# Massive Spinning Relativistic Particle: Revisited under BRST and Supervariable Approaches

Authors: A. Tripathi; B. Chauhan; A. K. Rao; R. P. Malik
Journal: Advances in High Energy Physics
 (2020)

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## 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 one0+1-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.


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## Body

## 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. Sections2 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 Lf 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., ψμ=ψ5=χ=0), 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 Lb and Lb¯ (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. AppendixD).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 Section2, 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 AppendicesA, 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.,ṗμ=0) massive spinning relativistic particle is embedded in a D-dimensional flat Minkowskian spacetime manifold that is characterized by a metric tensor ημν=diag+1,−1,−1⋯ where the Greek indices μ,ν,λ⋯=0,1,2⋯D−1. We adopt the convention of the left-derivative w.r.t. the fermionic variables χ,ψμ,ψ5,γ,c,c¯. We denote the (anti-)BRST fermionic sab2=0symmetry transformations by the symbol sab which anticommutes (i.e., χsab=−sabχ,sabψμ=−ψμsab, etc.) with all the fermionic variables χ,ψμ,ψ5,γ,c,c¯ and commutes (i.e., xμsab=sabxμ,pμsab=sabpμ,esab=sabe,sabb¯=b¯sab, etc.) with all the bosonic variables of our theory. We also denote the (anti-)BRST charges by the symbol Qab.


## 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])
(1)L0=mẋμ+iχψμẋμ+iχψμ−imχψ5,Lf=pμẋμ+i2ψμψ̇μ−ψ5ψ̇5−e2p2−m2+iχpμψμ−mψ5,Ls=12eẋμ+iχψμẋμ+iχψμ+e2m2+i2ψμψ̇μ−ψ5ψ̇5−imχψ5,where L0 is the Lagrangian with a square root, Lf is the first-order Lagrangian, and Ls is the second-order Lagrangian. Our one 0+1-dimensional (1D) system is embedded in a flat Minkowskian D-dimensional target space where (xμ,pμ) are the canonically conjugate bosonic coordinates and momenta (with μ=0,1,2⋯D−1). The trajectory of the particle is parameterized by an evolution parameter τ, and generalized velocities (ẋμ=dxμ/dτ,ψ̇μ=dψμ/dτ) are defined w.r.t. it. We have fermionic (ψμψν+ψνψμ=0,χψμ+ψμχ=0,ψμψ5+ψ5ψμ=0,χψ5+ψ5χ=0,χ2=0, etc.) variables in our theory which commute (ψμe−eψμ=0,ψμxν−xνψμ=0,ψμpν−pνψμ=0,ẋνψμ−ψμẋν=0, etc.) with all the bosonic variables xμ,pμ,e of our theory. It should be noted that ψμ is the superpartner of xμ and ψ5 variable has been invoked in the theory to incorporate a mass term m so that the mass-shell condition p2−m2=0 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 (Lf) is the only Lagrangian that is mentioned in [33] and there is absence of L0 as well as Ls.The LagrangianL0 has a square root, and its massless limit is not defined. On the other hand, the second-order Lagrangian Ls is endowed with a variable (i.e., einbein) which is located in the denominator. Thus, the Lagrangians L0 and Ls have their own limitations. We shall focus on the first-order Lagrangian Lf for our discussions where variables eτ and χτ 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 ψμ corresponds to the Rarita-Schwinger field and eτ becomes the vierbein field. The mass m, 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 LagrangianLf respects the following gauge δg and supergauge δsg symmetry transformations, namely,
(2)δgxμ=ξpμ,δgpμ=0,δge=ξ̇,δgψμ=0,δgψ5=0,δgχ=0,δsgxμ=κψμ,δsgψμ=iκpμ,δsgpμ=0,δsgψ5=iκm,δsgχ=iκ̇,δsge=2κχ,where ξτ and κτ are the infinitesimal gauge and supergauge symmetry transformation parameters, respectively. It is straightforward to note that ξτ is a bosonic and κτ is a fermionic (i.e., κ2=0) transformation parameter. Furthermore, the transformation δsg 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 Lf transforms to the following total derivatives:
(3)δgLf=ddτξ2p2+m2,δsgLf=ddτκ2pμψμ+mψ5.As a consequence, it is clear that the action integralS=∫−∞+∞dτLf, under the transformations δg and δsg, would be equal to zero (i.e., δpS=0,p=g,sg) due to the fact that all the physical variables vanish off at τ=±∞. There is a reparameterization symmetry, too, in our theory due to the basic infinitesimal transformation τ→τ′=τ−ετ where ετ is an infinitesimal transformation parameter. In fact, the physical variables of our 1D system transform under the infinitesimal reparameterization transformation δr as
(4)δrxμ=εẋμ,δrpμ=εṗμ,δrψμ=εψ̇μ,δrψ5=εψ̇5,δre=ddτεe,δrχ=ddτεχ.The above transformations are symmetry transformations for the action integralS=∫−∞+∞dτLf because of the following transformation property of Lf, namely,
(5)δrLf=ddτεLf⟹δrS=0.It is evident thatδrS=0 due to the fact that ϵτ and Lf vanish off at τ=±∞.The reparameterization symmetry transformationδr and gauge symmetry transformation δg are equivalent under the following limits:
(6)ξ=ϵe,ẋμ=epμ,ṗμ=0,provided we set all the fermionic variables χ,ψ5,ψμ of our theory equal to zero. In the above, we have used equations of motion: ẋμ=epμ and ṗμ=0, and we have identified the gauge symmetry transformation parameter ξτ with the combination of the reparameterization transformation parameter ϵτ and the einbein variable eτ. In an exactly similar fashion, we note that δr and δg+δsg are also equivalent. In this context, first of all, we note that there are two primary constraints (i.e., Πe≈0,Πχ≈0) and two secondary constraints (i.e., p2−m2≈0,pμψμ−mψ5≈0) on our theory where Πe and Πχ are the canonical conjugate momenta w.r.t. the Lagrange multiplier variables e and χ, 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 δ=δg+δsg for the physical variables of our theory as (see, e.g., [27])
(7)δxμ=ξpμ+κψμ,δpμ=0,δψμ=iκpμ,δe=ξ̇+2κχ,δχ=iκ̇,δψ5=iκm,under which the first-order Lagrangian Lf transforms to a total “time” derivative as
(8)δLf=ddτξ2p2+m2+κ2pμψμ+mψ5.As a consequence of the above observation, it is evident thatδS=0 where S=∫−∞+∞dτLf is the action integral. If we use the following equations of motion:
(9)ṗμ=0,ẋμ=epμ−iχψμ,ψ̇μ=χpμ,ψ̇5=mχ,and identify the transformation parameters as
(10)ξ=eε,κ=−iεχ,we find that the reparameterization symmetry transformation (4) (emerging due to the basic transformation: τ→τ′=τ−ετ) and the combined gauge and supergauge symmetry transformations (i.e., δ=δg+δsg), quoted in Equation (7), are equivalent to each other. It is worthwhile to note that, under the identifications (10), the transformation δe=ξ̇+2κχ becomes δe=d/dτεe as we note that 2κχ=−2ieχ2=0.We end this section with the following remarks. First of all, we note that the canonical Hamiltonians, derived fromL0 and Lf (as well as Ls), are
(11)Hc0=−iχpμψμ−mψ5,Hc=e2p2−m2−iχpμψμ−mψ5,where Hc0 is the canonical Hamiltonian corresponding to the Lagrangian L0. It is straightforward to note that the primary constraints Πe≈0, Πχ≈0 lead to the derivation of the secondary constraints p2−m2≈0, pμψμ−mψ5≈0 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 Subsection3.1, 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 Lb and Lb¯. 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 sab where the classical gauge symmetry parameter ξτ would be replaced by the fermionic (c2=c¯2=0,cc¯+c¯c=0, etc.) (anti-)ghost variables c¯c and the classical supergauge symmetry transformations parameter κτ would be replaced by a pair of bosonic β2=β¯≠0 (anti-)ghost variables β¯β. These off-shell nilpotent sab2=0, infinitesimal and continuous (anti-)BRST symmetry transformations sab, in their full blaze of glory for our 1D system of the massive spinning relativistic particle, are (see, e.g., [25])
(12)sabxμ=c¯pμ+β¯ψμ,e=c¯̇+2β¯χ,sabψμ=iβ¯pμ,sabc¯=−iβ¯2,sabc=ib¯,sabβ¯=0,sabβ=−iγ,sabpμ=0,sabγ=0,sabb¯=0,sabχ=iβ¯̇,sabb=2iβ¯γ,sabψ5=iβ¯m,sbxμ=cpμ+βψμ,sbe=ċ+2βχ,(13)sbψμ=iβpμ,sbc=−iβ2,sbc¯=ib,sbβ=0,sbβ¯=iγ,sbpμ=0,sbγ=0,sbb=0,sbχ=iβ̇,sbb¯=−2iβγ,sbψ5=iβm,where b and b¯ are the Nakanishi-Lautrup-type auxiliary variables, fermionic (χ2=0,c2=c¯2=0,γ2=0) variables χ,c,c¯,γ are present in our theory, and the rest of the symbols have already been explained earlier. As far as the absolute anticommutativity sbsab+sabsb=0 property is concerned, it can be checked that
(14)sb,sabxμ=ib+b¯+2ββ¯pμ,sb,sabe=iddτb+b¯+2ββ¯,are equal to zero only after imposing the CF-type restriction: b+b¯+2ββ¯=0 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., sabb+b¯+2ββ¯=0) quantity. Except for the variables (xμ,e), it is straightforward to check that the following is true for the other variables of our theory, namely,
(15)sb,sabΦ=0,Φ=pμ,ψμ,ψ5,χ,β,β¯,c,c¯,b,b¯,γ,where Φτ 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 sab2=0 and absolutely anticommuting sbsab+sabsb=0 in nature provided the whole theory is considered on a submanifold of space of quantum variables where the CF-type restriction: b+b¯+2ββ¯=0, 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(16)Lb=Lf+sbsabie22+cc¯+χψ5,Lb¯=Lf−sabsbie22+cc¯+χψ5,where Lf 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
(17)Lb=Lf+b2+bė+2β¯β−ic¯̇ċ+β¯2β2+2iχβc¯̇−β¯ċ−2eβ¯β̇+γχ+2γβc¯−β¯c+mβ¯β̇−β¯̇β+γχ−γ̇ψ5,(18)Lb¯=Lf+b¯2−b¯ė−2β¯β−ic¯̇ċ+β¯2β2+2iχβc¯̇−β¯ċ+2eβ¯̇β−γχ+2γβc¯−β¯c+mβ¯β̇−β¯̇β+γχ−γ̇ψ5,where, as pointed out earlier, b and b¯ are the Nakanishi-Lautrup-type auxiliary variables which lead to the derivation of EL-EOMs (from Lb and Lb¯) as
(19)2b+ė+2ββ¯=0,2b¯−ė+2ββ¯=0.It is elementary to note that the above relationships lead to the derivation of the CF-type restriction:b+b¯+2β¯β=0, 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 LagrangiansLb and Lb¯. In this context, we observe the following:
(20)sbLb=ddτc2p2+m2+β2pμψμ+mψ5+bċ+2βχ,(21)sabLb¯=ddτc¯2p2+m2+β¯2pμψμ+mψ5−b¯c¯̇+2β¯χ.It is clear from the above observations that the action integralsS1=∫−∞∞dτLb and S2=∫−∞∞dτLb¯ remain invariant (i.e., sbS1=0,sabS2=0) 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: b+b¯+2ββ¯=0, is satisfied. In other words, mathematically, we observe the following:
(22)sbLb¯=ddτc2p2+m2+β2pμψμ+mψ5−b¯ċ+2βχ+2ieβγ+ċ+2βχddτb+b¯+2ββ¯−2iβγb+b¯+2ββ¯,(23)sabLb=ddτc¯2p2+m2+β¯2pμψμ+mψ5+bc¯̇+2β¯χ+2ieβ¯γ−c¯̇+2β¯χddτb+b¯+2ββ¯+2iβ¯γb+b¯+2ββ¯.A close look at the above transformations demonstrates that if we impose the (anti-)BRST invariantsabb+b¯+2ββ¯=0 quantum CF-type restriction b+b¯+2ββ¯=0 from outside, we obtain the following BRST symmetry transformation of the Lagrangian Lb¯ and anti-BRST symmetry transformation of the Lagrangian Lb, namely,
(24)sbLb¯=ddτc2p2+m2+β2pμψμ+mψ5−b¯ċ+2βχ+2ieβγ,(25)sabLb=ddτc¯2p2+m2+β¯2pμψμ+mψ5+bc¯̇+2β¯χ+2ieβ¯γ.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., Lb and Lb¯) 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 “χψ5” 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 Lb and Lb¯. Finally, we note that the absolute anticommutativity property of the (anti-)BRST symmetries and equivalence of Lb and Lb¯ owe their origins to the CF-type restriction: b+b¯+2ββ¯=0.


### 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.,b+b¯+2ββ¯=0) 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 sbS1=0,sabS2=0 of the action integrals S1=∫−∞+∞dτLb and S2=∫−∞+∞dτLb¯ under the (anti-)BRST symmetry transformations (sab) (as quoted in Equations (20) and (21)) lead to the derivation of the Noether conserved (anti-)BRST charges (Qab1) as follows:
(26)Qab1=c¯2p2−m2+β¯pμψμ−mψ5−b¯c¯̇−2b¯β¯χ−imβ¯γ−β¯2ċ−2ββ¯2χ,(27)Qb1=c2p2−m2+βpμψμ−mψ5+bċ+2bβχ−imβγ+β2c¯̇+2β¯β2χ.The conservation law (i.e.,Q̇b1=0,Q̇ab1=0) can be proven by using the EL-EOMs derived from the Lagrangians Lb and Lb¯ (cf. Appendix B). We have used the superscript (1) on the (anti-)BRST charges (Qab1) 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 Lb and Lb¯. At this stage, it can be noted that the Noether conserved charges Qab1 are not off-shell nilpotent (Qab12≠0) of order two without any use of EL-EOMs. In other words, we note that the following is true, namely,
(28)sbQb1=−iQb1,Qb1≠0,sabQab1=−iQab1,Qab1≠0,unless we use the EL-EOMs from Lb and Lb¯. Thus, we lay emphasis on the fact that Qab1 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 fromLb¯ and Lb) to recast the Noether conserved charges Qab1 in a different form. For instance, the BRST charge Qb1 can be written in a different form by using the following EL-EOMs:
(29)ḃ=−12p2−m2−2β¯β̇+γχ,pμψμ−mψ5=2ieγ−2βc¯̇−β¯ċ,which are derived from Lb w.r.t. the e and χ variables. The ensuing expression for the conserved BRST charge (due to EL-EOMs (29)) is
(30)Qb2=bċ−ḃc+2ieβγ+2ββ¯ċ−2cβ¯β̇+γχ+2bβχ−2imγβ−β2c¯̇+2χβ2β¯.Here, the superscript (2) denotes that the expression for the BRST charge in Equation (30) has been derived from the Noether conserved BRST charge Qb1 by using the EL-EOMs quoted in Equation (29). It is now straightforward to check that the following is true, namely,
(31)sbQb2=−iQb2,Qb2=0⇒Qb22=0,where we have directly applied the BRST symmetry transformation (13) on the expression for Qb2 (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 sb and its generator Qb2. 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 Qb22=0 of the Qb2 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 (Qab). For this purpose, we use the following EL-EOMs:
(32)b¯̇=12p2−m2−2β¯̇β−γχ,pμψμ−mψ5=2ieγ−imγ−2βc¯̇−β¯ċ,that emerge out from the Lagrangian Lb¯ (when we consider the variables e and χ for their derivation) to recast the Noether conserved charge Qab1 as
(33)Qab2=b¯̇c¯−b¯c¯̇+2ieβ¯γ−2ββ¯c¯̇+2c¯β¯̇β−γχ−2b¯β¯χ−2imβ¯γ+β¯2ċ−2ββ¯2χ,where the superscript (2) on the anti-BRST charge Qab2 denotes the fact that it has been derived from the Noether conserved charge Qab1. We apply, at this stage, the anti-BRST symmetry transformations (12) directly on the anti-BRST charge Qab2 to obtain
(34)sabQab2=−iQab2,Qab2=0⇒Qab22=0.The above observation proves the off-shell nilpotency of the anti-BRST chargeQab2 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 (Qab22=0) of the (anti-)BRST charges Qab2. 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 chargesQab2. Toward this goal in mind, we first concentrate on the expression for Qb2 (cf. Equation (30)). Applying directly the anti-BRST symmetry transformations (12) on it, we obtain the following:
(35)sabQb2=ib+b¯+2ββ¯b¯̇+2χγ+2ββ¯̇−ib¯ddτb+b¯+2ββ¯.In the terminology of the standard relationship between the continuous symmetry transformationsab and its generator Qab2, it is evident that the l.h.s. of Equation (35) can be written in an explicit fashion as
(36)sabQb2=−iQb2,Qab2.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: b+b¯+2ββ¯=0, 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 nilpotentQab22=0 anti-BRST charge Qab2 in Equation (33). The direct application of the BRST symmetry transformation (sb) of Equation (13) on the anti-BRST charge Qab2 in (33) yields the following:
(37)sbQab2=ibddτb+b¯+2ββ¯−ib+b¯+2ββ¯ḃ+2β̇β¯+2γχ.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 b+b¯+2ββ¯=0 from outside. Exploiting the beauty of the standard relationship between continuous symmetry transformation (sb) and its generator (conserved and nilpotent BRST charge Qb2), we note that the l.h.s. of the above equation can be written as
(38)sbQab2=−iQab2,Qb2=0,provided, as stated earlier, we confine ourselves on the submanifold of the quantum Hilbert space of variables where the CF-type restriction b+b¯+2ββ¯=0 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 Lb and Lb¯. 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 Lb and Lb¯ both respect both the (anti-)BRST symmetries due to the existence of CF-type restriction.


## 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 sab where the classical gauge symmetry parameter ξτ would be replaced by the fermionic (c2=c¯2=0,cc¯+c¯c=0, etc.) (anti-)ghost variables c¯c and the classical supergauge symmetry transformations parameter κτ would be replaced by a pair of bosonic β2=β¯≠0 (anti-)ghost variables β¯β. These off-shell nilpotent sab2=0, infinitesimal and continuous (anti-)BRST symmetry transformations sab, in their full blaze of glory for our 1D system of the massive spinning relativistic particle, are (see, e.g., [25])
(12)sabxμ=c¯pμ+β¯ψμ,e=c¯̇+2β¯χ,sabψμ=iβ¯pμ,sabc¯=−iβ¯2,sabc=ib¯,sabβ¯=0,sabβ=−iγ,sabpμ=0,sabγ=0,sabb¯=0,sabχ=iβ¯̇,sabb=2iβ¯γ,sabψ5=iβ¯m,sbxμ=cpμ+βψμ,sbe=ċ+2βχ,(13)sbψμ=iβpμ,sbc=−iβ2,sbc¯=ib,sbβ=0,sbβ¯=iγ,sbpμ=0,sbγ=0,sbb=0,sbχ=iβ̇,sbb¯=−2iβγ,sbψ5=iβm,where b and b¯ are the Nakanishi-Lautrup-type auxiliary variables, fermionic (χ2=0,c2=c¯2=0,γ2=0) variables χ,c,c¯,γ are present in our theory, and the rest of the symbols have already been explained earlier. As far as the absolute anticommutativity sbsab+sabsb=0 property is concerned, it can be checked that
(14)sb,sabxμ=ib+b¯+2ββ¯pμ,sb,sabe=iddτb+b¯+2ββ¯,are equal to zero only after imposing the CF-type restriction: b+b¯+2ββ¯=0 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., sabb+b¯+2ββ¯=0) quantity. Except for the variables (xμ,e), it is straightforward to check that the following is true for the other variables of our theory, namely,
(15)sb,sabΦ=0,Φ=pμ,ψμ,ψ5,χ,β,β¯,c,c¯,b,b¯,γ,where Φτ 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 sab2=0 and absolutely anticommuting sbsab+sabsb=0 in nature provided the whole theory is considered on a submanifold of space of quantum variables where the CF-type restriction: b+b¯+2ββ¯=0, 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(16)Lb=Lf+sbsabie22+cc¯+χψ5,Lb¯=Lf−sabsbie22+cc¯+χψ5,where Lf 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
(17)Lb=Lf+b2+bė+2β¯β−ic¯̇ċ+β¯2β2+2iχβc¯̇−β¯ċ−2eβ¯β̇+γχ+2γβc¯−β¯c+mβ¯β̇−β¯̇β+γχ−γ̇ψ5,(18)Lb¯=Lf+b¯2−b¯ė−2β¯β−ic¯̇ċ+β¯2β2+2iχβc¯̇−β¯ċ+2eβ¯̇β−γχ+2γβc¯−β¯c+mβ¯β̇−β¯̇β+γχ−γ̇ψ5,where, as pointed out earlier, b and b¯ are the Nakanishi-Lautrup-type auxiliary variables which lead to the derivation of EL-EOMs (from Lb and Lb¯) as
(19)2b+ė+2ββ¯=0,2b¯−ė+2ββ¯=0.It is elementary to note that the above relationships lead to the derivation of the CF-type restriction:b+b¯+2β¯β=0, 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 LagrangiansLb and Lb¯. In this context, we observe the following:
(20)sbLb=ddτc2p2+m2+β2pμψμ+mψ5+bċ+2βχ,(21)sabLb¯=ddτc¯2p2+m2+β¯2pμψμ+mψ5−b¯c¯̇+2β¯χ.It is clear from the above observations that the action integralsS1=∫−∞∞dτLb and S2=∫−∞∞dτLb¯ remain invariant (i.e., sbS1=0,sabS2=0) 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: b+b¯+2ββ¯=0, is satisfied. In other words, mathematically, we observe the following:
(22)sbLb¯=ddτc2p2+m2+β2pμψμ+mψ5−b¯ċ+2βχ+2ieβγ+ċ+2βχddτb+b¯+2ββ¯−2iβγb+b¯+2ββ¯,(23)sabLb=ddτc¯2p2+m2+β¯2pμψμ+mψ5+bc¯̇+2β¯χ+2ieβ¯γ−c¯̇+2β¯χddτb+b¯+2ββ¯+2iβ¯γb+b¯+2ββ¯.A close look at the above transformations demonstrates that if we impose the (anti-)BRST invariantsabb+b¯+2ββ¯=0 quantum CF-type restriction b+b¯+2ββ¯=0 from outside, we obtain the following BRST symmetry transformation of the Lagrangian Lb¯ and anti-BRST symmetry transformation of the Lagrangian Lb, namely,
(24)sbLb¯=ddτc2p2+m2+β2pμψμ+mψ5−b¯ċ+2βχ+2ieβγ,(25)sabLb=ddτc¯2p2+m2+β¯2pμψμ+mψ5+bc¯̇+2β¯χ+2ieβ¯γ.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., Lb and Lb¯) 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 “χψ5” 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 Lb and Lb¯. Finally, we note that the absolute anticommutativity property of the (anti-)BRST symmetries and equivalence of Lb and Lb¯ owe their origins to the CF-type restriction: b+b¯+2ββ¯=0.


## 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.,b+b¯+2ββ¯=0) 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 sbS1=0,sabS2=0 of the action integrals S1=∫−∞+∞dτLb and S2=∫−∞+∞dτLb¯ under the (anti-)BRST symmetry transformations (sab) (as quoted in Equations (20) and (21)) lead to the derivation of the Noether conserved (anti-)BRST charges (Qab1) as follows:
(26)Qab1=c¯2p2−m2+β¯pμψμ−mψ5−b¯c¯̇−2b¯β¯χ−imβ¯γ−β¯2ċ−2ββ¯2χ,(27)Qb1=c2p2−m2+βpμψμ−mψ5+bċ+2bβχ−imβγ+β2c¯̇+2β¯β2χ.The conservation law (i.e.,Q̇b1=0,Q̇ab1=0) can be proven by using the EL-EOMs derived from the Lagrangians Lb and Lb¯ (cf. Appendix B). We have used the superscript (1) on the (anti-)BRST charges (Qab1) 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 Lb and Lb¯. At this stage, it can be noted that the Noether conserved charges Qab1 are not off-shell nilpotent (Qab12≠0) of order two without any use of EL-EOMs. In other words, we note that the following is true, namely,
(28)sbQb1=−iQb1,Qb1≠0,sabQab1=−iQab1,Qab1≠0,unless we use the EL-EOMs from Lb and Lb¯. Thus, we lay emphasis on the fact that Qab1 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 fromLb¯ and Lb) to recast the Noether conserved charges Qab1 in a different form. For instance, the BRST charge Qb1 can be written in a different form by using the following EL-EOMs:
(29)ḃ=−12p2−m2−2β¯β̇+γχ,pμψμ−mψ5=2ieγ−2βc¯̇−β¯ċ,which are derived from Lb w.r.t. the e and χ variables. The ensuing expression for the conserved BRST charge (due to EL-EOMs (29)) is
(30)Qb2=bċ−ḃc+2ieβγ+2ββ¯ċ−2cβ¯β̇+γχ+2bβχ−2imγβ−β2c¯̇+2χβ2β¯.Here, the superscript (2) denotes that the expression for the BRST charge in Equation (30) has been derived from the Noether conserved BRST charge Qb1 by using the EL-EOMs quoted in Equation (29). It is now straightforward to check that the following is true, namely,
(31)sbQb2=−iQb2,Qb2=0⇒Qb22=0,where we have directly applied the BRST symmetry transformation (13) on the expression for Qb2 (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 sb and its generator Qb2. 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 Qb22=0 of the Qb2 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 (Qab). For this purpose, we use the following EL-EOMs:
(32)b¯̇=12p2−m2−2β¯̇β−γχ,pμψμ−mψ5=2ieγ−imγ−2βc¯̇−β¯ċ,that emerge out from the Lagrangian Lb¯ (when we consider the variables e and χ for their derivation) to recast the Noether conserved charge Qab1 as
(33)Qab2=b¯̇c¯−b¯c¯̇+2ieβ¯γ−2ββ¯c¯̇+2c¯β¯̇β−γχ−2b¯β¯χ−2imβ¯γ+β¯2ċ−2ββ¯2χ,where the superscript (2) on the anti-BRST charge Qab2 denotes the fact that it has been derived from the Noether conserved charge Qab1. We apply, at this stage, the anti-BRST symmetry transformations (12) directly on the anti-BRST charge Qab2 to obtain
(34)sabQab2=−iQab2,Qab2=0⇒Qab22=0.The above observation proves the off-shell nilpotency of the anti-BRST chargeQab2 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 (Qab22=0) of the (anti-)BRST charges Qab2. 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 chargesQab2. Toward this goal in mind, we first concentrate on the expression for Qb2 (cf. Equation (30)). Applying directly the anti-BRST symmetry transformations (12) on it, we obtain the following:
(35)sabQb2=ib+b¯+2ββ¯b¯̇+2χγ+2ββ¯̇−ib¯ddτb+b¯+2ββ¯.In the terminology of the standard relationship between the continuous symmetry transformationsab and its generator Qab2, it is evident that the l.h.s. of Equation (35) can be written in an explicit fashion as
(36)sabQb2=−iQb2,Qab2.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: b+b¯+2ββ¯=0, 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 nilpotentQab22=0 anti-BRST charge Qab2 in Equation (33). The direct application of the BRST symmetry transformation (sb) of Equation (13) on the anti-BRST charge Qab2 in (33) yields the following:
(37)sbQab2=ibddτb+b¯+2ββ¯−ib+b¯+2ββ¯ḃ+2β̇β¯+2γχ.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 b+b¯+2ββ¯=0 from outside. Exploiting the beauty of the standard relationship between continuous symmetry transformation (sb) and its generator (conserved and nilpotent BRST charge Qb2), we note that the l.h.s. of the above equation can be written as
(38)sbQab2=−iQab2,Qb2=0,provided, as stated earlier, we confine ourselves on the submanifold of the quantum Hilbert space of variables where the CF-type restriction b+b¯+2ββ¯=0 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 Lb and Lb¯. 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 Lb and Lb¯ 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 Lb and b¯τ as follows:
(39)xμτ→Xμτ,θ¯=xμτ+θ¯Rμ1τ,pμτ→Pμτ,θ¯=pμτ+θ¯Rμ2τ,eτ→Eτ,θ¯=eτ+θ¯f1τ,cτ→Fτ,θ¯=cτ+θ¯b1τ,c¯τ→F¯τ,θ¯=c¯τ+θ¯b2τ,βτ→β~τ,θ¯=βτ+θ¯f2τ,β¯τ→β¯~τ,θ¯=β¯τ+θ¯f3τ,ψμτ→Ψμτ,θ¯=ψμτ+θ¯b3τ,ψ5τ→Ψ5τ,θ¯=ψ5τ+θ¯b4τ,χτ→χ~τ,θ¯=χτ+θ¯b5τ,γτ→Γτ,θ¯=γτ+θ¯b6τ,bτ→Bτ,θ¯=bτ+θ¯f4τ,b¯τ→B¯τ,θ¯=b¯τ+θ¯f5τ.In the above, we have taken the super expansions along the Grassmannianθ¯-direction of the antichiral 1,1-dimensional super submanifold which is parameterized by the superspace coordinates τ,θ¯. We note that, in the above super expansions, the secondary variables Rμ1,Rμ2,f1,f2,f3,f4,f5 are fermionic and the rest of the secondary variables b1,b2,b3,b4,b5,b6 are bosonic in nature due to the fermionic θ¯2=0 nature of the Grassmannian variable θ¯. It is elementary to state that, in the limit θ¯=0, we retrieve ordinary variables of our theory described by the Lagrangian Lb and b¯τ.The trivial BRST invariant quantities:sbpμ=0,sbγ=0,sbβ=0,sbb=0, imply that the secondary variables Rμ2=b6=f2=f4=0. 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 θ¯ (which is a mathematical artifact in the superspace formalism). In other words, we have the following:
(40)Pμbτ,θ=pμτ+θ¯0≡pμτ+θ¯sbpμτ⇒Pμbτ,θ=pμτ,Γbτ,θ=γτ+θ¯0≡γτ+θ¯sbγτ⇒Γbτ,θ=γτ,Bbτ,θ=bτ+θ¯0≡bτ+θ¯sbbτ⇒Bbτ,θ=bτ,β~bτ,θ=βτ+θ¯0≡βτ+θ¯sbβτ⇒β~bτ,θ=βτ,where the superscript b on the antichiral supervariables denotes the supervariables that have been obtained after the application of the BRST invariant sbpμ=sbγ=sbb=sbβ=0 restrictions so that the coefficients of θ¯, in the expansions (39), becomes zero. This is due to the fact that there is a mapping (i.e., sb↔∂θ¯,sab↔∂θ) between the (anti-)BRST symmetry transformations sab and the translational operators ∂θ,∂θ¯ 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 θ¯ 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 transformationssb as the coefficient of θ¯ 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,
(41)sbβ¯γ=0,sbċ+2βχ=0,sbeγχ+eβ¯β̇−iβ¯ċχ=0,sbβ2β¯+cγ=0,sbcpμ+βψμ=0,sbβxμ−icψμ=0,sbbβ¯+γc¯=0,sbb¯+2ββ¯=0,sbψ̇5−χm=0.The basic tenets of ACSA to BRST formalism require that the above quantities, at the quantum level, should be independent of the Grassmannian variableθ¯ when these are generalized onto the 1,1-dimensional antichiral super submanifold of the general 1,2-dimensional supermanifold. As a consequence, we have the following restrictions on the specific combinations of the antichiral supervariables, namely,
(42)β¯~τ,θ¯Γbτ,θ¯=β¯τγτ,Ḟτ,θ¯+2β~bτ,θ¯χ~τ,θ¯=ċτ+2βτχτ,Eτ,θ¯Γbτ,θ¯χ~τ,θ¯+Eτ,θ¯β¯~τ,θ¯β~̇bτ,θ¯−iβ¯~τ,θ¯Ḟτ,θ¯χ~τ,θ¯=eτγτχτ+eτβ¯τβ̇τ−iβ¯τċτχτ,β~2bτ,θ¯β¯~τ,θ¯+Fτ,θ¯Γbτ,θ¯=β2τβ¯τ+cτγτ,Fτ,θ¯Pμbτ,θ¯+β~bτ,θ¯Ψμτ,θ¯=cτpμτ+βτψμτ,β~bτ,θ¯Xμτ,θ¯−iFτ,θ¯Ψμτ,θ¯=βτxμτ−icτψμτ,Bbτ,θ¯β¯~τ,θ¯+Γbτ,θ¯F¯τ,θ¯=bτβ¯τ+γτc¯τ,B¯τ,θ¯+2β~bτ,θ¯β¯~τ,θ¯=b¯τ+2βτβ¯τ,Ψ̇5τ,θ¯−χ~τ,θ¯m=ψ̇5τ−χτm.The above restrictions are quantum gauge (i.e., BRST) invariant conditions on the antichiral supervariables where the supervariables with superscriptb 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 Lb and Lb¯ (cf. Equations (17) and (18)), namely,
(43)Rμ1=cpμ+βψμ,f1=ċ+2βχ,b1=iβ2,b2=ib,f3=iγ,b3=iβpμ,b4=iβm,b5=iβ̇,f5=−2iβγ.Ultimately, we obtain the super expansions of (39) in terms of the off-shell nilpotent (sb2=0) BRST transformations (13) of our theory as follows:
(44)Xμbτ,θ¯=xμ+θ¯cpμ+βψμ≡xμτ+θ¯sbxμ,Ebτ,θ¯=eτ+θ¯ċ+2βχ≡eτ+θ¯sbe,Fbτ,θ¯=cτ+θ¯iβ2≡cτ+θ¯sbc,F¯bτ,θ¯=c¯τ+θ¯ib≡c¯τ+θ¯sbc¯,β¯~bτ,θ¯=β¯τ+θ¯iγ≡β¯τ+θ¯sbβ¯,Ψμbτ,θ¯=ψμτ+θ¯iβpμ≡ψμτ+θ¯sbψμ,Ψ5bτ,θ¯=ψ5τ+θ¯iβm≡ψ5τ+θ¯sbψ5,χ~bτ,θ¯=χτ+θ¯iβ̇≡χτ+θ¯sbχ,B¯bτ,θ¯=b¯τ+θ¯−2iβγ≡b¯τ+θ¯sbb¯,which are besides the super expansions in (40) (that determine the trivial BRST symmetry transformations as sbpμ=0,sbγ=0,sbβ=0,sbb=0). The superscript b 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 θ,θ¯. 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 ofθ¯ in the super expansions (40) and (44) are nothing but the BRST transformations (13). Second, it is evident that ∂θ¯Ωbx,θ¯=sbωτ where Ωbx,θ¯ is the generic antichiral supervariable that is located on the l.h.s. of Equations (40) and (44) and the symbol ωτ corresponds to the generic ordinary variable that is present in the Lagrangians Lb and Lb¯. Finally, we observe that, due to the mapping sb↔∂θ¯, the off-shell nilpotency sb2=0 of the BRST symmetry transformations (13) is deeply connected with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ along the θ¯-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 Lb¯ (and the auxiliary variable bτ) onto (1, 1)-dimensional chiral super submanifold of the general (1, 2)-dimensional supermanifold (on which our 1D ordinary theory is generalized) as
(45)xμτ→Xμτ,θ=xμτ+θR¯μ1τ,pμτ→Pμτ,θ=pμτ+θR¯μ2τ,eτ→Eτ,θ=eτ+θf¯1τ,cτ→Fτ,θ=cτ+iθb¯1τ,c¯τ→F¯τ,θ=c¯τ+iθb¯2τ,βτ→β~τ,θ=βτ+θf¯2τ,β¯τ→β¯~τ,θ=β¯τ+θf¯3τ,ψμτ→Ψμτ,θ=ψμτ+θb¯3τ,ψ5τ→Ψ5τ,θ=ψ5τ+θb¯4τ,χτ→χ~τ,θ=χτ+θb¯5τ,γτ→Στ,θ=γτ+θb¯6τ,bτ→Bτ,θ=bτ+θf¯4τ,b¯τ→B¯τ,θ=b¯τ+θf¯5τ,where the 1,1-dimensional chiral super submanifold is parameterized by the superspace coordinates τ,θ and all the chiral supervariables on the l.h.s. of (45) are a function of these superspace coordinates. The fermionic θ2=0 nature of the Grassmannian variable θ implies that the secondary variables R¯μ1,R¯μ2,f¯1,f¯2,f¯3,f¯4,f¯5 are fermionic and b¯1,b¯2,b¯3,b¯4,b¯5,b¯6 are bosonic in nature. It is straightforward to note that, in the limit θ=0, we retrieve our ordinary variables of Lagrangian Lb¯ and the variable bτ.We note that there are trivially anti-BRST invariant quantities (cf. Equation (12)) such as sabpμ=0,sabγ=0,sabb¯=0,sabβ¯=0. As a consequence, we have the following trivial chiral super expansions (with inputs: R¯μ2=b¯6=f¯5=f¯3=0), namely,
(46)pμτ→Pμabτ,θ=pμτ+θ0≡pμτ+θsabpμτ,γτ→Γabτ,θ=γτ+θ0≡γτ+θsabγτ,b¯τ→B¯abτ,θ=b¯τ+θ0≡b¯τ+θsabb¯τ,β¯τ→β¯~abτ,θ=β¯τ+θ0≡β¯τ+θsabβ¯τ,where the superscript ab on the supervariables denotes the chiral supervariables where the coefficient of θ yields the anti-BRST symmetry transformations (12) in view of the mapping: sab↔∂θ [10–12], which becomes transparent when we observe that ∂θΩabτ,θ=sabωτ for the generic supervariable Ωabτ,θ and the corresponding ordinary generic variable ωτ. 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 variableθ 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 Lb¯. Toward this aim in our mind, we note that the following anti-BRST invariant quantities:
(47)sabβγ=0,sabc¯̇+2β¯χ=0,sabeγχ−eβ¯̇β+iβc¯̇χ=0,sabββ¯2−c¯γ=0,sabc¯pμ+β¯ψμ=0,sabβ¯xμ−ic¯ψμ=0,sabb¯β−γc=0,sabb+2β¯β=0,sabψ̇5−χm=0,are found to be very useful and interesting because their generalizations onto the (1, 1)-dimensional chiral super submanifold, namely,
(48)β~τ,θΓabτ,θ=βτγτ,F¯̇τ,θ+2β¯~abτ,θχ~τ,θ=c¯̇τ+2β¯τχτ,Eτ,θΓabτ,θχ~τ,θ−Eτ,θβ¯~̇τ,θβ~abτ,θ+iβ~τ,θF¯̇τ,θχ~τ,θ=eτγτχτ−eτβ¯̇τβτ+iβτc¯̇τχτ,β~τ,θβ¯~2abτ,θ−F¯τ,θΓabτ,θ=βτβ¯2τ−c¯τγτ,F¯τ,θPμabτ,θ+β¯~abτ,θΨμτ,θ=c¯τpμτ+β¯τψμτ,β¯~abτ,θXμτ,θ−iF¯τ,θΨμτ,θ=β¯τxμτ−ic¯τψμτ,B¯abτ,θβ~τ,θ−Γabτ,θFτ,θ=b¯τβτ−γτcτ,Bτ,θ+2β¯~abτ,θβ~τ,θ=bτ+2β¯τβτ,Ψ̇5τ,θ−χ~τ,θm=ψ̇5τ−χτm,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
(49)R¯μ1=c¯pμ+β¯ψμ,f¯1=c¯̇+2β¯χ,b¯1=ib¯,b¯2=−iβ¯2,f¯2=−iγ,b¯3=iβ¯pμ,b¯4=iβ¯m,b¯5=iβ¯̇,f¯4=2iβ¯γ.Thus, we have determined precisely the expressions for the secondary variables in terms of the basic and auxiliary variables ofLb¯ by requiring that the quantum gauge (i.e., anti-BRST) invariant quantities must be independent of θ 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:
(50)Xμabτ,θ=xμ+θc¯pμ+β¯ψμ≡xμτ+θsabxμ,Eabτ,θ=eτ+θc¯̇+2β¯χ≡eτ+θsabe,Fabτ,θ=cτ+θib¯≡cτ+θsabc,F¯abτ,θ=c¯τ+θ−iβ¯2≡c¯τ+θsabc¯,β~abτ,θ=β¯τ+θ−iγ≡βτ+θsabβ,Ψμabτ,θ=ψμτ+θiβ¯pμ≡ψμτ+θsabψμ,Ψ5abτ,θ=ψ5τ+θiβ¯m≡ψ5τ+θsabψ5,χ~abτ,θ=χτ+θ¯iβ¯̇≡χτ+θsabχ,Babτ,θ=bτ+θ2iβ¯γ≡bτ+θsabb.In the above equation, the superscriptab on the chiral supervariables (cf. the l.h.s. of (46) and (50)) denotes the super expansions that have been derived after the applications of the anti-BRST invariant restrictions in (48). We note that the coefficients of θ, in the above expansions, are nothing but the anti-BRST symmetry transformations (12) of our 1D system of a massive spinning relativistic particle.We wrap up this section with the following comments. First and foremost, we observe that the trivial anti-BRST invariant (e.g.,sabpμ=sabγ=sabb¯=sabβ¯=0) variables have been incorporated in the super expansions in (46). Second, the nontrivial anti-BRST symmetry transformations (12) have been incorporated in the super expansions (50). Finally, we have exploited the basic idea of ACSA to BRST formalism where we have demanded that the anti-BRST (i.e., quantum gauge) invariant quantities must be independent of the Grassmannian variable θ when they are generalized onto the 1,1-dimensional chiral super submanifold of the general (1, 2)-dimensional supermanifold.


## 6. Symmetry Invariance of Lagrangians: ACSA

In this section, we capture the (anti-)BRST symmetry invariance of the coupled (but equivalent)Lb and Lb¯ (cf. Equations (20)–(23)) within the framework of ACSA to BRST formalism. In this context, it is pertinent to point out that the CF-type condition (b+b¯+2ββ¯=0) is responsible for the existence of the coupled (but equivalent) Lagrangians Lb and Lb¯ and it is also responsible for the absolute anticommutativity (i.e., sb,sab=0) of the (anti-)BRST symmetries (sab) and the absolute anticommutativity (i.e., Qb,Qab=0) of the corresponding conserved Q̇ab=0 and off-shell nilpotent (i.e., Q̇ab2=0) (anti-)BRST charges Qab. Thus, it is important for us to capture the existence of the (anti-)BRST invariant (i.e., sabb+b¯+2ββ¯=0) CF-type restriction in the context of symmetry considerations of the coupled (but equivalent) Lagrangians Lb and Lb¯ for our 1D system of a reparameterization invariant massive model of spinning relativistic particle.Against the backdrop of the above statements, first of all, we consider the (anti-)BRST symmetry invariance (cf. Equations (20) and (21)) of the Lagrangians Lb and Lb¯. Toward this goal in mind, we generalize these Lagrangians on the (1, 1)-dimensional chiral and antichiral super submanifolds (in terms of the corresponding supervariables) as
(51)Lb¯→L~b¯cτ,θ=L~fcτ,θ+B¯abτ,θB¯abτ,θ−B¯abτ,θĖabτ,θ−2β¯~abτ,θβ~abτ,θ−iF¯̇abτ,θḞabτ,θ+β¯~abτ,θβ¯~abτ,θβ~abτ,θβ~abτ,θ+2iχ~abτ,θβ~abτ,θF¯̇abτ,θ−β¯~abτ,θḞabτ,θ+2Eabτ,θβ¯~̇abτ,θβ~abτ,θ−Γabτ,θχ~abτ,θ+2Γabτ,θβ~abτ,θF¯abτ,θ−β¯~abτ,θFabτ,θ+mβ¯~abτ,θβ~̇abτ,θ−β¯~̇abτ,θβ~abτ,θ+Γabτ,θχ~abτ,θ−Γ̇abτ,θΨ5abτ,θ,Lb→L~bacτ,θ¯=L~facτ,θ¯+Bbτ,θ¯Bbτ,θ¯+Bbτ,θ¯Ėbτ,θ¯+2β¯~bτ,θ¯β~bτ,θ¯−iF¯̇bτ,θ¯Ḟbτ,θ¯+β¯~bτ,θ¯β¯~bτ,θ¯β~bτ,θ¯β~bτ,θ¯+2iχ~bτ,θ¯β~bτ,θ¯F¯̇bτ,θ¯−β¯~bτ,θ¯Ḟbτ,θ¯−2Ebτ,θ¯β¯~bτ,θ¯β~̇bτ,θ¯+Γbτ,θ¯χ~bτ,θ¯+2Γbτ,θ¯β~bτ,θ¯F¯bτ,θ¯−β¯~bτ,θ¯Fbτ,θ¯+mβ¯~bτ,θ¯β~̇bτ,θ¯−β¯~̇bτ,θ¯β~bτ,θ¯+Γbτ,θ¯χ~bτ,θ¯−Γ̇bτ,θ¯Ψ5bτ,θ¯,where the superscripts c and ac on the super Lagrangians (i.e., L~b¯c,L~bac) denote that these Lagrangians incorporate chiral and antichiral supervariables that have been obtained after the (anti-)BRST invariant restrictions (cf. Equations (40), (44), (46), and (50)). Furthermore, we note that the super first-order Lagrangians are
(52)L~fcτ,θ=Pμabτ,θẊμabτ,θ−12Eabτ,θPμabτ,θPμabτ,θ−m2+i2Ψμabτ,θΨ̇μabτ,θ−Ψ5abτ,θΨ̇5abτ,θ+iχ~abτ,θPμabτ,θΨμabτ,θ−mΨ5abτ,θ,L~facτ,θ¯=Pμbτ,θ¯Ẋμbτ,θ¯−12Ebτ,θ¯Pμbτ,θ¯Pμbτ,θ¯−m2+i2Ψμbτ,θ¯Ψ̇μbτ,θ¯−Ψ5bτ,θ¯Ψ̇5bτ,θ¯+iχ~bτ,θ¯Pμbτ,θ¯Ψμbτ,θ¯−mΨ5bτ,θ¯.It is, at this stage, very essential to point out that some of the supervariables are (anti-)BRST invariant and, hence, they are merely ordinary variables. For instance, we note thatPμabτ,θ=Pμbτ,θ¯=pμτ,Γbτ,θ¯=Γabτ,θ=γτ,β¯~abτ,θ=β¯τ,β~bτ,θ¯=βτ,B¯abτ,θ=b¯τ,Bbτ,θ¯=bτ.In view of the mappings:sb↔∂θ¯,sab↔∂θ [10–12], we can now capture the (anti-)BRST invariance of Lb¯ and Lb (cf. Equations (20) and (21)) as
(53)∂∂θ¯L~bacτ,θ¯=ddτc2p2+m2+β2pμψμ+mψ5+bċ+2βχ≡sbLb,∂∂θL~b¯cτ,θ=ddτc¯2p2+m2+β¯2pμψμ−mψ5−b¯c¯̇+2β¯χ≡sabLb¯.Thus, we have captured the (anti-)BRST invariance of the LagrangiansLb¯ and Lb (cf. Equations (21) and (20)) within the framework of ACSA to BRST formalism. Geometrically, the chiral super Lagrangian L~b¯cτ,θ is a unique sum of the combination of (super)variables that have been obtained after the quantum gauge (i.e., anti-BRST) invariant restrictions. The translation of this unique sum, along the θ-direction of (1, 1)-dimensional chiral super submanifold, generates a total “time” derivative (cf. Equation (21)) in the ordinary space. As a consequence, the action integral S=∫−∞+∞dτLb¯ remains invariant under the anti-BRST symmetry transformations (sab). In exactly similar fashion, we can discuss the BRST invariance of the Lagrangian Lb (cf. Equation (20)) within the framework of ACSA to BRST formalism and provide the geometrical interpretation for the super antichiral Lagrangian L~bacτ,θ¯ and its connection with the BRST-invariance (cf. Equation (20)) in the ordinary space.Now let us capture Equations (22) and (23), where the BRST symmetry transformation operates on Lb¯ and the anti-BRST symmetry transformation acts on Lb, within the purview of ACSA to BRST formalism. In this context, let us, first of all, generalize the Lagrangian Lb onto the chiral (1, 1)-dimensional super submanifold such that chiral supervariables (with the superscript ab) appear in it. In other words, we have the following generalization:
(54)Lb→L~bcτ,θ=L~fcτ,θ+Babτ,θBabτ,θ+Babτ,θĖabτ,θ+2β¯~abτ,θβ~abτ,θ−iF¯̇abτ,θḞabτ,θβ¯~abτ,θβ¯~abτ,θβ~abτ,θβ~abτ,θ+2iχ~abτ,θβ~abτ,θF¯̇abτ,θ−β¯~abτ,θḞabτ,θ−2Eabτ,θβ¯~abτ,θβ~̇abτ,θ+Γabτ,θχ~abτ,θ+2Γabτ,θβ~abτ,θF¯abτ,θ−β¯~abτ,θFabτ,θ+mβ¯~abτ,θβ~̇bτ,θ−β¯~̇abτ,θβ~abτ,θ+Γabτ,θχ~abτ,θ−Γ̇abτ,θΨ5abτ,θ.It should be noted that some of the chiral supervariables with the superscriptab are, primarily, the ordinary variables. For instance, we note that all the chiral supervariables on the l.h.s. of (46) are actually such variables (i.e., Pμabτ,θ=pμτ,Γabτ,θ=γτ,β¯~abτ,θ=β¯τ). Keeping in our mind the mapping: sab⇔∂θ, it is clear that we can operate ∂θ on the above chiral Lagrangian L~bcτ,θ to yield the following:
(55)∂∂θL~bcτ,θ=ddτc¯2p2+m2+β¯2pμψμ+mψ5+bc¯̇+2β¯χ+2ieβ¯γ−c¯̇+2β¯χddτb+b¯+2ββ¯+2iβ¯γb+b¯+2ββ¯≡sabLb.The above observation establishes the fact that LagrangianLb also respects the anti-BRST symmetry transformation (12) provided we invoke the CF-type restriction b+b¯+2ββ¯=0 from outside. In other words, we have captured the existence of the (anti-)BRST invariant CF-type restriction within the framework of ACSA to BRST formalism and have proved that the Lagrangian Lb (which is perfectly BRST invariant (cf. Equation (20))) is also invariant w.r.t. the anti-BRST symmetry transformation (12) provided we confine ourselves to the submanifold of the quantum variable where the CF-type restriction is satisfied.At this juncture, we generalize the ordinary LagrangianLb¯ to its counterpart antichiral Lagrangian L~b¯acτ,θ¯ on the (1, 1)-dimensional antichiral super submanifold as
(56)Lb¯→L~b¯acτ,θ¯=L~facτ,θ¯+B¯bτ,θ¯B¯bτ,θ¯−B¯bτ,θ¯Ėbτ,θ¯−2β¯~bτ,θ¯β~bτ,θ¯−iF¯̇bτ,θ¯Ḟbτ,θ¯+β¯~bτ,θ¯β¯~bτ,θ¯β~bτ,θ¯β~bτ,θ¯+2iχ~bτ,θ¯β~bτ,θ¯F¯̇bτ,θ¯−β¯~bτ,θ¯Ḟbτ,θ¯+2Ebτ,θ¯β¯~̇bτ,θ¯β~bτ,θ¯−Γbτ,θ¯χ~bτ,θ¯+2Γbτ,θ¯β~bτ,θ¯F¯bτ,θ¯−β¯~bτ,θ¯Fbτ,θ¯+mβ¯~bτ,θ¯β~̇bτ,θ¯−β¯~̇bτ,θ¯β~bτ,θ¯+Γbτ,θ¯χ~bτ,θ¯−Γ̇bτ,θ¯Ψ5bτ,θ¯.It will be noted that some of the above antichiral supervariables (cf. Equation (40)) are basically ordinary variables (e.g., Pμbτ,θ¯=pμτ,β~bτ,θ¯=βτ,Γbτ,θ¯=γτ). As far as the dependence on the Grassmannian variable of L~bacτ,θ¯ is concerned, it is straightforward to note that we have θ¯-dependence. Thus, the mapping: sb↔∂θ¯, allows us to apply, on the super Lagrangian L~b¯acτ,θ¯, a derivative (∂θ¯) w.r.t. the Grassmannian variable θ¯. The ensuing mathematical expression, as the outcome of the above operation, is as follows:
(57)∂∂θ¯L~bacτ,θ¯=ddτc2p2+m2+β2pμψμ+mψ5−b¯ċ+2βχ+2ieβγ+ċ+2βχddτb+b¯+2ββ¯−2iβγb+b¯+2ββ¯≡sbLb¯.At this stage, it is an elementary exercise to state that the perfectly anti-BRST invariant LagrangianLb¯ (cf. Equation (21)) also respects the BRST symmetry transformations (13) provided the whole theory is considered on a submanifold of the Hilbert space of quantum variables where the CF-type restriction b+b¯+2ββ¯=0 is satisfied. In other words, the Lagrangian Lb¯ respects the BRST symmetry transformations (13) provided we impose the CF-type restriction from outside. Thus, we have derived the CF-type restriction (cf. Equation (57)) within the ambit of ACSA to BRST formalism.We end this section with the remark that the CF-type restriction is the hallmark [5, 6] of a quantum theory (discussed within the framework of BRST formalism). We have shown its existence on our theory within the framework of ACSA to BRST formalism. Hence, we have achieved a proper BRST quantization of our theory of the 1D system.


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

We have already seen that the (anti-)BRST symmetry transformations (13) and (12) are off-shell nilpotent sab2=0 and absolutely anticommuting (cf. Equation (15)) in nature provided the CF-type restriction (cf. Equation (14)) is imposed from outside on our theory. As the off-shell nilpotent (anti-)BRST symmetry transformations are generated by the conserved (anti-)BRST charges, the above off-shell nilpotency and absolute anticommutativity are also respected by the conserved and off-shell nilpotent (anti-)BRST charges. In our present section, we capture these properties of the conserved (Q̇ab=0) fermionic (anti-)BRST charges Qab within the framework of ACSA to BRST formalism.We have already demonstrated that the Noether conserved chargesQab1 (cf. Equations (26) and (27)) are not off-shell nilpotent. In fact, they are on-shell nilpotent. Using the EL-EOMs, we have recast these conserved charges into another forms (cf. Equations (30) and (33)) and denoted them by Qab2. These latter forms of the charges turn out to be off-shell nilpotent of order two (i.e., Qab22=0). We now concentrate on Qb2 and generalize the BRST charge Qb2 onto the 1,1-dimensional antichiral super submanifold as
(58)Qb2→Q~b2τ,θ¯=Bbτ,θ¯Ḟbτ,θ¯−Ḃbτ,θ¯Fbτ,θ¯+2iEbτ,θ¯β~bτ,θ¯Γbτ,θ¯+2β~bτ,θ¯β¯~bτ,θ¯Ḟbτ,θ¯−2Fbτ,θ¯β¯~bτ,θ¯β~̇bτ,θ¯+Γbτ,θ¯χ~bτ,θ¯+2Bbτ,θ¯β~bτ,θ¯χ~bτ,θ¯−2imΓbτ,θ¯β~bτ,θ¯−β~bτ,θ¯β~bτ,θ¯F¯̇bτ,θ¯+2χ~bτ,θ¯β~bτ,θ¯β~bτ,θ¯β¯~bτ,θ¯,where all the antichiral supervariables with the superscript b have been derived in Equations (40) and (44). It is evident that there are some supervariables in the above expression for Q~b2τ,θ¯ which are actually ordinary variables (cf. Equation (40)). Keeping in our mind the mapping: sb↔∂θ¯ [10–12], it can be explicitly checked that
(59)∂∂θ¯Q~b2τ,θ¯≡∫dθ¯Q~b2τ,θ¯=0⇔sbQb2=0.The above relationship is nothing but the proof of the off-shell nilpotencyQb22=0 of the conserved BRST charge Qb2. To corroborate this statement, we note that, in the ordinary space, the observation sbQb2=0 can be mathematically stated as Equation (31) which proves the off-shell nilpotency Qb22=0 of the conserved charge Qb2.We now focus on the proof of the nilpotencyQab22=0 of the anti-BRST charge Qab2 (cf. Equation (33)) within the ambit of ACSA to BRST formalism. In this context, we note that Qab2 can be generalized onto the 1,1-dimensional chiral super submanifold as
(60)Qab2→Q~ab2τ,θ=B¯̇abτ,θF¯abτ,θ−B¯abτ,θF¯̇abτ,θ+2iEabτ,θβ¯~abτ,θΓabτ,θ−2β~abτ,θβ¯~abτ,θF¯̇abτ,θ+2F¯abτ,θβ¯~̇abτ,θβ~abτ,θ−Γabτ,θχ~abτ,θ−2B¯abτ,θβ¯~abτ,θχ~abτ,θ−2imΓabτ,θβ¯~abτ,θ+β¯~abτ,θβ¯~abτ,θḞabτ,θ−2χ~abτ,θβ¯~abτ,θβ¯~abτ,θβ~abτ,θ,where all the supervariables with superscript ab are chiral expansions that have been quoted in Equations (46) and (50). It is pertinent to point out that some of the chiral supervariables, on the r.h.s. of (60), are actually ordinary variables (cf. Equation (46)) because they are anti-BRST invariant (e.g., sabβ¯=sabb¯=sabpμ=0) variables. In view of the mapping: sab↔∂θ [10–12], we are in the position to operate a derivative w.r.t. the Grassmannian variable θ on the expression for the super anti-BRST charge to show that
(61)∂∂θQ~ab2τ,θ≡∫dθQ~ab2τ,θ=0⇔sabQab2=0,where, as pointed out earlier, we have to substitute the chiral super expansion (46) and (50) into the r.h.s. of (60) and, then only, we have to operate ∂θ. The above equation (61) is nothing but the proof for the off-shell nilpotency Qab22=0 of the anti-BRST charge which becomes transparent when we exploit the beauty of the relationship between the continuous symmetries and their generators as we have shown in Equation (34). Thus, we have captured the off-shell nilpotency of the anti-BRST charge within the framework of ACSA.At this juncture, we pay our attention to capture the absolute anticommutativity of the BRST charge with the anti-BRST charge using the theoretical power of ACSA to BRST formalism. In this context, we note that the BRST chargeQb2 (cf. Equation (30)] can be also generalized onto the (1, 1)-dimensional chiral super submanifold as
(62)Qb2→Q~b2τ,θ=Babτ,θḞabτ,θ−Ḃabτ,θFabτ,θ+2iEabτ,θβ~abτ,θΓabτ,θ+2β~abτ,θβ¯~abτ,θḞabτ,θ−2Fabτ,θβ¯~abτ,θβ~̇abτ,θ+Γabτ,θ¯χ~abτ,θ+2Babτ,θβ~abτ,θχ~abτ,θ−2imΓabτ,θβ~abτ,θ−β~abτ,θβ~abτ,θF¯̇abτ,θ+2χ~abτ,θβ~abτ,θβ~abτ,θβ¯~abτ,θ,where the chiral supervariables on the r.h.s. are nothing but the super expansions that have been quoted in (46) and (50). It is worthwhile to point out that some of the supervariables are ordinary variables in the true sense of the word (cf. Equation (46)). We can now operate by the Grassmannian derivative ∂θ on (62) to produce
(63)∂∂θQ~b2τ,θ≡∫dθQ~b2τ,θ¯=ib+b¯+2ββ¯b¯̇+2χγ+2ββ¯̇−ib¯ddτb+b¯+2ββ¯≡sabQb2=−iQb2,Qab2.In the above, we have utilized the mapping∂θ⇔sab to express the l.h.s. in the ordinary space. The expression sabQb2 is nothing but the absolute anticommutativity (i.e., Qb2,Qab2) of the BRST charge with the anti-BRST charge Qab2. It is crystal clear that the absolute anticommutativity property (i.e., Qb2,Qab2=0) is satisfied if and only if we impose the condition b+b¯+2ββ¯=0. In other words, we have been able to derive the CF-type restriction: b+b¯+2ββ¯=0 (which characterizes a BRST quantized theory) by exploiting the theoretical tricks and techniques of ASCA to BRST formalism.Ultimately, we concentrate to capture the absolute anticommutativity of the anti-BRST chargeQab2 with the BRST charge Qb2 within the framework of ACSA to BRST formalism. Toward this goal in mind, we generalize the anti-BRST charge Qab2 (cf. Equation (33)) onto the (1, 1)-dimensional antichiral super submanifold as
(64)Qab2→Q~ab2τ,θ¯=B¯̇bτ,θ¯F¯bτ,θ¯−B¯bτ,θ¯F¯̇bτ,θ¯+2iEbτ,θ¯β¯~bτ,θ¯Γbτ,θ¯−2β~bτ,θ¯β¯~bτ,θ¯F¯̇bτ,θ¯+2F¯bτ,θ¯β¯~̇bτ,θ¯β~bτ,θ¯−Γbτ,θ¯χ~bτ,θ¯−2B¯bτ,θ¯β¯~bτ,θ¯χ~bτ,θ¯−2imΓbτ,θ¯β¯~bτ,θ¯+β¯~bτ,θ¯β¯~bτ,θ¯Ḟbτ,θ¯−2χ~bτ,θ¯β¯~bτ,θ¯β¯~bτ,θ¯β~bτ,θ¯,where the antichiral supervariable on the r.h.s. are nothing but the super expansions (40) and (44) that have been derived after the applications of the quantum gauge (i.e., BRST) invariant restrictions on the antichiral supervariables. It goes without saying that some of the supervariables on the r.h.s. of (64) are actually ordinary variables (cf. Equation (40)). In view of our understanding that we have sab⇔∂θ¯ [10–12], we can operate a derivation w.r.t. the Grassmannian variable θ¯ on the Q~ab2τ,θ¯ as
(65)∂∂θ¯Q~ab2τ,θ¯≡∫dθ¯Q~ab2τ,θ¯=0=ibddτb+b¯+2ββ¯−ib+b¯+2ββ¯ḃ+2ββ¯+2γχ⇔sbQab2=−iQab2,Qb2.In other words (cf. Equation (38)), we have captured the absolute anticommutativity property of the anti-BRST charge with the BRST charge within the framework of ACSA to BRST formalism. A close look at (65) demonstrates that the r.h.s. of ∂θ¯Q~b2τ,θ¯ is zero if and only if the CF-type restriction is imposed from outside. In other words, we have proven the existence of the CF-type restriction: b+b¯+2ββ¯=0 (on our BRST quantized theory) within the framework of ACSA to BRST formalism.We end this section with the following remarks. First and foremost, we observe that the mappings:sb↔∂θ¯,sab↔∂θ [10–12], play a crucial role in the proof of the two decisive properties of the (anti-)BRST conserved charges. Second, it is interesting to point out that the conserved charges Qab2 can be generalized onto 1,1-dimensional (anti-)chiral super submanifolds of the general 1,2-dimensional supermanifold. Finally, the operations of the translational generators ∂θ,∂θ¯ (along the chiral and antichiral directions of the super submanifolds) on the generalized forms of the supercharges Q~ab2 lead to the proof of the off-shell nilpotency (cf. Equations (59) and (61)) and absolute anticommutativity properties of the (anti-)BRST charges as well as the deduction of the (anti-)BRST invariant CF-type restriction (cf. Equations (63) and (65)) within the framework of ACSA to BRST formalism.


## 8. Conclusions

In our present endeavor, we have done a thread-bare analysis of the classical gauge, supergauge, and reparameterization symmetries of the first-order Lagrangian for the 1D system of a massive spinning relativistic particle. We have demonstrated that, in specific limits and identifications, the reparameterization symmetries incorporate the gauge and (super)gauge symmetries (cf. Section2). We have established that the constraints of our theory (described by the first-order Lagrangian) are of first-class variety in the terminology of Dirac’s prescription for the classification scheme [36, 37]. We have obtained the secondary constraints from the equivalent Lagrangians and corresponding canonical Hamiltonians of our 1D system of spinning relativistic particle (cf. Section 2 and Appendix A).We have elevated the combined classical gauge and supergauge symmetry transformations (cf. Equation (7)) to its counterpart quantum (anti-)BRST symmetry transformations which are respected by the coupled (but equivalent) (anti-)BRST invariant Lagrangians. The hallmark of a quantum theory (discussed within the purview of the BRST approach) is the existence of the CF-type restriction(s). We have demonstrated the existence of a single CF-type restriction on our theory by demanding the absolute anticommutativity of the off-shell nilpotent (anti-)BRST symmetry transformations as well as by proving the equivalence of the coupled Lagrangians with respect to the quantum gauge (i.e., (anti-)BRST) symmetry transformations (cf. Equations (22) and (23)). In other words, the absolute anticommutativity of the (anti-)BRST symmetry transformations and the existence of the coupled (but equivalent) Lagrangians owe their origins to the CF-type restriction which defines a submanifold in the quantum Hilbert space of variables that is defined by the equation: b+b¯+2ββ¯=0.To corroborate the sanctity of our (anti-)BRST symmetry transformations, coupled (but equivalent) Lagrangians, and their invariance(s), we have exploited the theoretical potential and power of ACSA to BRST formalism [20–24] where only the (anti-)chiral supervariables and their suitable expansion(s) along the Grassmannian direction(s) have been considered. One of the novel observations, in this context, has been the proof of absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges within the framework of ACSA to BRST formalism where we have considered only the (anti-)chiral super expansions. This proof, it should be emphasized, is obvious when the full expansions of the supervariables (defined on the full supermanifold) are taken into account. The importance of the ACSA to BRST formalism lies in its simplicity and its dependence on the quantum gauge (i.e., (anti-)BRST) invariant restrictions on the supervariables which are defined on the (anti-)chiral super submanifolds of the general (full) supermanifold. Whereas the (anti-)chiral super submanifolds are characterized by a single Grassmannian variable, the general supermanifold is defined by the superspace coordinates that incorporate a pair of Grassmannian variables. The quantum (anti-)BRST symmetry transformations are found to be associated with the translational generators ∂θ,∂θ¯ along the Grassmannian directions θ,θ¯.We have devoted a great of discussion on the derivation and proof of the existence of CF-type restriction (see, also, e.g., [25]) on our theory because the hallmark [5, 6] of a quantum theory (described and discussed within the framework of BRST formalism) is its presence. We have shown its appearance in the context of absolute anticommutativity of the (anti-)BRST symmetries (cf. Equation (14)), invariance and equivalence of the coupled (but equivalent) Lagrangians (cf. Equations (23)–(25)), and absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges (cf. Equations (35) and (37)) in the ordinary space. These features have also been captured in the superspace by exploiting the theoretical potential and power of ACSA to BRST formalism (cf. Sections 6 and 7).We would like to comment on the various kinds of superfield approaches (e.g., USFA, AVSA, and ACSA) to BRST formalism that have been developed over the years. The USFA is the one where mathematically beautiful HC has been exploited to derive the (anti-)BRST symmetry transformations for the gauge and associated (anti-)ghost fields in the case of a (non-)Abelian 1-form gauge theory (see, e.g., [10–12]). In addition, it has led to the systematic derivation of the (anti-)BRST invariant CF condition [7]. The AVSA is a minor extension of the USFA (developed in [10–12]) where the HC and gauge-invariant restriction(s) play an important role together for the derivation of the proper (anti-)BRST symmetry transformations of the gauge, (anti-)ghost, and matter fields together for an interacting gauge theory. The ACSA is a simplified version of USFA where the quantum gauge (i.e., (anti-)BRST) invariant restrictions on the superfields/supervariables lead to the derivation of the (anti-)BRST symmetry transformations for all the fields. Within the framework of ACSA, the (anti-)BRST invariant CF-type restriction(s) arise in the proof of (i) the invariance of the coupled (but equivalent) Lagrangian densities and (ii) the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges.In our present investigation, we have performed the BRST and supervariable analysis of a toy model (i.e., 1D system) of a massive spinning relativistic particle where only the (super)gauge symmetry transformations (7) have been exploited for the BRST analysis. We have not devoted any time on the BRST analysis corresponding to the infinitesimal reparameterization symmetries (4). It would be a nice future endeavor to exploit the latter symmetry transformations for the BRST analysis in view of the fact that such an exercise has already been performed by us in the case of a scalar relativistic particle [26]. We plan to extend the richness of our theoretical analysis to the realm of interesting systems of the quantum field theory as well as diffeomorphism invariant theories of gravitation and (super)strings. It is worthwhile to mention here that, in a recent set of papers (see, e.g., [38–40]), the BRST analysis has been performed for the celebrated ABJM theory. It would be, therefore, a very nice future project for us to apply our present theoretical analysis to the ABJM theory. We are currently very seriously involved with the classical diffeomorphism symmetry and its elevation to the quantum (anti-)BRST symmetries for the system of scalars, vectors, and metric tensor. Our results, in this direction, would be reported elsewhere [41].



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