Massive Spinning Relativistic Particle: Revisited Under the BRST and Supervariable Approaches

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 (0+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 anticommutatvity 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 anticommutatvity 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 anticommutatvity 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.


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][2][3][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 CFtype 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][9][10][11][12][13][14][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][11][12]. These approaches, however, lead to the derivation of the (anti-)BRST symmetries for the gauge field and associated (anti-)ghost fields only [10][11][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][17][18][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 compliment and supplement each-other in a beautiful fashion. This approach has been christened as the augmented version of superfield approach (AVSA) to BRST formalism.
In the recent set of papers [20][21][22][23][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][9][10][11][12][13][14][15][16][17][18][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 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 anticommuating 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).
In our present endeavor, we have shown the existence of the three classical level symmetries which are the gauge, supergauge and reparameterization transformations [cf. Eqs. (2), (4)] under which the first-order Lagrangian (L f ) remains invariant. We have further established that the reparameterization symmetry transformations contain (i) the gauge symmetry transformations [cf. Eq. (2)] provided all the fermionic variables are set equal to zero (i.e. ψ µ = ψ 5 = χ = 0), and (ii) the combination of gauge and supergauge symmetry transformations [cf. Eq. (7)] under specific limit when some appropriate equations of motion and identifications of transformation parameters are taken into account [cf. Eqs. (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 L b and Lb [cf. Eqs. (17), (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 sub-manifold of the quantum Hilbert space of variables where the CF-type restriction is satisfied [cf. Eqs. (20)- (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. Eqs. (35)- (38)]. In our present investigation, 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 sub-manifolds of the general (1, 2)-dimensional supermanifold have been taken into account 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 anti-chiral (1, 1)-dimensional super sub-manifolds within the framework of ACSA to BRST formalism (cf. Appendix D below).
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-Popove ghost terms within the framework of the BRST formalism [cf. Eq. (16) below]. 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 super-symmetrization of horizontalitiy 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 achieved all these goals in our present endeavor. Finally, the 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][26][27][28][29][30][31][32].
The theoretical material of our present endeavor is organized as follows. In Sec. 2, we discuss the gauge, supergauge and reparameterization symmetries of the Lagrangian that describes the 1D massive spinning relativistic particle. Our Sec. 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 Sec. 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 anti-chiral supervariables play a crucial role. Our Sec. 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 Sec. 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 anticommutatvity of the (anti-)BRST charges by applying the key techniques of ACSA to BRST formalism in Sec. 7. Finally, in Sec. 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 taking the help of ACSA to BRST formalism.

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 transformations 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 L 0 is the Lagrangian with a square-root, L f is the first-order Lagrangian * and L s is the second-order Lagrangian. Our one (0+1)-dimensional (1D) system is embedded in a flat Minkowskian D-dimensional target space where (x µ , p µ ) 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 ( ) with all the bosonic variables (x µ , p µ , e) of our theory. It should be noted that ψ µ is the superpartner of x µ and ψ 5 variable has been invoked in the theory to incorporate a mass term m so that the mass-shell condition (p 2 − m 2 = 0) for the free particle could be satisfied. The Lagrangian L 0 has a square-root and its massless limit is not defined. On the other hand, the second-order Lagrangian (L s ) is endowed with a variable (i.e. einbein) which is located in the denominator. Thus, the Lagrangians L 0 and L s have their own limitations. We shall focus on the first-order Lagrangian (L f ) for our discussions where variables e(τ ) and χ(τ ) are not purely Lagrange multiplier variables but their transformations are such that they behave like the "gauge" and "supergauge" variables [cf. Eq. (2) below]. Our 1D system is a model of supersymmetric gauge theory and its generalization to 4D theory provides a model for the supergravity theory where ψ µ corresponds to the Rarita-Schwinger field and e(τ ) becomes the vierbein field. The mass m, in the supergravity theory, represents the cosmological constant term. In a nut-shell, 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 L f respects the following gauge (δ g ) and supergauge (δ sg ) symmetry transformations, namely; where ξ(τ ) and κ(τ ) are the infinitesimal gauge and supergauge symmetry transformation parameters, respectively. It is straightforward to note that ξ(τ ) is a bosonic and κ(τ ) is a fermionic (i.e. κ 2 = 0) transformation parameter. Furthermore, the transformation δ sg is a supersymmetric transformation because it transforms a bosonic variable to a fermionic * 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 (L f ) is the only Lagrangian that is mentioned in [33].
variable and vice-versa. The transformations in Eq. (2) are symmetry transformations because the first-order Lagrangian L f transforms to the following total derivatives: As a consequence, it is clear that the action integral S = +∞ −∞ d τ L f , under the transformations δ g and δ sg , would be equal to zero 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 − ǫ(τ ) where ǫ(τ ) is an infinitesimal transformation parameter. In fact, the physical variables of our 1D system transform under the infinitesimal reparameterization transformation (δ r ) as: The above transformations are symmetry transformations for the action integral S = +∞ −∞ d τ L f because of the following transformation property of L f , namely; It is evident that δ r S = 0 due to the fact that ǫ(τ ) and L f vanish-off at τ = ± ∞. The reparameterization symmetry transformation (δ r ) and gauge symmetry transformation (δ g ) are equivalent under the following limits ξ = ǫ e,ẋ µ = e p µ ,ṗ µ = 0, provided we set all the fermionic variables (χ, ψ 5 , ψ µ ) of our theory equal to zero. In the above, we have used equations of motion:ẋ µ = e p µ andṗ µ = 0 and we have identified the gauge symmetry transformation parameter ξ(τ ) with the combination of the reparameterization transformation parameter ǫ(τ ) and the einbein variable e(τ ). In exactly similar fashion, we note that (δ r ) and (δ g + δ sg ) are also equivalent. In this context, first of all, we note that there are two primary constraints (i.e. Π e ≈ 0, Π χ ≈ 0) and two secondary constraints (i.e. p 2 − m 2 ≈ 0, p µ ψ µ − m ψ 5 ≈ 0) on our theory where Π e and Π χ are the canonical conjugate momenta w.r.t. the Lagrange multiplier variables e and χ, respectively † . These constraints generate the combined (super)gauge symmetry transformations δ = δ g + δ sg for the physical variables of our theory as (see, e.g. [33]) under which the first-order Lagrangian L f transforms to a total "time" derivative as: As a consequence of the above observation, it is evident that δS = 0 where S = +∞ −∞ dτ L f is the action integral. If we use the following equations of motion: and identify the transformations parameters as we find that the reparameterization symmetry transformation (4) [emerging due to the basic transformation: τ → τ ′ = τ − ǫ(τ )] and the combined gauge and supergauge symmetry transformations (i.e. δ = δ g + δ sg ), quoted in Eq. (7), are equivalent to each-other. It is worthwhile to note that, under the identifications (10), the transformation δe =ξ + 2 κ χ becomes δe = d dτ (ǫ e) as we note that 2 κ χ = − 2 i e χ 2 = 0. We end this section with the following remarks. First of all, we note that the canonical Hamiltonians, derived from L 0 and L f (as well as L s ), are where H (0) c is the canonical Hamiltonian corresponding to the Lagrangian L 0 . It is straightforward to note that the primary constraints Π e ≈ 0, Π χ ≈ 0 lead to the derivation of the secondary constraints (p 2 − m 2 ) ≈ 0, (p µ ψ µ − m ψ 5 ) ≈ 0 from the Hamiltonians (11) as well as from all the three equivalent Lagrangians (1) (cf. Appendix A below). Second, we have explicitly demonstrated that the (super)gauge symmetry transformations and reparameterization symmetry transformations are equivalent under specific conditions [cf. Eqs. (9), (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.

(Anti-)BRST Symmetries: Lagrangian Formulation
Our present section is divided into two parts. In the 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 L b and Lb. In the subsection 3.2, we establish the existence of the same by requiring the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges. , in their full blaze of glory for our 1D system of the massive spinning relativistic particle, are (see, e.g. [25])

(Anti-)BRST Invariance and CF-Type Restriction
s ab x µ =c p µ +β ψ µ , s ab e =ċ + 2β χ, s ab ψ µ = iβ p µ , s abc = −iβ 2 , s ab c = ib, s abβ = 0, s ab β = −i γ, s ab p µ = 0, where b andb are the Nakanishi-Lautrup type auxiliary variables, fermionic (χ 2 = 0, c 2 = c 2 = 0, γ 2 = 0) variables (χ, c,c, γ) are present in our theory and rest of the symbols have already been explained earlier. As far as the absolute anticommutativity (s b s ab +s ab s b = 0) property is concerned, it can be checked that are equal to zero only after imposing the CF-type restriction: b +b + 2 ββ = 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. s (a)b [b+b+2 ββ] = 0) quantity. Except for the variables (x µ , e), it is straightforward to check that the following is true for the other variables of our theory, namely; where Φ(τ ) 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 [(s (a)b ) 2 = 0] and absolutely anticommuting (s b s ab +s ab s b = 0) in nature provided the whole theory is considered on a submanifold of space of quantum variables where the CF-type restriction: b +b + 2 ββ = 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: where L f is the first-order Lagrangian that has been quoted in Eq. (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, b andb are the Nakanishi-Lautrup type auxiliary variables which lead to the derivation of EL-EOMs (from L b and Lb) as: 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 Lagrangians L b and Lb. In this context, we observe the following: It is clear from the above results that the action integrals under the quantum BRST and anti-BRST symmetry transformations that have been listed in Eqs. (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: A close look at the above transformations demonstrates that if we impose the (anti-)BRST invariant s (a)b (b +b + 2 ββ) = 0] quantum restriction (b +b + 2 ββ = 0) from outside, we obtain the following BRST symmetry transformation of the Lagrangian Lb and anti-BRST symmetry transformation of the Lagrangian L b , namely; It is crystal clear now that the observations in Eqs. (20), (21), (22), (23), (24) and (25) imply, in a straightforward manner, that both the Lagrangians (i.e. L b 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 sub-section with the following remarks. First and foremost, we observe that the presence of the term "χ ψ 5 " in the square-bracket of Eq. (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 Eqs. (14), (19), (22) and (23) where we have concentrated on the quantum (anti-)BRST symmetries which are respected by the coupled Lagrangians L b and Lb. Finally, we note that the absolute anticommutativity property of the (anti-)BRST symmetries and equivalence of L b and Lb owe their origins to the CF-type restriction: b +b + 2 ββ = 0.

(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 + 2 ββ = 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 (20) and (21)] lead to the derivation of the Noether conserved (anti-)BRST charges (Q (1) (a)b ) as follows: The conservation law (i.e.Q ab = 0) can be proven by using the EL-EOMs derived from the Lagrangians L b and Lb (cf. Appendix B below). We have used the superscript (1) on the (anti-)BRST charges (Q (1) (a)b ) 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 L b and Lb. At this stage, it can be noted that the Noether conserved charges Q 0) 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 L b and Lb. Thus, we lay emphasis on the fact that Q 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 Lb and L b ) to recast the Noether conserved charges Q can be written in a different form by using the following EL-EOMṡ which are derived from L b w.r.t. the e and χ variables. The ensuing expression for the conserved BRST charge [due to EL-EOM (29)] is: Here the superscript (2) denotes that the expression for the BRST charge in Eq. (30) has been derived from the Noether conserved BRST charge Q b by using the EL-EOMs quoted in Eq. (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 Q for the computation of the l.h.s. of Eq. (31). We would like to lay emphasis on the fact that Eq. (31) is nothing but the standard relationship between the continuous symmetry transformation s b and its generator Q (2) b . 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 ([Q (2) b ] 2 = 0) of the Q (2) b 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 (Q ab ). For this purpose, we use the following EL-EOMṡ that emerge out from the Lagrangian Lb (when we consider the variables e and χ for their derivation) to recast the Noether conserved charge Q where the superscript (2) on the anti-BRST charge Q (2) ab denotes the fact that it has been derived from the Noether conserved charge Q (1) ab . We apply, at this stage, the anti-BRST symmetry transformations (12) directly on the anti-BRST charge Q (2) ab to obtain: The above observation proves the off-shell nilpotency of the anti-BRST charge Q (2) ab because we do not use EL-EOMs and/or CF-type restriction in its proof. In Eq. (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 ([Q 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/onshell 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 Q (2) (a)b . Toward this goal in mind, we first concentrate on the expression for Q 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 (s ab ) and its generator Q ab , it is evident that the l.h.s. of Eq. (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: b +b + 2 ββ = 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 nilpotent (Q ab ) 2 = 0 anti-BRST charge (Q (2) ab ) in Eq. (33). The direct application of the BRST symmetry transformation (s b ) of Eq.(13) on the anti-BRST charge Q (2) ab 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 (b +b + 2 ββ = 0) from outside. Exploiting the beauty of the standard relationship between continuous symmetry transformation (s b ) and its generator (conserved and nilpotent BRST charge Q b ), we note that the l.h.s. of the above equation can be written as provided, as stated earlier, we confine ourselves on the sub-manifold of the quantum Hilbert space of variables where the CF-type restriction (b +b + 2 ββ = 0) is satisfied. 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 L b 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 L b and Lb both respect both the (anti-)BRST symmetries due to the existence of CF-type restriction.

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 anti-chiral supervariables [defined on the (1, 1)-dimensional anti-chiral super sub-manifold of the general (1, 2)-dimensional supermanifold]. The above anti-chiral supervariables are the generalizations of the ordinary variables of Lagrangian L b andb(τ ) as follows: In the above, we have taken the super expansions along the Grassmannianθ-direction of the anti-chiral (1, 1)-dimensional super sub-manifold which is parameterized by the superspace coordinates (τ,θ). We note that, in the above super expansions, the secondary variables (R are bosonic in nature due to the fermionic (θ 2 = 0) nature of the Grassmannian variableθ. It is elementary to state that, in the limitθ = 0, we retrieve ordinary variables of our theory described by the Lagrangian L b andb(τ ).
The trivial BRST invariant quantities: This is due to the fact that the basic tenets of ACSA requires that the BRST invariant quantities should be independent of the § It will be noted that the BRST symmetry transformations have been mentined in Ref. [33] for the massless spinning relativistic particle. However, the full (anti-)BRST symmetry transformations and the corresponding (anti-)BRST invariant CF-type restriction have been derived in our earlier work [25].
Grassmannian variableθ (which is a mathematical artifact in the superspace formalism). In other words, we have the following where the superscript (b) on the anti-chiral supervariables denotes the supervariables that have been obtained after the application of the BRST invariant ( restrictions so that the coefficients ofθ, in the expansions (39), becomes zero. This is due to the fact that there is a mapping (i.e. s b ↔ ∂θ, s ab ↔ ∂ θ ) between the (anti-) BRST symmetry transformations (s (a)b ) and the translational operators (∂ θ , ∂θ) along the Grassmannian directions of the (1, 2)-dimensional supermanifold that has been established in Refs. [10][11][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 ofθ 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 transformation (s b ) as the coefficient ofθ in the anti-chiral super expansions (39). Toward this goal in our mind, we have to find out the specific combinations of the non-trivial 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 requires that the above quantities, at the quantum level, should be independent of the Grassmannian variable (θ) when these are generalized onto the (1, 1)-dimensional anti-chiral super sub-manifold of the general (1, 2)dimensional supermanifold. As a consequence, we have the following restrictions on the specific combinations of the anti-chiral supervariables, namely; Ultimately, we obtain the super expansions of (39) in terms of the off-shell nilpotent (s 2 b = 0) BRST transformations (13) of our theory as follows In our Appendix C, we collect the step-by-step computations that lead to the derivation of (43) from (42). 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). θ) is the generic anti-chiral supervariable that is located on the l.h.s. of Eqs. (40) and (44) and the symbol ω(τ ) corresponds to the generic ordinary variable that is present in the Lagrangians L b and Lb. Finally, we observe that, due to the mapping s b ↔ ∂θ, the off-shell nilpotency (s 2 b = 0) of the BRST symmetry transformations (13) is deeply connected with the nilpotency (∂ 2 θ = 0) of the translational generator (∂θ) along theθ-direction of (1, 1)-dimensional anti-chiral super sub-manifold on which the anti-chiral supervariables are defined.

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 L b (and the auxiliary variable b(τ )) onto (1, 1)-dimensional chiral super sub-manifold of the general (1, 2)-dimensional supermanifold as: where the (1, 1)-dimensional chiral super sub-manifold is parameterized by the superspace coordinates (τ, θ) and all the chiral supervariables on the l.h.s. of (45) are function of these superspace coordinates. The fermionic (θ 2 = 0) nature of the Grassmannian variable θ implies that the secondary variables (R µ ,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 b(τ ).
The basic ingredient of the ACSA to BRST formalism requires that the non-trivial anti-BRST invariant quantities must be independent of the Grassmannian variable θ when these quantities are generalized onto the (1, 1)-dimensional chiral super sub-manifold. 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 are found to be very useful and interesting because their generalizations on the chiral super sub-manifold, 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 Lb by requiring that the quantum gauge [i.e. anti-BRST] invariant quantities must be independent of θ as the Grassmannian variables 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. Eq. (49)] into the expansions in (45) lead to the following: In the above equation, the superscript (ab) on the chiral supervariables (on the l.h.s.) denotes the super expansions that have been derived after the applications of the 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 theory. We wrap up this section with the following comments. First and foremost, we observe that the trivial anti-BRST invariant (e.g. s ab p µ = s ab γ = s abb = s abβ = 0) variables have been incorporated in the super expansions in (46). Second, the non-trivial 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 sub-manifold of the general (1, 2)-dimensional supermanifold.
where the superscript (c) and (ac) on the super Lagrangians (i.e.L 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 that: P (ab) In view of the mappings: s b ↔ ∂θ, s ab ↔ ∂ θ  22) and (23), where the BRST symmetry transformation operates on Lb and the anti-BRST symmetry transformation acts on L b , within the purview of ACSA to BRST formalism. In this context, let us, first of all, generalize the Lagrangian L b onto the chiral (1, 1)-dimensional super sub-manifold such that chiral supervariable [with the superscript (ab)] appear in it. In other words, we have the following generalization: It should be noted that some of the chiral supervariables with the superscript (ab) 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. P (ab) µ (τ, θ) = p µ (τ ), Γ (ab) (τ, θ) = γ(τ ),β (ab) (τ, θ) =β(τ )]. Keeping in our mind the mapping: s ab ⇔ ∂ θ , it is clear that we can operate ∂ θ on the above chiral LagrangianL The above observation establishes the fact that Lagrangian L b also respects the anti-BRST symmetry transformation (12) provided we apply the CF-type restriction (b+b + 2 ββ = 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 L b (which is perfectly BRST invariant) is also invariant w.r.t. the anti-BRST symmetry transformation (12) provided we confine ourselves to the sub-manifold of the quantum variable where the CF-type restriction is satisfied. At this juncture, we generalize the ordinary Lagrangian Lb to its counterpart anti-chiral LagrangianL (ac) b (τ,θ) on the (1, 1)-dimensional anti-chiral super sub-manifold as: It will be noted that some of the above anti-chiral supervariables [cf. Eq. (40)] are basically ordinary variables [e.g. P As far as the dependence on the Grassmannian variable ofL (ac) b (τ,θ) is concerned, it is straightforward to note that we haveθ-dependence. Thus, the mapping: s b ↔ ∂θ allows us to apply on the super LagrangianL  21)] also respects the BRST symmetry transformations (12) provided the whole theory is considered on a (1, 1)-dimensional anti-chiral super submanifold of the Hilbert space of variables where the CF-type restriction (b +b + 2 ββ = 0) is satisfied. In other words, the Lagrangian Lb respects BRST symmetry transformations (12) provided we impose the CF-type restriction from outside. Thus, we have derived the CF-type restriction [cf. Eq. (59)] 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 a proper BRST quantization of our theory of the 1D system.  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 (Q (a)b = 0) fermionic (anti-)BRST charges within the framework of ACSA to BRST formalism.
We have already demonstrated that the Noether conserved charges Q onto the (1, 1)-dimensional anti-chiral super sub-manifold as: where all the anti-chiral supervariables with the superscript (b) have been derived in Eqs.
(40) and (44). It is evident that there are some supervariables in the above expression for Q (2) b (τ,θ) which are actually ordinary variables [cf. Eq. (40)]. Keeping in our mind the mapping: s b ↔ ∂θ [10][11][12], it can be explicitly checked that: The above relationship is nothing but the proof of the off-shell nilpotency (Q where all the supervariables with superscript (ab) are chiral expansions that have been quoted in Eqs. (46) and (50). It is pertinent to point out that some of the chiral supervariables, on the r.h.s. of (62), are actually ordinary variables [cf. Eq. (46)] because they are anti-BRST invariant (e.g. s abβ = s abb = s ab p µ = 0) variables. In view of the mapping: s ab ↔ ∂ θ , 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: where, as pointed out earlier, we have to substitute the chiral super expansion (46) and (50) into the r.h.s. of (62) and, then only, we have to operate ∂ θ . The above equation (63) is nothing but the proof for the off-shell nilpotency (Q ab ) 2 = 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 Eq. (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 charge Q  30)) can be also generalized onto the (1, 1)-dimensional chiral super sub-manifold as 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. Eq. (46)]. We can now operate by the Grassmannian derivative ∂ θ on (64) to produce: (ab) }.
In the above, we have utilize the mapping ∂ θ ⇔ s ab to express the l.h.s. in the ordinary space. The expression s ab Q is nothing but the absolute anticommutativity (i.e. {Q (2) b , Q (2) ab }) of the BRST charge with the anti-BRST charge Q (2) ab . It is crystal clear that the absolute anticommutativity property (i.e. {Q (2) b , Q ab } = 0) is satisfied if and only if we imposes the condition (b +b + 2 ββ = 0). In other words, we have been able to derive the CF-type restriction: b +b + 2 ββ = 0 by exploiting the theoretical tricks and techniques of ASCA to BRST formalism.
Ultimately, we concentrate to capture the absolute anticommutativity of the anti-BRST charge Q (2) ab with the BRST charge Q (2) b within the framework of ACSA to BRST formalism. Toward this goal in mind, we generalize the anti-BRST charge Q (2) ab [cf. Eq. (33)] onto the (1, 1)-dimensional ant-chiral super sub-manifold as where the anti-chiral supervariable on the r.h.s. are nothing but the super expansions (40) and (44)  In view of our understanding that we have: s ab ⇔ ∂θ [10][11][12], we can operate a derivation w.r.t. the Grassmannian variableθ on theQ (2) ab (τ,θ) as: ∂ ∂θQ (2) ab (τ,θ) ≡ dθQ (2) ab (τ,θ) = 0 In other words [cf. Eq. (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 (67) demonstrates that the r.h.s. of ∂θQ (2) b (τ,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 + 2 ββ = 0 (on our 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: s b ↔ ∂θ, s ab ↔ ∂ θ [10][11][12] play 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 Q

Conclusions
In our present endeavour, we have done 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. Sec. 2). 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 [30,31]. We have obtained the secondary-constraints from the equivalent Lagrangians and corresponding canonical Hamiltonians of our 1D system of spinning relativistic particle (cf. Sec. 2 and Appendix A).
We have elevated the combined classical gauge and supergauge symmetry transformations [cf. Eq. (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 BRST approach) is the existence of the CF-type restriction(s). We have demonstrated the existence of a single CFtype 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. Eqs. (22), (23)]. In other words, the absolute anticommutativity of the (anti-)BRST symmetry transformations and existence of the coupled (but equivalent) Lagrangians owe their origins to the CF-type restriction which defines a sub-manifold in the quantum Hilbert space of variables that is satisfied by the equation: b +b + 2 ββ = 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][21][22][23][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 sub-manifolds of the general (full) supermanifold. Whereas the (anti-)chiral super sub-manifolds 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. Eq. (14)], invariance and equivalence of the coupled (but equivalent) Lagrangians [cf. Eqs. (23)- (25)] and absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges [cf. Eqs. (35), (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. Secs. 6, 7).
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. We plan to extend the richness of our theoretical analysis to the realm of interesting systems of quantum field theory as well as diffeomorphism invariant theories of gravitation and (super)strings. We are currently 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 in our future publications [38].
which leads to the derivation of the secondary constraint as (p µ ψ µ − m ψ 5 ) ≈ 0. The same result is also obtained from the canonical Hamiltonian H (0) c as we note that the Heisenberg EOMs for the time derivative on the conjugate momenta operators are: Hence, we have derived the secondary constraints (p 2 − m 2 ) ≈ 0 and (p µ ψ µ − m ψ 5 ) ≈ 0 from the Lagrangian L 0 with square-root and the corresponding canonical Hamiltonian H (0) c (which is nothing but the secondary constraint on our theory). Now the stage is set to derive the primary and secondary constraints from the Lagrangians L f and L s [cf. Eq. (1)] and corresponding canonical Hamiltonian H c [cf. Eq. (11)]. It is evident that the expressions for the canonical conjugate momenta w.r.t. the variables e and χ, from L f and L s , are Hence, we have primary constraints Π e ≈ 0 and Π χ ≈ 0 (i.e. weekly zero). As per the Dirac prescription (see, e.g. [36,37]), we are allowed to take a single-order time derivative on these primary constraints. Using the EL-EOMs w.r.t. e and χ variables, we find that: It is clear, from the above, that we have already derived the secondary constraints p 2 −m 2 ≈ 0 and p µ ψ µ − m ψ 5 ≈ 0. As far as the Lagrangian L s is concerned, we have the expression for the canonical conjugate momenta (p µ ) w.r.t. the coordinates (x µ ) as: The EL-EOM w.r.t. e (from the second-order Lagrangian L s ) yields d dτ which produces the secondary constraint p 2 − m 2 ≈ 0. Similarly, the EL-EOM w.r.t. χ is d dτ Hence, we have derived both the secondary constraints p 2 − m 2 ≈ 0 and p µ ψ µ − m ψ 5 ≈ 0 from the first-and second-order Lagrangians L f and L s .
Against the backdrop of the existence of the primary constraints Π e ≈ 0 and Π χ ≈ 0, we derive the secondary constraints from the canonical Hamiltonian H c [ cf. Eq. (11)] as follows (with the natural units = c = 1), namely; where we have used the canonical commutator [e, Π e ] = i and canonical anticommutator as: {χ, Π χ } = i in the natural units = c = 1. We end this Appendix with the remark that we have derived the secondary constraints p 2 − m 2 ≈ 0 and p µ ψ µ − m ψ 5 ≈ 0 from all the three equivalent Lagrangian (1) as well as from the canonical Hamiltonian (11). As a passing remark, we note that the whole dynamics of our theory is governed by the secondary constrains because a close look at H c [cf. Eq. (11)] demonstrates that the Hamiltonian is a linear combination of the constraints p 2 − m 2 ≈ 0 and p µ ψ µ − m ψ 5 ≈ 0. Last but not least, we note that the constraint p µ ψ µ − m ψ 5 ≈ 0 is the square-root of the mass-shell condition p 2 − m 2 ≈ 0 because we observe the following: It is straightforward to note, from the first-order and second-order Lagrangians, that we have the following explicit expressions, namely; as the canonical conjugate momenta w.r.t. the fermionic variables ψ µ and ψ 5 . As a consequence, we have the following canonical anticommutators: (A.14) Using the above anticommutators (A.14), we find that the r.h.s. of (A.12) is nothing but the mass-shell condition: p 2 − m 2 = 0. This observation establishes the fact that the two secondary constraints (i.e. p 2 − m 2 ≈ 0, p µ ψ µ − m ψ 5 ≈ 0) of the theory are inter-related.

Appendix B: On the Derivation of Conserved Noether Charges
The central goal of our present Appendix is to derive the (anti-)BRST charges Q  where we have taken into account the transformation property of the Lagrangian [cf. Eq. (21)] under the anti-BRST symmetry transformation (s ab ) that is quoted in Eq. (12). Furthermore, we have utilized our knowledge of the transformations: s abβ = 0 and s ab γ = 0.
In addition, we have also noted that ∂Lb/∂χ = 0 and ∂Lb/∂ḃ = 0. We would like to lay emphasis on the fact that we have taken into account the convention of the left-derivative w.r.t. the fermionic variables (ψ µ , ψ 5 , c,c). Hence, the expression for the anti-BRST charge in Eq. (B.1) is correct according to our adopted convention of the derivatives. The substitutions of the transformations (12) and the proper expressions for the derivatives into (B.1) lead to the exact expression for the anti-BRST charge (Q (1) ab ) that has been quoted in Eq. (26). The conservation law (i.eQ (1) ab = 0) can be proven by taking into account the following EL-EOMs that emerge out from the Lagrangian Lb, namely; It would be noted that we have not incorporated, in the above, the EL-EOMs w.r.t. the variables e and χ because these have been quoted in Eq. (32). At this stage we focus on the derivation of the Noether conserved charge Q Eq. (27)]. Applying the basic concept behind the Noether theorem, we note that we have the following expression for the BRST charge, namely; We would like to point out that, in the above, we have not incorporated Eq. (29) which are the EL-EOMs w.r.t. the variables e and χ that emerge out from L b . We end this Appendix with the remarks that we have derived the Noether conserved (anti-)BRST charges Q (1) (a)b [cf. Eqs. (26), (27)] which are only on-shell nilpotent of order two. To accomplish the off-shell nilpotency (without any use of EL-EOMs and/or CF-type restriction), it is essential for us to use the EL-EOMs w.r.t. e and χ [cf. Eqs. (29), (32)] to recast the Noether conserved (anti-)BRST charges (Q Eqs. (30), (33)].

Appendix C: On the Step-by-Step Derivation of the BRST Symmetry
Transformations by Using the ACSA to BRST Formalism In our present Appendix, we derive the BRST symmetry transformations (13) by exploiting the theoretical potential and power of ACSA to BRST formalism in a systematic fashion.
In other words, we obtain the relationships in (43) from the BRST invariant restrictions on the anti-chiral supervariables in (42). In this context, we note that: In other words, we obtain: f 3 (τ ) = k γ(τ ) where k is a numerical constant. Now we focus on the following restriction on the combination of the anti-chiral supervariables, namely; which is precisely the generalization of the BRST invariant quantity: s b (bβ + γc) = 0. The substitution of the expression forβ(τ,θ) =β(τ ) +θ [k γ(τ )] in the above and use of Eq.
(40) yield the following relationship: It is clear, from the above, that we obtain the following: From this relationship, it is clear that if we wish to have s bc = i b in our theory, the constant k would turn out to be: k = i. It would be recalled that s bc = i b is a standard transformation w.r.t. the BRST symmetry (s b ) within the framework of BRST formalism. Thus, ultimately, we have derived the following super expansions in terms of (13), namely; Thus, we note that coefficients ofθ are nothing but the BRST symmetry transformations: s b c = i b, s bβ = i γ. Now we concentrate on the BRST invariant quantity: s b (β 2β + c γ) = 0. This can be generalized onto the (1, 1)-dimensional anti-chiral super sub-manifold as: The substitutions ofβ (b) (τ,θ) = β(τ ), Γ b (τ,θ) = γ(τ ), F (τ,θ) = c(τ ) +θ b 1 (τ ) and (C.5) lead to the following explicit expressions for the l.h.s. and r.h.s., namely; The straightforward algebra yields: b 1 (τ ) = − i β 2 (τ ). Hence, we have the following super expansion for the anti-chiral supervariable F (τ,θ), namely; We note that the coefficient ofθ is nothing but s b c = − i β 2 . It is now straightforward to note that, from the BRST invariant quantities: s b (ċ + 2 β χ) = 0 and s b (b + 2 ββ) = 0 and their generalizations onto the (1, 1)-dimensional anti-chiral super sub-manifold [cf. Eq. (42)] (with the inputs from (C.5) and (C.8)), lead to the following: The above values of the secondary variables yield the following super expansions: are, precisely speaking, the quantum gauge invariant restrictions which lead to provide the correct relationships among the secondary variables and basic as well as auxiliary variables of the (anti-)BRST invariant theory. Second, the determination of the secondary variables [that lead to the derivation of the anti-BRST symmetry transformations (12) as the coefficient of θ in the expansion (50)] has been carried out on exactly similar kind of procedure as we have obtained the relationship (43) from (42) in the case of determination of the BRST symmetry transformations (13) as the coefficient ofθ in the expansions (44). Finally, the ACSA to BRST formalism is a simple but beautiful symmetry-based theoretical technique which is applicable to all kinds of physical systems of theoretical interest.
of the (anti-)BRST invariant [i.e. s (a)b (b+b+2 ββ) = 0] CF-type restriction: b+b+2 ββ = 0 in the ordinary space as well as in the superspace by exploiting the theoretical potential and power of ACSA to BRST formalism. In this context, we note that the off-shell nilpotent ([Q where the (anti-)BRST symmetry transformations have been quoted in Eqs. (13) and (12), respectovely. In other words, it is obvious that the BRST charge Q (a)b is deeply and intimately connected with the off-shell nilpotency of the (anti-)BRST symmetry transformations s (a)b in the ordinary space. It is worthwhile to point out that these connection can not be drawn from our earlier proof of the off-shell nilpotency [cf. Eqs. (38) and (36)].