Research Article | Open Access

A. K. Rao, A. Tripathi, R. P. Malik, "Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System", *Advances in High Energy Physics*, vol. 2021, Article ID 5593434, 20 pages, 2021. https://doi.org/10.1155/2021/5593434

# Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

**Academic Editor:**Mariana Frank

#### Abstract

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

#### 1. Introduction

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

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

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

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

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

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

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

##### 1.1. Convention and Notations

We adopt the convention of the left derivative w.r.t. *all* the fermionic variables of our theory. In the whole body of the text, we denote the fermionic (anti-)BRST transformations by the symbols and corresponding conserved and nilpotent charges carry the notations and in different contexts. The general -dimensional supermanifold is parameterized by , and its *chiral* and *antichiral* super submanifolds are characterized by and , respectively, where the *bosonic* coordinate is represented by the evolution parameter and the Grassmannian variables obey the *fermionic* relationships: , , and .

#### 2. Preliminaries: Lagrangian Formulation

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

##### 2.1. Classical Infinitesimal Symmetries

We begin with the *three* equivalent reparameterization invariant Lagrangians for the free nonrelativistic and non-SUSY particle as (see, e.g., [32] for details)
where the trajectory of the free nonrelativistic particle is embedded in a 2D configuration space characterized by the coordinates and the parameter specifies the trajectory of the particle as an evolution parameter. The momenta variables are defined by and , where and are the generalized “velocities” w.r.t. the coordinates and stands for any of the *three* Lagrangians of Equation (1). It is self-evident that the 4D phase space, corresponding to the 2D configuration space, is characterized by . In the above Equation (1), the mass of the nonrelativistic particle is denoted by and is the Lagrange multiplier variable that incorporates the constraint in the Lagrangians and . It is straightforward to note that and contain variables (and their first-order derivative) in the denominator *but* (i.e., the first-order Lagrangian) *does* not incorporate any variable (and/or its derivative) in *its* denominator. Furthermore, the starting Lagrangian (For a free massive NR particle, the standard action integral is , where and time “” is the evolution parameter. This action has *no* reparameterization invariance. If the evolution parameter is , then the action integral is which leads [31, 32] to the *final* action integral as . Hence, the starting Lagrangian becomes , where and .) does *not* permit the *massless* limit but the massless limits are well defined for and the second-order Lagrangian . We would like to stress that the Lagrange multiplier variable behaves like the “gauge” variable due to *its* transformation property.It will be worthwhile to dwell a bit on the derivation of the top entry in Equation (1). For a free massive NR particle, the standard action integral is , where and time “” is the evolution parameter. This action has *no* repa-rameterization invariance. If the evolution parameter is , then the action integral is which leads [31, 32] to the *fi-nal* action integral as . Hence, the starting Lagrangian becomes where and .

For our further discussions, we shall concentrate on the first-order Lagrangian because it has the maximum number of variables (i.e., , , , , and ) and allows the massless limit and there are *no* variables (and/or their first-order derivative w.r.t. ) in *its* denominator. It is straightforward to check that under the following infinitesimal and continuous 1D diffeomorphism (i.e., reparameterization) symmetry transformations , namely,
the action integral remains invariant (i.e., ) for the physically well-defined variables in and the infinitesimal diffeomorphism transformation parameter in , where is a *physically* well-defined function of such that it is *finite* at and vanishes off at . In fact, the infinitesimal and continuous reparameterization symmetry transformation is defined as for the generic variable of our present theory.

The above infinitesimal and continuous reparameterization symmetry transformations (2) encompass in their folds the gauge symmetry transformations which are generated (see, e.g., [32] for details) by the first-class constraints , where is the canonical conjugate momentum corresponding to the variable . Using the following Euler-Lagrange equations of motion (EL-EOMs) from , namely,
and identifying the transformation parameters , we obtain the infinitesimal and continuous *gauge* symmetry transformations , from the infinitesimal and continuous reparameterization symmetry transformations (2), as follows:

It is elementary now to check that the first-order Lagrangian transforms to a total derivative under the infinitesimal and continuous gauge transformations , namely, thereby rendering the action integral invariant (i.e., ) under the infinitesimal and continuous gauge symmetry transformations .

We end this section with the following decisive comments. First, the 1D diffeomorphism (i.e., reparameterization) transformations (2) are *more* general than the infinitesimal and continuous *gauge* symmetry transformations (4). Second, the Lagrange multiplier variable behaves like a “gauge” variable due to its transformation in (4). Third, all the *three* Lagrangians in (1) are *equivalent* and *all* of them respect the infinitesimal and continuous gauge and reparameterization symmetry transformations [32]. Fourth, all the Lagrangians describe the free motion of the NR particle. Hence, our system is a nonrelativistic free particle. Fifth, the first-order Lagrangian is theoretically more interesting to handle because, as pointed out earlier, it incorporates the maximum number of variables. Finally, we can exploit the reparameterization *and* gauge symmetry transformations (2) and (4) for the BRST quantization. Following the usual BRST prescription, we note that the (anti-)BRST symmetry transformations, corresponding to the *classical* reparameterization symmetry transformations (2), are
where are the fermionic (anti)ghost variables corresponding to the *classical* infinitesimal diffeomorphism transformation parameter (cf. Equation (2)). In an exactly similar fashion, the (anti-)BRST symmetry transformations, corresponding to the *classical* gauge symmetry transformations (4), are
where the fermionic variables are the (anti)ghost variables corresponding to the *classical* gauge symmetry transformation parameter of Equation (4). In addition to the (anti-)BRST symmetry transformations in (6) and (7), we have the following *standard* (anti-)BRST transformations:
where the pairs *and* are the Nakanishi-Lautrup auxiliary variables in the context of the BRST quantization of our reparameterization *and* gauge invariant system by exploiting the *classical* reparameterization and gauge transformations, respectively.

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

We have listed the quantum (anti-)BRST symmetries corresponding to the *classical* gauge symmetry transformations (4) in our Equations (7) and (8). It is elementary to check that these *quantum* symmetries are off-shell nilpotent of order two. The requirement of the absolute anticommutativity leads to the restriction . As a consequence, we have the *full* set of (anti-)BRST symmetry transformations (corresponding to the *classical* gauge symmetry transformations (4)) as follows:

It is straightforward to check that the above (anti-)BRST symmetry transformations are off-shell nilpotent and absolutely anticommuting in nature. The (anti-)BRST invariant Lagrangian (which is the generalization of the *classical* to its *quantum* level) can be written as (The structure of gauge-fixing and FP ghost terms is *exactly* like the Abelian 1-form gauge theory where we have the BRST invariant Lagrangian density: . Here, is the vector potential, is the field strength tensor, and the rest of the symbols are the same as in Equations (10) and (11). Note that the -form defines the field strength tensor (where in stands for the exterior derivative of the differential geometry).):

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

It should be noted that we have dropped the total derivative terms in obtaining from (10). The above equation demonstrates that we have obtained a *unique* (anti-)BRST invariant Lagrangian. This has happened because the CF-type restriction is trivial (i.e., ) in our *simple* case of the NR system. We can explicitly check that
which lead to the derivation of the conserved (anti-)BRST charges as follows:

In the last step, we have used which emerges out as the EL-EOM from w.r.t. the Lagrange multiplier variable . The structure of gauge-fixing and FP ghost terms in Equation (10) is exactly like the Abelian 1-form gauge theory where we have the BRST invariant Lagrangian density: . Here, is the vector potential, is the field strength tensor, and the rest of the symbols are the same as in Equation (10) and (11). Note that the 2-form defines the field strength tensor (where in which stands for the exterior derivative of the differential geometry).

We close this section with a few crucial and decisive remarks. First, we can check that the (anti-)BRST charges are conserved by using the EL-EOMs. Second, the (anti-)BRST charges are off-shell nilpotent of order *two* due to the *direct* observations that and which encode in their folds *and*. Third, the above nilpotency is *also* encoded in implying that due to *and* we *also* point out that due to the nilpotency of because . Fourth, we observe that and which explicitly lead to the conclusion that the off-shell nilpotent charges are *also* absolutely anticommuting in nature. Fifth, the above observation of the absolute anticommutativity can be also expressed in terms of the nilpotency property because we observe that and which imply that and (due to the off-shell nilpotency of the anti-BRST as well as the off-shell nilpotency of the BRST symmetry transformations). Sixth, it can be seen that the physical space (i.e., ) in the *total* Hilbert space of states is defined by which implies that and . In other words, the Dirac quantization conditions (with the first-class constraints ) are beautifully satisfied. Finally, the physicality criterion implies that the two physical states and belong to the *same* cohomological class w.r.t. the nilpotent BRST charge if they differ by a BRST *exact* state (i.e., for nonnull ).

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

This section is devoted to the derivation of the transformations , , , , , , , and by exploiting the theoretical tricks of MBTSA. Before we set out to perform this exercise, it is essential to pinpoint the off-shell nilpotency and absolute anticommutativity properties of the (anti-)BRST symmetry transformations on the phase variables (cf. Equation (6)). It can be easily checked that the off-shell nilpotency requirement (i.e., ) leads to the (anti-)BRST symmetry transformations for the (anti)ghost variables as

Furthermore, the absolute anticommutativity requirement for the generic phase variable leads to the following:

In other words, the absolute anticommutativity property () is satisfied if and only if we invoke the sanctity of the CF-type restriction . It goes without saying that the *above* cited requirements of the off-shell nilpotency and absolute anticommutativity properties are very *sacrosanct* within the framework of the BRST approach to gauge and/or reparameterization invariant theories.

Against the backdrop of the above discussions, we set out to deduce the (anti-)BRST symmetry transformations and (with ) and the CF-type restrictions within the framework of MBTSA. Towards this end in our mind, first of all, we generalize the *classical* function (in ) onto a -dimensional supermanifold as
where variables are the (anti)ghost variables of Equation (6) and is a secondary variable that has to be determined from the consistency conditions (that include the off-shell nilpotency as well as absolute anticommutativity requirements). It will be noted that, due to the mappings and [14–16], we have taken the coefficients of and in Equation (16) as the (anti)ghost variables . This has been done due to our observation in the infinitesimal reparameterization symmetry transformation (where ) at the *classical* level. Following the basic tenet of BRST formalism, the infinitesimal parameter has been replaced (in the BRST-*quantized* theory) by the (anti)ghost variables, thereby leading to the (anti-)BRST symmetry transformations and .

For our present 1D diffeomorphism (i.e., reparameterization) invariant theory, the generic variable can be generalized to a supervariable on the -dimensional supermanifold [22] with the following super expansion along *all* the Grassmannian directions of the -dimensional supermanifold, namely,
where the expression for is given in Equation (16). It should be noted that *all* the primary as well as secondary supervariables on the r.h.s. of (17) are a function of the -dimensional super infinitesimal diffeomorphism transformation (16). At this stage, we can perform the Taylor expansions for *all* the supervariables as

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

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

The above equality implies that *all* the coefficients of , , and of Equation (19) should be set equal to zero. In other words, we have the following:

Substitutions of the values of and into the expression for lead to the following:

As explained before Equation (20) (i.e., exploiting the key properties of *scalars*), it is evident that (17) can be *finally* written with as
where, due to the well-known mappings and [14–16], the coefficients of and are the anti-BRST and BRST symmetry transformations (cf. Equation (6)). We point out that the key properties of *scalars* on the r.h.s. of Equation (17) imply that we have , , , and .

A comparison between (21) and (23) implies that we have already derived the nilpotent (anti-)BRST symmetry transformations and . In other words, we have obtained , , , , , , , and . Furthermore, it is evident that

The requirement of the absolute anticommutativity implies that which, in turn, leads to the following relationships:

The explicit computations of the following, using the (anti-)BRST symmetry transformations of the phase variables in Equations (6) and (14), are

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

In other words, it is the consistency conditions of the BRST formalism that lead to the determination of in Equation (16) within the ambit of MBTSA. A close look at Equations (25)–(27) establishes that a precise determination of in (23) leads to (i) the validity of the absolute anticommutativity (i.e., ) of the off-shell nilpotent (anti-)BRST symmetries and (ii) the deduction of the (anti-)BRST invariant (This statement is *true* only when the whole theory is considered on a *submanifold* of the Hilbert space of the quantum variables where the CF-type restriction is satisfied. In other words, we explicitly compute and which imply that is true *only* on the abovementioned submanifold.) CF-type restriction on our theory. This statement (i.e., the (anti-)BRST invariance) is *true* only when the whole theory is considered on a *submanifold* of the Hilbert space of the quantum variables where the CF-type restriction is satisfied. In other words, we explicitly compute and which imply that is true *only* on the abovementioned submanifold.

We conclude this section with the following useful and crucial remarks. First, we set out to derive the (anti-)BRST symmetry transformations (corresponding to the classical reparameterization symmetry transformations) for the phase variables (cf. Equation (6)). We have accomplished this goal in Equation (21). Second, we have derived the CF-type restriction within the purview of MBTSA (cf. Equation (27)) which is actually hidden in the determination of in Equation (23). Third, for the application of the theoretical potential of MBTSA, we have taken the *full* super expansion of the *generic* supervariable (cf. Equation (17)) along *all* the possible Grassmannian directions of the -dimensional supermanifold. Fourth, unlike the application of the BT superfield/supervariable approach to the *gauge* theories [14–16] where spacetime does *not* change, in the case of MBTSA, the super diffeomorphism transformation (16) has been taken into account in all the *basic* as well as *secondary* supervariables. Fifth, taking into account the inputs from Equations (21) and (26), we obtain the following super expansion of the *generic* variable , namely,
where and the superscript on the supervariable denotes that this supervariable has been obtained after the application of HC. Finally, the standard nilpotent (anti-)BRST symmetry transformations (8) dictate that we can have the following (anti)chiral super expansions for the supervariables corresponding to , namely,
where the superscripts and denote the *chiral* and *antichiral* supervariables. The above observation gives us a clue that we should exploit the theoretical strength of ACSA to BRST formalism for our further discussions.

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

In addition to the quantum (anti-)BRST symmetries in (6), (8), and (14), we derive *all* the other off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries corresponding to the *classical* infinitesimal and continuous reparameterization symmetry transformations (2). We exploit the strength of the *sacrosanct* requirements of off-shell nilpotency and absolute anticommutativity properties. In this context, we point out that we have already derived and by invoking the sanctity of the off-shell nilpotency property for the phase variables (i.e., ). It is interesting to note that the following absolute anticommutativity requirements, namely,
lead to the derivation of the and . We can readily check that and are satisfied due to our knowledge of the BRST and anti-BRST symmetry transformations and *and* the fermionic nature of the (anti)ghost variables . We further note that and . The requirement of the absolute anticommutativity on the variable leads to

Thus, we emphasize that the absolute anticommutativity property on the phase variables (cf. Equation (15)) as well as on the Lagrange multiplier variable (cf. Equation (31)) is satisfied if and only if the CF-type restriction is invoked. In the full blaze of glory, the *quantum* (anti-)BRST symmetry transformations (corresponding to the infinitesimal reparameterization symmetry transformations (2)) are as follows:

The above *fermionic* symmetry transformations are off-shell nilpotent and absolutely anticommuting provided that the whole theory is considered on a submanifold of the space of quantum variables where the CF-type restriction is satisfied.

The existence of the *above* CF-type restriction leads to the derivation of the coupled (but equivalent) Lagrangians (i.e., and ) as follows:

We point out that the terms inside the square brackets are the *same* as in Equation (10) for the BRST analysis of the *classical* gauge symmetry transformations (4). Furthermore, in contrast to the *unique* (anti-)BRST invariant Lagrangian (cf. Equation (11)) (corresponding to the *classical* gauge symmetry transformations), we have obtained here a set of coupled (but equivalent) (anti-)BRST invariant Lagrangians in Equation (34). This has happened because of the fact that the CF-type restriction is *trivial* in the case of the *former* while it is a *nontrivial* restriction in the context of the *latter*.

One can readily compute the operation of on the quantities in the square brackets of Equation (34). In the full blaze of their glory, the coupled (but equivalent) Lagrangians and are as follows (It will be worthwhile to mention *here* that the *form* of the gauge-fixing and Faddeev-Popov ghost terms is the *same* as in the cases of NSUSY (i.e., scalar) and SUSY (i.e., spinning) relativistic particles [23, 24].):
where the subscripts and on the Lagrangians are appropriate because depends *uniquely* on the Nakanishi-Lautrup auxiliary variable (where is *not* present at all). Similarly, the Lagrangian is *uniquely* dependent on . They are coupled because the EL-EOMs with respect to and from and , respectively, yield
which lead to the deduction of the CF-type restrictions . Furthermore, the condition also demonstrates the existence of the CF-type restriction on our theory (cf. Appendix A below). It will be worthwhile to mention *here* that the *form* of the gauge-fixing and Faddeev-Popov ghost terms in Equation (35) is the *same* as in the cases of NSUSY (i.e., scalar) and SUSY (i.e., spinning) relativistic particles [23, 24].

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

The above observations demonstrate that the action integrals and remain invariant under the SUSY-type (i.e., fermionic) off-shell nilpotent, continuous, and infinitesimal (anti-)BRST symmetry transformations for the physical variables that vanish off at . At this crucial juncture, we establish the *equivalence* of the coupled Lagrangians and w.r.t. the (anti-)BRST symmetry transformations . In this context, we apply