Advances in High Energy Physics

Advances in High Energy Physics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5593434 | https://doi.org/10.1155/2021/5593434

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
Received08 Feb 2021
Revised03 Jun 2021
Accepted24 Jun 2021
Published23 Jul 2021

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 [14] 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 [1021] 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., [1416]) 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 [2529] 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 [1416], 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 [1416], 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 [1416] 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