)b}$, and (ii)
the appearance of the coupled (but equivalent) Lagrangians $L_{B}$ and
$L_{\bar{B}}$ [cf. Eq. (24) below]. The proper (anti-)BRST transformations of
the rest of the variables have been derived by using the ACSA to BRST
formalism. It is worth pointing out that, in the case of MBTSA, we have taken
into account the full super expansions of the supervariables along all the
possible Grassmannian directions of the general (1, 2)-dimensional
supermanifold. On the other hand, we have performed only the (anti-)chiral
super expansions of the supervariables in the context of ACSA to BRST
formalism. We have derived the exact expression for the CF-type restriction by
demanding (i) the invariance of the coupled (but equivalent) Lagrangians, and
(ii) the validity of the absolute anticommutatvity of the off-shell nilpotent
(anti-)BRST symmetries as well as conserved (anti-)BRST charges in the
ordinary space and in the superspace (within the framework of ACSA to BRST
formalism). These derivations of the CF-type restrictions, by various
theoretical methods, are novel results in our present investigation.

The following key factors have been at the heart of our curiosity to pursue
our present investigation. First, we have been able to apply the MBTSA to a
reparameterization invariant model of the scalar relativistic particle to
derive the (anti-)BRST symmetry transformations as well as the (anti-)BRST
invariant CF-type of restriction [27]. Thus, it has been very important for us
to apply the same mathematical technique (i.e. MBTSA) to a physically
interesting SUSY model of a reparameterization invariant theory where the
fermionic as well as bosonic variables exist. Second, we have been very
curious to verify the universality of the CF-type restriction in the context
of BRST quantization of the 1D diffeomorphism (i.e. reparameterization)
invariant theories. We find that the nature and form of the CF-type
restriction is the same for the SUSY as well as non-SUSY theories. Third, it
has been very interesting to note that the gauge-fixing and Faddeev-Popov
ghost terms together are same for the reparameterization invariant scalar and
SUSY relativistic particles. Finally, our present investigation (as well as
the previous one [27]) is our modest initial steps to apply the MBTSA as well
as the ACSA to BRST formalism together to physically interesting 4D (and
higher dimensional) diffeomorphism invariant theories which are important from
the point of view of the modern developments in gravitational as well as
(super)string theories (and related extended objects) of the high energy
physics.

The theoretical contents of our present endeavor are organised as follows. In
Sec. 2, we discuss a couple of continuous and infinitesimal symmetry
transformations and establish their relationship with the infinitesimal and
continuous 1D diffeomorphism (i.e. reparameterization) symmetry
transformations. Our Sec. 3 is devoted to the upgradation of the classical
infinitesimal reparameterization symmetry transformations to the quantum off-
shell nilpotent and absolutely anticommuting (anti-)BRST symmetry
transformations. The latter property is satisfied due to the existence of an
(anti-)BRST invariant CF-type restriction. This section also contains the
coupled (but equivalent) Lagrangians that respect both the (anti-)BRST
symmetry transformations on the submanifold (of the subspace of quantum
variables) where the CF-type restriction is satisfied. In Sec. 4, we derive
the (anti-)BRST transformations for the target space fermionic as well as
bosonic variables. In addition, we deduce the CF-type of restriction by
exploiting the theoretical strength of MBTSA. We utilize the potential of ACSA
to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations
for the other variables of our BRST-invariant theory in Sec. 5. We capture the
(anti-)BRST invariance of the coupled (but equivalent) Lagrangians within the
ambit of ACSA and establish, once again, the existence of our quantum
(anti-)BRST invariant CF-type restriction in Sec. 6. Sec. 7 of our present
endeavor contains theoretical proof of the nilpotency and absolute
anticommutativity of the conserved and off-shell nilpotent (anti-)BRST charges
in the ordinary space as well as in the superspace by exploiting the
theoretical strength of ACSA. Finally, in Sec. 8, we summarize our key
accomplishments and point out a few directions for further investigations.

In our Appendices A, B and C, we collect some explicit computations that
corroborate a few key claims and statements that have been made in the main
body of the text of our present endeavor. Our Appendix D is devoted to provide
an alternative proof of the existence of an (anti-)BRST invariant CF-type
restriction on our theory.

Convention and Notations: We take the flat metric tensor of the D-dimensional
target spacetime manifold as $\eta_{\mu\nu}=(+1,-1,-1,-1...)$ so that the dot
product between two non-null vectors $(P_{\mu},Q_{\mu})$ is: $P\cdot
Q=\eta_{\mu\nu}\,P^{\mu}\,Q^{\nu}=P_{0}\,Q_{0}-P_{i}\,Q_{i}$ where the Greek
indices $\mu,\nu,\lambda...=0,1,2...D-1$ and Latin indices
$i,j,k...=1,2,3...D-1$. We adopt the convention of the left-derivative w.r.t.
all the fermionic variables. We always denote the nilpotent (anti-)BRST
transformations by the notations $s_{(a)b}$. Our 1D model is generalized onto
a (1, 2)-dimensional supermanifold which is parameterized by the superspace
coordinates ${{Z}}^{M}=(\tau,\theta,\bar{\theta})$ where $\tau$ is the bosonic
evolution parameter and a pair of Grassmannian variable
$(\theta,\bar{\theta})$ satisfy:
$\theta^{2}={\bar{\theta}}^{2}=0,\;\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$.
In our present investigation, we shall focus only on the (1, 1)-dimensional
(anti-)chiral super sub-manifolds of our chosen general (1, 2)-dimensional
supermanifold in the context of ACSA.

##  2 Preliminaries: Continuous and Infinitesimal Reparameterization Symmetry
Transformations

Our present section is divided into two parts. In subsection 2.1, we discuss
some classical infinitesimal continuous symmetries and their relationships
with the classical infinitesimal reparameterization symmetry transformations.
Our subsection 2.2 is devoted to a concise discussion and description of the
quantum (anti-)BRST symmetry transformations.

###  2.1 Some Classical Infinitesimal and Continuous symmetries

We begin with the following first-order Lagrangian $(L_{f})$ for the free (0 +
1)-dimensional (1D) massive spinning (i.e. supersymmetric) relativistic
particle [28, 29]

$\displaystyle
L_{f}=p_{\mu}\,\dot{x}^{\mu}+\frac{i}{2}(\psi_{\mu}\,\dot{\psi}^{\mu}-\psi_{5}\,\dot{\psi}_{5})-\frac{e}{2}\;(p^{2}-m^{2})+i\,\chi\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5}),$
(1)

where the trajectory of the particle is parameterized by $\tau$ and the
“generalized” velocities
$\big{(}\dot{x}^{\mu}=d\,x^{\mu}/{d\,\tau},\;\dot{\psi}^{\mu}=d\,\psi^{\mu}/{d\,\tau}\big{)}$
are defined w.r.t. to it. This 1D trajectory is embedded in the D-dimensional
flat Minkowskian target spacetime manifold where $(x_{\mu},p^{\mu})$ (with
$\mu=0,1,2,...D-1$) are the canonical conjugate pair of spacetime and momenta
variables which are function of the evolution parameter $\tau$. We have the
fermionic variables $(\chi,\psi_{\mu},\psi_{5})$ in our theory. The Lagrangian
(1) also has the bosonic variable $e$ and fermionic variable $\chi$ as the
Lagrange multiplier variables. These latter variables behave like the “gauge”
and “superaguge” variables due to their transformation properties under the
gauge and supersymmetric gauge transformations. In fact, the fermionic
variable $\psi_{\mu}$ is the superpartner of $x_{\mu}$ and the other fermionic
variable $\psi_{5}$ has been introduced to incorporate the mass $m$ into the
Lagrangian $L_{f}$ where the mass-shell condition $(p^{2}\,-\,m^{2}=0)$ is
satisfied by the free ($\dot{p}_{\mu}=0$) massive spinning relativistic
particle. Our present 1D model is interesting because it provides a prototype
example of a supersymmetric (SUSY) gauge theory and its generalization to the
4D theory provides an example of the supergravity theory where fermionic
variable $\psi_{\mu}$ becomes the Rarita-Schwinger Lorentz vector spin 3/2
field and the einbein variable $e$ turns itself into the vierbein field.

The action integral $S=\int_{-\infty}^{+\infty}d\,\tau\,L_{f}$ respects the
following infinitesimal and continuous reparameterization symmetry
transformations $(\delta_{r})$

$\displaystyle\delta_{r}\,x_{\mu}=\epsilon\,{\dot{x}}_{\mu},\;\;\;\qquad\delta_{r}\,p_{\mu}=\epsilon\,{\dot{p}}_{\mu},\;\;\;\;\qquad\quad\delta_{r}\psi_{\mu}=\epsilon\,{\dot{\psi}}_{\mu},$
$\displaystyle\delta_{r}\,\psi_{5}=\epsilon\,{\dot{\psi}}_{5},\qquad\;\;\;\delta_{r}\,e=\frac{d}{d\,\tau}\,(\epsilon\,e),\qquad\;\;\;\delta_{r}\,\chi=\frac{d}{d\,\tau}\,(\epsilon\,\chi),$
(2)

because the first-order Lagrangian $(L_{f})$ transforms, under the above
infinitesimal reparameterization symmetry transformation $(\delta_{r})$, as

$\displaystyle\delta_{r}\,L_{f}=\frac{d}{d\tau}\,[\epsilon\,L_{f}]\,\,\quad\Longrightarrow\,\,\quad\delta_{r}\,S=0,$
(3)

where $\delta_{r}$ basically corresponds to the infinitesimal 1D
diffeomorphism/reparameterization transformation:
$\tau\longrightarrow\tau^{\prime}=\tau\,-\,\epsilon(\tau)$. Here the
transformation parameter $\epsilon(\tau)$ is infinitesimal. It is an
elementary exercise to note that, if we set all the fermionic variables equal
to zero (i.e. $\chi,\psi_{\mu},\psi_{5}=0$), we obtain an infinitesimal gauge
symmetry transformation $(\delta_{g})$ from the infinitesimal
reparameterization symmetry transformation (2) as follows

$\displaystyle\delta_{g}x_{\mu}=\xi\,p_{\mu},\;\;\;\quad\delta_{g}p_{\mu}=0,\;\;\;\quad\delta_{g}e=\frac{d}{d\,\tau}(\xi)=\dot{\xi},\;\;\;\quad\delta_{g}\psi_{\mu}=\delta_{g}\psi_{5}=\delta_{g}\chi=0,$
(4)

where we have identified $e\,\epsilon=\xi$ and used the Eular-Lagrange
eqations of motions (EL-EOMs):
${\dot{x}}_{\mu}=e\,p_{\mu},\;{\dot{p}}_{\mu}=0$. In equation (4), the bosonic
infinitesimal transformation parameter $\xi(\tau)$ is nothing but the gauge
transformation parameter. It can be readily checked that we have the following
transformation for $L_{f}$ and S under $\delta_{g}$:

$\displaystyle\delta_{g}\,L_{f}=\frac{d}{d\tau}\,\Big{[}\frac{\xi}{2}\,(p^{2}+m^{2})\Big{]}\quad\implies\quad\delta_{g}\,S=0,\qquad
S=\int_{-\infty}^{+\infty}d\tau\;L_{f}.$ (5)

We have an infinitesimal classical supergauge symmetry transformations
$(\delta_{sg})$ in our theory which transforms the fermionic variables into
their bosonic counterparts and vice-versa. These continuous and infinitesimal
symmetry transformations are

$\displaystyle\delta_{sg}\,x_{\mu}=\kappa\,\psi_{\mu},\qquad\qquad\delta_{sg}\,\psi_{\mu}=i\,\kappa\,p_{\mu},\qquad\qquad\delta_{sg}\,p_{\mu}=0,$
$\displaystyle\delta_{sg}\,\psi_{5}=i\,\kappa\,m,\qquad\qquad\delta_{sg}\,\chi=i\,\dot{\kappa},\qquad\qquad\delta_{sg}\,e=2\,\kappa\,\chi,$
(6)

where the infinitesimal transformation parameter $\kappa(\tau)$ is fermionic
(i.e. $\kappa^{2}=0$) in nature. It is straightforward to observe that we have
the following:

$\displaystyle\delta_{sg}\,L_{f}=\frac{d}{d\tau}\,\Big{[}\frac{\kappa}{2}\,(p_{\mu}\,\psi^{\mu}+m\,\psi_{5})\Big{]}\quad\implies\quad\delta_{sg}\,S=0.$
(7)

Under the combined $\delta=\delta_{g}+\delta_{sg}$ classical symmetry
transformation $(\delta)$, we have the following continuous and infinitesimal
symmetry transformations $(\delta)$, namely;

$\displaystyle\delta\,x_{\mu}=\xi\,p_{\mu}+\kappa\,\psi_{\mu},\qquad\delta
p_{\mu}=0,\qquad\delta\psi_{\mu}=i\,\kappa\,p_{\mu},$ $\displaystyle\delta
e=\dot{\xi}+2\,\kappa\,\chi,\qquad\delta\chi=i\,\dot{\kappa},\qquad\delta\psi_{5}=i\,\kappa\,m,$
(8)

which lead to the transformation of the first-order Lagrangian $L_{f}$ as

$\displaystyle\delta
L_{f}=\frac{d}{d\tau}\,\Big{[}\frac{\xi}{2}\,(p^{2}+m^{2})+\frac{\kappa}{2}\,(p_{\mu}\,\psi^{\mu}+m\,\psi_{5})\Big{]}\quad\implies\quad\delta\,S=0.$
(9)

Thus, the continuous and infinitesimal transformation $\delta$ is indeed a
symmetry transformation for the action integral
$S=\int_{-\infty}^{\infty}d\,\tau\,L_{f}$ due to Gauss’s divergence theorem.

As the gauge symmetry transformation (4) can be incorporated into the
reparameterization symmetry transformations (2) with the help of some EL-EOMs
and some identification of the transformation parameters, in exactly similar
fashion, the combined (super)gauge symmetry transformations (8) can be
incorporated into the reparameterization symmetry transformations (2) if we
take the help of the following EL-EOMs, namely;

$\displaystyle\dot{p}_{\mu}=0,\qquad\dot{x}_{\mu}=e\,p_{\mu}-i\,\chi\,\psi_{\mu},\qquad\dot{\psi}_{\mu}=\chi\,p_{\mu},\qquad\dot{\psi}_{5}=m\,\chi,$
(10)

and identify the transformation parameters as: $e\,\epsilon=\xi$ and
$-\,i\,\epsilon\,\chi=\kappa$ (see, e.g. [29] for details). Thus, we note that
the classical infinitesimal reparameterization symmetry transformations (2)
are a set of very general kind of symmetry transformations whose special cases
are the continuous and infinitesimal symmetry transformations (4) and (8).

###  2.2 Quantum (Anti-)BRST Symmetries Corresponding to the Classical
(Super)gauge Symmetry Transformations

The classical continuous and infinitesimal (super)gauge symmetry
transformations (8) can be elevated to their counterpart quantum nilpotent
$(s_{(a)b}^{2}=0)$, infinitesimal and continuous (anti-)BRST transformations
$s_{(a)b}$ as follows [5, 6, 28, 29]

$\displaystyle
s_{ab}\;x_{\mu}={\bar{c}}\;p_{\mu}+\bar{\beta}\;\psi_{\mu},\quad\qquad
s_{ab}\;e=\dot{\bar{c}}+2\;\bar{\beta}\;\chi,\;\quad\qquad
s_{ab}\;\psi_{\mu}=i\;\bar{\beta}\;p_{\mu},$ $\displaystyle
s_{ab}\;\bar{c}=-i\;{\bar{\beta}}^{2},\quad s_{ab}\;c=i\;\bar{b},\quad
s_{ab}\;\bar{\beta}=0,\;\quad s_{ab}\;\beta=-i\;\gamma,\quad
s_{ab}\;p_{\mu}=0,$ $\displaystyle s_{ab}\;\gamma=0,\quad
s_{ab}\;\bar{b}=0,\quad s_{ab}\;\chi=i\;\dot{\bar{\beta}},\quad
s_{ab}\;b=2\;i\;\bar{\beta}\;\gamma,\quad s_{ab}\,\psi_{5}=i\,\bar{\beta}\,m,$
(11) $\displaystyle s_{b}\;x_{\mu}=c\;p_{\mu}+\beta\;\psi_{\mu},\quad\qquad
s_{b}\;e=\dot{c}+2\;\beta\;\chi,\quad\qquad
s_{b}\;\psi_{\mu}=i\;\beta\;p_{\mu},$ $\displaystyle
s_{b}\;c=-i\;\beta^{2},\;\quad s_{b}\;{\bar{c}}=i\;b,\;\quad
s_{b}\;\beta=0,\;\quad s_{b}\;\bar{\beta}=i\;\gamma,\;\quad s_{b}\;p_{\mu}=0,$
$\displaystyle s_{b}\;\gamma=0,\quad s_{b}\;b=0,\quad
s_{b}\;\chi=i\;\dot{\beta},\qquad s_{b}\;\bar{b}=-2\;i\;\beta\;\gamma,\quad
s_{b}\,\psi_{5}=i\,\beta\,m,$ (12)

where the fermionic $(c^{2}={\bar{c}}^{2}=c\,{\bar{c}}+{\bar{c}}\,c=0)$
(anti-)ghost variables $(\bar{c})c$ and the bosonic
$(\beta^{2}={\bar{\beta}}^{2}\neq
0,\;\beta\,{\bar{\beta}}={\bar{\beta}}\,\beta)$ (anti-)ghost variables
$(\bar{\beta})\beta$ correspond to the bosonic and fermionic gauge and
supergauge transformation parameters $\xi$ and $\kappa$ of Eq. (8),
respectively. The variables $(\bar{b})b$ are the Nakanishi-Lautrup type
auxiliary variables and $\gamma$ is an additional fermionic $(\gamma^{2}=0)$
variable in our BRST quantized as well as invariant theory.

It is straightforward to note that the anticommutativity property of the off-
shell nilpotent $(s_{(a)b}^{2}=0)$ (anti-)BRST symmetry transformations
($s_{(a)b}$), namely;

$\displaystyle\\{s_{b},s_{ab}\\}\,x_{\mu}=i\,(b+\bar{b}+2\,\beta\,\bar{\beta})\,p_{\mu},\qquad\quad\\{s_{b},s_{ab}\\}\,e=i\,\frac{d}{d\,\tau}\,(b+\bar{b}+2\,\beta\,\bar{\beta}),$
(13)

is true if and only if the CF-type restriction:
$b+\bar{b}+2\,\beta\,\bar{\beta}=0$ is invoked from outside. However, this
restriction is a physical constraint because it is an (anti-)BRST invariant
(i.e. $s_{(a)b}\,[b+\bar{b}+2\,\beta\,\bar{\beta}]=0$) quantity. It can be
readily checked that:

$\displaystyle\\{s_{b},s_{ab}\\}\,\Phi=0,\qquad\quad\Phi=p_{\mu},\psi_{\mu},\psi_{5},b,\bar{b},\beta,\bar{\beta},c,\bar{c},\gamma.$
(14)

In other words, we observe that the off-shell nilpotent $(s_{(a)b}^{2})=0$
(anti-)BRST symmetry transformations $(s_{(a)b})$ are absolutely anticommuting
(i.e. $\\{s_{b},s_{ab}\\}=s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$) provided the entire
theory is considered on the quantum submanifold where the CF-type restriction
$b+\bar{b}+2\,\beta\,\bar{\beta}=0$ is satisfied. It is the existence of this
physical restriction that leads to the existence of coupled (but equivalent)
Lagrangians

$\displaystyle
L_{b}=L_{f}+s_{b}\,s_{ab}\,\Big{[}\frac{i}{2}\,e^{2}-\frac{1}{2}\bar{c}\,c+\chi\,\psi_{5}\Big{]},$
$\displaystyle
L_{\bar{b}}=L_{f}-s_{ab}\,s_{b}\,\Big{[}\frac{i}{2}\,e^{2}-\frac{1}{2}\bar{c}\,c+\chi\,\psi_{5}\Big{]},$
(15)

which incorporate the gauge-fixing and Faddeev-Popov ghost terms in addition
to the first-order Lagrangian $(L_{f})$ of Eq. (1). In the full blaze of
glory, the above Lagrangians (in terms of all the appropriate variables) are
as follows [29]

$\displaystyle L_{b}$ $\displaystyle=$ $\displaystyle
L_{f}+b^{2}+b\,({\dot{e}}+2\,{\bar{\beta}}\,\beta)-i\,\dot{\bar{c}}\,{\dot{c}}+{\bar{\beta}}^{2}\,{\beta}^{2}+2\,i\,\chi\,(\beta\,\dot{\bar{c}}-\bar{\beta}\,\dot{c})-2\,e\,(\bar{\beta}\,\dot{\beta}+\gamma\,\chi)$
(16) $\displaystyle+$ $\displaystyle
2\,\gamma\,(\beta\,\bar{c}-\bar{\beta}\,c)+m\,(\bar{\beta}\,\dot{\beta}-\dot{\bar{\beta}}\,\beta+\gamma\,\chi)-\dot{\gamma}\,\psi_{5},$
$\displaystyle L_{\bar{b}}$ $\displaystyle=$ $\displaystyle
L_{f}+{\bar{b}}^{2}-\bar{b}\,({\dot{e}}-2\,{\bar{\beta}}\,\beta)-i\,\dot{\bar{c}}\,{\dot{c}}+{\bar{\beta}}^{2}\,{\beta}^{2}+2\,i\,\chi\,(\beta\,\dot{\bar{c}}-\bar{\beta}\,\dot{c})+2\,e\,(\dot{\bar{\beta}}\,\beta-\gamma\,\chi)$
(17) $\displaystyle+$ $\displaystyle
2\,\gamma\,(\beta\,\bar{c}-\bar{\beta}\,c)+m\,(\bar{\beta}\,\dot{\beta}-\dot{\bar{\beta}}\,\beta+\gamma\,\chi)-\dot{\gamma}\,\psi_{5},$

where the subscripts $b$ and $\bar{b}$ are appropriate because the Lagrangian
$L_{b}$ depends on the Nakanishi-Lautrup auxiliary variable $b$ but the
Lagrangian $L_{\bar{b}}$ contains the auxiliary variable $\bar{b}$ in its full
expression. It is straightforward to check that $L_{b}$ and $L_{\bar{b}}$ of
our theory respect the perfect BRST and anti-BRST transformations because we
have:

$\displaystyle
s_{b}\,L_{b}=\frac{d}{d\,\tau}\,\Big{[}\frac{c}{2}\,(p^{2}+m^{2})+\frac{\beta}{2}\,(p_{\mu}\,\psi^{\mu}+m\,\psi_{5})+b\,(\dot{c}+2\,\beta\,\chi)\Big{]},$
(18) $\displaystyle
s_{ab}\,L_{\bar{b}}=\frac{d}{d\,\tau}\,\Big{[}\frac{\bar{c}}{2}\,(p^{2}+m^{2})+\frac{\bar{\beta}}{2}\,(p_{\mu}\,\psi^{\mu}+m\,\psi_{5})-\bar{b}\,(\dot{\bar{c}}+2\,\bar{\beta}\,\chi)\Big{]}.$
(19)

As a consequence, 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
BRST and anti-BRST symmetry transformations (12) and (11), respectively. We
define a perfect symmetry as the one under which the action integral remains
invariant without any use of the CF-type restriction and/or EL-EOMs.

The BRST quantization of the massive spinning particle can be performed using
the (anti-)BRST transformations (11) and (12) which correspond to the
classical (super)gauge symmetry transformations (8). In our recent publication
[29], we have discussed all the details of this quantization scheme. However,
we have not touched the continuous and infinitesimal reparameterization
transformations (2). We focus on the latter classical symmetry transformations
in the next section for the BRST analysis as it is our modest first step
towards our main goal to discuss the diffeomorphism invariant SUSY theories in
the physical (3 + 1)-dimensional (4D) and higher dimensional spacetime.

##  3 Quantum (Anti-)BRST Symmetries Corresponding to the Classical
Reparameterization Symmetry

In this section, we discuss the quantum (anti-)BRST symmetry transformations
corresponding to the classical infinitesimal reparameterization symmetry
transformations (2). This is essential and important because we wish to
perform the BRST quantization of a 1D diffeomorphism (i.e. reparameterization)
invariant SUSY theory. We exploit the standard techniques and tricks of the
BRST formalism to elevate the above classical symmetry to its counterparts
quantum symmetries. In fact, the off-shell nilpotent $(s_{(a)b}^{2}=0)$
(anti-)BRST symmetry transformations [corresponding to the classical Eq. (2)]
are

$\displaystyle s_{ab}\;x_{\mu}={\bar{C}}\;{\dot{x}}_{\mu},\qquad
s_{ab}\;p_{\mu}={\bar{C}}\;{\dot{p}}_{\mu},\qquad
s_{ab}\;e=\frac{d}{d\,\tau}\,({\bar{C}}\,e),\qquad
s_{ab}\;\psi_{\mu}={\bar{C}}\;{\dot{\psi}}_{\mu},$ $\displaystyle
s_{ab}\,\psi_{5}={\bar{C}}\;{\dot{\psi}}_{5},\qquad
s_{ab}\;\chi=\frac{d}{d\,\tau}\,({\bar{C}}\,\chi),\qquad
s_{ab}\;\bar{C}=\bar{C}\,\dot{\bar{C}},\;\;\;\qquad s_{ab}\;C=i\,\bar{B},$
$\displaystyle s_{ab}\;\bar{B}=0,\qquad\qquad
s_{ab}\;B={\dot{B}}\,{\bar{C}}-B\,\dot{\bar{C}},$ (20) $\displaystyle
s_{b}\;x_{\mu}={C}\;{\dot{x}}_{\mu},\qquad
s_{b}\;p_{\mu}={C}\;{\dot{p}}_{\mu},\qquad
s_{b}\;e=\frac{d}{d\,\tau}\,({C}\,e),\qquad
s_{b}\;\psi_{\mu}={C}\;{\dot{\psi}}_{\mu},$ $\displaystyle
s_{b}\,\psi_{5}={C}\;{\dot{\psi}}_{5},\qquad
s_{b}\;\chi=\frac{d}{d\,\tau}\,({C}\,\chi),\qquad
s_{b}\;\bar{C}=i\,B,\;\;\;\qquad s_{b}\;C=C\,{\dot{C}},$ $\displaystyle
s_{b}\;B=0,\qquad\qquad s_{b}\;\bar{B}=\dot{\bar{B}}\,{C}-\bar{B}\,\dot{C},$
(21)

where $B$ and $\bar{B}$ are the Nakanishi-Lautrup auxiliary and $({\bar{C}})C$
are the (anti-)ghost variables of our theory. As far as the absolute
anticommutativity property (i.e. $\\{s_{b},s_{ab}\\}=0$) of the above
transformations is concerned, we note the following

$\displaystyle\\{s_{b},s_{ab}\\}\,s_{\mu}=i\,\big{[}B+{\bar{B}}+i\,({\bar{C}}\,{\dot{C}}-\dot{\bar{C}}\,C)\big{]}\,{\dot{s}}_{\mu},$
$\displaystyle\\{s_{b},s_{ab}\\}\,\Phi=i\,\frac{d}{d\,\tau}\Big{[}\big{\\{}B+{\bar{B}}+i\,({\bar{C}}\,{\dot{C}}-\dot{\bar{C}}\,C\big{\\}}\,\Phi\Big{]},$
$\displaystyle\\{s_{b},s_{ab}\\}\,\Psi=0,\qquad\Psi=B,\bar{B},C,\bar{C},$ (22)

where
$s_{\mu}=x_{\mu}(\tau),\,p_{\mu}(\tau),\,\psi_{\mu}(\tau),\,\psi_{5}(\tau)$
and $\Phi=e(\tau),\,\chi(\tau)$. Thus, we note that the absolute
anticommutativity property of the (anti-)BRST symmetry transformations [cf.
Eqs. (20),(21)] is satisfied (i.e.
$\\{s_{b},s_{ab}\\}=s_{b}\,s_{ab}\,+\,s_{ab}\,s_{b}=0$ ) if and only if we
invoke the (anti-)BRST invariant (i.e.
$s_{(a)b}\,[B+{\bar{B}}+i\,({\bar{C}}\,{\dot{C}}-\dot{\bar{C}}\,C)]=0$) CF-
type constraint $[B+{\bar{B}}+i\,({\bar{C}}\,{\dot{C}}-\dot{\bar{C}}\,C)=0]$.
We note, therefore, that a CF-type constraint exists on our theory which is
the root-cause behind the absolute anticommutativity (i.e.
$\\{s_{b},s_{ab}\\}=0$) of the (anti-)BRST symmetry transformations and the
existence of coupled (but equivalent) (anti-)BRST invariant Lagrangians as

$\displaystyle
L_{B}=L_{f}+s_{b}\,s_{ab}\,\Big{[}\frac{i}{2}\,e^{2}-\frac{1}{2}\,\bar{C}\,C+\chi\,\psi_{5}\Big{]},$
$\displaystyle
L_{\bar{B}}=L_{f}-s_{ab}\,s_{b}\,\Big{[}\frac{i}{2}\,e^{2}-\frac{1}{2}\,\bar{C}\,C+\chi\,\psi_{5}\Big{]},$
(23)

where the (anti-)BRST symmetry transformations $(s_{(a)b})$ are the quantum
symmetries [cf. Eqs. (20), (21)] corresponding to the classical infinitesimal
reparameterization symmetry transformations (2). It will be noted that the
quantities in the square brackets of (23) are the same as quoted in Eq. (15).
However, the (anti-)BRST symmetry transformations in (23) are different from
(15) as are the notations for the Nakanishi-Lautrup type auxiliary and
(anti-)ghost variables [cf. Eqs. (11), (12), (20) and (21) for details].

The above coupled (but equivalent) Lagrangians (23) can be expressed in terms
of the auxiliary and basic variables in an explicit form as:

$\displaystyle
L_{B}=L_{f}+B\,\Big{[}e\,\dot{e}-i\,(2\,\dot{\bar{C}}\,C+{\bar{C}}\,\dot{C})\Big{]}+\frac{B^{2}}{2}-\,i\,e^{2}\,\dot{\bar{C}}\,\dot{C}-i\,e\,\dot{e}\,\dot{\bar{C}}\,C-\dot{\bar{C}}\,{\bar{C}}\,\dot{C}\,C,$
$\displaystyle
L_{\bar{B}}=L_{f}-\bar{B}\,\Big{[}e\,\dot{e}-i\,(2\,{\bar{C}}\,\dot{C}+\dot{\bar{C}}\,C)\Big{]}+\frac{{\bar{B}}^{2}}{2}-\,i\,e^{2}\,\dot{\bar{C}}\,\dot{C}-i\,e\,\dot{e}\,{\bar{C}}\,\dot{C}-\dot{\bar{C}}\,{\bar{C}}\,\dot{C}\,C.$
(24)

We note that the pure Faddeev-Popov ghost part (i.e.
$-\,\dot{\bar{C}}\,{\bar{C}}\,\dot{C}\,C$) of the above coupled (but
equivalent) Lagrangians is the same. It can be readily checked that the EL-
EOMs from Lagrangians $L_{B}$ and $L_{\bar{B}}$, w.r.t. the auxiliary
variables $B$ and $\bar{B}$, lead to the following

$\displaystyle
B=-\,e\,\dot{e}+2\,i\,\dot{\bar{C}}\,C+i\,\bar{C}\,\dot{C},\qquad\bar{B}=e\,\dot{e}-2\,i\,{\bar{C}}\,\dot{C}-i\,\dot{\bar{C}}\,C,$
(25)

which are responsible for the derivation of the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$. The above Lagrangians
$L_{B}$ and $L_{\bar{B}}$ of our theory respect the BRST and anti-BRST
transformations in a precise and perfect manner because it is interesting to
check that:

$\displaystyle
s_{b}\,L_{B}=\frac{d}{d\,\tau}\Big{[}C\,L_{f}+e^{2}\,B\,\dot{C}+e\,\dot{e}\,B\,C-i\,B\,\bar{C}\,\dot{C}\,C+B^{2}\,C\Big{]},$
(26) $\displaystyle
s_{ab}\,L_{\bar{B}}=\frac{d}{d\,\tau}\Big{[}\bar{C}\,L_{f}-e^{2}\,\bar{B}\,\dot{\bar{C}}-e\,\dot{e}\,\bar{B}\,\bar{C}-i\,\bar{B}\,\dot{\bar{C}}\,\bar{C}\,C+{\bar{B}}^{2}\,\bar{C}\Big{]}.$
(27)

As a consequence, the action integrals:
$S_{1}=\int_{-\infty}^{\infty}d\,\tau\,L_{B}$ and
$S_{2}=\int_{-\infty}^{\infty}d\,\tau\,L_{\bar{B}}$ respect the continuous and
nilpotent symmetries $s_{b}$ and $s_{ab}$ because of the Gauss’s divergence
theorem (where all the physical variables of our theory vanish off at
$\tau=\pm\,\infty$). We can also apply $s_{b}$ on $L_{\bar{B}}$ and $s_{ab}$
on $L_{B}$. The ensuing results are as follows

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

which demonstrate that the coupled Lagrangians $L_{B}$ and $L_{\bar{B}}$ of
our theory are equivalent in the sense that both of them respect both (i.e.
BRST and anti-BRST) transformations due to the validity of the physical CF-
type restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$.

It is very interesting to point out that the contributions of the term
$``\chi\,\psi_{5}"$ in Eq. (23) turn out to be total derivatives because we
observe that:

$\displaystyle s_{b}\,s_{ab}\,(\chi\,\psi_{5})$ $\displaystyle=$
$\displaystyle\frac{d}{d\,\tau}\,\Big{[}(i\,B\,\chi-{\bar{C}}\,\dot{C}\,{\chi}-\bar{C}\,C\,\dot{\chi})\,\psi_{5}-\bar{C}\,C\,\chi\,\dot{\psi}_{5}\Big{]},$
$\displaystyle-\,s_{ab}\,s_{b}\,(\chi\,\psi_{5})$ $\displaystyle=$
$\displaystyle-\,\frac{d}{d\,\tau}\,\Big{[}(i\,\bar{B}\,\chi+\dot{\bar{C}}\,C\,{\chi}+\bar{C}\,C\,\dot{\chi})\,\psi_{5}+\bar{C}\,C\,\chi\,\dot{\psi}_{5}\Big{]}.$
(30)

As a consequence, the gauge-fixing and Faddeev-Popov ghost terms of the
Lagrangians $L_{B}$ and $L_{\bar{B}}$ of Eq. (24) originate from the same
terms as the ones derived in our earlier work [6] on the massless spinning
relativistic particle. Thus, we note that the variable (i.e. $\psi_{5})$,
corresponding to the mass term for a massive spinning relativistic particle,
does not contribute anything new to the gauge-fixing and Faddeev-Popov ghost
terms. In other words, the dynamics of our theory (at the BRST quantized
level) is unaffected by the presence of the $``\chi\,\psi_{5}"$ term. This is
a novel observation in our theory which is radically different from our
earlier work [29] where the $``\chi\,\psi_{5}"$ term contributes to the
dynamics. The observations in Eq. (30) also imply that the absolute
anticommutativity property
$\\{s_{b},s_{ab}\\}\,(\chi\,\psi_{5})=i\,\frac{d}{d\tau}[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]$
of the (anti-)BRST symmetries $(s_{(a)b})$ is satisfied (i.e.
$\\{s_{b},s_{ab}\\}=0$) only when the CF-type restriction is imposed from
outside.

According to Noether’s theorem, the continuous symmetry invariance of the
action integrals, corresponding to the transformations (26) and (27), leads to
the derivation of the conserved currents (i.e. conserved charges for our 1D
system) as:

$\displaystyle
J_{B}=i\,B\,\bar{C}\,C\,\dot{C}+B^{2}\,C+B\,e^{2}\,\dot{C}+B\,e\,\dot{e}\,C+\frac{1}{2}\,e\,C\,(p^{2}-m^{2})+i\,\chi\,C\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5}),$
(31) $\displaystyle
J_{\bar{B}}=i\,\bar{B}\,\bar{C}\,\dot{\bar{C}}\,C+{\bar{B}}^{2}\,\bar{C}-\bar{B}\,e^{2}\,\dot{\bar{C}}-\bar{B}\,e\,\dot{e}\,\bar{C}+\frac{1}{2}\,e\,\bar{C}\,(p^{2}-m^{2})+i\,\chi\,\bar{C}\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5}).$
(32)

The conservation law $({dJ_{r}}/{d\tau})=0$ (with $r=B,\bar{B}$) can be proven
by using the EL-EOMs that derived from the Lagrangians $L_{B}$ and
$L_{\bar{B}}$. For instance, we point out that the following EL-EOMs w.r.t.
the variables
($x_{\mu},p_{\mu},\psi_{\mu},\psi_{5},\chi,e,B,\bar{B},C,\bar{C}$), namely;

$\displaystyle{\dot{p}}_{\mu}=0,\qquad{\dot{x}}_{\mu}=e\,p_{\mu}-i\,\chi\,\psi_{\mu},\qquad\psi_{\mu}=\chi\,p_{\mu},\qquad{\dot{\psi}}_{5}=m\,\chi,$
$\displaystyle p_{\mu}\,\psi^{\mu}=m\,\psi_{5},\qquad
B-i\,(2\,\dot{\bar{C}}\,C+i\,e\,\dot{e}+\bar{C}\,\dot{C})=0,$ $\displaystyle
e\,\dot{B}+i\,e\,\dot{\bar{C}}\,\dot{C}-i\,e\,\ddot{\bar{C}}\,C+\frac{1}{2}\,(p^{2}-m^{2})=0,$
$\displaystyle
i\,\dot{B}\,\bar{C}-i\,B\,\dot{\bar{C}}+i\,e\,\dot{e}\,\dot{\bar{C}}+i\,e^{2}\,\ddot{\bar{C}}+\bar{C}\,\ddot{\bar{C}}\,C+2\,\bar{C}\,\dot{\bar{C}}\,\dot{C}=0,$
$\displaystyle-\,i\,B\,\dot{C}-2\,i\,\dot{B}\,C-3\,i\,e\,\dot{e}\,\dot{C}-i\,e^{2}\,\ddot{C}-i\,{\dot{e}}^{2}\,C-i\,e\,\ddot{e}\,C+\bar{C}\,C\,\ddot{C}+2\,\dot{\bar{C}}\,C\,\dot{C}=0,$
(33)

are obtained from $L_{B}$. The equations of motion that are different from
(33) and emerge out from $L_{\bar{B}}$ (as the EL-EOMs) are as follows:

$\displaystyle\bar{B}+i\,(2\,{\bar{C}}\,\dot{C}+i\,e\,\dot{e}+\dot{\bar{C}}\,C)=0,\quad-\,e\,\dot{\bar{B}}+i\,e\,\dot{\bar{C}}\,\dot{C}-i\,e\,{\bar{C}}\,\ddot{C}+\frac{1}{2}\,(p^{2}-m^{2})=0,$
$\displaystyle
i\,\dot{\bar{B}}\,C-i\,\bar{B}\,\dot{C}-i\,e\,\dot{e}\,\dot{C}-i\,e^{2}\,\ddot{C}+\bar{C}\,C\,\ddot{C}+2\,\dot{\bar{C}}\,C\,\dot{C}=0,$
$\displaystyle-\,i\,\bar{B}\,\dot{\bar{C}}-2\,i\,\dot{\bar{B}}\,\bar{C}+3\,i\,e\,\dot{e}\,\dot{\bar{C}}+i\,e^{2}\,\ddot{\bar{C}}+i\,{\dot{e}}^{2}\,\bar{C}+i\,e\,\ddot{e}\,\bar{C}+\bar{C}\,\ddot{\bar{C}}\,C+2\,{\bar{C}}\,\dot{\bar{C}}\,\dot{C}=0.$
(34)

These conserved currents $(J_{B},J_{\bar{B}})$ lead to the definition of the
conserved charges $Q_{B}$ and $Q_{\bar{B}}$ which are same as the conserved
currents quoted in Eqs. (31) and (32). This is due to the fact that we are
dealing with a 1D system of a massive SUSY relativistic particle.

We close this section with the following key comments. First, we observe that
the off-shell nilpotent (anti-)BRST symmetry transformations (20) and (21) are
absolutely anticommuting in nature provided we invoke the (anti-)BRST
invariant CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ from outside. Second,
this restriction can be obtained from the coupled (but equivalent) Lagrangians
$L_{B}$ and $L_{\bar{B}}$ if we use the EL-EOMs [cf. Eqs. (25)] w.r.t. the
Nakanishi-Lautrup type auxiliary variables $B$ and $\bar{B}$. Third, we
observe that the term $(\chi\,\psi_{5})$ in the square bracket of Eq. (23)
does not contribute anything to the dynamics as well as to the gauge-fixing
and Faddeev-Popov ghost terms. Fourth, the coupled Lagrangians $L_{B}$ and
$L_{\bar{B}}$ are equivalent in the sense that both of them respect both off-
shell nilpotent (anti-)BRST symmetries on a submanifold of the quantum
variables where the CF-type constraint:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is satisfied. This key
observation is an alternative proof of the existence of CF-type restriction on
our theory. Finally, we observe that the absolute anticommutativity (i.e.
$\\{Q_{B},Q_{\bar{B}}\\}=0$) of the conserved (i.e.
${\dot{Q}}_{(\bar{B})B}=0$) and off-shell nilpotent (i.e.
$Q_{(\bar{B})B}^{2}=0$) (anti-)BRST charges $(Q_{(\bar{B})B})$ is satisfied
only due to the validity of the existence of the CF-type restriction (cf. Sec.
6 below). The CF-type restrictions on our present theory is important as it is
connected with the geometrical object called gerbs [30, 31] which physically
imply the independent existence of the BRST and anti-BRST symmetries and their
corresponding nilpotent (anti-)BRST charges.

##  4 Off-Shell Nilpotent Symmetries of the Target Space Variables and CF-Type
Restriction: MBTSA

In this section, we derive the off-shell nilpotent (anti-)BRST symmetry
transformations for the target space variables
$(x_{\mu},\,p_{\mu},\,\psi_{\mu},\,\psi_{5})$ which are scalars w.r.t. the 1D
space of the trajectory for the massive spinning relativistic particle that is
embedded in the D-dimensional target space. For this purpose, we exploit the
theoretical power and potential of MBTSA (see e.g. [26]). Towards this goal in
mind, first of all, we generalize the 1D diffeomorphism transformation***Here
$f(\tau)$ is any arbitrary function of the evolution parameter $\tau$ such
that it is finite at $\tau=0$ and vanishes off at $\tau=\pm\infty$. In other
words, $f(\tau)$ is a physically well-defined function of $\tau$.
$\tau\,\longrightarrow\tau^{{}^{\prime}}=f\,(\tau)\equiv\tau-\epsilon\,(\tau)$
[where $\epsilon\,(\tau)$ is the infinitesimal transformation parameter
corresponding to the 1D diffeomorphism (i.e. reparameterization) symmetry
transformations] to its counterpart on the (1, 2)-dimensional supermanifold as

$\displaystyle f\,(\tau)\,\longrightarrow
f\,(\tau,\,\theta,\,\bar{\theta})=\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,h\,(\tau),$
(35)

where the general (1, 2)-dimensional supermanifold is parameterized by a
bosonic (i.e. evolution) coordinate $\tau$ and a pair of Grassmannian
variables $(\theta,\bar{\theta})$ that satisfy:
$\theta^{2}=\bar{\theta}^{2}=0,\,\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$.
It is worth pointing out that the coefficients of the Grassmannian variables
$(\theta,\bar{\theta})$, in Eq. (35), are nothing but the fermionic (i.e.
$C^{2}={\bar{C}}^{2}=0,\,C\,\bar{C}+\bar{C}\,C=0$) (anti-)ghost variables
$(\bar{C})\,C$ of the (anti-)BRST symmetry transformations (20) and (21)
corresponding to the infinitesimal reparameterization symmetry transformations
(2). In other words, the infinitesimal reparameterization bosonic
transformation parameter $\epsilon(\tau)$ has been replaced by the fermionic
(anti-)ghost variables $(\bar{C})C$ of the (anti-)BRST symmetry
transformations. This has been done purposely, in view of the fact that, in an
earlier work (see e.g. [10-12]), it has been established that the
translational generators $(\partial_{\theta},\partial_{\bar{\theta}})$, along
the Grassmannian directions $(\theta,\bar{\theta})$, are intimately connected
with the nilpotent (anti-)BRST transformations $s_{(a)b}$ in the ordinary
space. In other words, we have already taken into account
$s_{ab}\,\tau=-\,\bar{C},s_{b}\,\tau=-\,C$ which are the generalization of the
classical infinitesimal 1D diffeomorphism symmetry transformation:
$\delta_{r}\,\tau=-\,\epsilon(\tau)$ to its quantum counterparts
$(s_{(a)\,b})$ within the framework of BRST formalism. It is worthwhile to
point out that the secondary variable $h(\tau)$ of the expansion (37) has to
be determined from other consistency conditions which we elaborate in our
forthcoming paragraphs.

According to the basic tenets of MBTSA, we have to generalize all the ordinary
variables of the Lagrangians (24) onto the (1, 2)-dimensional supermanifold as
their counterparts supervariables where the generalization of the 1D
diffeomorphism transformation [cf. Eq. (35)] has to be incorporated (in a
suitable fashion) into the expressions for the supervariables. For instance,
we shall have the following generalization as far as the generic target space
variable $s_{\mu}(\tau)$ [cf. Eq. (22)] is concerned, namely;

$\displaystyle
s_{\mu}(\tau)\quad\longrightarrow\quad{\tilde{S}}_{\mu}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})\equiv{\tilde{S}}_{\mu}(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,h,\theta,\bar{\theta}),$
(36)

where the pair of variables $(\theta,\bar{\theta})$, as pointed out earlier,
are the Grassmannian variables (i.e.
$\theta^{2}={\bar{\theta}}^{2}=0,\;\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$)
of the superspace coordinates $Z^{M}=(\tau,\theta,\bar{\theta})$ that
characterize the (1, 2)-dimensional supermanifold on which our 1D ordinary
theory of the reparameterization invariant massive spinning particle is
considered. Now, following the techniques of MBTSA, we have the following
super expansion of (36) along all the possible directions of the Grassmannian
variables of the $(1,2)$-dimensional supermanifold, namely;

$\displaystyle{\tilde{S}}_{\mu}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})=S_{\mu}(f(\tau,\theta,\bar{\theta}))+\theta\,{\bar{R}}_{\mu}(f(\tau,\theta,\bar{\theta}))+\bar{\theta}\,R_{\mu}(f(\tau,\theta,\bar{\theta}))+\theta\,\bar{\theta}\,P_{\mu}(f(\tau,\theta,\bar{\theta})),$
(37)

where, on the r.h.s., we have the secondary supervariables which are also
function of $f(\tau,\theta,\bar{\theta})$. As a consequence, we can have the
following Taylor expansions (for those secondary variables that are present on
the r.h.s.), namely;

$\displaystyle
S_{\mu}\,(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,h)=s_{\mu}\,(\tau)-\theta\,\bar{C}\,{\dot{s}_{\mu}\,(\tau)-\bar{\theta}\,C\dot{s}_{\mu}\,(\tau)+\theta\bar{\theta}\,(h\,\dot{s}_{\mu}-\bar{C}\,C\,\ddot{s}_{\mu})},$
$\displaystyle\theta\,\bar{R}_{\mu}\,(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,h)\,\equiv\,\theta\,\bar{R}_{\mu}(\tau)-\theta\bar{\theta}\,\dot{\bar{R_{\mu}}}\,(\tau),$
$\displaystyle\bar{\theta}\,R_{\mu}\,(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,h)\,\equiv\,\bar{\theta}\,R_{\mu}(\tau)+\theta\bar{\theta}\,\dot{R_{\mu}}\,(\tau),$
$\displaystyle\theta\,\bar{\theta}\,P_{\mu}\,(\tau-\theta\,\bar{C}-\bar{\theta}\,C+\theta\,\bar{\theta}\,h)\,\equiv\,\theta\,\bar{\theta}\,P_{\mu}(\tau).$
(38)

At this juncture, we would like to lay stress on the fact that, in the super
expansion (37), all the supervariables on the r.h.s. have to be ordinary
variables as all of them are Lorentz scalars w.r.t. the 1D trajectory of the
particle†††A pure Lorentz (bosonic or fermionic) scalar is the one which does
not transform at all under any kind of spacetime and/or internal
transformations. (that is embedded in the D-dimensional flat Minkowskian
target space). As a result, the expansion (37) can be written as:

$\displaystyle\tilde{S}_{\mu}\,(f\,(\tau,\,\theta,\,\bar{\theta}),\,\theta,\,\bar{\theta})$
$\displaystyle=$ $\displaystyle
s_{\mu}\,(\tau)+\theta\,\bar{R}_{\mu}\,(\tau)+\bar{\theta}\,R_{\mu}\,(\tau)+\theta\,\bar{\theta}\,P_{\mu}\,(\tau)$
(39) $\displaystyle\equiv$ $\displaystyle
s_{\mu}\,(\tau)+\theta\,(s_{ab}\,s_{\mu}\,(\tau))+\bar{\theta}\,(s_{b}\,s_{\mu}\,(\tau))+\theta\bar{\theta}\,(s_{b}\,s_{ab}\,s_{\mu}\,(\tau)),$

where $s_{(a)b}$ are the (anti-)BRST symmetry transformations (20) and (21).
This is due to fact that the (anti-)BRST symmetry transformations $s_{(a)b}$
have been shown to be deeply connected with the translational generators
$(\partial_{\theta},\partial_{\bar{\theta}})$ along the
$(\theta,\bar{\theta})$-directions of the (1, 2)-dimensional supermanifold
(see e.g. [10-12] for details).

It is evident that we have to compute the values of $R_{\mu},\bar{R}_{\mu}$
and $P_{\mu}$ [in terms of the auxiliary and basic variables of the
Lagrangians (24)] so that we can obtain the off-shell nilpotent (anti-)BRST
symmetry transformations for the generic variable $s_{\mu}(\tau)$. At this
stage, the so called “horizontality condition” (HC) comes to our help where we
demand that:
$\tilde{S}_{\mu}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})=s_{\mu}(\tau).$
This relationship can be explicitly expressed as

$\displaystyle
s_{\mu}(\tau)+\theta\,(\bar{R}_{\mu}-\bar{C}\,\dot{s}_{\mu})+\bar{\theta}\,(R_{\mu}-C\,\dot{s}_{\mu})$
$\displaystyle+\theta\,\bar{\theta}\,[h\,\dot{s}_{\mu}-\bar{C}\,C\,\ddot{x}_{\mu}-C\,\dot{\bar{R_{\mu}}}+\bar{C}\,\dot{R}_{\mu}+P_{\mu}]\equiv
s_{\mu}(\tau),$ (40)

where we have collected all the terms from Eq. (38) to express (36).
Physically, the above requirement corresponds to the fact that a Lorentz-
scalar does not transform under any kind of physically well-defined spacetime
transformations . Needless to say, the relationship (40) implies that we have
the following explicit relationships:

$\displaystyle
R_{\mu}=C\,\dot{s}_{\mu},\quad\bar{R}_{\mu}=\bar{C}\,{\dot{s}}_{\mu},\quad
P_{\mu}=C\,\dot{\bar{R_{\mu}}}-\bar{C}\,\dot{R}_{\mu}+\bar{C}\,C\,\dot{s}_{\mu}-h\,\ddot{s}_{\mu}.$
(41)

It is straightforward to note that we have already obtained
$s_{b}\,s_{\mu}=C\,{\dot{s}}_{\mu}$ and
$s_{ab}\,s_{\mu}=\bar{C}\,{\dot{s}}_{\mu}$ as is evident from Eq. (39). The
requirement of the absolute anticommutativity (that is one of the sacrosanct
properties of the (anti-)BRST symmetry transformations) implies that we have
the following equalities, namely;

$\displaystyle
s_{b}\,s_{ab}\,s_{\mu}=-\,s_{ab}\,s_{b}\,s_{\mu}\quad\implies\quad\\{s_{b},s_{ab}\\}\,s_{\mu}=0.$
(42)

On the other hand, the requirement of the off-shell nilpotency (that is
another sacrosanct property of the (anti-)BRST symmetry transformations) leads
to the following:

$\displaystyle s_{b}\,C=C\,\dot{C},\qquad\qquad
s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}}.$ (43)

On top of the already obtained off-shell nilpotent (anti-)BRST symmetry
transformations:
$s_{b}\,s_{\mu}=C\,{\dot{s}}_{\mu},\,s_{ab}\,s_{\mu}=\bar{C}\,{\dot{s}}_{\mu},\,s_{b}\,C=C\,\dot{C},\,s_{ab}\,\bar{C}=\bar{C}\,\dot{\bar{C}},$
we take into account the standard (anti-)BRST symmetry transformations
$s_{b}\,\bar{C}=i\,B$ and $s_{ab}\,C=i\,\bar{B}$ in terms of the Nakanishi-
Lautrup auxiliary variables. These standard inputs lead to the determination
of the l.h.s. and r.h.s. of the first equality in Eq. (42) as:

$\displaystyle
s_{b}\,s_{ab}\,s_{\mu}=(i\,B-\bar{C}\,\dot{C})\,{\dot{s}}_{\mu}-\bar{C}\,C\,{\ddot{s}}_{\mu}\equiv
P_{\mu}(\tau),$
$\displaystyle-\,s_{ab}\,s_{b}\,s_{\mu}=(-i\,\bar{B}-\dot{\bar{C}}\,C)\,{\dot{s}}_{\mu}-\bar{C}\,C\,{\ddot{s}}_{\mu}\equiv
P_{\mu}(\tau),$ (44)

where $P_{\mu}\,(\tau)$ is present in the expansion (39). A close look at Eq.
(44) implies that we have:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ which is nothing but the
CF-type restriction. In addition, the observation of Eq. (41) implies that
there is an explicit expression for $P_{\mu}$ in terms of $h(\tau)$ [that is
present in the expansion of $f(\tau,\theta,\bar{\theta})$ in Eq. (35)].
Plugging in the values of $R_{\mu}=C\,\dot{s}_{\mu}$ and
$\bar{R}_{\mu}=\bar{C}\,\dot{s}_{\mu}$ into Eq. (41) leads to

$\displaystyle
P_{\mu}\,(\tau)=-\,[(\dot{\bar{C}}\,C+\bar{C}\,\dot{C}+h)\,\dot{s}_{\mu}+\bar{C}\,C\,\ddot{s}_{\mu}].$
(45)

Comparison of (44) and (45) yields:

$\displaystyle
h=-\,i\,B-\dot{\bar{C}}\,C\,\equiv\,+\,i\,\bar{B}-\bar{C}\,\dot{C}\,\implies\,B+\bar{B}+i\,(\bar{C}\,\dot{C}+\dot{\bar{C}}\,C)=0.$
(46)

Thus, we note that the comparison of the values of $h\,(\tau)$ [that is
determined from the comparison between Eq. (44) and Eq. (45)] leads to the
deduction of the (anti-)BRST invariant (i.e.
$s_{(a)b}\,[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)]=0$) CF-type
restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ which plays
an important role in the proof: $\\{s_{b},s_{ab}\\}=0$.

We wrap-up this section with the following useful and important remarks.
First, we have taken into account the standard choice in the BRST formalism
which is: $s_{b}\,\bar{C}=i\,B,\,s_{ab}\,C=i\,\bar{B}$. In other words, we
have made the following (anti-)chiral super expansions for the (anti-)chiral
supervariables (in view of $\partial_{\bar{\theta}}\leftrightarrow
s_{ab},\,\partial_{\theta}\leftrightarrow s_{b},$), namely;

$\displaystyle
C\,(\tau)\,\longrightarrow\,F^{(c)}\,(\tau,\,\theta)=\,C\,(\tau)+\theta\,[i\,\bar{B}(\tau)]\equiv
C\,(\tau)+\theta\,[s_{ab}\,C(\tau)],$
$\displaystyle\bar{C}\,(\tau)\,\longrightarrow\,\bar{F}^{(ac)}\,(\tau,\,\bar{\theta})=\,\bar{C}\,(\tau)+\bar{\theta}\,[i\,B(\tau)]\equiv\bar{C}\,(\tau)+\bar{\theta}\,[s_{b}\,\bar{C}(\tau)],$
(47)

where $F^{(c)}\,(\tau,\,\theta)$ and $\bar{F}^{(ac)}\,(\tau,\,\bar{\theta})$
are the chiral and anti-chiral supervariables that have been defined on the
(1, 1)-dimensional suitably chosen chiral and anti-chiral super sub-manifolds
of the general (1, 2)-dimensional supermanifold. Second, we have seen that
[cf. Eq. (22)] the absolute anticommutativity property (i.e.
$\\{s_{b},s_{ab}\\}=0$) of the (anti-)BRST transformations $(s_{(a)b})$ is
satisfied if and only if the CF-type restriction (46) is satisfied. Third, we
point out that the requirement of the following

$\displaystyle\\{s_{b},s_{ab}\\}\,C=0\;\implies\;s_{b}\,{\bar{B}}=\dot{\bar{B}}\,{C}-\bar{B}\,\dot{C},$
$\displaystyle\\{s_{b},s_{ab}\\}\,{\bar{C}}=0\;\implies\;s_{ab}\,{B}=\dot{B}\,{\bar{C}}-B\,\dot{\bar{C}}.$
(48)

leads to the derivation of (anti-)BRST symmetry transformations on the
Nakanishi-Lautrup auxiliary variables $(B)\bar{B}$. Fourth, within the
framework of MBTSA, the CF-type restriction (46) is derived from the
expression for $h(\tau)$ due to the consistency condition (i.e.
$s_{b}s_{ab}\,s_{\mu}=-\,s_{ab}s_{b}\,s_{\mu}\equiv P_{\mu}$). Fifth, the
(anti-)BRST symmetry transformations
($s_{b}s_{\mu}={C}\,{\dot{s}}_{\mu},\,\,s_{ab}s_{\mu}={\bar{C}}\,{\dot{s}}_{\mu}$)
on the generic variable $s_{\mu}\equiv
x_{\mu},\,p_{\mu},\,\psi_{\mu},\,\psi_{5}$ imply that we have already obtained
the following (anti-)BRST symmetry transformations

$\displaystyle s_{ab}\;x_{\mu}={\bar{C}}\;{\dot{x}}_{\mu},\qquad
s_{ab}\;p_{\mu}={\bar{C}}\;{\dot{p}}_{\mu},\qquad
s_{ab}\;\psi_{\mu}={\bar{C}}\;{\dot{\psi}}_{\mu},\qquad
s_{ab}\,\psi_{5}={\bar{C}}\;{\dot{\psi}}_{5},$ $\displaystyle
s_{b}\;x_{\mu}={C}\;{\dot{x}}_{\mu},\qquad\;s_{b}\;p_{\mu}={C}\;{\dot{p}}_{\mu},\qquad\;s_{b}\;\psi_{\mu}={C}\;{\dot{\psi}}_{\mu},\qquad
s_{b}\,\psi_{5}={C}\;{\dot{\psi}}_{5},$ (49)

for the target space variables $(x_{\mu},p_{\mu},\psi_{\mu},\psi_{5})$ of our
theory that are present in the first-order Lagrangian $L_{f}$ [cf. Eq. (1)]
for the massive relativistic particle. Finally, the explicit form of Eq. (39)
can be written, after the application of HC, as follows

$\displaystyle X_{\mu}^{(h)}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})$
$\displaystyle=$ $\displaystyle
x_{\mu}(\tau)+\theta\,(\bar{C}\,{\dot{x}}_{\mu})+\bar{\theta}\,(C\,{\dot{x}}_{\mu})+\theta\,\bar{\theta}\,[(i\,B-\bar{C}\,\dot{C})\,{\dot{x}}_{\mu}-\bar{C}\,C\,{\ddot{x}}_{\mu}],$
$\displaystyle\equiv$ $\displaystyle
x_{\mu}(\tau)+\theta\,(s_{ab}\,x_{\mu})+\bar{\theta}\,(s_{b}\,x_{\mu})+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,x_{\mu}),$
$\displaystyle P_{\mu}^{(h)}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})$
$\displaystyle=$ $\displaystyle
p_{\mu}(\tau)+\theta\,(\bar{C}\,{\dot{p}}_{\mu})+\bar{\theta}\,(C\,{\dot{p}}_{\mu})+\theta\,\bar{\theta}\,[(i\,B-\bar{C}\,\dot{C})\,{\dot{p}}_{\mu}-\bar{C}\,C\,{\ddot{p}}_{\mu}],$
$\displaystyle\equiv$ $\displaystyle
p_{\mu}(\tau)+\theta\,(s_{ab}\,p_{\mu})+\bar{\theta}\,(s_{b}\,p_{\mu})+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,p_{\mu}),$
$\displaystyle\Psi_{\mu}^{(h)}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})$
$\displaystyle=$
$\displaystyle\psi_{\mu}(\tau)+\theta\,(\bar{C}\,{\dot{\psi}}_{\mu})+\bar{\theta}\,(C\,{\dot{\psi}}_{\mu})+\theta\,\bar{\theta}\,[(i\,B-\bar{C}\,\dot{C})\,{\dot{\psi}}_{\mu}-\bar{C}\,C\,{\ddot{\psi}}_{\mu}],$
$\displaystyle\equiv$
$\displaystyle\psi_{\mu}(\tau)+\theta\,(s_{ab}\,\psi_{\mu})+\bar{\theta}\,(s_{b}\,\psi_{\mu})+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,\psi_{\mu}),$
$\displaystyle\Psi_{5}^{(h)}(f(\tau,\theta,\bar{\theta}),\theta,\bar{\theta})$
$\displaystyle=$
$\displaystyle\psi_{5}(\tau)+\theta\,(\bar{C}\,{\dot{\psi}}_{5})+\bar{\theta}\,(C\,{\dot{\psi}}_{5})+\theta\,\bar{\theta}\,[(i\,B-\bar{C}\,\dot{C})\,{\dot{\psi}}_{5}-\bar{C}\,C\,{\ddot{\psi}}_{5}],$
(50) $\displaystyle\equiv$
$\displaystyle\psi_{5}(\tau)+\theta\,(s_{ab}\,\psi_{5})+\bar{\theta}\,(s_{b}\,\psi_{5})+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,\psi_{5})$

where the superscript $(h)$ denotes the full expansion of the supervariables
after the application of HC. A straightforward comparison of (39) with (50)
shows that we have already derived the (anti-)BRST symmetry transformations
(49) as the coefficients of $(\theta)\bar{\theta}$ in the super expansions
(50) along with
$s_{b}\,s_{ab}\,x_{\mu},s_{b}\,s_{ab}\,\psi_{\mu},s_{b}\,s_{ab}\,p_{\mu},s_{b}\,s_{ab}\,\psi_{5}$
which are the coefficients of $\theta\bar{\theta}$. We also note, from Eq.
(50), that we have a mapping:
$s_{b}\,\leftrightarrow\,\partial_{\bar{\theta}}|_{\theta=0},\,s_{ab}\,\leftrightarrow\,\partial_{\theta}|_{\bar{\theta}=0}$.
This observation is consistent with results obtained in the Refs. [10-12].

##  5 Off-Shell Nilpotent (Anti-)BRST Symmetries for Other Variables of Our
Theory: ACSA

In this section, we exploit the basic principle behind ACSA to BRST formalism
to obtain, first of all, the off-shell nilpotent BRST symmetry transformations
(21) by generalizing the basic and auxiliary variables of the Lagrangian
$L_{B}$ [cf. Eq. (24)] on the anti-chiral $(1,1)$-dimensional super sub-
manifold [of the most general (1, 2)-dimensional supermanifold] as

$\displaystyle B(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad\tilde{B}(\tau,\bar{\theta})=B(\tau)+\bar{\theta}\,f_{1}(\tau),$
$\displaystyle e(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad E(\tau,\bar{\theta})=e(\tau)+\bar{\theta}\,f_{2}(\tau),$
$\displaystyle\chi(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad
K(\tau,\bar{\theta})=\chi(\tau)+\bar{\theta}\,b_{1}(\tau),$ $\displaystyle
C(\tau)\quad$ $\displaystyle\longrightarrow$ $\displaystyle\quad
F(\tau,\bar{\theta})=C(\tau)+\bar{\theta}\,b_{2}(\tau),$
$\displaystyle\bar{C}(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad\bar{F}(\tau,\bar{\theta})=\bar{C}(\tau)+\bar{\theta}\,b_{3}(\tau),$
$\displaystyle\bar{B}(\tau)\quad$ $\displaystyle\longrightarrow$
$\displaystyle\quad\tilde{\bar{B}}(\tau,\bar{\theta})=\bar{B}(\tau)+\bar{\theta}\,f_{3}(\tau),$
(51)

where $(f_{1},f_{2},f_{3})$ are the fermionic and $(b_{1},b_{2},b_{3})$ are
bosonic secondary variables which are to be determined in terms of the basic
and auxiliary variables of $L_{B}$ by exploiting the quantum gauge (i.e. BRST)
invariant restrictions. It will be noted that the anti-chiral (1,
1)-dimensional super sub-manifold is parameterized by the superspace
coordinates $Z^{M}=(\tau,\,\bar{\theta})$ where $\tau$ is the bosonic
evolution parameter and Grassmannian variable $\bar{\theta}$ is fermionic
(${\bar{\theta}}^{2}=0$) in nature. In addition to (51), we have the anti-
chiral limit (i.e. $\theta=0$) of the expansions (50) as follows

$\displaystyle
X_{\mu}^{(ha)}(\tau,\bar{\theta})=x_{\mu}(\tau)+\bar{\theta}\,(C\,{\dot{x}}_{\mu}),\quad\qquad\Psi_{\mu}^{(ha)}(\tau,\bar{\theta})=\psi_{\mu}(\tau)+\bar{\theta}\,(C\,{\dot{\psi}}_{\mu}),$
$\displaystyle
P_{\mu}^{(ha)}(\tau,\bar{\theta})=p_{\mu}(\tau)+\bar{\theta}\,(C\,{\dot{p}}_{\mu}),~{}\quad\qquad\Psi_{5}^{(ha)}(\tau,\bar{\theta})=\psi_{5}(\tau)+\bar{\theta}\,(C\,{\dot{\psi}}_{5}),$
(52)

where the superscript $(ha)$ denotes the anti-chiral limit of the super
expansions (of the supervariables [cf. Eq. (50)]) that have been obtained
after the application of HC. It is straightforward to note that the BRST
invariance ($s_{b}\,B=0$) of the variable $B$ implies that we have the
following (with $f_{1}(\tau)=0$), namely;

$\displaystyle{\tilde{B}}(\tau,\bar{\theta})\quad\longrightarrow\quad{\tilde{B}}^{(b)}(\tau,\bar{\theta})=B(\tau)+\bar{\theta}\,(0)=B(\tau)+\bar{\theta}\,(s_{b}\,B),$
(53)

where the superscript $(b)$ stands for the anti-chiral supervariable that has
been obtained after the BRST invariant $(s_{b}\,B=0)$ restriction. In other
words, we have already obtained the BRST symmetry transformation $s_{b}\,B=0$
as the coefficient of $\bar{\theta}$ in (53) due to our knowledge of:
$s_{b}\leftrightarrow\partial_{\bar{\theta}}$ [i.e.
$\partial_{\bar{\theta}}\,{\tilde{B}}^{(b)}(\tau,\bar{\theta})=s_{b}\,B=0$].

The off-shell nilpotency of the BRST symmetry transformations (21) ensures
that we have the following BRST invariant quantities:

$\displaystyle s_{b}\,(C\,{\dot{x}}_{\mu})=0,\quad
s_{b}\,(C\,{\dot{p}}_{\mu})=0,\quad s_{b}\,(C\,\dot{C})=0,\quad
s_{b}\,(C\,\dot{e}+\dot{C}\,e)=0,$ $\displaystyle
s_{b}\,(\dot{C}\,\chi+C\,\dot{\chi})=0,\quad
s_{b}\,(C\,{\dot{\psi}}_{\mu})=0,\quad s_{b}(C\,{\dot{\psi}}_{5})=0,\quad
s_{b}\,(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})=0.$ (54)

The above quantum gauge (i.e. BRST) invariant quantities must be independent
of the Grassmannian variable $\bar{\theta}$ when they are generalized onto
$(1,1)$-dimensional anti-chiral super sub-manifold [of the most general
$(1,2)$-dimensional supermanifold)] on which our 1D ordinary theory has been
generalized. In other words, we have the validity of the following equalities
in terms of the supervariables and ordinary variables; namely;

$\displaystyle
F(\tau,\bar{\theta})\,{\dot{X}}_{\mu}^{(ha)}(\tau,\bar{\theta})=C(\tau)\,{\dot{x}}_{\mu}(\tau),\qquad\quad
F(\tau,\bar{\theta})\,{\dot{F}}(\tau,\bar{\theta})=C(\tau)\,{\dot{C}}(\tau)$
$\displaystyle
F(\tau,\bar{\theta})\,{\dot{\Psi}}_{\mu}^{(ha)}(\tau,\bar{\theta})=C(\tau)\,{\dot{\psi}}_{\mu}(\tau),\qquad\quad
F(\tau,\bar{\theta})\,{\dot{\Psi}}_{5}^{(ha)}(\tau,\bar{\theta})=C(\tau)\,{\dot{\psi}}_{\mu}(\tau),$
$\displaystyle
F(\tau,\bar{\theta})\,{\dot{P}}_{\mu}^{(ha)}(\tau,\bar{\theta})=C(\tau)\,{\dot{p}}_{\mu}(\tau),\qquad\quad{\tilde{B}}^{(b)}(\tau,\bar{\theta})=B(\tau),$
$\displaystyle\dot{F}(\tau,\bar{\theta})\,E(\tau,\bar{\theta})+F(\tau,\bar{\theta})\,\dot{E}(\tau,\bar{\theta})=\dot{C}(\tau)\,e(\tau)+C(\tau)\,\dot{e}(\tau),$
$\displaystyle\dot{F}(\tau,\bar{\theta})\,K(\tau,\bar{\theta})+F(\tau,\bar{\theta})\,\dot{K}(\tau,\bar{\theta})=\dot{C}(\tau)\,\chi(\tau)+C(\tau)\,\dot{\chi}(\tau),$
$\displaystyle\dot{\tilde{\bar{B}}}(\tau,\bar{\theta})\,F(\tau,\bar{\theta})-\tilde{\bar{B}}(\tau,\bar{\theta})\,{\dot{F}}(\tau,\bar{\theta})=\dot{\bar{B}}(\tau)\,C(\tau)-{\bar{B}}(\tau)\,\dot{C}(\tau),$
(55)

where the supervariables with superscripts $(ha)$ and $(b)$ have been already
explained and derived in Eqs. (52) and (53). The substitutions of the
expansions from (52) and (51) lead to the determination of the secondary
variables of the latter equation [cf. Eq. (51)] as:

$\displaystyle f_{2}(\tau)=\dot{C}\,e+C\,\dot{e},\qquad
b_{1}(\tau)=\dot{C}\,\chi+C\,\dot{\chi},\qquad b_{2}(\tau)=C\,\dot{C}$
$\displaystyle b_{3}(\tau)=i\,B,\qquad
f_{3}(\tau)=\dot{\bar{B}}\,C-\bar{B}\,\dot{C}.$ (56)

The above relationships demonstrate that we have already obtained the
secondary variables of the super expansion (51) in terms of the auxiliary and
basic variables of $L_{B}$ (and the Nakanishi-Lautrup auxiliary variable
$\bar{B}(\tau)$ of the Lagrangian $L_{\bar{B}}$ [cf. Eq. (24)]).

The substitutions of the above expressions for the secondary variables into
the super expansions (51) [besides Eqs. (52), (53)] are as follows

$\displaystyle
E^{(b)}(\tau,\bar{\theta})=e(\tau)+\bar{\theta}\,(e\,\dot{C}+\dot{e}\,C)\equiv
e(\tau)+\bar{\theta}\,(s_{b}\,e),$ $\displaystyle
K^{(b)}(\tau,\bar{\theta})=\chi(\tau)+\bar{\theta}\,(C\,\dot{\chi}+\dot{C}\,\chi)\equiv\chi(\tau)+\bar{\theta}\,(s_{b}\,\chi),$
$\displaystyle
F^{(b)}(\tau,\bar{\theta})=C(\tau)+\bar{\theta}\,(C\,\dot{C})\equiv
C(\tau)+\bar{\theta}\,(s_{b}\,C),$
$\displaystyle{\bar{F}}^{(b)}(\tau,\bar{\theta})=\bar{C}(\tau)+\bar{\theta}\,(i\,B)\equiv\bar{C}(\tau)+\bar{\theta}\,(s_{b}\,\bar{C}),$
$\displaystyle{\tilde{\bar{B}}}^{(b)}(\tau,\bar{\theta})=\bar{B}(\tau)+\bar{\theta}\,(\dot{\bar{B}}\,C-\bar{B}\,\dot{C})\equiv\bar{B}(\tau)+\bar{\theta}\,(s_{b}\,\bar{B}),$
(57)

where the superscript $(b)$ on the supervariables denotes the anti-chiral
supervariables that have been obtained after the imposition of the BRST (i.e.
quantum gauge) invariant restrictions in Eq. (55). It is clear from (57) that
we have a mapping: $\partial_{\bar{\theta}}\leftrightarrow s_{b}$ which
demonstrates that the off-shell nilpotency $(\partial_{\bar{\theta}}^{2}=0)$
of the translational generator $\partial_{\bar{\theta}}$ along
$\bar{\theta}$-direction of the $(1,1)$-dimensional anti-chiral super sub-
manifold and off-shell nilpotency $(s_{b}^{2}=0)$ of the BRST transformations
(21) in the ordinary space are interrelated. A careful look at Eqs. (50) and
(57) demonstrate that we have already derived the BRST symmetry
transformations (21) for all the variables of $L_{B}$ as the coefficients of
$\bar{\theta}$.

Now we dwell a bit on the derivation of the anti-BRST transformations (20)
within the framework of ACSA. Towards this end in mind, we note that the
following are the chiral (i.e. $\bar{\theta}=0$) limit of the full super
expansions in (50), namely;

$\displaystyle
X_{\mu}^{(hc)}(\tau,\theta)=x_{\mu}(\tau)+\theta\,(\bar{C}\,{\dot{x}}_{\mu})\equiv
x_{\mu}(\tau)+\theta\,(s_{ab}\,x_{\mu}),$ $\displaystyle
P_{\mu}^{(hc)}(\tau,\theta)=p_{\mu}(\tau)+\theta\,(\bar{C}\,{\dot{p}}_{\mu})\equiv
p_{\mu}(\tau)+\theta\,(s_{ab}\,p_{\mu}),$
$\displaystyle\Psi_{\mu}^{(hc)}(\tau,\theta)=\psi_{\mu}(\tau)+\theta\,(\bar{C}\,{\dot{\psi}}_{\mu})\equiv\psi_{\mu}(\tau)+\theta\,(s_{ab}\,\psi_{\mu}),$
$\displaystyle\Psi_{5}^{(hc)}(\tau,\theta)=\psi_{5}(\tau)+\theta\,(\bar{C}\,{\dot{\psi}}_{5})\equiv\psi_{5}(\tau)+\theta\,(s_{ab}\,\psi_{5}),$
(58)

where the superscript $(hc)$ stands for the chiral limit of the supervariables
that have been derived after the application of HC in Eq. (50). The above
expansions in (58) would be utilized in the anti-BRST invariant restrictions
on the chiral supervariables which we are going to discuss now. It can be
readily checked that we have the following interesting anti-BRST invariant
quantities

$\displaystyle s_{ab}\,(\bar{C}\,{\dot{x}}_{\mu})=0,\quad
s_{ab}\,(\bar{C}\,{\dot{p}}_{\mu})=0,\quad
s_{ab}\,(\bar{C}\,\dot{\bar{C}})=0,\quad
s_{ab}\,(\dot{\bar{C}}\,e+\bar{C}\,\dot{e})=0,$ $\displaystyle
s_{ab}\,(\dot{\bar{C}}\,\chi+\bar{C}\,\dot{\chi})=0,\quad
s_{ab}\,(\bar{C}\,{\dot{\psi}}_{\mu})=0,\quad
s_{ab}(\bar{C}\,{\dot{\psi}}_{5})=0,\quad
s_{ab}\,(\dot{B}\,\bar{C}-B\,\dot{\bar{C}})=0,$ (59)

where the anti-BRST symetry transformations $(s_{ab})$ are the ones that have
been listed in Eq. (20). Keeping in our mind the mapping:
$s_{ab}\leftrightarrow\partial_{\theta}$ and the observation
$s_{ab}\,\bar{B}=0$,we have the following

$\displaystyle{\tilde{\bar{B}}}(\tau,\theta)\quad\longrightarrow\quad{\tilde{\bar{B}}}^{(ab)}(\tau,\theta)=\bar{B}(\tau)+\theta\,(0)=\bar{B}(\tau)+\theta\,(s_{ab}\,\bar{B}),$
(60)

where the superscript $(ab)$ denotes the expansions of the supervariables that
has been obtained after the application of the anti-BRST invariant
restriction: $\tilde{\bar{B}}(\tau,\theta)=\bar{B}(\tau)$ that is obtained due
to the anti-BRST invariance $(s_{ab}\,\bar{B}=0)$. We also note that
$\partial_{\theta}\,{\tilde{\bar{B}}}^{(ab)}(\tau,\theta)=s_{ab}\,\bar{B}=0$.
Exploiting the basic principle of ACSA to BRST formalism, we obtain the
following equalities in terms of chiral and ordinary variables, namely;

$\displaystyle\bar{F}(\tau,\theta)\,{\dot{X}}_{\mu}^{(hc)}(\tau,\theta)=\bar{C}(\tau)\,{\dot{x}}_{\mu}(\tau),\qquad\quad\bar{F}(\tau,\theta)\,{\dot{P}}_{\mu}^{(hc)}(\tau,\theta)=\bar{C}(\tau)\,{\dot{p}}_{\mu}(\tau),$
$\displaystyle\bar{F}(\tau,\theta)\,{\dot{\Psi}}_{\mu}^{(hc)}(\tau,\theta)=\bar{C}(\tau)\,{\dot{\psi}}_{\mu}(\tau),\qquad\quad\bar{F}(\tau,\theta)\,{\dot{\Psi}}_{5}^{(hc)}(\tau,\theta)=\bar{C}(\tau)\,{\dot{\psi}}_{5}(\tau),$
$\displaystyle\bar{F}(\tau,\theta)\,{\dot{\bar{F}}}(\tau,\theta)=\bar{C}(\tau)\,\dot{\bar{C}}(\tau),\qquad\quad\quad\;\;{\tilde{\bar{B}}}^{(ab)}(\tau,\theta)=\bar{B}(\tau),$
$\displaystyle\dot{\bar{F}}(\tau,\theta)\,E(\tau,\theta)+\bar{F}(\tau,\theta)\,\dot{E}(\tau,\theta)=\dot{\bar{C}}(\tau)\,e(\tau)+\bar{C}(\tau)\,\dot{e}(\tau),$
$\displaystyle\dot{\bar{F}}(\tau,\theta)\,K(\tau,\theta)+\bar{F}(\tau,\theta)\,\dot{K}(\tau,\theta)=\dot{\bar{C}}(\tau)\,\chi(\tau)+\bar{C}(\tau)\,\dot{\chi}(\tau),$
$\displaystyle\dot{\tilde{B}}(\tau,\theta)\,\bar{F}(\tau,\theta)-\tilde{B}(\tau,\theta)\,\dot{\bar{F}}(\tau,\theta)=\dot{B}(\tau)\,\bar{C}(\tau)-B(\tau)\,\dot{\bar{C}}(\tau),$
(61)

where the chiral supervariables with superscripts $(hc)$ and $(ab)$ have been
discussed and explained in Eqs. (58) and (60). It is worth pointing out that
the equalities in (61) are nothing but the generalization of our observations
in (59) where the anti-BRST invariant quantities have been obtained [because
of the off-shell nilpotency of the anti-BRST symmetry transformations (20)].
At this stage, it is crucial to point out that, besides our chiral
supervariables in (58) and (60), we have the following generalizations:

$\displaystyle B(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad\tilde{B}(\tau,\theta)=B(\tau)+\theta\,{\bar{f}}_{1}(\tau),$
$\displaystyle e(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad E(\tau,\theta)=e(\tau)+\theta\,{\bar{f}}_{2}(\tau),$
$\displaystyle\chi(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad K(\tau,\theta)=\chi(\tau)+\theta\,{\bar{b}}_{1}(\tau),$
$\displaystyle C(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad F(\tau,\theta)=C(\tau)+\theta\,{\bar{b}}_{2}(\tau),$
$\displaystyle\bar{C}(\tau)\qquad$ $\displaystyle\longrightarrow$
$\displaystyle\qquad\bar{F}(\tau,\theta)=\bar{C}(\tau)+\theta\,{\bar{b}}_{3}(\tau).$
(62)

The above chiral supervariables are defined and their expansions have been
carried out on a $(1,1)$-dimensional super sub-manifold that is characterized
by the superspace coordinates $Z^{M}=(\tau,\theta)$ where the Grassmannian
coordinate $\theta$ is fermionic $(\theta^{2}=0)$ in nature. It is
straightforward to draw the conclusion that the secondary variables
$({\bar{b}}_{1},{\bar{b}}_{2},{\bar{b}}_{3})$ and
$({\bar{f}}_{1},{\bar{f}}_{2},{\bar{f}}_{3})$, on the r.h.s. of Eq. (62) are
bosonic and fermionic sets, respectively.

The stage is now set to utilize the equalities (61) where we have to plug in
the chiral super expansions (58) as well as the chiral generalizations (62).
This exercise leads to the following relationships between the secondary
variables of (62) and the basic as well as auxiliary variables of Lagrangian
$L_{\bar{B}}$ (and the Nakanishi-Lautrup auxiliary variable $B$ of the
perfectly BRST invariant Lagrangian $L_{B}$), namely;

$\displaystyle{\bar{f}}_{1}=\dot{B}\,{\bar{C}}-B\,\dot{\bar{C}},\qquad\quad{\bar{f}}_{2}=\bar{C}\,\dot{e}+\dot{\bar{C}}\,e,$
$\displaystyle{\bar{b}}_{1}=\bar{C}\,\dot{\chi}+\dot{\bar{C}}\,\chi,\quad\qquad{\bar{b}}_{2}=i\,\bar{B},\quad\qquad{\bar{b}}_{3}=\bar{C}\,\dot{\bar{C}}.$
(63)

In other words, we have already determined the secondary variables (i.e. the
coefficients of $\theta$) of the super expansions (62). The substitutions of
(63) into the chiral super expansion (62) on the (1,1)-dimensional chiral
super sub-manifold leads to the following

$\displaystyle{\tilde{B}}^{(ab)}(\tau,\theta)=B(\tau)+\theta\,(\dot{B}\,\bar{C}-B\,\dot{\bar{C}})\equiv
B(\tau)+\theta\,(s_{ab}\,B),$ $\displaystyle
E^{(ab)}(\tau,\theta)=e(\tau)+\theta\,(e\,\dot{\bar{C}}+\dot{e}\,\bar{C})\equiv
e(\tau)+\theta\,(s_{ab}\,e),$ $\displaystyle
K^{(ab)}(\tau,\theta)=\chi(\tau)+\theta\,(\dot{\bar{C}}\,\chi+\bar{C}\,\dot{\chi})\equiv\chi(\tau)+\theta\,(s_{ab}\,\chi),$
$\displaystyle F^{(ab)}(\tau,\theta)=C(\tau)+\theta\,(i\,\bar{B})\equiv
C(\tau)+\theta\,(s_{ab}\,C),$
$\displaystyle{\bar{F}}^{(ab)}(\tau,\theta)=\bar{C}(\tau)+\theta\,(\bar{C}\,\dot{\bar{C}})\equiv\bar{C}(\tau)+\theta\,(s_{ab}\,\bar{C}),$
(64)

where the superscript $(ab)$ denotes the chiral supervariables that have been
obtained after the application of the anti-BRST (i.e. quantum gauge) invariant
restrictions in Eq. (61). It is evident, from the equation (64), that we have
already obtained the anti-BRST symmetry transformations [cf. Eq. (20)] of the
variables $(B,e,\chi,C,\bar{C})$ as the coefficients of $\theta$ in the chiral
super expansions (62). We also observe, in the above chiral super expansions,
that there is a mapping: $\partial_{\theta}\leftrightarrow s_{ab}$ which
agrees with the result of Refs. [10-12]. For the sake of completeness, we
perform step-by-step computations of the mathematical relationships of the
equation (63) in our Appendix A.

##  6  Symmetry Invariance of the Lagrangians: ACSA

In this section, we use the results of the previous section to capture the
(anti-)BRST invariance of the Lagrangians [cf. Eqs. (26)-(29)] within the
framework of ACSA. To accomplish this goal, first of all, we generalize the
Lagrangians (24) to their counterparts (anti-)chiral super Lagrangians as
follows

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

where the superscripts $(ac)$ and $(c)$, on the r.h.s., denote the anti-chiral
and chiral versions of the first-order Lagrangian $L_{f}$ [cf. Eq. (1)]. In
other words, we have the following

$\displaystyle L_{f}\rightarrow{\tilde{L}}_{f}^{(ac)}(\tau,\bar{\theta})$
$\displaystyle=$ $\displaystyle
P_{\mu}^{(b)}(\tau,\bar{\theta})\,{\dot{X}}^{\mu{(b)}}(\tau,\bar{\theta})-\frac{1}{2}\,E^{(b)}(\tau,\bar{\theta})\,\Big{[}P_{\mu}^{(b)}(\tau,\bar{\theta})\,P^{\mu{(b)}}(\tau,\bar{\theta})-m^{2}\Big{]}$
(67) $\displaystyle+$
$\displaystyle\frac{i}{2}\,\Big{[}\Psi_{\mu}^{(b)}(\tau,\bar{\theta})\,{\dot{\Psi}}^{\mu{(b)}}(\tau,\bar{\theta})-\Psi_{5}^{(b)}(\tau,\bar{\theta})\,{\dot{\Psi}}_{5}^{(b)}(\tau,\bar{\theta})\Big{]}$
$\displaystyle+$ $\displaystyle
i\,{\tilde{\chi}}^{(b)}(\tau,\bar{\theta})\,\Big{[}P_{\mu}^{(b)}(\tau,\bar{\theta})\,\Psi^{\mu{(b)}}(\tau,\bar{\theta})-m\,\Psi_{5}^{(b)}(\tau,\bar{\theta})\Big{]},$
$\displaystyle L_{f}\rightarrow{\tilde{L}}_{f}^{(c)}(\tau,\theta)$
$\displaystyle=$ $\displaystyle
P_{\mu}^{(ab)}(\tau,\theta)\,{\dot{X}}^{\mu{(ab)}}(\tau,\theta)-\frac{1}{2}\,E^{(ab)}(\tau,\theta)\,\Big{[}P_{\mu}^{(ab)}(\tau,\theta)\,P^{\mu{(ab)}}(\tau,\theta)-m^{2}\Big{]}$
(68) $\displaystyle+$
$\displaystyle\frac{i}{2}\,\Big{[}\Psi_{\mu}^{(ab)}(\tau,\theta)\,{\dot{\Psi}}^{\mu{(ab)}}(\tau,\theta)-\Psi_{5}^{(ab)}(\tau,\theta)\,{\dot{\Psi}}_{5}^{(ab)}(\tau,\theta)\Big{]}$
$\displaystyle+$ $\displaystyle
i\,{\tilde{\chi}}^{(ab)}(\tau,\theta)\,\Big{[}P_{\mu}^{(ab)}(\tau,\theta)\,\Psi^{\mu{(ab)}}(\tau,\theta)-m\,\Psi_{5}^{(ab)}(\tau,\theta)\Big{]},$

where the superscripts $(b)$ and $(ab)$ on the supervariables of
${\tilde{L}}_{f}^{(ac)}$ and ${\tilde{L}}_{f}^{(c)}$ have been already
explained in the previous section. The superscripts $(ac)$ and $(c)$ on the
super Lagrangians, on the l.h.s. of the equations (65) and (66) denote the
anti-chiral and chiral generalizations of the ordinary Lagrangians (24).
Keeping in our mind the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$,
we observe the (anti-)BRST invariance of the first-order Lagrangian in the
language of ACSA to BRST formalism as follows:

$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,{\tilde{L}}_{f}^{(ac)}=\frac{d}{d\,\tau}\big{[}C\,L_{f}\big{]}\equiv
s_{b}\,L_{f},\qquad\quad\frac{\partial}{\partial\,\theta}\,{\tilde{L}}_{f}^{(c)}=\frac{d}{d\,\tau}\big{[}\bar{C}\,L_{f}\big{]}\equiv
s_{ab}\,L_{f}.$ (69)

In other words, we have accomplished the objective of establishing a precise
connection between the (anti-)BRST invariance of $L_{f}$ in the ordinary space
[cf. Eqs. (3), (27), (26)] and superspace within the ambit of ACSA to BRST
formalism. The results in (69) will be useful in the proof of the (anti-)BRST
invariance [cf. Eqs. (27),(26)] of the coupled (but equivalent) Lagrangians
$L_{B}$ and $L_{\bar{B}}$. Geometrically, it is clear from our observation in
(69) that super Lagrangians ${\tilde{L}}_{f}^{(ac,c)}$ are the unique sum of
(anti-)chiral supervariables [obtained after the (anti-)BRST invariant
restrictions] such that their translations along
$(\theta,\bar{\theta})$-directions in the superspace produce the total
derivatives in the ordinary space.

We now focus on the BRST and anti-BRST invariance of $L_{B}$ and $L_{\bar{B}}$
within the purview of ACSA. In the explicit expressions of (65) and (66), we
substitute the super expansions of (52), (57), (58) and (64) and apply the
derivatives $(\partial_{\bar{\theta}},\partial_{\theta})$ on them due to the
mappings: $\partial_{\bar{\theta}}\leftrightarrow
s_{b},\,\,\partial_{\theta}\leftrightarrow s_{ab}$. It is straightforward to
check that we have the following explicit relationships between the
invariances in the superspace and ordinary space:

$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,{\tilde{L}}_{B}^{(ac)}=\frac{d}{d\,\tau}\,\Big{[}C\,L_{f}+e^{2}\,B\,\dot{C}+e\,\dot{e}\,B\,C-i\,B\,\bar{C}\,\dot{C}\,C+B^{2}\,C\Big{]}=s_{b}\,L_{B},$
$\displaystyle\frac{\partial}{\partial\,\theta}\,{\tilde{L}}_{\bar{B}}^{(c)}=\frac{d}{d\,\tau}\,\Big{[}\bar{C}\,L_{f}-e^{2}\,\bar{B}\,\dot{\bar{C}}-e\,\dot{e}\,\bar{B}\,\bar{C}-i\,\bar{B}\,\dot{\bar{C}}\,\bar{C}\,C+{\bar{B}}^{2}\,\bar{C}\Big{]}=s_{ab}\,L_{\bar{B}}.$
(70)

We would like to emphasize that the super Lagrangian ${\tilde{L}}_{B}^{(ac)}$
is a unique sum of anti-chiral supervariables (derived after the applications
of the BRST invariant restrictions) such that its translation along
$\bar{\theta}$-direction of the $(1,1)$-dimensional anti-chiral super sub-
manifold leads to a total derivative in the ordinary space. The latter is
nothing but the BRST invariance of the ordinary Lagrangian $L_{B}$ [cf. Eq.
(26)]. In exactly similar fashion, we can provide a geometrical interpretation
for the anti-BRST invariance of $L_{\bar{B}}$ [cf. Eq. (27)] in the
terminology of the superspace translational generator $(\partial_{\theta})$
along the $\theta$-direction of the suitably chosen chiral $(1,1)$-dimensional
super sub-manifold.

At this juncture, we concentrate on the deduction of the CF-type restriction
$[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0]$ in the proof of the
equivalence between the Lagrangians $L_{B}$ and $L_{\bar{B}}$ [cf. Eq. (24)]
within the ambit of ACSA to BRST formalism. In other words, we capture the
transformations $s_{ab}\,L_{B}$ [cf. Eq. (29)] and $s_{b}\,L_{\bar{B}}$ [cf.
Eq. (28)] in the terminology of the ACSA. Towards this central goal in our
mind, first of all, we generalize the ordinary Lagrangian $L_{B}$ to its
counterpart chiral super Lagrangian as

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

where the superscripts $(c)$ and $(ab)$ have already been explained in our
earlier discussions. It is very interesting to observe that we have the
following:

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

The above equation shows that the Lagrangian $L_{B}$ respects the anti-BRST
symmetry transformations (20) only when the CF-type restriction is invoked
from outside. In a subtle manner, we have obtained the CF-type restriction
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ within the ambit of ACSA
while proving the anti-BRST invariance of the Lagrangian $L_{B}$. We now
demonstrate the BRST invariance of the Lagrangian $L_{\bar{B}}$ and existence
of the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ within the purview of
theoretical tricks and techniques of ACSA to BRST formalism. Towards these
aims in our mind, we generalize the Lagrangian $L_{\bar{B}}$ onto
$(1,1)$-dimensional anti-chiral super submanifold as follows

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

where the superscripts $(ac)$ and $(b)$ have been already explained in our
earlier discussions. Keeping in our mind the mapping:
$\partial_{\bar{\theta}}\leftrightarrow s_{b}$, we observe the following
interesting relationship between $s_{b}\,L_{\bar{B}}$ and its counterpart in
superspace, namely;

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

where the r.h.s. is nothing but the operation of $s_{b}$ on the Lagrangian
$L_{\bar{B}}$ [cf. Eq. (28)]. In other words, we have established an intimate
relationship between the BRST symmetry transformation on $L_{\bar{B}}$ and
operation of the translational generator $\partial_{\bar{\theta}}$ on the
anti-chiral super Lagrangian ${\tilde{L}}_{\bar{B}}^{(ac)}(\tau,\bar{\theta})$
[defined on the $(1,1)$-dimensional (anti-)chiral super sub-manifold]. A
careful and close look on the r.h.s. of (74) demonstrates that we have derived
the CF-type restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$
while proving the BRST invariance of $L_{\bar{B}}$ within the purview of ACSA.

We end this section with the following concluding remarks. First, we have
captured the (anti-)BRST invariance [cf. Eq. (69)] of the first-order
Lagrangian $L_{f}$ within the ambit of ACSA. Second, we have been able to
express the (anti-)BRST invariance [cf. Eqs. (26), (27)] of the Lagrangians
$L_{B}$ and $L_{\bar{B}}$ of our theory within the purview of ACSA to BRST
formalism [cf. Eq. (70)]. Third, we have been able to demonstrate that our
observations in the equations (28) and (29) can also be expressed in
superspace [cf. Eqs. (72), (74)] within the ambit of ACSA to BRST formalism.
Finally, we have derived the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$, in a subtle manner, by
expressing the transformations $s_{b}\,L_{\bar{B}}$ and $s_{ab}\,L_{B}$ in the
language of ACSA to BRST formalism [cf. Eqs. (72), (74) for details]. In other
words, in the ordinary space, whatever we have seen in the proof of the
absolute anticommutativity of (anti-)BRST symmetry transformations [cf. Eq.
(22)], the same restriction appears when we discuss $s_{b}\,L_{\bar{B}}$ and
$s_{ab}\,L_{B}$ in the superspace by using the tricks and techniques of ACSA
to BRST formalism.

##  7 Off-Shell Nilpotency and Absolute Anticommutativity of the Conserved
(Anti-)BRST Charges

In this section, we prove the off-shell nilpotency $[Q_{(\bar{B})B}^{2}=0]$
and absolute anticommutativity $\big{[}\\{Q_{B},Q_{\bar{B}}\\}=0\big{]}$ of
the (anti-)BRST charges $Q_{(\bar{B})B}$ which have already been derived in
our Sec. 3 where $J_{B}=Q_{B}$ and $J_{\bar{B}}=Q_{\bar{B}}$ [cf. Eqs. (31),
(32)]. In sub-section 7.1, we discuss the above properties of the (anti-)BRST
charges $[Q_{(\bar{B})B}]$ in the ordinary space. Our sub-section 7.2 contains
the theoretical material related with the techniques of capturing the
nilpotency and anticommutativity properties of the above charges within the
ambit of ACSA. It is quite interesting to state that the CF-type restriction
appears when we prove the absolute anticommutativity property of the charges
in the ordinary space as well as in the super space (within the ambit of
ACSA).

###  7.1 Off-Shell Nilpotency and Absolute Anticommutativity Properties:
Ordinary Space

In this sub-section, we primarily exploit the theoretical potential of the
well-known relationship between the continuous symmetry transformations and
their generators. In other words, we can prove the off-shell nilpotency
$(Q_{B}^{2}=Q_{\bar{B}}^{2}=0)$ of the (anti-)BRST charges $Q_{(\bar{B})B}$ in
the ordinary space by using the standard relationship between the
infinitesimal continuous (anti-)BRST transformations $(s_{(a)b})$ and their
generators ($Q_{(\bar{B})B}$) as:

$\displaystyle s_{b}\,Q_{B}=-\,i\\{Q_{B},Q_{B}\\}=0\quad\implies\quad
Q_{B}^{2}=0,$ $\displaystyle
s_{ab}\,Q_{\bar{B}}=-\,i\\{Q_{\bar{B}},Q_{\bar{B}}\\}=0\quad\implies\quad
Q_{\bar{B}}^{2}=0.$ (75)

The above proofs of the off-shell nilpotency of the conserved charges are
nothing but the reflection of the off-shell nilpotency $(s_{(a)b}^{2}=0)$ of
the (anti-)BRST symmetry transformations (20) and (21) in the ordinary space.
It would be worthwhile to point out the fact that, in the computation of the
l.h.s. of (75), we have directly applied the (anti-)BRST symmetry
transformations (20) and (21) on the appropriate form of the conserved (i.e.
$\dot{Q}_{(\bar{B})B}=0$) (anti-)BRST charges [cf. Eqs. (31), (32)]. As far as
the validity of Eq. (75) is concerned, we discuss the step-by-step derivation
of $s_{b}\,Q_{B}=0$ and $s_{ab}\,Q_{\bar{B}}=0$ in our Appendix B and
demonstrate that the Noether conserved charges [cf. Eqs. (31), (32)] are off-
shell nilpotent of order two (i.e. $Q_{(\bar{B})B}^{2}=0$) in the ordinary
space.

The above explicit proof of the off-shell nilpotency of the (anti-)BRST
charges ensures that they should be able to be written as an exact quantity
w.r.t. the off-shell nilpotent $[s_{(a)b}^{2}=0]$ (anti-)BRST symmetry
transformations $[s_{(a)b}]$. Towards this goal in mind, we use the following
EL-EOMs (derived from the Lagrangians $L_{B}$ and $L_{\bar{B}}$), namely:

$\displaystyle p_{\mu}\,\psi^{\mu}=m\,\psi_{5},\qquad
B=-e\,\dot{e}+2\,i\,\dot{\bar{C}}\,C+i\,\bar{C}\,\dot{C},$ $\displaystyle
e\,\dot{B}+i\,e\,\dot{\bar{C}}\,\dot{C}-i\,e\,\ddot{\bar{C}}\,C+\frac{1}{2}\,(p^{2}-m^{2})=0,$
$\displaystyle
B\,\dot{C}+2\,\dot{B}\,C+3\,e\,\dot{e}\,\dot{C}+e^{2}\,\ddot{C}+{\dot{e}}^{2}\,C+e\,\ddot{e}\,C-i\,\bar{C}\,\ddot{C}\,C-2\,i\,\dot{\bar{C}}\,\dot{C}\,C=0,$
$\displaystyle\bar{B}=e\,\dot{e}-2\,i\,\bar{C}\,\dot{C}-i\,\dot{\bar{C}}\,C,\quad
e\,\dot{\bar{B}}-i\,e\,\dot{\bar{C}}\,\dot{C}+i\,e\,{\bar{C}}\,\ddot{C}-\frac{1}{2}\,(p^{2}-m^{2})=0,$
$\displaystyle\bar{B}\,\dot{\bar{C}}+2\,\dot{\bar{B}}\,\bar{C}-3\,e\,\dot{e}\,\dot{\bar{C}}-e^{2}\,\ddot{\bar{C}}-{\dot{e}}^{2}\,\bar{C}-e\,\ddot{e}\,\bar{C}-i\,\ddot{\bar{C}}\,\bar{C}\,C-2\,i\,\dot{\bar{C}}\,\bar{C}\,\dot{C}=0,$
(76)

to get rid of the constraints $(p^{2}-m^{2})\approx 0$ and
$(p_{\mu}\,\psi^{\mu}-m\,\psi_{5})\approx 0$ from the expressions for the
(anti-)BRST charges $Q_{(\bar{B})B}$ [cf. Eqs. (31), (32)] to recast them
as‡‡‡We have discussed the different forms of the (anti-)BRST charges in our
Appendix C where the emphasis is laid on the derivation of the expressions for
the (anti)BRST charges [cf. Eq. (77)].

$\displaystyle
Q_{\bar{B}}^{(1)}=e^{2}\,\Big{[}\dot{\bar{B}}\,\bar{C}-\bar{B}\,\dot{\bar{C}}+i\,(\ddot{\bar{C}}\,\bar{C}\,C+\dot{\bar{C}}\,\bar{C}\,\dot{C})\Big{]}+2\,i\,e\,\dot{e}\,\dot{\bar{C}}\,\bar{C}\,C,$
$\displaystyle
Q_{B}^{(1)}=e^{2}\,\Big{[}B\,\dot{C}-\dot{B}\,C-i\,(\bar{C}\,\ddot{C}\,C+\dot{\bar{C}}\,\dot{C}\,C)\Big{]}-2\,i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C.$
(77)

The above expressions of the conserved (anti-)BRST charges can be
mathematically expressed in the following exact forms w.r.t. the off-shell
nilpotent $[s_{(a)b}^{2}=0]$ (anti-)BRST symmetry transformations
$[s_{(a)b}]$, namely;

$\displaystyle
Q_{\bar{B}}^{(1)}=s_{ab}\,[i\,e^{2}\,({\bar{C}}\,\dot{C}-\dot{\bar{C}}\,C)],\qquad
Q_{B}^{(1)}=s_{b}\,[i\,e^{2}\,(\dot{\bar{C}}\,C-\bar{C}\,\dot{C})].$ (78)

Now it is straightforward to note that $s_{ab}\,Q_{\bar{B}}^{(1)}=0$ and
$s_{b}\,Q_{B}^{(1)}=0$ due to the off-shell nilpotency [$s_{(a)b}^{2}=0$] of
the (anti-)BRST symmetry transformations $[s_{(a)b}]$. Thus, we conclude, from
our observations in Eq. (78), that the nilpotency of the (anti-)BRST
transformations [$s_{(a)b}$] is deeply connected with the nilpotency of their
generators (anti-)BRST charges [$Q_{(\bar{B})B}$] which becomes completely
transparent from the direct observations of the following computations:

$\displaystyle s_{ab}\,Q_{\bar{B}}^{(1)}$ $\displaystyle=$
$\displaystyle-\,i\,\big{\\{}Q_{\bar{B}}^{(1)},Q_{\bar{B}}^{(1)}\big{\\}}=0\quad\Longleftrightarrow\quad\big{[}Q_{\bar{B}}^{(1)}\big{]}^{2}=0\quad\Longleftrightarrow\quad
s_{ab}^{2}=0,$ $\displaystyle s_{b}\,Q_{B}^{(1)}$ $\displaystyle=$
$\displaystyle-\,i\,\big{\\{}Q_{B}^{(1)},Q_{B}^{(1)}\big{\\}}=0\quad\Longleftrightarrow\quad\big{[}Q_{B}^{(1)}\big{]}^{2}=0\quad\Longleftrightarrow\quad
s_{b}^{2}=0.$ (79)

The above equation completes our discussion on the proof of the off-shell
nilpotency of the conserved (anti-)BRST charges in the ordinary space.

Now we dwell on the proof of the absolute anticommutativity (i.e.
$Q_{B}^{(1)}\,Q_{\bar{B}}^{(1)}+Q_{\bar{B}}^{(1)}\,Q_{B}^{(1)}=0$) of the
conserved off-shell nilpotent (anti-)BRST charges [$Q_{(\bar{B})B}^{(1)}$].
Towards this central objective in mind, first of all, we assume the sanctity
and validity of the CF-type restriction;
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$, right from the
beginning. As a result, we can express the (anti-) BRST charges
$Q^{(1)}_{(\bar{B})B}$, in the alternative forms, as follows:

$\displaystyle
Q_{\bar{B}}^{(1)}\;\rightarrow\;Q_{\bar{B}}^{(2)}=e^{2}\,(B\,\dot{\bar{C}}-\dot{B}\,\bar{C}+2\,i\,\dot{\bar{C}}\,\bar{C}\,\dot{C})+2\,i\,e\,\dot{e}\,\dot{\bar{C}}\,\bar{C}\,C,$
$\displaystyle
Q_{B}^{(1)}\;\rightarrow\;Q_{B}^{(2)}=e^{2}\,(\dot{\bar{B}}\,C-\bar{B}\,\dot{C}-2\,i\,\dot{\bar{C}}\,\dot{C}\,C)-2\,i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C.$
(80)

We point out that it is because of the use of the CF-type restriction
[$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ ] that we have been able
to express $Q_{\bar{B}}^{(2)}$ in terms of the Nakanishi-Lautrap auxiliary
variable $B(\tau)$ and $Q_{B}^{(2)}$ in the language of other Nakanishi-
Lautrap type auxiliary variable $\bar{B}(\tau)$. At this crucial stage, we
observe the following interesting relationships:

$\displaystyle
Q_{\bar{B}}^{(2)}=s_{b}\,\Big{[}i\,e^{2}\,\dot{\bar{C}}\,\bar{C}\Big{]},\qquad\qquad
Q_{B}^{(2)}=s_{ab}\,\Big{[}-i\,\,e^{2}\,\dot{C}\,C\Big{]}.$ (81)

In other words, we have been able to express the anti-BRST charge
[$Q_{\bar{B}}^{(2)}$] as the BRST exact quantity. On the other hand, we have
been able to write the BRST charge [$Q_{B}^{(2)}$] as an exact quantity w.r.t.
the nilpotent anti-BRST transformation $s_{ab}$. A close and careful
observation of (81) leads to the following (due to the well-known relationship
between the continuous (anti-)BRST symmetry transformations $[s_{(a)b}]$ and
their generators as conserved (anti-)BRST charges $[Q_{(\bar{B})B}^{(2)}]$),
namely;

$\displaystyle s_{b}\,Q_{\bar{B}}^{(2)}$ $\displaystyle=$
$\displaystyle-\,i\,\big{\\{}Q_{\bar{B}}^{(2)},Q_{B}^{(2)}\big{\\}}=0\quad\Longleftrightarrow\quad
s_{b}^{2}=0,$ $\displaystyle s_{ab}\,Q_{B}^{(2)}$ $\displaystyle=$
$\displaystyle-\,i\,\big{\\{}Q_{B}^{(2)},Q_{\bar{B}}^{(2)}\big{\\}}=0\quad\Longleftrightarrow\quad
s_{ab}^{2}=0.$ (82)

As a consequence, we note that the absolute anticommutativity of the
(anti-)BRST charges $[Q_{(\bar{B})B}^{(2)}]$ is connected with the off-shell
nilpotency $[s_{(a)b}^{2}]=0$ of the (anti-)BRST symmetries.

We would like to lay emphasis on the key observations that have been made in
Eq. (82). It is very interesting (due to the validity of the CF-type
restriction on our theory) to pinpoint that (i) the anticommutativity of the
BRST charge $Q_{B}^{(2)}$ with the anti-BRST charge $Q_{\bar{B}}^{(2)}$ is
intimately connected with the nilpotency $(s_{b}^{2}=0)$ of the BRST
transformations $(s_{b})$, and (ii) the anticommutativity property of the
anti-BRST charge $Q_{\bar{B}}^{(2)}$ with the BRST charge $Q_{B}^{(2)}$ owes
its origin to the nilpotency $(s_{ab}^{2}=0)$ of the anti-BRST transformations
$(s_{ab})$. We conclude this sub-section with the following remarks. First, we
have shown that the nilpotency of the (anti-)BRST charges $[Q_{(\bar{B})B}]$
is deeply related with the nilpotency of the (anti-)BRST transformations
$[s_{(a)b}]$. Second, we have been able to express the modified form of the
BRST charge $[Q_{B}^{(1)}]$ and anti-BRST charge $Q_{\bar{B}}^{(1)}$ as the
exact expressions w.r.t. the BRST transformations $(s_{b})$ and anti-BRST
transformations $(s_{ab})$ [cf. Eq. (78)], respectively. Third, it is due to
the existence of the CF-type restriction on our theory that we have been able
to express another modified form of the BRST charge $Q_{B}^{(2)}$ as an exact
expression w.r.t. the anti-BRST transformations $(s_{ab})$ and the anti-BRST
charge $Q_{\bar{B}}^{(2)}$ in the BRST-exact form. This exercise has enabled
us to prove the absolute anticommutativity [i.e.
$\\{Q_{B}^{(2)},Q_{\bar{B}}^{(2)}\\}=0$] of the nilpotent (anti-)BRST charges
$Q_{(\bar{B})B}^{(2)}$. Finally, the proof of the absolute anticommutativity
property [cf. Eq. (82)] crucially depends on the existence of the CF-type
restriction. Thus, in a subtle manner, we have derived and corroborated the
sanctity of the existence of the CF-type restriction
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ on our theory§§§We have
provided an alternative proof for the appearance of the CF-type restriction
(on our 1D reparameterization invariant SUSY theory) in our Appendix D.. This
completes our discussions on the absolute anticommutativity of the (anti-)BRST
charges in the ordinary space.

###  7.2 Off-Shell Nilpotency and Absolute Anticommutativity Properties: ACSA
to BRST Formalism in Superspace

In this sub-section, we capture the properties of the nilpotency (i.e.
fermionic nature) and absolute anticommutativity (i.e. linear independence) of
the (anti-)BRST charges within the purview of ACSA where the superspace
consideration on the (1, 1)-dimensional (anti-)chiral super submanifolds has
been taken into account. First of all, we focus on the off-shell nilpotency
[$Q_{(\bar{B})B}^{2}=0$] of the (anti-)BRST charges [$Q_{(\bar{B})B}$]. In
this context, keeping in our knowledge the mappings:
$\partial_{\theta}\leftrightarrow s_{ab}$,
$\partial_{\bar{\theta}}\leftrightarrow s_{b}$, it can be readily seen that
the expressions for the (anti-)BRST charges that have been quoted in Eq. (78)
can be translated into the superspace as

$\displaystyle
Q_{\bar{B}}^{(1)}=\frac{\partial}{\partial\,\theta}\,\Big{[}i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\big{\\{}\bar{F}^{(ab)}(\tau,\theta)\,\dot{F}^{(ab)}(\tau,\theta)-\dot{\bar{F}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta)\big{\\}}\Big{]}$
$\displaystyle\equiv\int
d\,\theta\,\Big{[}i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\big{\\{}\bar{F}^{(ab)}(\tau,\theta)\,\dot{F}^{(ab)}(\tau,\theta)-\dot{\bar{F}}^{(ab)}(\tau,\theta)\,F^{(ab)}(\tau,\theta)\big{\\}}\Big{]},$
(83) $\displaystyle
Q_{B}^{(1)}=\frac{\partial}{\partial\,\bar{\theta}}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\big{\\{}\bar{F}^{(b)}(\tau,\bar{\theta})\,\dot{F}^{(b)}(\tau,\bar{\theta})-\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta})\big{\\}}\Big{]}$
$\displaystyle\equiv\int
d\,\bar{\theta}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\big{\\{}\bar{F}^{(b)}(\tau,\bar{\theta})\,\dot{F}^{(b)}(\tau,\bar{\theta})-\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,F^{(b)}(\tau,\bar{\theta})\big{\\}}\Big{]},$
(84)

where the supervariables with the superscripts $(ab)$ and $(b)$ have been
obtained in Eqs. (64) and (57), respectively. At this stage, the off-shell
nilpotency $[(Q_{(\bar{B})B}^{(1)})^{2}=0]$ of the conserved (anti-)BRST
charges $[Q_{(\bar{B})B}^{(1)}]$ can be written in the superspace (by using
the theoretical techniques and tricks of ACSA to BRST formalism) as:

$\displaystyle\partial_{\theta}\,Q_{\bar{B}}^{(1)}=0\quad\Longleftrightarrow\quad\partial_{\theta}^{2}=0,\qquad\qquad\partial_{\bar{\theta}}\,Q_{B}^{(1)}=0\quad\Longleftrightarrow\quad\partial_{\bar{\theta}}^{2}=0.$
(85)

Thus, we conclude that the off-shell nilpotency of the anti-BRST charge
$(Q_{\bar{B}}^{(1)})$ is deeply related to the nilpotency
$(\partial_{\theta}^{2}=0)$ of the translational generator
$(\partial_{\theta})$ along the $\theta$-direction of the $(1,1)$\-
dimensional chiral super sub-manifold of the general $(1,2)$-dimensional
supermanifold. Similar type of comments can be made in the context of the off-
shell nilpotency of the BRST charge $Q_{B}^{(1)}$ and its intimate
relationship with the nilpotency $(\partial_{\bar{\theta}}^{2}=0)$ of the
translational generator $(\partial_{\bar{\theta}})$ on the anti-chiral super
sub-manifold.

We concentrate now on capturing the absolute anticommutativity of the
(anti-)BRST charges within the purview of ACSA where the superspace of the
$(1,1)$-dimensional (anti-) chiral super sub-manifolds are taken into
consideration. Towards this central goal in our mind, we express the modified
forms of the (anti-)BRST charges $Q_{(\bar{B})B}^{(2)}$ of Eq. (81) in the
following mathematical expression within the framework of ACSA, namely;

$\displaystyle Q_{\bar{B}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\bar{F}}^{(b)}(\tau,\bar{\theta})\Big{]}$
(86) $\displaystyle\equiv$ $\displaystyle\int
d\,\bar{\theta}\,\Big{[}i\,E^{(b)}(\tau,\bar{\theta})\,E^{(b)}(\tau,\bar{\theta})\,\dot{\bar{F}}^{(b)}(\tau,\bar{\theta})\,{\bar{F}}^{(b)}(\tau,\bar{\theta})\Big{]},$
$\displaystyle Q_{B}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\,\theta}\Big{[}-\,i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\dot{F}^{(ab)}(\tau,\theta)\,{F}^{(ab)}(\tau,\theta)\Big{]}$
(87) $\displaystyle\equiv$ $\displaystyle\int
d\,\theta\,\Big{[}-\,i\,E^{(ab)}(\tau,\theta)\,E^{(ab)}(\tau,\theta)\,\dot{F}^{(ab)}(\tau,\theta)\,{F}^{(ab)}(\tau,\theta)\Big{]},$

where the supervariables with the superscripts $(ab)$ and $(b)$ have been
quoted in Eqs. (64) and (57), respectively. At this crucial juncture, we note
the following:

$\displaystyle\partial_{\bar{\theta}}\,Q_{\bar{B}}^{(2)}=0\quad\Longleftrightarrow\quad\partial_{\bar{\theta}}^{2}=0,\qquad\qquad\partial_{\theta}\,Q_{B}^{(2)}=0\quad\Longleftrightarrow\quad\partial_{\theta}^{2}=0.$
(88)

The above relations are nothing but the explicit proof of the
anticommutativity property of the (anti-)BRST charges $Q_{(\bar{B})B}^{(2)}$
(within the ambit of ACSA).

We wrap-up this sub-section with the following comments. First, the off-shell
nilpotency $[Q_{(\bar{B})B}^{2}=0]$ of the (anti-)BRST charges
$Q_{(\bar{B})B}$ is intimately connected with the nilpotency
$(\partial_{\theta}^{2}=0,\partial_{\bar{\theta}}^{2}=0)$ of the translational
generators $(\partial_{\theta},\partial_{\bar{\theta}})$ along the
$(\theta,\bar{\theta})$-directions of the $(1,1)$-dimensional chiral and anti-
chiral super sub-manifolds. Second, in the ordinary space, the above
statements of the off-shell nilpotency are captured in the equations (75) and
(79). Third, the absolute anticommutativity of the BRST charge with the anti-
BRST charge is related to the nilpotency $(\partial_{\theta}^{2}=0)$ of the
translational generator $(\partial_{\theta})$ along the $\theta$-direction of
the chiral super submanifold. On the other hand, the absolute
anticommutativity of the anti-BRST charge with the BRST charge is connected
with the nilpotency $(\partial_{\bar{\theta}}^{2}=0)$ of the translational
generator $(\partial_{\bar{\theta}})$ along the $\bar{\theta}$-direction of
the anti-chiral super submanifold. Fourth, the above statements have been
corroborated in the ordinary space by the equation (82) where the off-shell
nilpotency $[s_{(a)b}^{2}=0]$ of the (anti-)BRST transformations $[s_{(a)b}]$
and the anticommutativity $[\\{Q_{B}^{(2)},Q_{\bar{B}}^{(2)}\\}=0]$ of the
(anti-)BRST charges $[Q_{(\bar{B})B}^{(2)}]$ are found to be inter-connected
in an intimate manner.

##  8 Conclusions

The USFA to BRST formalism (see, e.g. [10-12]) is useful in the context of the
gauge theories where the spacetime coordinates do not change. Thus, it was a
challenge to include the diffeomorphism (i.e. the general spacetime
transformations) within the framework of Bonora-Tonin (BT) superfield approach
to BRST formalism (see, e.g. [10-12]). This was achieved by Bonora in Ref.
[26] which has been christened by us as the MBTSA where the generalization of
the 1D diffeomorphism [i.e.
$\tau\rightarrow\tau\,^{\prime}=f\,(\tau)\,\equiv\,\tau-\varepsilon\,(\tau)$]
to the (1, 2)-dimensional supermanifold [cf. Eq. (35)] has played an important
role in the derivation of the (anti-)BRST symmetry transformations [cf. Eq.
(50)] for the target space variables
($x_{\mu},\,p_{\mu},\,\psi_{\mu},\,\psi_{5}$). In addition, this approach has
enabled us to deduce the (anti-)BRST invariant CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ that is responsible for
the absolute anticommutativity of the (anti-)BRST symmetry transformations
[cf. Eq. (22)] and existence of the coupled (but equivalent) Lagrangians [cf.
Eq. (24)] for our theory.

We have taken into account the standard (anti-)BRST symmetry transformations
($s_{b}\,\bar{C}=i\,B,\,s_{ab}\,C=i\,\bar{B}$) for the (anti-)ghost variables
$(\bar{C})C$ which have, in a subtle manner, forced us to consider the
(anti-)chiral super expansions [cf. Eq. (47)]. This has provided us the clue
to adopt the ACSA to BRST formalism for the deduction of the proper
(anti-)BRST transformations for the rest of the variables of our theory (cf.
Sec. 5). Within the purview of ACSA, we derive the CF-type restriction when we
prove the equivalence of the coupled (but equivalent) Lagrangians (cf. Sec.
6). Furthermore, it is the validity of the CF-type restriction:
$B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ that has enabled us to
write (i) the BRST charge as an exact quantity w.r.t. the nilpotent anti-BRST
transformation, and (ii) the anti-BRST charge as an exact expression w.r.t.
the nilpotent BRST transformation. These observations have been responsible
for the proof of the absolute anticommutativity of the (anti-)BRST charges
(cf. Sec. 7). In other words, it is the proof of the anticommutativity of the
conserved and nilpotent charges $[Q_{(\bar{B})B}^{(2)}]$ which leads to the
existence of the CF-type restriction on our SUSY theory (cf. Sec. 7).

We would like to lay emphasis on the fact that the observation of the absolute
anticommutativity property, in the context of the conserved (anti-)BRST
charges, is a novel and interesting result in view of the fact that we have
taken into account only the (anti-)chiral super expansions of the
supervariables¶¶¶In our earlier works (see, e.g. [32-36]), we have applied the
techniques and tricks of ACSA to obtain the nilpotent symmetries of the ${\cal
N}=2$ SUSY quantum mechanical models of interest. However, we have found that
the conserved charges are not absolutely anticommuting. Thus, our observation
of the absolute anticommutativity property in the context of (anti-)BRST
charges is novel and interesting.. This observation of the absolute
anticommutativity property is obvious when one takes into account the full
super expansions of the supervariables along all the possible Grassmannian
directions of the suitably chosen supermanifold on which the ordinary theory
is generalized. Furthermore, the appearance of the CF-type restriction in the
computations of $s_{b}\,L_{\bar{B}}$ and $s_{ab}\,L_{B}$ [cf. Eqs. (28), (29)]
in the ordinary space and its analogue in the superspace (where ACSA plays an
important role) are very interesting observations in our present endeavor (cf.
Sec. 6). The other observation that merits a clear and special mention is the
universality of the CF-type restriction:
$B+\bar{B}+i(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ in the context of the
reparameterization (i.e. 1D diffeomorphism) invariant non-SUSY theory of a
scalar relativistic particle [27] and our present SUSY system of a spinning
relativistic particle (where the nilpotent SUSY transformations exist between
the bosonic and fermionic variables of our SUSY theory).

It is worthwhile to mention that, for the D-dimensional diffeomorphism
invariant theory [26, 37] where the infinitesimal diffeomorphism symmetry
transformation is: $x_{\mu}\rightarrow
x_{\mu}^{\prime}=x_{\mu}-\epsilon_{\mu}(x)$ (with $\mu=0,1,2\,...\,D-1$), the
general form of the CF-type restriction has been obtained as:
$B_{\mu}+\bar{B}_{\mu}+i\,(\bar{C}^{\rho}\,\partial_{\rho}\,C_{\mu}+C^{\rho}\,\partial_{\rho}\,\bar{C}_{\mu})=0$
where the (anti-)ghost fields $(\bar{C}_{\mu})C_{\mu}$ correspond to the
infinitesimal transformation parameter $\epsilon_{\mu}(x)$ in the general
coordinate transformation: $x_{\mu}^{\prime}=x_{\mu}-\epsilon_{\mu}(x)$ and
the Nakanishi-Lautrup fields $(\bar{B}_{\mu})B_{\mu}$ appear in the
(anti-)BRST symmetry transformations: $s_{b}\,\bar{C}_{\mu}=i\,B_{\mu}$,
$s_{ab}\,C_{\mu}=i\,\bar{B}_{\mu}$. It is straightforward to note that the CF-
type restriction: $B+\bar{B}+i(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ is the
limiting case of the above general D-dimensional CF-type restriction in the
case of the BRST approach to D-dimensional diffeomorphism invariant theory.
Thus, our theoretical treatments of the reparameterization (i.e. 1D
diffeomorphism) invariant theories of the scalar and spinning relativistic
particles are correct (where the CF-type restriction is:
$B+\bar{B}+i(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$).

One of the highlights of ACSA to BRST formalism is the observation that it
distinguishes between the suitably chosen $(1,1)$-dimensional chiral and anti-
chiral super sub-manifolds in the proof of the absolute anticommutativity of
the conserved (anti-)BRST charges. For instance, we note that the
anticommutativity of the BRST charge $[Q_{B}^{(2)}]$ with the anti-BRST charge
$[Q_{\bar{B}}^{(2)}]$ is connected with the nilpotency
$(\partial_{\theta}^{2}=0)$ of the translational generator
$(\partial_{\theta})$ along the $\theta$-direction of the chiral super
submanifold [cf. Eq. (85)]. On the other hand, the anticommutativity of the
anti-BRST charge $[Q_{\bar{B}}^{(2)}]$ with the BRST charge $[Q_{B}^{(2)}]$
crucially depends on the nilpotency $(\partial^{2}_{\bar{\theta}}=0)$ of the
translational generator $(\partial_{\bar{\theta}})$ [cf. Eq. (88)] along the
$\bar{\theta}$-direction of the anti-chiral super sub-manifold (cf. Sec. 7 for
details). This observation is a reflection of our discussion on the absolute
anticommutativity property of the (anti-)BRST charges in the ordinary space
(cf. Sec. 7) where the off-shell nilpotency of the (anti-)BRST transformations
[cf. Eq. (82)] play a decisive role.

We plan to extend our present study to the physical (3+1)-dimensional (4D)
theories of the gravitation and higher dimensional (super)string theories
where there is existence of the diffeomorphism invariance. In our earlier work
[38], we have discussed the BRST approach to a bosonic string theory where
there is a 2D diffeomorphism symmetry on the world-sheet. We have also
discussed the reparameterization symmetry transformations for the non-
relativistic particle where space and time variables have been treated as
operators [39]. It would be interesting to discuss the (anti-)BRST symmetries
for the classical reparameterization symmetry of this theory, too. In other
words, we plan to apply the ideas of MBTSA and ACSA together to find out the
(anti-)BRST symmetries of the above mentioned theories. The mathematical
elegance, rigor and beauty of the MBTSA [26, 37] should find more applications
to some physical systems of interest in theoretical high energy physics. We
envisage to take up these challenges in our future investigations [40].

Acknowledgments

B. Chauhan thankfully acknowledges the financial support from DST (Govt. of
India) under its INSPIRE-fellowship scheme and A. Tripathi as well as A. K.
Rao are grateful to the general BHU-fellowship [from Banaras Hindu University
(BHU), Varanasi] under which the present investigation has been performed.

Appendix A: On the Step-by-Step Computation of the Secondary Variables for the
Off-shell Nilpotent Anti-BRST Transformations

In this Appendix, we concentrate on the clear-cut derivation of the secondary
variables [cf. Eq. (63)] in terms of the basic and auxiliary variables of the
Lagrangians (24). For this purpose, we invoke the basic principle of ACSA
which states that the anti-BRST invariant expressions [cf. Eq. (59)] must be
free from the Grassmannian variable $\theta$ when these quantities are
generalized onto the $(1,1)$-dimensional chiral super sub-manifold. First of
all, we consider $s_{ab}\,(\bar{C}\,{\dot{x}}_{\mu})=0$ which leads to the
following anti-BRST invariant restriction:

$\bar{F}(\tau,\theta)\,{\dot{X}}_{\mu}^{(hc)}(\tau,\theta)=\bar{C}(\tau)\,{\dot{x}}_{\mu}(\tau).$
$None$

At this stage, we substitute the super expansions from (62) and (58) which
leads to the precise determination of the secondary variable
${\bar{b}}_{3}=\bar{C}\,\dot{\bar{C}}$. As a consequence, we have now the
super expansion of the chiral supervariable $\bar{F}(\tau,\theta)$ as

${\bar{F}}^{(ab)}(\tau,\theta)=\bar{C}(\tau)+\theta\,(\bar{C}\,\dot{\bar{C}})\equiv\bar{C}(\tau)+\theta\,(s_{ab}\,\bar{C}),$
$None$

where the superscript $(ab)$ denotes the chiral super expansion of
$\bar{F}(\tau,\theta)$ after the application of the anti-BRST restriction
(A.1). We note that the coefficient of $\theta$ is nothing but the anti-BRST
symmetry transformation of $\bar{C}$ [cf. Eq. (20)]. In other words, we find
that $\partial_{\theta}\,{\bar{F}}^{(ab)}(\tau,\theta)=s_{ab}\bar{C}$ which
agrees with the mapping: $\partial_{\theta}\leftrightarrow s_{ab}$ of Refs.
[10-12].

At this juncture, we take up the anti-BRST invariant quantities
$\big{(}i.e.s_{ab}\,[\,d/d\,\tau\,(\bar{C}\,e)]=0,\,s_{ab}\,[\,d/d\,\tau\,(\bar{C}\,\chi)]=0\big{)}$
which leads to the following restrictions:

$\frac{d}{d\,\tau}\,\Big{[}\bar{F}^{(ab)}(\tau,\theta)\,E(\tau,\theta)\Big{]}=\frac{d}{d\,\tau}\Big{[}\bar{C}(\tau)\,e(\tau)\Big{]},\\\
$
$\frac{d}{d\,\tau}\,\Big{[}\bar{F}^{(ab)}(\tau,\theta)\,K(\tau,\theta)\Big{]}=\frac{d}{d\,\tau}\Big{[}\bar{C}(\tau)\,\chi(\tau)\Big{]}.$
$None$

Substitutions from (A.2) and (62) lead to the following equations in terms of
the fermionic (anti-)ghost variables as well as secondary variables
$\bar{f}_{2}$ and $\bar{b}_{1}$, namely;

$\dot{\bar{C}}\,\bar{f}_{2}+\bar{C}\,\dot{\bar{f}}_{2}-\bar{C}\,\ddot{\bar{C}}\,e-\bar{C}\,\dot{\bar{C}}\,\dot{e}=0,$
$\dot{\bar{C}}\,\bar{b}_{1}+\bar{C}\,\dot{\bar{b}}_{1}-\bar{C}\,\ddot{\bar{C}}\,\chi-\bar{C}\,\dot{\bar{C}}\,\dot{\chi}=0$
$None$

It is straightforward to verify that we have the following solutions:

$\bar{f}_{2}=\bar{C}\,\dot{e}+\dot{\bar{C}}\,e,\qquad\bar{b}_{1}=\bar{C}\,\dot{\chi}+\dot{\bar{C}}\,\chi.$
$None$

We take now the anti-BRST invariance:
$s_{ab}\,(\dot{B}\,\bar{C}-B\,\dot{\bar{C}})=0$. This leads to the following
restrictions on the chiral supervariables

$\dot{\tilde{B}}(\tau,\theta)\,\bar{F}^{(ab)}(\tau,\theta)-{\tilde{B}}(\tau,\theta)\,\dot{\bar{F}}^{(ab)}(\tau,\theta)=\dot{B}(\tau)\,\bar{C}(\tau)-B(\tau)\,\dot{\bar{C}}(\tau).$
$None$

Substitutions of the super expansions from (62) and (A.2) lead to the
following relationship

$\dot{B}\,\bar{C}\,\dot{\bar{C}}-B\,\bar{C}\,\ddot{\bar{C}}+\dot{\bar{f_{1}}}\,\bar{C}-\bar{f}_{1}\,\dot{\bar{C}}=0.$
$None$

It is evident that the precise solution is
$\bar{f_{1}}=\dot{B}\,\bar{C}-B\,\dot{\bar{C}}$. Thus far, we have been able
to determine the precise forms of the secondary variables
($\bar{b}_{1},\bar{b}_{3},\bar{f}_{1},\bar{f}_{3}$) in terms of the basic,
auxiliary and (anti-)ghost variables of the Lagrangians (24).

In our present Appendix, we have followed the tricks and techniques of ACSA to
BRST formalism which was motivated by our standard assumption that:
$s_{b}\bar{C}=i\,B,\,s_{ab}\,C=i\,\bar{B}$ in the BRST approach to gauge
and/or diffeomorphism invariant theories. This standard assumption led to the
(anti-)chiral super expansions of the supervariables in Eq. (47). This implies
that we have already determined the remaining secondary variable of Eq. (63)
as: $\bar{b}_{2}=i\,\bar{B}$. This completes our discussion on the step-by-
step determination of the secondary variables of Eq. (63) which are present in
the chiral super expansions (62).

Appendix B: On the Proof of the Off-Shell Nilpotency of the Noether Conserved
(Anti-)BRST Charges

For the sake of completeness, we provide here the explicit proof of the
nilpotency of the Noether conserved charges $J_{B}=Q_{B}$ and
$J_{\bar{B}}=Q_{\bar{B}}$ [cf. Eqs. (31), (32)] in our present Appendix. For
this purpose, first of all, we take the expression for the BRST charge
($Q_{B}$) which can be re-expressed as follows:

$Q_{B}=B^{2}\,C-i\,B\,\bar{C}\,\dot{C}\,C+e^{2}\,B\,\dot{C}+e\,\dot{e}\,B\,C+\frac{1}{2}\,e\,C\,(p^{2}-m^{2})+i\,\chi\,C\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5}).$
$None$

We apply directly the BRST symmetry transformations $(s_{b})$, quoted in Eq.
(21), on the above charge. It can be readily checked that the first two terms
cancel out due to: $s_{b}\,\bar{C}=i\,B$ and the nilpotency [i.e.
$s_{b}\,(C\,\dot{C})=0$] of the transformation $s_{b}\,C=C\,\dot{C}$. The last
term in (B.1) contributes to zero due to $s_{b}\,(\chi\,C)=0$ and fermionic
nature $(C^{2}=0)$ of the ghost variable $C(\tau)$. In exactly similar
fashion, the last but one term [i.e. $\frac{1}{2}\,e\,C\,(p^{2}-m^{2})$]
contributes to zero due to our observation: $s_{b}\,(e\,C)=0$ and the
fermionic $(C^{2}=0)$ nature of the ghost variable $C(\tau)$. It is
straightforward to check that the contributions from the rest of the terms are
also equal to zero, namely;

$s_{b}\,\big{[}e^{2}\,B\,\dot{C}+e\,\dot{e}\,B\,C\big{]}=B\,e^{2}\,\ddot{C}\,C+2\,B\,e\,\dot{e}\,\dot{C}\,C+B\,e^{2}\,C\,\ddot{C}+2\,B\,e\,\dot{e}\,C\,\dot{C}=0.$
$None$

In the above, we have used the fermionic ($C^{2}=0$) nature of the ghost
variable $C(\tau)$ which leads to:
$C\,\dot{C}+\dot{C}\,C=0,\;C\,\ddot{C}+\ddot{C}\,C=0$. Hence, we observe
explicitly that $s_{b}\,Q_{B}=s_{b}\,J_{B}\equiv-\,i\,\,\\{Q_{B},Q_{B}\\}=0$
implies the nilpotency ($Q^{2}_{B}=0$) of the BRST charge $Q_{B}$ due to the
explicit computation of $s_{b}\,Q_{B}$ where $s_{b}$ is quoted in (21) and
$Q_{B}$ is the Noether conserved charge that is given in (31).

We follow exactly similar kinds of theoretical techniques and tricks to
evaluate $s_{ab}\,Q_{\bar{B}}\equiv s_{ab}\,J_{\bar{B}}$ whare $s_{ab}$ is
given in (20) and $J_{\bar{B}}=Q_{\bar{B}}$ is quoted in (32). It turns out
that $s_{ab}\,Q_{\bar{B}}=-\,i\,\,\\{Q_{\bar{B}},Q_{\bar{B}}\\}=0$ due to
explicit computation $(s_{ab}\,Q_{\bar{B}}=0)$ of the value of
$s_{ab}\,Q_{\bar{B}}$. In other words, we have proven the nilpotency (i.e.
fermionic nature) of the anti-BRST charge $Q_{\bar{B}}$ by explicitly
demonstrating that $s_{ab}\,Q_{\bar{B}}=0\Longrightarrow Q_{\bar{B}}^{2}=0$.

Appendix C: On the Different Forms of the Conserved (Anti-)BRST Charges

Besides the expressions for the (anti-)BRST charges in (31) and (32), we
require different forms of these charges to prove their nilpotency (i.e.
fermionic nature) and anticommutativity (i.e. linear independence) in a
straightforward manner. This exercise has been found to be advantageous in the
context of our discussion on ACSA, too. First and foremost, we concentrate on
the derivation of the BRST charge in (77) and its usefulness. In this
connection, we note that the Noether charge (31) can be re-written as follows

$Q_{B}\rightarrow\tilde{Q}_{B}=e^{2}\,[B\,\dot{C}-\dot{B}\,C-i\,\dot{\bar{C}}\,\dot{C}\,C]+B^{2}\,C+e\,\dot{e}\,B\,C-i\,B\,\bar{C}\,\dot{C}\,C$
$None$

where we have used the following EL-EOMs (and their modified version), namely;

$p_{\mu}\,\psi^{\mu}=m\,\psi_{5},\qquad\quad\frac{e\,C}{2}\,(p^{2}-m^{2})=-e^{2}\,\dot{B}\,C-i\,e^{2}\,\dot{\bar{C}}\,\dot{C}\,C.$
$None$

At this juncture, we note that
$B=-e\,\dot{e}+i\,(2\,\dot{\bar{C}}\,C+\bar{C}\,\dot{C})$. The substitution of
this expression leads to the following observations, namely;

$B^{2}\,C+e\,\dot{e}\,B\,C=-\,i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C,\qquad\quad-\,i\,B\,\bar{C}\,\dot{C}\,C=i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C.$
$None$

Thus, we obtain the modified form of the BRST charge $(\tilde{Q}_{B})$ as

$\tilde{Q}_{B}\rightarrow\tilde{Q}_{B}^{(1)}=e^{2}\,[B\,\dot{C}-\dot{B}\,C-i\,\dot{\bar{C}}\,\dot{C}\,C].$
$None$

This is due to the fact that the sum of last three terms in (C.1) is zero. It
can be readily checked that the above form of the charge [cf. Eq. (C.4)] is
not off-shell nilpotent. Hence, it is not suitable for our further
discussions.

Let us focus on the derivation of an alternative form of this BRST charge. In
the expression for the charge $\tilde{Q}_{B}$ [cf. Eq. (C.1)], we can express
$[-\,i\,B\,\bar{C}\,\dot{C}\,C]$, from the third equation from the top in (76)
which yields the following

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

Using (C.3) and (C.5), we obtain

${\tilde{Q}}_{B}^{(1)}\quad\longrightarrow\quad\tilde{Q}_{B}^{(2)}=e^{2}\,[B\,\dot{C}-\dot{B}\,C+i\,(\bar{C}\,\ddot{C}\,C-\dot{\bar{C}}\,\dot{C}\,C)]+2\,i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C.$
$None$

It is elementary exercise to check that the above expression for the BRST
charge is still not off-shell nilpotent of order two. Hence, it cannot be
expressed as an exact quantity w.r.t. the BRST transformations $(s_{b})$. At
this crucial point, we re-express (C.5) in a different type of mathematical
form as follows

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

From the relationship:
$B=-\,e\,\dot{e}+i\,(2\,\dot{\bar{C}}\,C+\bar{C}\,\dot{C})$, it is clear that
$i\,\bar{C}\,\dot{C}=B+e\,\dot{e}-2\,i\,\dot{\bar{C}}\,C$. As a result, we
have the following

$i\,B\,\bar{C}\,\dot{C}\,C=B\,(B+e\,\dot{e}-2\,i\,\dot{\bar{C}}\,C)\,C\quad\equiv\quad
B^{2}\,C+e\,\dot{e}\,B\,C.$ $None$

Substituting the above equality into (C.7), we obtain

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

However, we also note that
$i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C=-\,i\,B\,\bar{C}\,\dot{C}\,C$ [cf. Eq.
(C.3)] due to the fact that
$B=-e\,\dot{e}+i\,(2\,\dot{\bar{C}}\,C+\bar{C}\,\dot{C})$. Thus, we observe
that the equality in (C.9) reduces to:

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

The above equation leads to the following

${\tilde{Q}}_{B}\rightarrow
Q_{B}^{(1)}=e^{2}\,\Big{[}B\,\dot{C}-\dot{B}\,C-i\,(\dot{\bar{C}}\,\dot{C}\,C+\bar{C}\,\ddot{C}\,C)\Big{]}-2\,i\,e\,\dot{e}\,\bar{C}\,\dot{C}\,C,$
$None$

where ${\tilde{Q}}_{B}$ is quoted in (C.1) and $Q_{B}^{(1)}$ is written in Eq.
(77) of the main body of our text. The importance of (C.11) is the observation
that it can be written as an exact quantity w.r.t. the BRST transformations
$(s_{b})$. As a result, the charge $Q_{B}^{(1)}$ is off-shell nilpotent of
order two (i.e.
$s_{b}\,Q_{B}^{(1)}=-\,i\,\\{Q_{B}^{(1)},Q_{B}^{(1)}\\}=0\Longrightarrow[Q_{B}^{(1)}]^{2}=0)$.
Exactly similar kinds of arguments can be provided for the derivation of
$Q_{\bar{B}}^{(1)}$ in Eq. (77). As a consequence, we observe that:
$s_{ab}\,Q_{\bar{B}}^{(1)}=-\,i\,\\{Q_{\bar{B}}^{(1)},Q_{\bar{B}}^{(1)}\\}=0\Longrightarrow[Q_{\bar{B}}^{(1)}]^{2}=0$.
This observation proves the off-shell nilpotency of $Q_{\bar{B}}^{(1)}$ in a
straightforward fashion.

Appendix D: On an Alternative Proof of the Existence of the CF-Type
Restriction on Our SUSY System

Our present Appendix provides an alternative proof of the existence of the CF-
type restriction: $B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0$ which
is straightforward and different from our derivation in Sec. 7 where the proof
is a bit subtle. In this context, we apply, first of all, directly the BRST
symmetry transformations $(s_{b})$ on the expression for the anti-BRST charge
$Q_{\bar{B}}^{(1)}$ [cf. Eq. (77)]. In other words, we derive
$s_{b}\,Q_{\bar{B}}^{(1)}=-\,i\,\\{Q_{\bar{B}}^{(1)},Q_{B}^{(1)}\\}$ which is
nothing but the anticommutator of the anti-BRST charge $(Q_{\bar{B}}^{(1)})$
with the BRST charge $(Q_{B}^{(1)})$. The outcome of this exercise can be
explicitly expressed as:

$s_{b}\,Q_{\bar{B}}^{(1)}=2\,e^{2}\,\bar{B}\,\dot{\bar{C}}\,C-2\,e^{2}\,\dot{\bar{B}}\,\bar{C}\,\dot{C}-2\,e\,\dot{e}\,\dot{\bar{B}}\,\bar{C}\,C+2\,e\,\dot{e}\,\bar{B}\,\dot{\bar{C}}\,C-e^{2}\,\ddot{\bar{B}}\,\bar{C}\,C+e^{2}\,\bar{B}\,\bar{C}\,\ddot{C}$
$+2\,i\,e\,\dot{e}\,\dot{\bar{C}}\,\bar{C}\,\dot{C}\,C+e^{2}\,B\,\ddot{\bar{C}}\,C-e^{2}\,\ddot{B}\,\bar{C}\,C+2\,e\,\dot{e}\,B\,\dot{\bar{C}}\,C-2\,e\,\dot{e}\,\dot{B}\,\bar{C}\,C+i\,e^{2}\,\ddot{\bar{C}}\,\bar{C}\,\dot{C}\,C$
$+i\,e^{2}\,\dot{\bar{C}}\bar{C}\,\ddot{C}\,C+e^{2}\,B\,\dot{\bar{C}}\,\dot{C}-e^{2}\,\dot{B}\,\bar{C}\,\dot{C}+i\,e^{2}\,B\,\dot{\bar{B}}-i\,e^{2}\,\bar{B}\,\dot{B}+e^{2}\,\dot{\bar{B}}\,\dot{\bar{C}}\,C-e^{2}\,\bar{B}\,\dot{\bar{C}}\,\dot{C}.$
$None$

The above expression can be re-arranged in such a manner that we shall have
the coefficients of $e^{2}$ and $2\,e\,\dot{e}$ separately and independently
as illustrated below: