#### Abstract

We discuss the continuous and infinitesimal gauge, supergauge, reparameterization, nilpotent Becchi-Rouet-Stora-Tyutin (BRST), and anti-BRST symmetries and derive corresponding nilpotent charges for the one -dimensional (1D) massive model of a spinning relativistic particle. We exploit the theoretical potential and power of the BRST and supervariable approaches to derive the (anti-)BRST symmetries and coupled (but equivalent) Lagrangians for this system. In particular, we capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of the newly proposed (anti-)chiral supervariable approach (ACSA) to BRST formalism where only the (anti-)chiral supervariables (and their suitable super expansions) are taken into account along the Grassmannian direction(s). One of the novel observations of our present investigation is the derivation of the Curci-Ferrari- (CF-) type restriction by the requirement of the absolute anticommutativity of the (anti-)BRST charges in the ordinary space. We obtain the same restriction within the framework of ACSA to BRST formalism by (i) the symmetry invariance of the coupled Lagrangians and (ii) the proof of the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges. The observation of the anticommutativity property of the (anti-)BRST charges is a novel result in view of the fact that we have taken into account only the (anti-)chiral super expansions.

#### 1. Introduction

The basic concepts behind the local gauge theories are at the heart of a precise theoretical description of three out of four fundamental interactions of nature. Becchi-Rouet-Stora-Tyutin (BRST) formalism [1–4] is one of the most intuitive and beautiful approaches to quantize the local gauge theories where the unitarity and quantum gauge (i.e., (anti-)BRST) invariance are respected together at any arbitrary order of perturbative computations for a given physical process that is permitted by the local (i.e., interacting) gauge theory at the quantum level. A couple of decisive features of the BRST formalism are the nilpotency of the (anti-)BRST symmetries as well as the existence of the absolute anticommutativity property between the BRST and anti-BRST symmetry transformations for a given local classical gauge transformation. The hallmark of the quantum (anti-)BRST symmetries is the existence of the (anti-)BRST invariant Curci-Ferrari- (CF-) type restriction(s) [5, 6] that ensure the absolute anticommutativity property of the (anti-)BRST symmetry transformations and the existence of the coupled (but equivalent) Lagrangian densities for the quantum gauge theories. The Abelian 1-form gauge theory is an exception where the CF-type restriction is trivial and the Lagrangian density is unique (but that is a limiting case of the non-Abelian 1-form gauge theory where the CF condition [7] exists).

The usual superfield approach (USFA) to BRST formalism [8–15] sheds light on the geometrical origin for the off-shell nilpotency and absolute anticommutativity of the (anti-)BRST symmetry transformations where the horizontality condition (HC) plays an important and decisive role [10–12]. These approaches, however, lead to the derivation of the (anti-)BRST symmetries for the gauge field and associated (anti-)ghost fields only [10–12]. The above USFA does not shed any light on the (anti-)BRST symmetries, associated with the matter fields, in an interacting gauge theory. In our earlier works (see, e.g., [16–19]), we have systematically and consistently generalized the USFA where in addition to the HC, we exploit the potential and power of the gauge-invariant restrictions (GIRs) to obtain the (anti-)BRST symmetry transformations for the matter, (anti-)ghost, and gauge fields of an interacting gauge theory together. There is no conflict between the HC and GIRs as they complement and supplement each other in a beautiful fashion. This approach has been christened as the augmented version of superfield approach (AVSA) to BRST formalism. The USFA with HC, developed in [10–12], is mathematically very elegant and, in one stroke, it leads to the derivation of proper (anti-)BRST symmetry transformations for the gauge and associated (anti-)ghost fields along with the derivation of the (anti-)BRST invariant CF-condition. The AVSA is a minor extension of [10–12] where HC and gauge invariant restrictions are exploited together (cf. Section 8).

In the recent set of papers [20–24], we have developed a simpler version of the AVSA where only the (anti-)chiral supervariables/superfields and their appropriate super expansions have been taken into consideration. This superfield approach to BRST formalism has been christened as the (anti-)chiral superfield/supervariable approach (ACSA). It may be mentioned here that, in all the earlier superfield approaches [8–19], the full super expansions of the superfields/supervariables, along all the Grassmannian directions of the (D, 2)-dimensional supermanifold, have been taken into account for the consideration of a D-dimensional local gauge-invariant theory (defined on the flat Minkowskian space). One of the decisive features of the ACSA to BRST formalism is its dependence on the quantum gauge (i.e., (anti-)BRST) invariant restrictions on the supervariables/superfields which lead to the derivation of appropriate (anti-)BRST symmetry transformations for all the fields/variables of the theory together with the deduction of the (anti-)BRST invariant CF-type restriction(s). The upshot of the results from ACSA is the observation that the conserved and nilpotent (anti-)BRST charges turn out to be absolutely anticommuting in nature despite the fact that only the (anti-)chiral super expansions of the supervariables/superfields are taken into account (within the framework of ACSA to BRST formalism).

The purpose of our present investigation is to apply the ACSA to BRST formalism to the 1D system of a massive spinning relativistic particle and derive the proper (anti-)BRST symmetry transformations for this system so that it can be discussed and described within the framework of BRST formalism. Our present 1D reparameterization invariant system is important in its own right as it provides a prototype model for the (super)gauge-invariant theory as well as an example for a toy model of the supergravity theory. Needless to say, its generalization leads to the theory of superstrings, too. If the existence of the continuous symmetries is the guiding principle for the definition of a beautiful theory in physics, the 1D model of a massive spinning particle represents one such example which encompasses in its folds a host of beautiful continuous symmetries (cf. Sections 2 and 3). In our present investigation, we lay a whole lot of emphasis on the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations of our 1D system and derive the corresponding conserved Noether charges. It is worthwhile to mention, at this stage, that, physically, the property of off-shell nilpotency of the (anti-)BRST symmetries and corresponding charges imply their fermionic nature and the absolute anticommutativity property encodes the linear independence of the above nilpotent symmetries and charges.

Against the backdrop of the above discussions, in our present endeavor, we have shown the existence of the three classical level symmetries which are the gauge, supergauge, and reparameterization transformations (cf. Equations (2) and (4)) under which the first-order Lagrangian for the 1D system of a massive spinning relativistic particle remains invariant. We have further established that the reparameterization symmetry transformations contain (i) the gauge symmetry transformations (cf. Equation (2)) provided all the fermionic variables are set equal to zero (i.e., ), and (ii) the combination of gauge and supergauge symmetry transformations (cf. Equation (7)) under specific conditions where the appropriate equations of motion and identifications of the transformation parameters are taken into account (cf. Equations (6), (9), and (10). We have elevated the classical (super)gauge symmetry transformations (7) to the quantum level within the framework of BRST formalism and derived the (anti-)BRST symmetries that are respected by the coupled (but equivalent) Lagrangians and (cf. Equations (17) and (18)). We have demonstrated that both the Lagrangians are equivalent because both of them respect both the BRST and anti-BRST symmetry transformations at the quantum level provided the whole theory is considered on the submanifold of the quantum Hilbert space of variables where the CF-type restriction is satisfied (cf. Equations (20), (21), (22), and (23)). We have further shown the existence of the (anti-)BRST invariant CF-type restriction at the level of the proof of absolute anticommutativity of the (anti-)BRST conserved charges in the ordinary space (cf. Equations (35), (36), (37), and (38)).

In our present endeavor, we have captured all the above key features within the framework of ACSA to BRST formalism where only the (anti-)chiral supervariables and their corresponding super expansion(s) along the Grassmannian direction(s) of the (1, 1)-dimensional (anti-)chiral super submanifolds of the general (1, 2)-dimensional supermanifold have been taken into consideration in a consistent and systematic fashion. One of the novel observations is the proof of the absolute anticommutativity property of the conserved and nilpotent (anti-)BRST charges within the ambit of ACSA to BRST formalism where only the (anti-)chiral super expansion(s) of the (anti-)chiral supervariables have been taken into account. Moreover, we note that the above proof also distinguishes between the chiral and antichiral (1, 1)-dimensional super submanifolds within the framework of ACSA to BRST formalism (cf. Appendix D).

Our present investigation is essential and interesting on the following counts. First and foremost, our 1D system of the massive spinning relativistic particle is more general than its massless counterpart which has been discussed in our earlier work [25]. Second, our present system is a toy model of a supersymmetric gauge theory whose generalization to 4D provides a model for the supergravity theory with a cosmological constant term. Hence, this toy model is interesting and important in its own right. Third, our present model is also a generalization of the scalar relativistic particle where the fermionic as well as bosonic (anti-)ghost variables appear within the framework of BRST formalism. Fourth, we have been curious to find out the contribution of the mass term (and its associated variable) in the determination of the gauge-fixing and Faddeev-Popov ghost terms within the framework of BRST formalism (cf. Equation (16)). Fifth, we have found out the CF-type restriction for the 1D massless spinning particle in our earlier work by exploiting the beauty of the supersymmetrization of horizontality condition [25]. Thus, we are now curious to find out its existence by proving the absolute anticommutativity of the conserved (anti-)BRST charges. Furthermore, we are interested in capturing its existence within the framework of ACSA to BRST formalism. We have accomplished all these goals in our present endeavor. Finally, a thorough study of our 1D system of a massive spinning relativistic has been a challenge for us as we have already studied a scalar relativistic particle and a massless spinning relativistic particle from various angels in our earlier works [25–32].

The theoretical material of our present endeavor is organized as follows. In Section 2, we discuss the gauge, supergauge, and reparameterization symmetries of the Lagrangian that describes the 1D massive spinning relativistic particle. Our Section 3 deals with the (anti-)BRST symmetries corresponding to the combined gauge and supergauge symmetries where the fermionic as well as the bosonic (anti-)ghost variables appear in the BRST analysis. The subject matter of Section 4 concerns itself with the derivation of the BRST symmetries within the framework of ACSA to BRST formalism where the quantum gauge (i.e., BRST) invariant restrictions on the antichiral supervariables play a crucial role. Our Section 5 is devoted to the derivation of anti-BRST symmetries by exploiting the anti-BRST invariant restrictions on the chiral supervariables within the purview of ACSA to BRST formalism. In Section 6, we prove the existence of the CF-type restriction by capturing the symmetry invariance of the Lagrangians within the ambit of ACSA. We capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges by applying the key techniques of ACSA to BRST formalism in Section 7. Finally, in Section 8, we make some concluding remarks and point out a few future directions for further investigations.

In our Appendices A, B, and C, we collect a few of the explicit computations which supplement as well as complement some of the crucial and key statements that have been made and emphasized in the main body of our present endeavor. Our Appendix D is devoted to the discussion of an alternative proof of the absolute anticommutativity of the (anti-)BRST charges and the existence of the CF-type restriction (i) in the ordinary space and (ii) in the superspace by exploiting the theoretical tricks and techniques of ACSA.

Convention and notations: the free (i.e., ) massive spinning relativistic particle is embedded in a D-dimensional flat Minkowskian spacetime manifold that is characterized by a metric tensor where the Greek indices . We adopt the convention of the left-derivative w.r.t. the fermionic variables . We denote the (anti-)BRST fermionic symmetry transformations by the symbol which anticommutes (i.e., etc.) with all the fermionic variables and commutes (i.e., etc.) with all the bosonic variables of our theory. We also denote the (anti-)BRST charges by the symbol .

#### 2. Preliminaries: Some Continuous and Infinitesimal Symmetries in Lagrangian Formulation

In this section, we discuss some infinitesimal and continuous symmetries and demonstrate their equivalence under some specific conditions where the usefulness of some appropriate equations of motion as well as identifications of a few transformation parameters has been exploited. We begin with the following three equivalent Lagrangians which describe the 1D system of a massive spinning relativistic particle (see, e.g., [33]) where is the Lagrangian with a square root, is the first-order Lagrangian, and is the second-order Lagrangian. Our one -dimensional (1D) system is embedded in a flat Minkowskian D-dimensional target space where () are the canonically conjugate bosonic coordinates and momenta (with ). The trajectory of the particle is parameterized by an evolution parameter , and generalized velocities () are defined w.r.t. it. We have fermionic ( etc.) variables in our theory which commute ( etc.) with all the bosonic variables of our theory. It should be noted that is the superpartner of and variable has been invoked in the theory to incorporate a mass term so that the mass-shell condition for the free particle could be satisfied. We would like to point out that, in Ref. [33], the emphasis is laid on the first-order Lagrangians and their usefulness. Hence, the first-order Lagrangian () is the only Lagrangian that is mentioned in [33] and there is absence of as well as .

The Lagrangian has a square root, and its massless limit is not defined. On the other hand, the second-order Lagrangian is endowed with a variable (i.e., einbein) which is located in the denominator. Thus, the Lagrangians and have their own limitations. We shall focus on the first-order Lagrangian for our discussions where variables and are not purely Lagrange multiplier variables but their transformations are such that they behave like the “gauge” and “supergauge” variables (cf. Equation (2)). Our 1D system is a model of supersymmetric gauge theory, and its generalization to 4D theory provides a model for the supergravity theory where corresponds to the Rarita-Schwinger field and becomes the vierbein field. The mass , in the supergravity theory, represents the cosmological constant term. In a nutshell, our present 1D model of a massive spinning relativistic particle is important and interesting in its own right because its generalization also becomes a model of the superstring theory (see, e.g., [34, 35]).

The Lagrangian respects the following gauge and supergauge symmetry transformations, namely, where and are the infinitesimal gauge and supergauge symmetry transformation parameters, respectively. It is straightforward to note that is a bosonic and is a fermionic (i.e., ) transformation parameter. Furthermore, the transformation is a supersymmetric transformation because it transforms a bosonic variable to a fermionic variable and vice versa. The transformations in Equation (2) are symmetry transformations because the first-order Lagrangian transforms to the following total derivatives:

As a consequence, it is clear that the action integral , under the transformations and , would be equal to zero (i.e., ) due to the fact that all the physical variables vanish off at . There is a reparameterization symmetry, too, in our theory due to the basic infinitesimal transformation where is an infinitesimal transformation parameter. In fact, the physical variables of our 1D system transform under the infinitesimal reparameterization transformation as

The above transformations are symmetry transformations for the action integral because of the following transformation property of , namely,

It is evident that due to the fact that and vanish off at .

The reparameterization symmetry transformation and gauge symmetry transformation are equivalent under the following limits: provided we set all the fermionic variables of our theory equal to zero. In the above, we have used equations of motion: and , and we have identified the gauge symmetry transformation parameter with the combination of the reparameterization transformation parameter and the einbein variable . In an exactly similar fashion, we note that and are also equivalent. In this context, first of all, we note that there are two primary constraints (i.e., ) and two secondary constraints (i.e., ) on our theory where and are the canonical conjugate momenta w.r.t. the Lagrange multiplier variables and , respectively. The above four constraints of our theory are first-class in the terminology of Dirac’s prescription for the classification scheme of constraints because they (anti)commute among themselves [36, 37]. These constraints generate the combined (super)gauge symmetry transformations for the physical variables of our theory as (see, e.g., [27]) under which the first-order Lagrangian transforms to a total “time” derivative as

As a consequence of the above observation, it is evident that where is the action integral. If we use the following equations of motion: and identify the transformation parameters as we find that the reparameterization symmetry transformation (4) (emerging due to the basic transformation: ) and the combined gauge and supergauge symmetry transformations (i.e., ), quoted in Equation (7), are equivalent to each other. It is worthwhile to note that, under the identifications (10), the transformation becomes as we note that

We end this section with the following remarks. First of all, we note that the canonical Hamiltonians, derived from and (as well as ), are where is the canonical Hamiltonian corresponding to the Lagrangian . It is straightforward to note that the primary constraints , lead to the derivation of the secondary constraints , from the Hamiltonians (11) as well as from all the three equivalent Lagrangians (1) (cf. Appendix A). Second, we have explicitly demonstrated that the (super)gauge symmetry transformations and reparameterization symmetry transformations are equivalent under specific conditions (cf. Equations (9) and (10)). Finally, the system under consideration is very interesting and important because it is endowed with many symmetries and it provides a prototype example for the supersymmetric gauge theory, superstrings, and a model for the supergravity theory.

#### 3. (Anti-)BRST Symmetries: Lagrangian Formulation

Our present section is divided into two parts. In Subsection 3.1, we show the existence of the CF-type restriction by the requirement of absolute anticommutativity of the (anti-)BRST symmetries and (anti-)BRST invariance of the coupled (but equivalent) Lagrangians and . In Subsection 3.2, we establish the existence of the same by requiring the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges.

##### 3.1. (Anti-)BRST Invariance and CF-Type Restriction

Corresponding to the combined classical (super)gauge symmetry transformations (cf. Equation (7)), we can write down the quantum (anti-)BRST symmetry transformations where the classical gauge symmetry parameter would be replaced by the fermionic ( etc.) (anti-)ghost variables and the classical supergauge symmetry transformations parameter would be replaced by a pair of bosonic (anti-)ghost variables . These off-shell nilpotent , infinitesimal and continuous (anti-)BRST symmetry transformations , in their full blaze of glory for our 1D system of the massive spinning relativistic particle, are (see, e.g., [25]) where and are the Nakanishi-Lautrup-type auxiliary variables, fermionic () variables are present in our theory, and the rest of the symbols have already been explained earlier. As far as the absolute anticommutativity property is concerned, it can be checked that are equal to zero only after imposing the CF-type restriction: from outside. It is worthwhile to mention that this CF-type restriction is a physical restriction within the realm of BRST formalism because it is an (anti-)BRST invariant (i.e., ) quantity. Except for the variables (), it is straightforward to check that the following is true for the other variables of our theory, namely, where is the generic variable of the (anti-)BRST invariant theory. Thus, it is crystal clear that the (anti-)BRST symmetry transformations in (12) and (13) are off-shell nilpotent and absolutely anticommuting in nature provided the whole theory is considered on a submanifold of space of quantum variables where the CF-type restriction: , is satisfied in the quantum Hilbert space (see, e.g., [25]).

The coupled (but equivalent) Lagrangians for our (anti-)BRST invariant system of the 1D massive spinning relativistic particle can be written as where is the first-order Lagrangian that has been quoted in Equation (1). The above Lagrangians for our 1D system of a massive spinning relativistic particle can be written, in their full glory incorporating the gauge-fixing and Faddeev-Popov ghost terms, as where, as pointed out earlier, and are the Nakanishi-Lautrup-type auxiliary variables which lead to the derivation of EL-EOMs (from and ) as

It is elementary to note that the above relationships lead to the derivation of the CF-type restriction: , which is the hallmark of a quantum gauge theory discussed within the framework of BRST formalism [5, 6].

At this juncture, we are in the position to focus on the symmetry properties of the coupled Lagrangians and . In this context, we observe the following:

It is clear from the above observations that the action integrals and remain invariant (i.e., ) under the quantum BRST and anti-BRST symmetry transformations that have been listed in Equations (13) and (12). The coupled (but equivalent) Lagrangian respect both (i.e., BRST and anti-BRST) quantum symmetries provided the whole theory is considered on a submanifold of the quantum Hilbert space of variables where the CF-type restriction: , is satisfied. In other words, mathematically, we observe the following:

A close look at the above transformations demonstrates that if we impose the (anti-)BRST invariant quantum CF-type restriction from outside, we obtain the following BRST symmetry transformation of the Lagrangian and anti-BRST symmetry transformation of the Lagrangian , namely,

It is crystal clear now that the observations in Equations (20), (21), (22), (23), (24), and (25) imply, in a straightforward manner, that both the Lagrangians (i.e., and ) respect both the quantum symmetries (i.e., BRST and anti-BRST symmetry transformations) in the space of quantum variables where the CF-type restriction is satisfied.

We end this subsection with the following remarks. First and foremost, we observe that the presence of the term “” in the square bracket of Equation (16) is due to the massive nature of the spinning relativistic particle. In the massless case, it disappears (see, e.g., Ref. [25]). Second, the hallmark of the quantum gauge theory (within the framework of the BRST formalism) is encoded in the existence of the CF-type restriction which we have demonstrated in Equations (14), (19), (22), and (23) where we have concentrated on the quantum (anti-)BRST symmetries which are respected by the coupled Lagrangians and . Finally, we note that the absolute anticommutativity property of the (anti-)BRST symmetries and equivalence of and owe their origins to the CF-type restriction: .

##### 3.2. (Anti-)BRST Charges and CF-Type Restriction

In this subsection, we demonstrate the existence of the (anti-)BRST invariant CF-type restriction (i.e., ) by demanding the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges of our present theory. In this context, first of all, we note that, according to Noether’s theorem, the invariances of the action integrals and under the (anti-)BRST symmetry transformations () (as quoted in Equations (20) and (21)) lead to the derivation of the Noether conserved (anti-)BRST charges () as follows:

The conservation law (i.e., ) can be proven by using the EL-EOMs derived from the Lagrangians and (cf. Appendix B). We have used the superscript (1) on the (anti-)BRST charges () to denote that these charges have been directly derived by using the basic principle behind Noether’s theorem. However, we have the option of expressing these charges in a different form by using the EL-EOMs that are derived from and . At this stage, it can be noted that the Noether conserved charges are not off-shell nilpotent () of order two without any use of EL-EOMs. In other words, we note that the following is true, namely, unless we use the EL-EOMs from and . Thus, we lay emphasis on the fact that are only the on-shell nilpotent conserved charges (even though we have used the off-shell nilpotent (anti-)BRST symmetry transformations (12) and (13) in their derivation).

We have the freedom to use the EL-EOMs (derived from and ) to recast the Noether conserved charges in a different form. For instance, the BRST charge can be written in a different form by using the following EL-EOMs: which are derived from w.r.t. the and variables. The ensuing expression for the conserved BRST charge (due to EL-EOMs (29)) is

Here, the superscript (2) denotes that the expression for the BRST charge in Equation (30) has been derived from the Noether conserved BRST charge by using the EL-EOMs quoted in Equation (29). It is now straightforward to check that the following is true, namely, where we have directly applied the BRST symmetry transformation (13) on the expression for (cf. Equation (30)) for the computation of the l.h.s. of Equation (31). We would like to lay emphasis on the fact that Equation (31) is nothing but the standard relationship between the continuous symmetry transformation and its generator . The latter is, to be precise, the conserved BRST charge which is the generator of the symmetry transformations (13). We, ultimately, note that the off-shell nilpotency of the has been proven in (31) where we have not used any EL-EOMs and/or CF-type restriction.

Let us now concentrate on the proof of the off-shell nilpotency of the anti-BRST charge (). For this purpose, we use the following EL-EOMs: that emerge out from the Lagrangian (when we consider the variables and for their derivation) to recast the Noether conserved charge as where the superscript (2) on the anti-BRST charge denotes the fact that it has been derived from the Noether conserved charge . We apply, at this stage, the anti-BRST symmetry transformations (12) directly on the anti-BRST charge to obtain

The above observation proves the off-shell nilpotency of the anti-BRST charge because we do not use EL-EOMs and/or CF-type restriction in its proof. In Equation (34), we have used the basic principle behind the continuous symmetries and their generators. There are other ways, too, to prove the off-shell nilpotency () of the (anti-)BRST charges . However, we have concentrated, in our present endeavor, only on the standard relationship between the continuous symmetries and their generators.

A couple of decisive features of the BRST formalism is the validity of the off-shell/on-shell nilpotency and absolute anticommutativity properties of the (anti-)BRST symmetries as well as the (anti-)BRST charges. We concentrate now on the proof of the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges . Toward this goal in mind, we first concentrate on the expression for (cf. Equation (30)). Applying directly the anti-BRST symmetry transformations (12) on it, we obtain the following:

In the terminology of the standard relationship between the continuous symmetry transformation and its generator , it is evident that the l.h.s. of Equation (35) can be written in an explicit fashion as

A close look at (35) and (36) demonstrates that the absolute anticommutativity of the conserved (anti-)BRST charges (that are off-shell nilpotent of order two) is true if and only if the CF restriction: , is imposed on the theory from outside. However, as discussed earlier, this restriction, on the quantum theory, is a physical condition because this CF-type restriction is an (anti-)BRST invariant quantity.

Let us now focus on the expression for the off-shell nilpotent anti-BRST charge in Equation (33). The direct application of the BRST symmetry transformation () of Equation (13) on the anti-BRST charge in (33) yields the following:

It is straightforward to note that the r.h.s. of (37) would be equal to zero if we impose the (anti-)BRST invariant CF-type restriction from outside. Exploiting the beauty of the standard relationship between continuous symmetry transformation () and its generator (conserved and nilpotent BRST charge ), we note that the l.h.s. of the above equation can be written as provided, as stated earlier, we confine ourselves on the submanifold of the quantum Hilbert space of variables where the CF-type restriction is satisfied. We have been able to establish an intimate connection between the CF-type restriction and the geometrical objects called gerbes [5, 6]. The existence of this restriction provides an independent identity to the BRST and anti-BRST symmetries and the corresponding (anti-)BRST charges.

We end this subsection with the following remarks. First and foremost, the existence of CF-type restriction is the hallmark of a quantum theory described within the framework of BRST formalism [5, 6]. Second, the CF-type restriction is responsible for the existence of the coupled (but equivalent) Lagrangians and . Third, the absolute anticommutativity of the (anti-)BRST symmetries and corresponding (anti-)BRST charges owe their origins to the CF-type restriction. Finally, we have been able to show that and both respect both the (anti-)BRST symmetries due to the existence of CF-type restriction.

#### 4. BRST Symmetry Transformations: ACSA

We exploit the basic tenets of ACSA to BRST formalism to derive the proper off-shell nilpotent BRST symmetry transformation (13) where we take into account the antichiral supervariables (defined on the (1, 1)-dimensional antichiral super submanifold of the general (1, 2)-dimensional supermanifold). The above antichiral supervariables are the generalizations of the ordinary variables of Lagrangian and as follows:

In the above, we have taken the super expansions along the Grassmannian -direction of the antichiral -dimensional super submanifold which is parameterized by the superspace coordinates . We note that, in the above super expansions, the secondary variables are fermionic and the rest of the secondary variables are bosonic in nature due to the fermionic nature of the Grassmannian variable . It is elementary to state that, in the limit , we retrieve ordinary variables of our theory described by the Lagrangian and .

The trivial BRST invariant quantities: , imply that the secondary variables . This is due to the fact that the basic tenets of ACSA require that the BRST invariant quantities should be independent of the Grassmannian variable (which is a mathematical artifact in the superspace formalism). In other words, we have the following: where the superscript on the antichiral supervariables denotes the supervariables that have been obtained after the application of the BRST invariant restrictions so that the coefficients of , in the expansions (39), becomes zero. This is due to the fact that there is a mapping (i.e., ) between the (anti-)BRST symmetry transformations and the translational operators along the Grassmannian directions of the (1, 2)-dimensional supermanifold that has been established in Refs. [10–12]. It is crystal clear, from our discussions in this paragraph, that we have to determine precisely all the secondary variables in terms of the basic and auxiliary variables of our theory so that we could know the coefficients of in the super expansions (39).

Against the backdrop of our earlier discussions, we have to obtain the precise expressions for the secondary variables so that we could obtain the BRST symmetry transformations as the coefficient of in the antichiral super expansions (39). Toward this goal in our mind, we have to find out the specific combinations of the nontrivial quantities that are BRST invariant. In this context, we note that the following useful and interesting quantities are BRST invariant, namely,

The basic tenets of ACSA to BRST formalism require that the above quantities, at the quantum level, should be independent of the Grassmannian variable when these are generalized onto the -dimensional antichiral super submanifold of the general -dimensional supermanifold. As a consequence, we have the following restrictions on the specific combinations of the antichiral supervariables, namely,

The above restrictions are quantum gauge (i.e., BRST) invariant conditions on the antichiral supervariables where the supervariables with superscript have been derived and explained in Equation (40) that corresponds to the trivial BRST symmetry transformations.

The substitutions of the antichiral super expansions (39) and the trivial expansions (40) into (42) lead to the following precise expressions for the secondary variables in terms of the basic and auxiliary variables of the coupled (but equivalent) (anti-)BRST invariant Lagrangians and (cf. Equations (17) and (18)), namely,

Ultimately, we obtain the super expansions of (39) in terms of the off-shell nilpotent () BRST transformations (13) of our theory as follows: which are besides the super expansions in (40) (that determine the trivial BRST symmetry transformations as ). The superscript on the antichiral supervariable on the l.h.s. of the above expansions denotes the fact that these supervariables have been determined after the quantum gauge (i.e., BRST) invariant restrictions have been imposed on the supervariables as quoted in Equation (42). In our Appendix C, we collect the step-by-step computations that lead to the derivation of (43) from (42). At the classical level, we know that the gauge invariant quantities (GIRs) are physical objects. Within the framework of BRST formalism, all the (anti-)BRST invariant quantities are physical objects at the quantum level. Hence, these quantities should be independent of the Grassmannian variables . In fact, this requirement is one of the basic tenets of ACSA to BRST formalism which is quite physical.

We end this section with the following remarks. First of all, we note that the coefficients of in the super expansions (40) and (44) are nothing but the BRST transformations (13). Second, it is evident that where is the generic antichiral supervariable that is located on the l.h.s. of Equations (40) and (44) and the symbol corresponds to the generic ordinary variable that is present in the Lagrangians and . Finally, we observe that, due to the mapping , the off-shell nilpotency of the BRST symmetry transformations (13) is deeply connected with the nilpotency of the translational generator along the -direction of (1, 1)-dimensional antichiral super submanifold on which the antichiral supervariables are defined. It will be noted that only the BRST symmetry transformations have been mentioned in Ref. [33] for the spinning relativistic particle. However, the full set of (anti-)BRST symmetry transformations and the corresponding (anti-)BRST invariant CF-type restriction have been derived in our earlier work [25].

#### 5. Anti-BRST Symmetry Transformations: ACSA

In this section, we derive the anti-BRST symmetry transformations (12) by exploiting the theoretical potential and power of ACSA to BRST formalism. Toward this objective in mind, first of all, we generalize the ordinary variables of (and the auxiliary variable ) onto (1, 1)-dimensional chiral super submanifold of the general (1, 2)-dimensional supermanifold (on which our 1D ordinary theory is generalized) as
where the -dimensional chiral super submanifold is parameterized by the superspace coordinates and all the chiral supervariables on the l.h.s. of (45) are a function of these superspace coordinates. The fermionic nature of the Grassmannian variable implies that the secondary variables are *fermionic* and are *bosonic* in nature. It is straightforward to note that, in the limit , we retrieve our ordinary variables of Lagrangian and the variable .

We note that there are trivially anti-BRST invariant quantities (cf. Equation (12)) such as . As a consequence, we have the following trivial chiral super expansions (with inputs: ), namely, where the superscript on the supervariables denotes the chiral supervariables where the coefficient of yields the anti-BRST symmetry transformations (12) in view of the mapping: [10–12], which becomes transparent when we observe that for the generic supervariable and the corresponding ordinary generic variable . The trivial super expansion (46) would be utilized in our further discussions.

The basic ingredient of the ACSA to BRST formalism requires that the nontrivial anti-BRST invariant quantities must be independent of the Grassmannian variable when these quantities are generalized onto the (1, 1)-dimensional chiral super submanifold. We exploit this idea to determine the secondary variables of the super expansion (45) in terms of the basic and auxiliary variables of . Toward this aim in our mind, we note that the following anti-BRST invariant quantities: are found to be very useful and interesting because their generalizations onto the (1, 1)-dimensional chiral super submanifold, namely, yield the precise values of the secondary variables of the expansion in (45). To be more precise, we note that the equalities in (48) lead to

Thus, we have determined precisely the expressions for the secondary variables in terms of the basic and auxiliary variables of by requiring that the quantum gauge (i.e., anti-BRST) invariant quantities must be independent of as the Grassmannian variable(s) are only mathematical artifact and they are not physical quantity in the real sense of the word.

The substitutions of all the expressions for the secondary variables (cf. Equation (49)) into the expansions in (45) lead to the following: