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

Abstract

We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of the one -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. Within the framework of the latter, we have shown the existence of the CF-type restriction by proving the (i) symmetry invariance of the coupled Lagrangians and (ii) the absolute anticommutativity property of the conserved (anti-)BRST charges. The application of the MBTSA to a physical SUSY system (i.e., a 1D model of a massive spinning particle) is a novel result in our present endeavor. In the application of ACSA, we have considered only the (anti-)chiral super expansions of the supervariables. Hence, the observation of the absolute anticommutativity of the (anti-)BRST charges is a novel result. The CF-type restriction is universal in nature as it turns out to be the same for the SUSY and non-SUSY reparameterization (i.e., 1D diffeomorphism) invariant models of the (non-)relativistic particles.

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 are fermionic (i.e., nilpotent) and absolutely anticommuting (i.e., ) in nature, (iii) the quantum gauge (i.e., BRST) invariance and unitarity are respected together in the perturbative computations at any arbitrary order, (iv) there is appearance 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, and (v) the (anti-)BRST symmetries transform a bosonic field/variable to a fermionic field/variable and vice-versa. Hence, these symmetries are supersymmetric type (i.e., SUSY type).

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. The former property encodes the SUSY-type 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 context of a 4D 1-form non-Abelian 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 symmetries that are associated with the gauge and (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., [16–18]) 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 (see, e.g., [19–21]), 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 [22], 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 (non-)relativistic particles [23, 24]. To be precise, judicious combination of MBTSA and ACSA has been very fruitful in our recent works [23, 24] 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 and Malik [25], the diffeomorphism symmetry transformations have been incorporated into the superfields which are defined on a -dimensional supermanifold on which a -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 investigation, 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 ( and ) and fermionic variables along with the (anti-)BRST invariant CF-type restrictions: which is responsible for (i) the validity (cf. Equation (22)) of the absolute anticommutativity (i.e., ) of the off-shell nilpotent () (anti-)BRST symmetry transformations and (ii) the derivation of and which are coupled (but equivalent) (cf. Equation (24)). 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-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 anticommutativity 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 [24]. 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 are 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 the same for the reparameterization invariant scalar and SUSY relativistic as well as a nonrelativistic particle [23]. Finally, our present investigation (as well as others [23, 24]) 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 theoretical high energy physics.

The theoretical contents of our present endeavor are organised as follows. In Section 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 Section 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 Section 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 Section 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 Section 6. Section 7 of our present endeavor contains theoretical proof of the nilpotency and absolute anticommutativity of the conserved and off-shell nilpotent (anti-)BRST charges. Finally, in Section 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 -dimensional target spacetime manifold as so that the dot product between two nonnull vectors is where the Greek indices and Latin indices . 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 . Our 1D model is generalized onto a -dimensional supermanifold which is parameterized by the superspace coordinates where is the bosonic evolution parameter and a pair of Grassmannian variable satisfy . In our present investigation, we shall focus only on the -dimensional (anti-)chiral super submanifolds of our chosen general-dimensional supermanifold in the context of ACSA.

2. Preliminaries: Continuous and Infinitesimal Reparameterization Symmetry Transformations

We divide our present section into two parts. We discuss, in our Section 2.1, a few classical infinitesimal continuous symmetries and their relationships with the classical infinitesimal reparameterization symmetry transformations. Our Section 2.2 is devoted to a concise description of the quantum (anti-)BRST symmetry transformations.

2.1. Some Classical Infinitesimal and Continuous Symmetries

We begin with the following first-order Lagrangian for the free one -dimensional (1D) massive spinning (i.e., SUSY) relativistic particle (see, e.g., [5, 26]) where we have parametrized the trajectory of the particle by and defined the “generalized” velocities w.r.t. to it. This 1D trajectory is embedded in the -dimensional flat Minkowskian target spacetime manifold where (with ) are the canonical conjugate pair of spacetime and momenta variables which are function of the evolution parameter . We have the fermionic variables in our theory. The Lagrangian (1) also has the bosonic variable and fermionic variable 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 is the superpartner of and the other fermionic variable has been introduced to incorporate the rest mass into the Lagrangian where the mass-shell condition is satisfied by the free () 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 becomes the Rarita-Schwinger Lorentz vector spin 3/2 field and the einbein variable turns itself into the vierbein field.

The action integral respects the following infinitesimal and continuous reparameterization symmetry transformations : because the first-order Lagrangian transforms, under the above infinitesimal reparameterization symmetry transformation , as where basically corresponds to the infinitesimal 1D diffeomorphism/reparameterization transformation: . Here, the transformation parameter is infinitesimal. It is an elementary exercise to note that, if we set all the fermionic variables equal to zero (i.e., ), we obtain an infinitesimal gauge symmetry transformation from the infinitesimal reparameterization symmetry transformation (2) as follows: where we have identified and used the Eular-Lagrange eqations of motions (EL-EOMs): . In Equation (4), the bosonic infinitesimal transformation parameter is nothing but the gauge transformation parameter. It can be readily checked that we have the following transformation for and S under :

We have infinitesimal classical supergauge symmetry transformations in our theory which transforms the fermionic variables into their bosonic counterparts and vice-versa. These continuous and infinitesimal symmetry transformations are where the infinitesimal transformation parameter is fermionic (i.e., ) in nature. It is straightforward to observe that we have the following:

Under the combined classical symmetry transformation , we have the following continuous and infinitesimal symmetry transformations , namely, which lead to the transformation of the first-order Lagrangian as

Thus, the continuous and infinitesimal transformation is indeed a symmetry transformation for the action integral 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, and identify the transformation parameters as and (see, e.g., [5, 26] 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 , infinitesimal, and continuous (anti-)BRST transformations as follows [5, 6, 26]: where the fermionic (anti-)ghost variables and the bosonic (anti-)ghost variables correspond to the bosonic and fermionic gauge and supergauge transformation parameters and of Equation (8), respectively. The variables are the Nakanishi-Lautrup-type auxiliary variables, and is an additional fermionic variable in our BRST-quantized (as well as invariant) theory.

It is straightforward to note that the anticommutativity property of the off-shell nilpotent (anti-)BRST symmetry transformations (), namely, is true if and only if the CF-type restriction is invoked from outside. However, this restriction is a physical constraint because it is an (anti-)BRST invariant (i.e., ) quantity. It can be readily checked that

In other words, we observe that the off-shell nilpotent (anti-)BRST symmetry transformations are absolutely anticommuting (i.e., ) provided the entire theory is considered on the quantum submanifold where the CF-type restriction is satisfied. It is the existence of this physical restriction that leads to the existence of coupled (but equivalent) Lagrangians which incorporate the gauge-fixing and Faddeev-Popov ghost terms in addition to the first-order Lagrangian of Equation (1). In the full blaze of glory, the above Lagrangians (in terms of all the appropriate variables) are as follows [6, 26]: where the subscripts and are appropriate because the Lagrangian depends on the Nakanishi-Lautrup auxiliary variable but the Lagrangian contains the auxiliary variable in its full expression. It is straightforward to check that and of our theory respect the perfect BRST and anti-BRST transformations because we have

As a consequence, the action integrals and 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 [26], 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 -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 counterpart quantum symmetries. In fact, the off-shell nilpotent (anti-)BRST symmetry transformations (corresponding to the classical Equation (2)) are where and are the Nakanishi-Lautrup auxiliary and are the (anti-)ghost variables of our theory. As far as the absolute anticommutativity property (i.e., ) of the above transformations is concerned, we note the following: where and . Thus, we note that the absolute anticommutativity property of the (anti-)BRST symmetry transformations (cf. Equations (20) and (21)) is satisfied (i.e., ) if and only if we invoke the (anti-)BRST invariant (i.e., ) CF-type restriction . We note, therefore, that a CF-type constraint exists on our theory which is the root cause behind the absolute anticommutativity (i.e., ) of the (anti-)BRST symmetry transformations and it leads to the existence of the coupled (but equivalent) (anti-)BRST invariant Lagrangians as where the (anti-)BRST symmetry transformations are the quantum symmetries (cf. Equations (20) and (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 Equation (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. Equation (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

We note that the pure Faddeev-Popov ghost part (i.e., ) of the above coupled (but equivalent) Lagrangians is the same. It can be readily checked that the EL-EOMs from Lagrangians and , w.r.t. the auxiliary variables and , lead to the following: which are responsible for the derivation of the CF-type restriction: . The above Lagrangians and of our theory respect the BRST and anti-BRST transformations in a precise and perfect manner because it is interesting to check that

As a consequence, the action integrals and respect the continuous and nilpotent symmetries and because of the Gauss’s divergence theorem (where all the physical variables of our theory vanish off as ). We can also apply on and on . The ensuing results are as follows: which demonstrate that the coupled Lagrangians and 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: .

It is very interesting to point out that the contributions of the term “” in Equation (23) turn out to be total derivatives because we observe that

As a consequence, the gauge-fixing and Faddeev-Popov ghost terms of the Lagrangians and of Equation (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., , 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 “” term. This is a novel observation in our theory which is radically different from our earlier work [26] where the “” term contributes to the dynamics. The observations in Equation (30) also imply that the absolute anticommutativity property of the (anti-)BRST symmetries is satisfied (i.e., ) 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

The conservation law (with ) can be proven by using the EL-EOMs that derived from the Lagrangians and . For instance, we point out that the following EL-EOMs w.r.t. the variables (), namely, are obtained from . The equations of motion that are different from (33) and emerge out from (as the EL-EOMs) are as follows:

These conserved currents lead to the definition of the conserved charges and which are the same (i.e., ) as the conserved currents quoted in Equations (31) and (32). This is due to the fact that we are dealing with a 1D system.

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 from outside. Second, this restriction can be obtained from the coupled (but equivalent) Lagrangians and if we use the EL-EOMs (cf. Equation (25)) w.r.t. the Nakanishi-Lautrup-type auxiliary variables and . Third, we observe that the term in the square bracket of Equation (23) does not contribute anything to the dynamics as well as to the gauge-fixing and Faddeev-Popov ghost terms. Fourth, the coupled Lagrangians and 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 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., ) of the conserved (i.e., ) and off-shell nilpotent (i.e., ) (anti-)BRST charges is satisfied only due to the validity of the existence of the CF-type restriction (cf. Section 6).

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 which are scalars w.r.t. the 1D space of the trajectory for the massive spinning relativistic particle that is embedded in the -dimensional target space. For this purpose, we exploit the theoretical power and potential of MBTSA (see, e.g., [22, 25]). In this context, first of all, we promote the 1D diffeomorphism transformation (where is the infinitesimal transformation parameter corresponding to the 1D diffeomorphism (i.e., reparameterization) symmetry transformation) to its counterpart on the -dimensional supermanifold as where the general -dimensional supermanifold is parameterized by a bosonic (i.e., evolution) coordinate and a pair of Grassmannian variables that satisfy . In the above diffeomorphism transformation, the function is any arbitrary function of the evolution parameter such that it is finite at and vanishes off at . In other words, is a physically well-defined function of . It is worth pointing out that the coefficients of the Grassmannian variables , in Equation (35), are nothing but the fermionic (i.e., ) (anti-)ghost variables 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 has been replaced by the fermionic (anti-)ghost variables of the (anti-)BRST symmetry transformations. This has been done purposely in view of the fact that in earlier works (see, e.g., [10–12]), it has been established that the translational generators , along the Grassmannian directions , are intimately connected with the nilpotent (anti-)BRST transformations in the ordinary space. In other words, we have already taken into account , which are the generalization of the classical infinitesimal 1D diffeomorphism symmetry transformation to its quantum counterparts within the framework of BRST formalism. It is worthwhile to point out that the secondary variable 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 -dimensional supermanifold as their counterpart supervariables where the generalization of the 1D diffeomorphism transformation (cf. Equation (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 (cf. Equation (22)) is concerned, namely, where the pair of variables , as pointed out earlier, is the Grassmannian variables (i.e., ) of the superspace coordinates that characterize the -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 -dimensional supermanifold, namely, where on the r.h.s., we have the secondary supervariables which are also function of . As a consequence, we can have the following Taylor expansions (for those secondary variables that are present on the r.h.s.), namely,

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 (that is embedded in the -dimensional flat Minkowskian target space). It is worthwhile to point out that 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. As a result, the expansion (37) can be written as where are the (anti-)BRST symmetry transformations (20) and (21). This is due to fact that the (anti-)BRST symmetry transformations have been shown to be deeply connected with the translational generators along the -directions of the -dimensional supermanifold (see, e.g., [10–12] for details).

It is evident that we have to compute the values of , , and (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 . At this stage, the so called “horizontality condition” (HC) comes to our help where we demand that This relationship can be explicitly expressed as where we have collected all the terms from Equation (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:

It is straightforward to note that we have already obtained and as is evident from Equation (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,

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:

On top of the already obtained off-shell nilpotent (anti-)BRST symmetry transformations: , we take into account the standard (anti-)BRST symmetry transformations and 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 Equation (42) as [22, 25] where is present in the expansion (39). A close look at Equation (44) implies that we have which is nothing but the CF-type restriction. In addition, the observation of Equation (41) implies that there is an explicit expression for in terms of (that is present in the expansion of in Equation (35)). Plugging in the values of and into Equation (41) leads to [22, 25]

Comparison of (44) and (45) yields (see, e.g., [22, 25] for details)

Thus, we note that the comparison of the values of (that is determined from the comparison between Equations (44) and (45)) leads to the deduction of the (anti-)BRST invariant (i.e., ) CF-type restriction which plays an important role in the proof: .

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 . In other words, we have made the following (anti-)chiral super expansions for the (anti-)chiral supervariables (in view of ), namely, where and are the chiral and antichiral supervariables that have been defined on the -dimensional suitably chosen chiral and antichiral super submanifolds of the general-dimensional supermanifold. Second, we have seen that (cf. Equation (22)) the absolute anticommutativity property (i.e., ) of the (anti-)BRST transformations is satisfied if and only if the CF-type restriction (46) is satisfied. Third, we point out that the requirement of the following leads to the derivation of (anti-)BRST symmetry transformations on the Nakanishi-Lautrup auxiliary variables . Fourth, within the framework of MBTSA, the CF-type restriction (46) is derived from the expression for due to the consistency condition (i.e., ). Fifth, the (anti-)BRST symmetry transformations () on the generic variable imply that we have already obtained the following (anti-)BRST symmetry transformations: for the target space variables of our theory that are present in the first-order Lagrangian (cf. Equation (1)) for the massive spinning (i.e., SUSY) relativistic particle. Finally, the explicit form of Equation (39) can be written, after the application of HC, as follows: where the superscript 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 in the super expansions (50) along with which are the coefficients of . We also note, from Equation (50), that we have a mapping: . This observation is consistent with results obtained in 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 (cf. Equation (24)) on the antichiral-dimensional super submanifold (of the most general-dimensional supermanifold) as where are the fermionic and are bosonic secondary variables. These variables have to be expressed in terms of the auxiliary and basic variables that are present in . For this purpose, we exploit the BRST-invariant restrictions. It will be noted that the antichiral -dimensional super submanifold is parameterized by the superspace coordinates where is the bosonic evolution parameter and Grassmannian variable is fermionic () in nature. In addition to (51), we have the antichiral limit (i.e., ) of the expansions (50) as follows: where the superscript denotes the antichiral limit of the super expansions (of the supervariables (cf. Equation (50))) that have been obtained after the application of HC. It is straightforward to note that the BRST invariance () of the variable implies that we have the following (with ), namely, where the superscript stands for the antichiral supervariable that has been obtained after the BRST invariant restriction. In other words, we have already obtained the BRST symmetry transformation as the coefficient of in (53) due to our knowledge of (i.e., ).

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

The above quantum gauge (i.e., BRST) invariant quantities must be independent of the Grassmannian variable when they are generalized onto -dimensional antichiral super submanifold (of the most general-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, where the supervariables with superscripts and have already been explained and derived in Equations (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. Equation (51)) as

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 (and the Nakanishi-Lautrup auxiliary variable of the Lagrangian (cf. Equation (24))).

The substitutions of the above expressions for the secondary variables into the super expansions (51) (besides Equations (52) and (53)) are as follows: where the superscript on the supervariables denotes the antichiral supervariables that have been obtained after the imposition of the BRST (i.e., quantum gauge) invariant restrictions in Equation (55). It is clear from (57) that we have a mapping which demonstrates that the off-shell nilpotency of the translational generator along -direction of the -dimensional antichiral super submanifold and off-shell nilpotency of the BRST transformations (21) in the ordinary space are interrelated. A careful look at Equations (50) and (57) demonstrates that we have already derived the BRST symmetry transformations (21) for all the variables of as the coefficients of .

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., ) limit of the full super expansions in (50), namely, where the superscript stands for the chiral limit of the supervariables that have been derived after the application of HC in Equation (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 as follows. It can be readily checked that we have the following interesting anti-BRST invariant quantities: where the anti-BRST symmetry transformations are the ones that have been listed in Equation (20). Keeping in our mind the mapping and the observation ,we have the following: where the superscript denotes the expansion of the supervariable that has been obtained after the application of the anti-BRST invariant restriction that is obtained due to the anti-BRST invariance . We also note that . Exploiting the basic principle of ACSA to BRST formalism, we obtain the following equalities in terms of chiral and ordinary variables, namely, where the chiral supervariables with superscripts and have been discussed and explained in Equations (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:

The above chiral supervariables are defined, and their expansions have been carried out on a -dimensional super submanifold that is characterized by the superspace coordinates where the Grassmannian coordinate is fermionic in nature. It is straightforward to draw the conclusion that the secondary variables and , on the r.h.s. of Equation (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 (and the Nakanishi-Lautrup auxiliary variable of the perfectly BRST invariant Lagrangian ), namely,

In other words, we have already determined the secondary variables (i.e., the coefficients of ) of the super expansions (62). The substitutions of (63) into the chiral super expansion (62) on the -dimensional chiral super submanifold leads to the following: where the superscript denotes the chiral supervariables that have been obtained after the application of the anti-BRST (i.e., quantum gauge) invariant restrictions in Equation (61). It is evident, from Equation (64), that we have already obtained the anti-BRST symmetry transformations (cf. Equation (20)) of the variables as the coefficients of in the chiral super expansions (62). We also observe, in the above chiral super expansions, that there is a mapping which agrees with the result of Refs. [10–12]. For the sake of completeness, we perform the step-by-step computations of the mathematical relationships obtained in 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. Equations (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: where the superscripts and , on the r.h.s., denote the antichiral and chiral versions of the first-order Lagrangian (cf. Equation (1)). In other words, we have the following: where the superscripts and on the supervariables of and have been already explained in the previous section. The superscripts and on the super Lagrangians, on the l.h.s. of Equations (65) and (66), denote the antichiral and chiral generalizations of the ordinary Lagrangians (24). Keeping in our mind the mappings: , we observe the (anti-)BRST invariance of the first-order Lagrangian in the language of ACSA to BRST formalism as follows:

In other words, we have accomplished the objective of establishing a precise connection between the (anti-)BRST invariance of in the ordinary space (cf. Equations (3), (27), and (26)) and superspace within the purview of ACSA. The results in (69) will be useful in the proof of the (anti-)BRST invariance (cf. Equations (27) and (26)) of the coupled (but equivalent) Lagrangians and . Geometrically, it is clear from our observation in (69) that super Lagrangians are the unique sum of (anti-)chiral supervariables (obtained after the (anti-)BRST invariant restrictions) such that their translations along -directions in the superspace produce the total derivatives in the ordinary space.

We now focus on the BRST and anti-BRST invariance of and 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 on them due to the mappings: . It is straightforward to check that we have the following explicit relationships between the invariances in the superspace and ordinary space:

We would like to emphasize that the super Lagrangian is a unique sum of antichiral supervariables (derived after the applications of the BRST invariant restrictions) such that its translation along -direction of the -dimensional antichiral super submanifold leads to a total derivative in the ordinary space. The latter is nothing but the BRST invariance of the ordinary Lagrangian (cf. Equation (26)). In exactly similar fashion, we can provide a geometrical interpretation for the anti-BRST invariance of (cf. Equation (27)) in the terminology of the superspace translational generator along the -direction of the suitably chosen chiral-dimensional super submanifold.

At this juncture, we concentrate on the deduction of the CF-type restriction in the proof of the equivalence between the Lagrangians and (cf. Equation (24)) within the ambit of ACSA to BRST formalism. In other words, we capture the transformations (cf. Equation (29)) and (cf. Equation (28)) in the terminology of the ACSA. Towards this central goal in our mind, first of all, we generalize the ordinary Lagrangian to its counterpart chiral super Lagrangian as where the superscripts and have already been explained in our earlier discussions. It is very interesting to observe that we have the following:

The above equation shows that the Lagrangian 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 within the ambit of ACSA while proving the anti-BRST invariance of the Lagrangian . We now demonstrate the BRST invariance of the Lagrangian and existence of the CF-type restriction within the purview of theoretical tricks and techniques of ACSA to BRST formalism. Towards these aims in our mind, we generalize the Lagrangian onto -dimensional antichiral super submanifold as follows: where the superscripts and have been already explained in our earlier discussions. Keeping in our mind the mapping: , we observe the following interesting relationship between and its counterpart in superspace, namely, where the r.h.s. is nothing but the operation of on the Lagrangian (cf. Equation (28)). In other words, we have established an intimate relationship between the BRST symmetry transformation on and operation of the translational generator on the antichiral super Lagrangian (defined on the -dimensional (anti-)chiral super submanifold). A careful and close look on the r.h.s. of (74) demonstrates that we have derived the CF-type restriction while proving the BRST invariance of within the purview of ACSA.

We end this section with the following concluding remarks. First, we have captured the (anti-)BRST invariance (cf. Equation (69)) of the first-order Lagrangian within the ambit of ACSA. Second, we have been able to express the (anti-)BRST invariance (cf. Equations (26) and (27)) of the Lagrangians and of our theory within the purview of ACSA (cf. Equation (70)). Third, we have been able to demonstrate that our observations in Equations (28) and (29) can also be expressed in superspace (cf. Equations (72) and (74)) within the ambit of ACSA. Finally, we have derived the CF-type restriction , in a subtle manner, by expressing the transformations and in the language of ACSA (cf. Equations (72) and (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. Equation (22)), the same restriction appears when we discuss and in the superspace by using the formal theoretical techniques of ACSA.

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

In this section, we prove the off-shell nilpotency and absolute anticommutativity of the (anti-)BRST charges which have already been derived in our Section 3 where and (cf. Equations (31) and (32)). In Section 7.1, we discuss the above properties of the (anti-)BRST charges in the ordinary space. Our 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 superspace (within the purview of ACSA). We describe explicitly the computation of in our Appendix B where the algebra is a bit more involved, and ultimately, we demonstrate the existence of the CF-type restriction in the ordinary space (cf. Equation (28)) which has also appeared in Equation (74) within the ambit of ACSA.

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

In this subsection, 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 of the (anti-)BRST charges in the ordinary space by using the standard relationship between the infinitesimal continuous (anti-)BRST transformations and their generators () as

The above proofs of the off-shell nilpotency of the conserved charges are nothing but the reflection of the off-shell nilpotency 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., ) (anti-)BRST charges (cf. Equations (31) and (32)). In other words, the straightforward application of on gives us a zero result. The same is the situation (i.e., ) when we apply the anti-BRST symmetry transformations on the anti-BRST charge . Hence, the Noether conserved charges (cf. Equations (31) and (32)) are off-shell nilpotent of order two (i.e., ) in the ordinary space due to the key relationship that is given in (75).

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 (anti-)BRST symmetry transformations . Towards this goal in mind, we use the following EL-EOMs (derived from the Lagrangians and ), namely, to get rid of the constraints and from the expressions for the (anti-)BRST charges (cf. Equations (31) and (32)) to recast them as

We have discussed 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. Equation (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 (anti-)BRST symmetry transformations , namely,

Now it is straightforward to note that and due to the off-shell nilpotency [] of the (anti-)BRST symmetry transformations . Thus, we conclude, from our observations in Equation (78), that the nilpotency of the (anti-)BRST transformations [] is deeply connected with the nilpotency of their generators (anti-)BRST charges [] which becomes completely transparent from the direct observations of the following computations:

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., ) of the conserved off-shell nilpotent (anti-)BRST charges []. Towards this central objective in mind, first of all, we assume the sanctity and validity of the CF-type restriction, , right from the beginning. As a result, we can express the (anti-)BRST charges , in the alternative forms, as follows:

We point out that it is because of the use of the CF-type restriction [] that we have been able to express in terms of the Nakanishi-Lautrup auxiliary variable and in the language of other Nakanishi-Lautrup-type auxiliary variable . At this crucial stage, we observe the following interesting relationships:

In other words, we have been able to express the anti-BRST charge [] as the BRST exact quantity. On the other hand, we have been able to write the BRST charge [] as an exact quantity w.r.t. the nilpotent anti-BRST transformation . A close and careful observation of (81) leads to the following (due to the well-known relationship between the continuous (anti-)BRST symmetry transformations and their generators as conserved (anti-)BRST charges ), namely,

As a result, we observe that the absolute anticommutativity of the (anti-)BRST charges is related to the nilpotency of the (anti-)BRST symmetries.

We would like to lay stress on the key results that have been seen in Equation (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 with the anti-BRST charge is intimately connected with the nilpotency of the BRST transformations and (ii) the anticommutativity property of the anti-BRST charge with the BRST charge owes its origin to the nilpotency of the anti-BRST transformations . We conclude this subsection with the following remarks. First, we have shown that the nilpotency of the (anti-)BRST charges is deeply related with the nilpotency of the (anti-)BRST transformations . Second, we have been able to express the modified form of the BRST charge and anti-BRST charge as the exact expressions w.r.t. the BRST transformations and anti-BRST transformations (cf. Equation (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 as an exact expression w.r.t. the anti-BRST transformations and the anti-BRST charge in the BRST-exact form. This exercise has enabled us to prove the absolute anticommutativity (i.e., ) of the nilpotent (anti-)BRST charges . Finally, the proof of the absolute anticommutativity property (cf. Equation (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 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 property of the conserved (anti-)BRST charges (in the ordinary Minkowski space).

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

In this subsection, 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 -dimensional (anti-)chiral super submanifolds has been taken into account. First of all, we focus on the off-shell nilpotency [] of the (anti-)BRST charges []. In this context, keeping in our knowledge the mappings: , , it can be readily seen that the expressions for the (anti-)BRST charges that have been quoted in Equation (78) can be translated into the superspace as follows: where the supervariables with the superscripts and have been obtained in Equations (64) and (57), respectively. At this stage, the off-shell nilpotency of the conserved (anti-)BRST charges can be written in the superspace (by using the theoretical techniques and tricks of ACSA to BRST formalism) as

Thus, we conclude that the off-shell nilpotency of the anti-BRST charge is deeply related to the nilpotency of the translational generator along the -direction of the -dimensional chiral super submanifold of the general-dimensional supermanifold. Similar type of comments can be made in the context of the off-shell nilpotency of the BRST charge and its intimate relationship with the nilpotency of the translational generator on the antichiral super submanifold.

We concentrate now on capturing the absolute anticommutativity of the (anti-)BRST charges within the purview of ACSA where the superspace of the -dimensional (anti-)chiral super submanifolds is taken into consideration. Towards this central goal in our mind, we express the modified forms of the (anti-)BRST charges of Equation (81) in the following mathematical expression within the framework of ACSA, namely, where the supervariables with the superscripts and have been quoted in Equations (64) and (57), respectively. At this crucial juncture, we note the following:

The above relations are nothing but the explicit proof of the absolute anticommutativity relations of the conserved (anti-)BRST charges (within the ambit of ACSA).

We wrap up this subsection with the following comments. First, the off-shell nilpotency of the (anti-)BRST charges is intimately connected with the nilpotency of the translational generators along the -directions of the -dimensional chiral and antichiral super submanifolds. Second, in the ordinary space, the above statements of the off-shell nilpotency are captured in Equations (75) and (79). Third, the absolute anticommutativity of the BRST charge with the anti-BRST charge is related to the nilpotency of the translational generator along the -direction of the chiral super submanifold. The absolute anticommutativity of the anti-BRST charge with the BRST charge, on the other hand, is connected with the nilpotency of the translational generator along the -direction of the antichiral super submanifold. Fourth, the above statements have been corroborated in the ordinary space by Equation (82) where the off-shell nilpotency of the (anti-)BRST transformations and the anticommutativity of the (anti-)BRST charges are found to be interconnected in an intimate and beautiful 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. [22] which has been christened by us as the MBTSA where the generalization of the 1D diffeomorphism [i.e., ] to the -dimensional supermanifold (cf. Equation (35)) has played an important role in the derivation of the (anti-)BRST symmetry transformations (cf. Equation (50)) for the target space variables (). In addition, this approach has enabled us to deduce the (anti-)BRST invariant CF-type restriction that is responsible for the absolute anticommutativity of the (anti-)BRST symmetry transformations (cf. Equations (20) and (21)) and existence of the coupled (but equivalent) Lagrangians (24) for our theory.

We have taken into account the standard (anti-)BRST symmetry transformations () for the (anti-)ghost variables which have, in a subtle manner, forced us to consider the (anti-)chiral super expansions (cf. Equation (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. Section 5). Within the purview of ACSA, we have derived the CF-type restriction when we have proven the equivalence of the coupled (but equivalent) Lagrangians (cf. Section 6). Furthermore, it is the validity of the CF-type restriction 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. Section 7). In other words, it is the proof of the anticommutativity of the conserved and nilpotent charges which lead to the existence of the CF-type restriction on our SUSY theory (cf. Section 7).

We would like to emphasize that the observation of the absolute anticommutativity property, in the context of the conserved (anti-)BRST charges, is a novel observation because of the fact that only the (anti-)chiral super expansions have been considered within the ambit of ACSA. 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 and (cf. Equations (28) and (29)) in the ordinary space and its analogue in the superspace are very interesting observations in our present endeavor (cf. Section 6). The other observation that merits a clear and special mention is the universality of the CF-type restriction in the context of the reparameterization (i.e., 1D diffeomorphism) invariant non-SUSY theory of a scalar relativistic particle as well as a nonrelativistic particle [23, 24] and our present SUSY system of a spinning relativistic particle (where the nilpotent SUSY transformations exist between the specific set of bosonic and fermionic variables of our SUSY theory).

It is worthwhile to mention that, for the -dimensional diffeomorphism invariant theory [22, 25] where the infinitesimal diffeomorphism symmetry transformation is (with ), the general form of the CF-type restriction has been obtained as where the (anti-)ghost fields correspond to the infinitesimal transformation parameter in the general coordinate transformation and the Nakanishi-Lautrup fields appear in the (anti-)BRST symmetry transformations: , . It is straightforward to note that the CF-type restriction is the limiting case of the above general-dimensional CF-type restriction in the case of the BRST approach to -dimensional diffeomorphism invariant theory [22, 25]. Thus, our theoretical treatments of the reparameterization (i.e., 1D diffeomorphism) invariant theories of the scalar and spinning relativistic particles are correct.

One of the highlights of ACSA to BRST formalism is the observation that it distinguishes between the suitably chosen -dimensional chiral and antichiral super submanifolds in the proof of the absolute anticommutativity of the conserved (anti-)BRST charges. For instance, we note that the anticommutativity of the BRST charge with the anti-BRST charge is connected with the nilpotency of the translational generator along the -direction of the chiral super submanifold (cf. Equation (88)). On the other hand, the anticommutativity of the anti-BRST charge with the BRST charge crucially depends on the nilpotency of the translational generator (cf. Equation (88)) along the -direction of the antichiral super submanifold (cf. Section 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. Section 7) where the off-shell nilpotency of the (anti-)BRST transformations (cf. Equation (82)) plays a decisive role.

We plan to extend our present study to the physical -dimensional (4D) theories of the gravitation and higher dimensional (super)string theories where there is existence of the diffeomorphism invariance. 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 [22, 25] 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. Before we end this section, it is worthwhile to point out that in our earlier works (see, e.g., [27, 28]), we have applied the techniques and tricks of ACSA to obtain the nilpotent symmetries of the 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.

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. Equation (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. Equation (59)) should not depend on (i.e., Grassmannian variable) when these quantities are promoted onto the -dimensional chiral version of super submanifold. First of all, we consider which leads to the following restriction:

At this stage, we substitute the super expansions from (62) and (58) which leads to the precise determination of the secondary variable . As a consequence, we have now the super expansion of the chiral supervariable as where the superscript denotes the chiral super expansion of after the application of the anti-BRST restriction (A.1). We find that the coefficient of is the anti-BRST symmetry transformation of (cf. Equation (20)). In other words, we find that which agrees with the mapping of Refs. [10–12].

At this juncture, we take up the anti-BRST invariant quantities (i.e., ) which leads to the following restrictions:

Substitutions from (A.2) and (62) lead to the following equations in terms of the fermionic (anti-)ghost variables as well as secondary variables and , namely,

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

We take now the anti-BRST invariance: . This leads to the following restrictions on the chiral supervariables:

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

It is evident that the precise solution is . Thus far, we have been able to determine the precise forms of the secondary variables () 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 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 Equation (47). This implies that we have already determined the remaining secondary variable of Equation (62) as . This completes our discussion on the step-by-step determination of the secondary variables of (cf. Equation (63)) which are present in the chiral super expansions (62).

B. On the Proof of Equations (28) and (74) in the Ordinary Space

For the sake of completeness, we provide here the explicit proof of Equations (28) and/or (74) in the ordinary space that leads to the derivation of the CF-type restriction which demonstrates, thereby, the equivalence of the Lagrangians and w.r.t. the (anti-)BRST symmetry transformations. This is due to our earlier observation (cf. Equation (27)) that has a perfect symmetry w.r.t. the anti-BRST symmetry transformations . The direct applications of on lead to the following:

It is straightforward to note (with the input ) that the coefficients of and can be expressed in terms of the CF-type restrictions as follows:

At this stage, we focus on the terms (from (B.1)) which can be expressed as a sum of a total derivative and other terms, namely,

Now, adding the left-over terms without the total derivative terms as well as the CF-type terms from (B.1) and (B.3), we obtain the following:

It can be readily seen that the following is completely true if we express the first two terms of the above equation, namely,

At this juncture, we add the terms from (B.4) and nonderivative terms from (B.5) to yield the following result: which can be reexpressed, in terms of the CF-type restriction, as follows:

The total sum of the contributions from (B.3), (B.5), and (B.7) is equal to

Now, adding all the terms of Equations (B.1), (B.2), and (B.8), we obtain the same result as given in Equations (28) and/or (74) in the ordinary space for the computation of .

We wrap up this appendix with the following concluding remarks. We have already demonstrated the existence of the CF-type restriction in the ordinary space (cf. Equation (28)) and superspace (cf. Equation (74)) by expressing the BRST symmetry transformation (i.e., ) of . Exactly in a similar manner, it can be shown that the quantity can be expressed in the ordinary space (cf. Equation (29)) and superspace (cf. Equation (72)) leading to the appearance of the CF-type restriction. It is interesting, furthermore, to point out that this restriction also appears when we prove the absolute anticommutativity of the conserved and off-shell nilpotent (anti-)BRST charges in the ordinary space as well as in the superspace (cf. Appendix D for details) using the ACSA to BRST formalism.

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 rewritten as follows: where we have used the following EL-EOMs (and their modified version), namely,

At this juncture, we note that . The substitution of this expression leads to the following observations, namely,

Thus, we obtain the modified form of the BRST charge as

This is because of 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. Equation (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 (cf. Equation (C.1)), we can express , from the third equation from the top in (76) which yields the following:

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

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 . At this crucial point, we reexpress (C.5) in a different type of mathematical form as follows:

From the relationship: , it is clear that . As a result, we have the following:

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

However, we also note that (cf. Equation (C.3)) due to the fact that . Thus, we observe that the equality in (C.9) reduces to

The above equation leads to the following: where is quoted in (C.1) and is written in Equation (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 . As a result, the charge is off-shell nilpotent of order two (i.e., . Exactly similar kinds of arguments can be provided for the derivation of in Equation (77). As a consequence, we observe that . This observation proves the off-shell nilpotency of in a straightforward fashion.

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 which is straightforward and different from our derivation in Section 7 where the proof is a bit subtle. In this context, we apply, first of all, directly the BRST symmetry transformations on the expression for the anti-BRST charge (cf. Equation (77)). In other words, we derive which is nothing but the anticommutator of the anti-BRST charge with the BRST charge . The outcome of this exercise can be explicitly expressed as

The above expression can be rearranged in such a manner that we shall have the coefficients of and separately and independently as illustrated below:

At this stage, the straightforward algebraic exercise produces the following result: which demonstrates explicitly that the absolute anticommutativity property is satisfied if and only if the CF-type restriction is invoked from outside. In other words, the requirement of the absolute anticommutativity of the (anti-)BRST charges leads to the existence of a CF-type restriction on our theory.

We now concentrate on the computation of which leads to the following as the sum of the coefficients of and , namely,

The above expression can be rearranged by performing some algebraic tricks in the following form (in terms of the CF-type restrictions ), namely,

The above expression demonstrates that the absolute anticommutativity property (i.e., ) is satisfied if and only if the validity of the CF-type restriction is invoked at the quantum level in our BRST quantized theory. There is another way of stating this observation. That is to say the fact that the sanctity of anticommutativity property (i.e., ) leads to the deduction of the CF-type restriction in a straightforward manner.

We close our appendix with the final remark that we can capture the derivation of the CF-type restriction within the purview of ACSA. Towards this central objective in our mind, first of all, we generalize the expressions for the BRST and anti-BRST charges (cf. Equation (77)) on the -dimensional chiral and antichiral supermanifold as where the superscripts and , on the BRST and anti-BRST charges, denote the chiral and antichiral versions of Equation (77). The other notations have already been explained earlier. It can now be checked that we have the following:

Thus, a careful and close look at the r.h.s. of (D.7) demonstrates that we have deduced the CF-type restriction within the ambit of ACSA in the proof of the absolute anticommutativity of the conserved (anti-)BRST charges. It is worthwhile to mention here that our Equation (D.7) captures the absolute anticommutativity of the (anti-)BRST charges in the ordinary space as well as in the superspace provided the whole theory is considered on the subspace (of the entire quantum Hilbert space) of variables where the CF-type restriction is satisfied.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

B. Chauhan thankfully acknowledges the financial support from DST, Government of India, under its INSPIRE fellowship scheme, and A. Tripathi and A. K. Rao are grateful to the general BHU fellowship (from Banaras Hindu University (BHU), Varanasi) under which the present investigation has been carried out.