# Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

Authors: A. K. Rao; A. Tripathi; R. P. Malik
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose spacex and time t variables are a function of an evolution parameter τ. The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameter τ. We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized by τ) is generalized onto a 1,2-dimensional supermanifold which is characterized by the superspace coordinates ZM=τ,θ,θ¯ where a pair of the Grassmannian variables satisfy the fermionic relationships: θ2=θ¯2=0, θθ¯+θ¯θ=0, and τ is the bosonic evolution parameter. In the context of ACSA, we take into account only the 1,1-dimensional (anti)chiral super submanifolds of the general1,2-dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too.


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## Body

## 1. Introduction

During the last few years, there has been an upsurge of interest in the study of diffeomorphism invariant theories because one of the key and decisive features of the gravitational and (super)string theories is the observation that they respect theclassical diffeomorphism symmetry transformations. The latter symmetry transformations can be exploited within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism [1–4] where the classical diffeomorphism symmetry transformation is elevated to the quantum (anti-)BRST symmetry transformations. In fact, it is the key feature of the BRST formalism that the classical diffeomorphism transformation parameter is traded with the fermionic (anti)ghost fields/variables at the quantum level. In other words, the (anti-)BRST transformations are of the supersymmetric (SUSY) kind under which the bosonic type of fields/variables transforms to the fermionic type of fields/variables and vice versa. Two of the key properties of the (anti-)BRST transformations are the on-shell/off-shell nilpotency and absolute anticommutativity. These key properties encompass in their folds the fermionic as well as independent natures of the quantum BRST and anti-BRST symmetries at the level of physical interpretation. The nilpotency property (i.e., fermionic nature) of the (anti-)BRST symmetries (and their corresponding charges) is also connected with some aspects of the cohomological properties of differential geometry and a few decisive features of supersymmetry.The BRST formalism has been exploited in the covariant canonicalquantization of the gauge and diffeomorphism invariant theories in the past. At the classical level, the gauge theories are characterized by the existence of the first-class constraints [5, 6] on them. This fundamental feature is translated, at the quantum level, into the language of the existence of the Curci-Ferrari- (CF-) type restriction(s) when the classical theory is quantized by exploiting the theoretical richness of BRST formalism. Hence, the existence of the CF-type restriction(s) is the key signature of a BRST-quantized version of the gauge and/or diffeomorphism invariant theory. The CF-type restrictions are (i) deeply connected with the geometrical objects called gerbes [7, 8], (ii) responsible for the absolute anticommutativity of the quantum (anti-)BRST transformations, and (iii) the root cause behind the existence of the coupled (but equivalent) Lagrangians/Lagrangian densities for the (anti-)BRST invariant quantum theories (corresponding to the classical gauge/diffeomorphism invariant theories). The Abelian 1-form gauge theory is an exception where we obtain a unique (anti-)BRST invariant Lagrangian density because the CF-type restriction is trivial in this case. However, this restriction turns out to be the limiting case of the non-Abelian 1-form gauge theory where the nontrivial CF condition exists [9].It is the supervariable/superfield approaches [10–21] to BRST formalism that provide the geometrical basis for the off-shell nilpotency and absolute anticommutativity of the (anti-)BRST symmetries as well as the existence of the CF-type restrictions for a BRST-quantized gauge/diffeomorphism invariant theory. In the usual superfield approaches (USFA), it is the horizontality condition (HC) that plays a decisive role as it leads to (i) the derivation of the (anti-)BRST symmetry transformations for only the gauge and (anti)ghost fields as well as (ii) the derivation of the CF-type restriction(s). The augmented version of the superfield approach (AVSA) is an extension of USFA where, in addition to the HC, the gauge (i.e., (anti-)BRST) invariant restrictions are exploited together which lead to the derivation of the (anti-)BRST symmetry transformations for the gauge, (anti)ghost, and matter fields together in an interacting gauge theory. It has been a challenging problem to incorporate the diffeomorphism transformation within the ambit of the superfield approach to gauge theories (see, e.g., [14–16]) so that one can discuss the gravitational and (super)string theories within the framework of USFA/AVSA. In this direction, a breakthrough has recently been made by Bonora [22] where the superfield approach has been applied to derive the proper (anti-)BRST transformations as well as the CF-type restriction for the D-dimensional diffeomorphism invariant theory. This approach has been christened by us as the modified Bonora-Tonin (BT) superfield approach (MBTSA) to BRST formalism. In a recent couple of papers [23, 24], we have applied the theoretical beauty of the MBTSA as well as ACSA (i.e., (anti)chiral superfield/supervariable approach) to BRST formalism [25–29] in the context of the 1D diffeomorphism (i.e., reparameterization) invariant theories of the non-SUSY (i.e., scalar) as well as SUSY (i.e., spinning) relativistic free particles.The central theme of our present investigation is to concentrate on the reparameterization (i.e., 1D diffeomorphism) invariant theory of a massive nonsupersymmetric (NSUSY) and nonrelativistic (NR) free particle where thestandard NR Lagrangian L0tx,ẋ=1/2mẋ2 (with ẋ=dx/dt) is rendered reparameterization invariant by treating the “time” variable on a par with the x variable [30] parameterized by an evolution parameter τ such that the new Lagrangian L0τx,ẋ,t,ṫ=mẋ2/2ṫ, where ẋ=dx/dτ and ṫ=dt/dτ. The latter Lagrangian respects the reparameterization symmetry [31, 32], and it has been discussed in different theoretical settings where the noncommutativity of the spacetime appears by the symmetry considerations, constraint analysis, redefinitions of variables, etc. This reparameterization invariant model of the free particle (ṗx=ṗt=0) has been discussed by us within the frameworks of BRST formalism as well as quantum groups [32]. However, in the BRST analysis, we have exploited the gauge symmetry of this NSUSY and NR system [32] without discussing anything about the reparameterization transformations. In our present investigation, we have applied the beautiful blend of theoretical ideas from MBTSA and ACSA to derive the proper (anti-)BRST symmetries and CF-type restriction for this NR system. This model is interesting in its own right as it is a NR system (unlike our earlier discussions [23, 24] on the relativistic systems), and “time” itself has been treated as a physical observable that depends on the evolution parameter τ. The latter property of our present NR system is important as “time” has also been treated as an observable in quantum mechanics instead of an evolution parameter (see, e.g., [30]).The following motivating factors have been at the heart of our curiosity in pursuing our present endeavor. First, so far, we have been able to apply the beautiful blend of theoretical ideas behind MBTSA and ACSA to BRST formalism in the cases of reparameterization invariant systems of the (i) relativistic nonsupersymmetric (NSUSY) scalar free particle and (ii) supersymmetric (SUSY) (i.e., spinning) relativistic free particle. Thus, it has been a challenging problem for us to apply thesame theoretical ideas to discuss the NSUSY and nonrelativistic (NR) system of a reparameterization invariant free particle. We have accomplished this goal in our present investigation. Second, we have shown the universality of the CF-type restriction in the cases of reparameterization (i.e., 1D diffeomorphism) invariant NSUSY as well as SUSY systems of the free relativistic particles. Thus, we have been motivated to see the existence of the same CF-type restriction in our present case of reparameterization invariant systems of NSUSY and NR free particles. We have been able to demonstrate that it is the same CF-type restriction that exists in the BRST approach to our present NSUSY and NR system. Third, we have found out that the gauge-fixing and Faddeev-Popov (FP) ghost terms for the systems of non-SUSY (i.e., scalar) and SUSY (i.e., spinning) relativistic particles are the same within the ambit of BRST formalism. Thus, we have been curious to find out the gauge-fixing and FP ghost terms for our present non-SUSY and NR system. It is surprising that the above terms are the same for our present system, too. Finally, our present work is another modest initial step towards our main goal of applying the theoretical potential of MBTSA and ACSA to the physical four 3+1-dimensional (4D) diffeomorphism invariant gravitational and (super)string theories in the higher dimensions (i.e., D>4) of spacetime.The contents of our present endeavor are organized as follows. In Section2, we recapitulate the bare essentials of the Lagrangian formulation of our reparameterization invariant nonrelativistic system and discuss the BRST quantization of this model by exploiting its infinitesimal gauge symmetry transformations. Section 3 is devoted to the application of MBTSA for the derivation of (i) the quantum (anti-)BRST symmetry transformations for the phase space variables and (ii) the underlying (anti-)BRST invariant Curci-Ferrari- (CF-) type restriction (corresponding to the classical infinitesimal reparameterization symmetry transformations). The theoretical content of Section 4 is related to the derivation of the full set of (anti-)BRST symmetry transformations by requiring the off-shell nilpotency and absolute anticommutativity properties. We also show the existence of the CF-type restriction and deduce the coupled (anti-)BRST invariant Lagrangians for our theory. In Section 5, we derive the (anti-)BRST symmetry transformations for the other variables (i.e., different from the phase variables) within the purview of ACSA. Section 6 deals with the proof of the equivalence of the coupled Lagrangians within the framework of ACSA to BRST formalism. In Section 7, we prove the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges in the ordinary space and superspace (by exploiting the theoretical richness of ACSA to BRST formalism). Finally, in Section 8, we discuss our key results and point out a few future directions for further investigation(s).In AppendicesA, B, and C, we perform some explicit computations that supplement as well as corroborate the key claims that have been made in the main body of our text.


### 1.1. Convention and Notations

We adopt the convention of the left derivative w.r.t.all the fermionic variables of our theory. In the whole body of the text, we denote the fermionic (anti-)BRST transformations by the symbols sab and corresponding conserved and nilpotent charges carry the notations Qab and QB¯B in different contexts. The general 1,2-dimensional supermanifold is parameterized by ZM=τ,θ,θ¯, and its chiral and antichiral super submanifolds are characterized by τ,θ and τ,θ¯, respectively, where the bosonic coordinate is represented by the evolution parameter τ and the Grassmannian variables θ,θ¯ obey the fermionic relationships: θ2=0, θ¯2=0, and θθ¯+θ¯θ=0.


## 1.1. Convention and Notations

We adopt the convention of the left derivative w.r.t.all the fermionic variables of our theory. In the whole body of the text, we denote the fermionic (anti-)BRST transformations by the symbols sab and corresponding conserved and nilpotent charges carry the notations Qab and QB¯B in different contexts. The general 1,2-dimensional supermanifold is parameterized by ZM=τ,θ,θ¯, and its chiral and antichiral super submanifolds are characterized by τ,θ and τ,θ¯, respectively, where the bosonic coordinate is represented by the evolution parameter τ and the Grassmannian variables θ,θ¯ obey the fermionic relationships: θ2=0, θ¯2=0, and θθ¯+θ¯θ=0.


## 2. Preliminaries: Lagrangian Formulation

Our present section is divided intotwo parts. In Section 2.1, we discuss the classical infinitesimal reparameterization and gauge symmetry transformations. Section 2.2 deals with the BRST quantization of our system by exploiting the classical gauge symmetry transformations (which are also infinitesimal and continuous).


### 2.1. Classical Infinitesimal Symmetries

We begin with thethree equivalent reparameterization invariant Lagrangians for the free nonrelativistic and non-SUSY particle as (see, e.g., [32] for details)
(1)L0x,ẋ,t,ṫ=mẋ22ṫ,Lfx,ẋ,t,ṫ,px,pt=pxẋ+ptṫ−12Epx2+2mpt,Lsx,ẋ,t,ṫ=ẋ22E+mẋ22ṫEmṫ−1,where the trajectory of the free nonrelativistic particle is embedded in a 2D configuration space characterized by the coordinates xτ,tτ and the parameter τ specifies the trajectory of the particle as an evolution parameter. The momenta variables px,pt are defined by px=∂L/∂ẋ and pt=∂L/∂ṫ, where ẋ=dx/dτ and ṫ=dt/dτ are the generalized “velocities” w.r.t. the coordinates xτ,tτ and L stands for any of the three Lagrangians of Equation (1). It is self-evident that the 4D phase space, corresponding to the 2D configuration space, is characterized by xτ,tτ,pxτ,ptτ. In the above Equation (1), the mass of the nonrelativistic particle is denoted by m and Eτ is the Lagrange multiplier variable that incorporates the constraint px2+2mpt≈0 in the Lagrangians Lf and Ls. It is straightforward to note that L0 and Ls contain variables (and their first-order derivative) in the denominator butLf (i.e., the first-order Lagrangian) does not incorporate any variable (and/or its derivative) in its denominator. Furthermore, the starting Lagrangian (For a free massive NR particle, the standard action integral is St=∫−∞+∞dt1/2mẋ2, where ẋ=dx/dt and time “t” is the evolution parameter. This action has no reparameterization invariance. If the evolution parameter is τ, then the action integral is Sτ=m/2∫−∞+∞dτdt/dτdx/dτdτ/dtdx/dτdτ/dt which leads [31, 32] to the final action integral as Sτ=m/2∫−∞+∞dτdx/dτ2dτ/dt≡∫−∞+∞dτL0. Hence, the starting Lagrangian becomes L0x,ẋ,t,ṫ=mẋ2/2ṫ, where ẋ=dx/dτ and ṫ=dt/dτ.) L0 does not permit the masslessm=0 limit but the massless m=0 limits are well defined for Lf and the second-order Lagrangian Ls. We would like to stress that the Lagrange multiplier variable behaves like the “gauge” variable due to its transformation property.It will be worthwhile to dwell a bit on the derivation of the top entry i.e.,L0x,t,ẋ,t in Equation (1). For a free massive NR particle, the standard action integral is St=∫−∞+∞dt1/2mẋ2, where ẋ=dx/dt and time “t” is the evolution parameter. This action has no repa-rameterization invariance. If the evolution parameter is τ, then the action integral is Sτ=m/2∫−∞+∞dτdt/dτdx/dτdτ/dtdx/dτdτ/dt which leads [31, 32] to the fi-nal action integral as Sτ=m/2∫−∞+∞dτdx/dτ2dτ/dt≡∫−∞+∞dτL0. Hence, the starting Lagrangian becomes L0x,ẋ,t,ṫ=mẋ2/2ṫ where ẋ=dx/dτ and ṫ=dt/dτ.For our further discussions, we shall concentrate on the first-order LagrangianLf because it has the maximum number of variables (i.e., x, px, t, pt, and E) and allows the massless limit and there are no variables (and/or their first-order derivative w.r.t. τ) in its denominator. It is straightforward to check that under the following infinitesimal and continuous 1D diffeomorphism (i.e., reparameterization) symmetry transformations δr, namely,
(2)δrx=εẋ,δrpx=εṗx,δrt=εṫ,δrpt=εṗt,δrE=ddτεE,δrLf=ddτεLf,the action integral S=∫−∞+∞dτLf remains invariant (i.e., δrS=0) for the physically well-defined variables in Lf and the infinitesimal diffeomorphism transformation parameter ετ in τ⟶τ′=gτ=τ−ετ, where gτ is a physically well-defined function of τ such that it is finite at τ=0 and vanishes off at τ=±∞. In fact, the infinitesimal and continuous reparameterization symmetry transformation δr is defined as δrϕτ=ϕ′τ−ϕτ for the generic variable ϕτ=xτ,pxτ,tτ,ptτ,Eτ of our present theory.The above infinitesimal and continuous reparameterization symmetry transformations (2) encompass in their folds the gauge symmetry transformations δg which are generated (see, e.g., [32] for details) by the first-class constraints ΠE≈0,px2+2mpt≈0, where ΠE is the canonical conjugate momentum corresponding to the variable Eτ. Using the following Euler-Lagrange equations of motion (EL-EOMs) from Lf, namely,
(3)ṗx=0,ṗt=0,ẋ=Epx,ṫ=Em,and identifying the transformation parameters ετEτ=ξτ, we obtain the infinitesimal and continuous gauge symmetry transformations δg, from the infinitesimal and continuous reparameterization symmetry transformations (2), as follows:
(4)δgx=ξpx,δgt=ξm,δgpx=0,δgpt=0,δgE=ξ̇.It is elementary now to check that the first-order LagrangianLf transforms to a total derivative under the infinitesimal and continuous gauge transformations δg, namely,
(5)δgLf=ddτξ2px2,thereby rendering the action integral S=∫−∞+∞dτLf invariant (i.e., δgS=0) under the infinitesimal and continuous gauge symmetry transformations δg.We end this section with the following decisive comments. First, the 1D diffeomorphism (i.e., reparameterization) transformations (2) are more general than the infinitesimal and continuous gauge symmetry transformations (4). Second, the Lagrange multiplier variable Eτ behaves like a “gauge” variable due to its transformation δgE=ξ̇ in (4). Third, all the three Lagrangians in (1) are equivalent and all of them respect the infinitesimal and continuous gauge and reparameterization symmetry transformations [32]. Fourth, all the Lagrangians describe the free motion ṗx=0,ṗt=0 of the NR particle. Hence, our system is a nonrelativistic free ṗx=ṗt=0 particle. Fifth, the first-order Lagrangian Lf is theoretically more interesting to handle because, as pointed out earlier, it incorporates the maximum number of variables. Finally, we can exploit the reparameterization and gauge symmetry transformations (2) and (4) for the BRST quantization. Following the usual BRST prescription, we note that the (anti-)BRST symmetry transformations, corresponding to the classical reparameterization symmetry transformations (2), are
(6)sabx=C¯ẋ,sabpx=C¯ṗx,sabt=C¯ṫ,sabpt=C¯ṗt,sabE=ddτC¯E,sbx=Cẋ,sbpx=Cṗx,sbt=Cṫ,sbpt=Cṗt,sbE=ddτCE,where C¯C are the fermionic C2=C¯2=0,CC¯+C¯C=0 (anti)ghost variables corresponding to the classical infinitesimal diffeomorphism transformation parameter ετ (cf. Equation (2)). In an exactly similar fashion, the (anti-)BRST symmetry transformations, corresponding to the classical gauge symmetry transformations (4), are
(7)sabx=c¯px,sabpx=0,sabt=c¯m,sabpt=0,sabE=c¯̇,sbx=cpx,sabpx=0,sbt=cm,sbpt=0,sbE=ċ,where the fermionic c2=c¯2=0,cc¯+c¯c=0 variables c¯c are the (anti)ghost variables corresponding to the classical gauge symmetry transformation parameter ξτ of Equation (4). In addition to the (anti-)BRST symmetry transformations in (6) and (7), we have the following standard (anti-)BRST transformations:
(8)sabC=iB¯,sabB¯=0,sbC¯=iB,sbB=0,sabc=ib¯,sabb¯=0,sbc¯=ib,sbb=0,where the pairs B,B¯andb,b¯ are the Nakanishi-Lautrup auxiliary variables in the context of the BRST quantization of our reparameterization and gauge invariant system by exploiting the classical reparameterization and gauge transformations, respectively.


### 2.2. Quantum Nilpotent (Anti-)BRST Symmetries Corresponding to the Classical Gauge Symmetry Transformations

We have listed the quantum (anti-)BRST symmetries corresponding to theclassical gauge symmetry transformations (4) in our Equations (7) and (8). It is elementary to check that these quantum symmetries are off-shell nilpotent sab2=0 of order two. The requirement of the absolute anticommutativity sbsab+sabsb=0 leads to the restriction b+b¯=0⇒b¯=−b. As a consequence, we have the full set of (anti-)BRST symmetry transformations (corresponding to the classical gauge symmetry transformations (4)) as follows:
(9)sabx=c¯px,sabpx=0,sabt=c¯m,sabpt=0,sabE=c¯̇,sabc¯=0,sabc=−ib,sabb=0,sbx=cpx,sbpx=0,sbt=cm,sbpt=0,sbE=ċ,sbc=0,sbc¯=ib,sbb=0.It is straightforward to check that the above (anti-)BRST symmetry transformations are off-shell nilpotentsab2=0 and absolutely anticommuting sbsab+sabsb=0 in nature. The (anti-)BRST invariant Lagrangian Lb (which is the generalization of the classicalLf to its quantum level) can be written as (The structure of gauge-fixing and FP ghost terms is exactly like the Abelian 1-form A1=dxμAμ gauge theory where we have the BRST invariant Lagrangian density: Lb=−1/4FμνFμν+sb−ic¯∂μAμ+b/2≡−1/4FμνFμν+sab−ic∂μAμ+b/2≡−1/4FμνFμν+sbsabi/2AμAμ−1/2c¯c. Here, Aμ is the vector potential, Fμν=∂μAν−∂νAμ is the field strength tensor, and the rest of the symbols are the same as in Equations (10) and (11). Note that the 2-form F2=dA1=1/2dxμ∧dxνFμν defines the field strength tensor Fμν (where d=dxμ∂μ in F2=dA1 stands for the exterior derivative of the differential geometry).):
(10)Lb=Lf+sb−ic¯Ė+b2≡Lf+sabicĖ+b2≡Lf+sbsabiE22−c¯c2≡Lf−sabsbiE22−c¯c2.In other words, we have expressed the gauge-fixing and Faddeev-Popov (FP) ghost terms inthree different ways which, ultimately, lead to the following expression for Lb, namely,
(11)Lb=Lf+bĖ+b22−ic¯̇ċ≡pxẋ+ptṫ−12Epx2+2mpt+bĖ+b22−ic¯̇ċ.It should be noted that we have dropped the total derivative terms in obtainingLb from (10). The above equation demonstrates that we have obtained a unique (anti-)BRST invariant Lagrangian. This has happened because the CF-type restriction is trivial (i.e., b+b¯=0) in our simple case of the NR system. We can explicitly check that
(12)sbLb=ddτc2px2+bċ,sabLb=ddτc¯2px2+bc¯̇,which lead to the derivation of the conserved (anti-)BRST charges Qab as follows:
(13)Qab=bc¯̇+12c¯px2+2mpt≡bc¯̇−ḃc¯,Qb=bċ+12cpx2+2mpt≡bċ−ḃc.In the last step, we have usedḃ=−1/2px2+2mpt which emerges out as the EL-EOM from Lb w.r.t. the Lagrange multiplier variable Eτ. The structure of gauge-fixing and FP ghost terms in Equation (10) is exactly like the Abelian 1-form A1=dxμAμ gauge theory where we have the BRST invariant Lagrangian density: Lb=−1/4FμvFμv+sb−ic¯∂μAμ+b/2≡−1/4FμvFμv+sab−ic∂μAμ+b/2≡−1/4FμvFμv+sbsab1/2AμAμ−1/2c¯c. Here, Aμ is the vector potential, Fμv=∂μAv−∂vAμ is the field strength tensor, and the rest of the symbols are the same as in Equation (10) and (11). Note that the 2-form F2=dA1=1/2dxμΛdxvFμv defines the field strength tensor Fμv(where d=dxμ∂μ in F2=dA1 which stands for the exterior derivative of the differential geometry).We close this section with a few crucial and decisive remarks. First, we can check that the (anti-)BRST charges are conservedQ̇ab=0 by using the EL-EOMs. Second, the (anti-)BRST charges Qab are off-shell nilpotent Qab2=0 of order two due to the direct observations that sbQb=sbbċ−ḃc=0 and sabQab=sabbc¯̇−ḃc¯=0 which encode in their folds sbQb=−iQb,Qb=0⇒Qb2=0andsabQab=−iQab,Qab=0⇒Qab2=0. Third, the above nilpotency is also encoded in Qb=sbbE+ic¯̇c implying that sbQb=0 due to sb2=0and we also point out that sabQab=0 due to the nilpotency sab2=0 of sab because Qab=sabbE+ic¯ċ. Fourth, we observe that sabQb=iQb,Qab≡−ibḃ+iḃb=0 and sbQab=−iQab,Qb=ibḃ−ibḃ=0 which explicitly lead to the conclusion that the off-shell nilpotent charges Qab are also absolutely anticommuting QbQab+QabQb=0 in nature. Fifth, the above observation of the absolute anticommutativity can be also expressed in terms of the nilpotency property because we observe that Qb=sab−iċc and Qab=sbic¯̇c¯ which imply that sabQb=−iQb,Qab=0 and sbQab=−iQab,Qb=0 (due to the off-shell nilpotency sab2=0 of the anti-BRST as well as the off-shell nilpotency sb2=0 of the BRST symmetry transformations). Sixth, it can be seen that the physical space (i.e., phys>) in the total Hilbert space of states is defined by Qbphys>=0 which implies that bphys>≡ΠEphys>=0 and ḃphys>≡px2+2mptphys>=0. In other words, the Dirac quantization conditions (with the first-class constraints ΠE≈0,px2+2mpt≈0) are beautifully satisfied. Finally, the physicality criterion Qbphys>=0 implies that the two physical states phys′> and phys> belong to the same cohomological class w.r.t. the nilpotent BRST charge Qb if they differ by a BRST exact state (i.e., phys′>=phys>+Qbχ> for nonnull χ>).


## 2.1. Classical Infinitesimal Symmetries

We begin with thethree equivalent reparameterization invariant Lagrangians for the free nonrelativistic and non-SUSY particle as (see, e.g., [32] for details)
(1)L0x,ẋ,t,ṫ=mẋ22ṫ,Lfx,ẋ,t,ṫ,px,pt=pxẋ+ptṫ−12Epx2+2mpt,Lsx,ẋ,t,ṫ=ẋ22E+mẋ22ṫEmṫ−1,where the trajectory of the free nonrelativistic particle is embedded in a 2D configuration space characterized by the coordinates xτ,tτ and the parameter τ specifies the trajectory of the particle as an evolution parameter. The momenta variables px,pt are defined by px=∂L/∂ẋ and pt=∂L/∂ṫ, where ẋ=dx/dτ and ṫ=dt/dτ are the generalized “velocities” w.r.t. the coordinates xτ,tτ and L stands for any of the three Lagrangians of Equation (1). It is self-evident that the 4D phase space, corresponding to the 2D configuration space, is characterized by xτ,tτ,pxτ,ptτ. In the above Equation (1), the mass of the nonrelativistic particle is denoted by m and Eτ is the Lagrange multiplier variable that incorporates the constraint px2+2mpt≈0 in the Lagrangians Lf and Ls. It is straightforward to note that L0 and Ls contain variables (and their first-order derivative) in the denominator butLf (i.e., the first-order Lagrangian) does not incorporate any variable (and/or its derivative) in its denominator. Furthermore, the starting Lagrangian (For a free massive NR particle, the standard action integral is St=∫−∞+∞dt1/2mẋ2, where ẋ=dx/dt and time “t” is the evolution parameter. This action has no reparameterization invariance. If the evolution parameter is τ, then the action integral is Sτ=m/2∫−∞+∞dτdt/dτdx/dτdτ/dtdx/dτdτ/dt which leads [31, 32] to the final action integral as Sτ=m/2∫−∞+∞dτdx/dτ2dτ/dt≡∫−∞+∞dτL0. Hence, the starting Lagrangian becomes L0x,ẋ,t,ṫ=mẋ2/2ṫ, where ẋ=dx/dτ and ṫ=dt/dτ.) L0 does not permit the masslessm=0 limit but the massless m=0 limits are well defined for Lf and the second-order Lagrangian Ls. We would like to stress that the Lagrange multiplier variable behaves like the “gauge” variable due to its transformation property.It will be worthwhile to dwell a bit on the derivation of the top entry i.e.,L0x,t,ẋ,t in Equation (1). For a free massive NR particle, the standard action integral is St=∫−∞+∞dt1/2mẋ2, where ẋ=dx/dt and time “t” is the evolution parameter. This action has no repa-rameterization invariance. If the evolution parameter is τ, then the action integral is Sτ=m/2∫−∞+∞dτdt/dτdx/dτdτ/dtdx/dτdτ/dt which leads [31, 32] to the fi-nal action integral as Sτ=m/2∫−∞+∞dτdx/dτ2dτ/dt≡∫−∞+∞dτL0. Hence, the starting Lagrangian becomes L0x,ẋ,t,ṫ=mẋ2/2ṫ where ẋ=dx/dτ and ṫ=dt/dτ.For our further discussions, we shall concentrate on the first-order LagrangianLf because it has the maximum number of variables (i.e., x, px, t, pt, and E) and allows the massless limit and there are no variables (and/or their first-order derivative w.r.t. τ) in its denominator. It is straightforward to check that under the following infinitesimal and continuous 1D diffeomorphism (i.e., reparameterization) symmetry transformations δr, namely,
(2)δrx=εẋ,δrpx=εṗx,δrt=εṫ,δrpt=εṗt,δrE=ddτεE,δrLf=ddτεLf,the action integral S=∫−∞+∞dτLf remains invariant (i.e., δrS=0) for the physically well-defined variables in Lf and the infinitesimal diffeomorphism transformation parameter ετ in τ⟶τ′=gτ=τ−ετ, where gτ is a physically well-defined function of τ such that it is finite at τ=0 and vanishes off at τ=±∞. In fact, the infinitesimal and continuous reparameterization symmetry transformation δr is defined as δrϕτ=ϕ′τ−ϕτ for the generic variable ϕτ=xτ,pxτ,tτ,ptτ,Eτ of our present theory.The above infinitesimal and continuous reparameterization symmetry transformations (2) encompass in their folds the gauge symmetry transformations δg which are generated (see, e.g., [32] for details) by the first-class constraints ΠE≈0,px2+2mpt≈0, where ΠE is the canonical conjugate momentum corresponding to the variable Eτ. Using the following Euler-Lagrange equations of motion (EL-EOMs) from Lf, namely,
(3)ṗx=0,ṗt=0,ẋ=Epx,ṫ=Em,and identifying the transformation parameters ετEτ=ξτ, we obtain the infinitesimal and continuous gauge symmetry transformations δg, from the infinitesimal and continuous reparameterization symmetry transformations (2), as follows:
(4)δgx=ξpx,δgt=ξm,δgpx=0,δgpt=0,δgE=ξ̇.It is elementary now to check that the first-order LagrangianLf transforms to a total derivative under the infinitesimal and continuous gauge transformations δg, namely,
(5)δgLf=ddτξ2px2,thereby rendering the action integral S=∫−∞+∞dτLf invariant (i.e., δgS=0) under the infinitesimal and continuous gauge symmetry transformations δg.We end this section with the following decisive comments. First, the 1D diffeomorphism (i.e., reparameterization) transformations (2) are more general than the infinitesimal and continuous gauge symmetry transformations (4). Second, the Lagrange multiplier variable Eτ behaves like a “gauge” variable due to its transformation δgE=ξ̇ in (4). Third, all the three Lagrangians in (1) are equivalent and all of them respect the infinitesimal and continuous gauge and reparameterization symmetry transformations [32]. Fourth, all the Lagrangians describe the free motion ṗx=0,ṗt=0 of the NR particle. Hence, our system is a nonrelativistic free ṗx=ṗt=0 particle. Fifth, the first-order Lagrangian Lf is theoretically more interesting to handle because, as pointed out earlier, it incorporates the maximum number of variables. Finally, we can exploit the reparameterization and gauge symmetry transformations (2) and (4) for the BRST quantization. Following the usual BRST prescription, we note that the (anti-)BRST symmetry transformations, corresponding to the classical reparameterization symmetry transformations (2), are
(6)sabx=C¯ẋ,sabpx=C¯ṗx,sabt=C¯ṫ,sabpt=C¯ṗt,sabE=ddτC¯E,sbx=Cẋ,sbpx=Cṗx,sbt=Cṫ,sbpt=Cṗt,sbE=ddτCE,where C¯C are the fermionic C2=C¯2=0,CC¯+C¯C=0 (anti)ghost variables corresponding to the classical infinitesimal diffeomorphism transformation parameter ετ (cf. Equation (2)). In an exactly similar fashion, the (anti-)BRST symmetry transformations, corresponding to the classical gauge symmetry transformations (4), are
(7)sabx=c¯px,sabpx=0,sabt=c¯m,sabpt=0,sabE=c¯̇,sbx=cpx,sabpx=0,sbt=cm,sbpt=0,sbE=ċ,where the fermionic c2=c¯2=0,cc¯+c¯c=0 variables c¯c are the (anti)ghost variables corresponding to the classical gauge symmetry transformation parameter ξτ of Equation (4). In addition to the (anti-)BRST symmetry transformations in (6) and (7), we have the following standard (anti-)BRST transformations:
(8)sabC=iB¯,sabB¯=0,sbC¯=iB,sbB=0,sabc=ib¯,sabb¯=0,sbc¯=ib,sbb=0,where the pairs B,B¯andb,b¯ are the Nakanishi-Lautrup auxiliary variables in the context of the BRST quantization of our reparameterization and gauge invariant system by exploiting the classical reparameterization and gauge transformations, respectively.


## 2.2. Quantum Nilpotent (Anti-)BRST Symmetries Corresponding to the Classical Gauge Symmetry Transformations

We have listed the quantum (anti-)BRST symmetries corresponding to theclassical gauge symmetry transformations (4) in our Equations (7) and (8). It is elementary to check that these quantum symmetries are off-shell nilpotent sab2=0 of order two. The requirement of the absolute anticommutativity sbsab+sabsb=0 leads to the restriction b+b¯=0⇒b¯=−b. As a consequence, we have the full set of (anti-)BRST symmetry transformations (corresponding to the classical gauge symmetry transformations (4)) as follows:
(9)sabx=c¯px,sabpx=0,sabt=c¯m,sabpt=0,sabE=c¯̇,sabc¯=0,sabc=−ib,sabb=0,sbx=cpx,sbpx=0,sbt=cm,sbpt=0,sbE=ċ,sbc=0,sbc¯=ib,sbb=0.It is straightforward to check that the above (anti-)BRST symmetry transformations are off-shell nilpotentsab2=0 and absolutely anticommuting sbsab+sabsb=0 in nature. The (anti-)BRST invariant Lagrangian Lb (which is the generalization of the classicalLf to its quantum level) can be written as (The structure of gauge-fixing and FP ghost terms is exactly like the Abelian 1-form A1=dxμAμ gauge theory where we have the BRST invariant Lagrangian density: Lb=−1/4FμνFμν+sb−ic¯∂μAμ+b/2≡−1/4FμνFμν+sab−ic∂μAμ+b/2≡−1/4FμνFμν+sbsabi/2AμAμ−1/2c¯c. Here, Aμ is the vector potential, Fμν=∂μAν−∂νAμ is the field strength tensor, and the rest of the symbols are the same as in Equations (10) and (11). Note that the 2-form F2=dA1=1/2dxμ∧dxνFμν defines the field strength tensor Fμν (where d=dxμ∂μ in F2=dA1 stands for the exterior derivative of the differential geometry).):
(10)Lb=Lf+sb−ic¯Ė+b2≡Lf+sabicĖ+b2≡Lf+sbsabiE22−c¯c2≡Lf−sabsbiE22−c¯c2.In other words, we have expressed the gauge-fixing and Faddeev-Popov (FP) ghost terms inthree different ways which, ultimately, lead to the following expression for Lb, namely,
(11)Lb=Lf+bĖ+b22−ic¯̇ċ≡pxẋ+ptṫ−12Epx2+2mpt+bĖ+b22−ic¯̇ċ.It should be noted that we have dropped the total derivative terms in obtainingLb from (10). The above equation demonstrates that we have obtained a unique (anti-)BRST invariant Lagrangian. This has happened because the CF-type restriction is trivial (i.e., b+b¯=0) in our simple case of the NR system. We can explicitly check that
(12)sbLb=ddτc2px2+bċ,sabLb=ddτc¯2px2+bc¯̇,which lead to the derivation of the conserved (anti-)BRST charges Qab as follows:
(13)Qab=bc¯̇+12c¯px2+2mpt≡bc¯̇−ḃc¯,Qb=bċ+12cpx2+2mpt≡bċ−ḃc.In the last step, we have usedḃ=−1/2px2+2mpt which emerges out as the EL-EOM from Lb w.r.t. the Lagrange multiplier variable Eτ. The structure of gauge-fixing and FP ghost terms in Equation (10) is exactly like the Abelian 1-form A1=dxμAμ gauge theory where we have the BRST invariant Lagrangian density: Lb=−1/4FμvFμv+sb−ic¯∂μAμ+b/2≡−1/4FμvFμv+sab−ic∂μAμ+b/2≡−1/4FμvFμv+sbsab1/2AμAμ−1/2c¯c. Here, Aμ is the vector potential, Fμv=∂μAv−∂vAμ is the field strength tensor, and the rest of the symbols are the same as in Equation (10) and (11). Note that the 2-form F2=dA1=1/2dxμΛdxvFμv defines the field strength tensor Fμv(where d=dxμ∂μ in F2=dA1 which stands for the exterior derivative of the differential geometry).We close this section with a few crucial and decisive remarks. First, we can check that the (anti-)BRST charges are conservedQ̇ab=0 by using the EL-EOMs. Second, the (anti-)BRST charges Qab are off-shell nilpotent Qab2=0 of order two due to the direct observations that sbQb=sbbċ−ḃc=0 and sabQab=sabbc¯̇−ḃc¯=0 which encode in their folds sbQb=−iQb,Qb=0⇒Qb2=0andsabQab=−iQab,Qab=0⇒Qab2=0. Third, the above nilpotency is also encoded in Qb=sbbE+ic¯̇c implying that sbQb=0 due to sb2=0and we also point out that sabQab=0 due to the nilpotency sab2=0 of sab because Qab=sabbE+ic¯ċ. Fourth, we observe that sabQb=iQb,Qab≡−ibḃ+iḃb=0 and sbQab=−iQab,Qb=ibḃ−ibḃ=0 which explicitly lead to the conclusion that the off-shell nilpotent charges Qab are also absolutely anticommuting QbQab+QabQb=0 in nature. Fifth, the above observation of the absolute anticommutativity can be also expressed in terms of the nilpotency property because we observe that Qb=sab−iċc and Qab=sbic¯̇c¯ which imply that sabQb=−iQb,Qab=0 and sbQab=−iQab,Qb=0 (due to the off-shell nilpotency sab2=0 of the anti-BRST as well as the off-shell nilpotency sb2=0 of the BRST symmetry transformations). Sixth, it can be seen that the physical space (i.e., phys>) in the total Hilbert space of states is defined by Qbphys>=0 which implies that bphys>≡ΠEphys>=0 and ḃphys>≡px2+2mptphys>=0. In other words, the Dirac quantization conditions (with the first-class constraints ΠE≈0,px2+2mpt≈0) are beautifully satisfied. Finally, the physicality criterion Qbphys>=0 implies that the two physical states phys′> and phys> belong to the same cohomological class w.r.t. the nilpotent BRST charge Qb if they differ by a BRST exact state (i.e., phys′>=phys>+Qbχ> for nonnull χ>).


## 3. Nilpotent and Anticommuting (Anti-)BRST Symmetries for the Phase Variables: MBTSA

This section is devoted to the derivation of the transformationssbx=Cẋ, sbpx=Cṗx, sbt=Cṫ, sbpt=Cṗt, sabx=C¯ẋ, sabpx=C¯ṗx, sabt=C¯ṫ, and sabpt=C¯ṗt by exploiting the theoretical tricks of MBTSA. Before we set out to perform this exercise, it is essential to pinpoint the off-shell nilpotency and absolute anticommutativity properties of the (anti-)BRST symmetry transformations on the phase variables (cf. Equation (6)). It can be easily checked that the off-shell nilpotency requirement (i.e., sab2S=0,S=x,px,t,pt) leads to the (anti-)BRST symmetry transformations for the (anti)ghost variables as
(14)sabC¯=C¯C¯̇,sbC=CĊ.Furthermore, the absolute anticommutativity requirementsb,sabS=0 for the generic phase variable S=x,px,t,pt leads to the following:
(15)sb,sabS=iB+B¯+iC¯Ċ−C¯̇CṠ⇒B+B¯+iC¯Ċ−C¯̇C=0.In other words, the absolute anticommutativity property (sbsab+sabsb=0) is satisfied if and only if we invoke the sanctity of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0. It goes without saying that the above cited requirements of the off-shell nilpotency and absolute anticommutativity properties are very sacrosanct within the framework of the BRST approach to gauge and/or reparameterization invariant theories.Against the backdrop of the above discussions, we set out to deduce the (anti-)BRST symmetry transformationssabS=C¯Ṡ and sbS=CṠ (with S=x,px,t,pt) and the CF-type restrictions B+B¯+iC¯Ċ−C¯̇C=0 within the framework of MBTSA. Towards this end in our mind, first of all, we generalize the classical function gτ (in τ⟶τ′=gτ≡τ−ετ) onto a 1,2-dimensional supermanifold as
(16)gτ⟶g~τ,θ,θ¯=τ−θC¯τ−θ¯Cτ+θθ¯kτ,where C¯C variables are the (anti)ghost variables of Equation (6) and kτ is a secondary variable that has to be determined from the consistency conditions (that include the off-shell nilpotency as well as absolute anticommutativity requirements). It will be noted that, due to the mappings sb↔∂θ¯θ=0 and sab↔∂θθ¯=0 [14–16], we have taken the coefficients of θ and θ¯ in Equation (16) as the (anti)ghost variables C¯C. This has been done due to our observation in the infinitesimal reparameterization symmetry transformation δr (where δrτ=−ετ) at the classical level. Following the basic tenet of BRST formalism, the infinitesimal parameter ετ has been replaced (in the BRST-quantized theory) by the (anti)ghost variables, thereby leading to the (anti-)BRST symmetry transformations sabτ=−C¯ and sbτ=−C.For our present 1D diffeomorphism (i.e., reparameterization) invariant theory, the generic variableSτ can be generalized to a supervariable S~g~τ,θ,θ¯,θ,θ¯ on the 1,2-dimensional supermanifold [22] with the following super expansion along all the Grassmannian directions of the 1,2-dimensional supermanifold, namely,
(17)S~g~τ,θ,θ¯,θ,θ¯=Sg~τ,θ,θ¯+θR¯g~τ,θ,θ¯+θ¯Rg~τ,θ,θ¯+θθ¯Qg~τ,θ,θ¯,where the expression for g~τ,θ,θ¯ is given in Equation (16). It should be noted that all the primary as well as secondary supervariables on the r.h.s. of (17) are a function of the 1,2-dimensional super infinitesimal diffeomorphism transformation (16). At this stage, we can perform the Taylor expansions for all the supervariables as
(18)θθ¯Qτ−θC¯−θ¯C+θθ¯k=θθ¯Qτ,θ¯Rτ−θC¯−θ¯C+θθ¯k=θ¯Rτ+θθ¯C¯τṘτ,θR¯τ−θC¯−θ¯C+θθ¯k=θR¯τ−θθ¯CτR¯̇τ,Sτ−θC¯−θ¯C+θθ¯k=Sτ−θC¯τṠτ−θ¯CτṠτ+θθ¯kτṠτ−C¯τCτS¨τ.Collecting all these terms and substituting them into (17), we obtain the following super expansion for the supervariable on the 1,2-dimensional supermanifold, namely,
(19)S~g~τ,θ,θ¯,θ,θ¯=Sτ+θR¯−C¯Ṡ+θ¯R−CṠ+θθ¯Q+C¯Ṙ−CR¯̇+kṠ+CC¯S¨.We now exploit the horizontality condition (HC) which physically implies thatall the scalar variables should not transform at all under any kind of spacetime, internal, supersymmetric, etc., transformations. With respect to the 1D space of the trajectory of the particle, all the supervariables on the l.h.s. and r.h.s. of Equation (18) are scalars. The HC, in our case, is
(20)S~g~τ,θ,θ¯,θ,θ¯=Sτ.The above equality implies thatall the coefficients of θ, θ¯, and θθ¯ of Equation (19) should be set equal to zero. In other words, we have the following:
(21)R=CṠ,R¯=C¯Ṡ,Q=CR¯̇−C¯Ṙ−kṠ+C¯CS¨.Substitutions of the values ofR and R¯ into the expression for Q lead to the following:
(22)Q=−C¯Ċ+C¯̇CṠ−kṠ−C¯CS¨.As explained before Equation (20) (i.e., exploiting the key properties of scalars), it is evident that (17) can be finally written with S~g~τ,θ,θ¯,θ,θ¯=S~τ,θ,θ¯ as
(23)S~τ,θ,θ¯=Sτ+θR¯τ+θ¯Rτ+θθ¯Qτ≡Sτ+θsabS+θ¯sbS+θθ¯sbsabS,where, due to the well-known mappings sb↔∂θ¯θ=0 and sab↔∂θθ¯=0 [14–16], the coefficients of θ and θ¯ are the anti-BRST and BRST symmetry transformations (cf. Equation (6)). We point out that the key properties of scalars on the r.h.s. of Equation (17) imply that we have Sg~τ,θ,θ¯=Sτ, Rg~τ,θ,θ¯=Rτ, R¯g~τ,θ,θ¯=R¯τ, and Qg~τ,θ,θ¯=Qτ.A comparison between (21) and (23) implies that we have already derived the nilpotent (anti-)BRST symmetry transformations R=sbS=CṠ and R¯=sabS=C¯Ṡ. In other words, we have obtained sbx=Cẋ, sbpx=Cṗx, sbt=Cṫ, sbpt=Cṗt, sabx=C¯ẋ, sabpx=C¯ṗx, sabt=C¯ṫ, and sabpt=C¯ṗt. Furthermore, it is evident that
(24)sbsabS=Q=−C¯Ċ+C¯̇CṠ−kṠ−C¯CS¨.The requirement of the absolute anticommutativitysb,sabS=0 implies that sbsabS=−sabsbS which, in turn, leads to the following relationships:
(25)sbsabS=sbR¯=Q≡−C¯Ċ+C¯̇CṠ−kṠ−C¯CS¨,−sabsbS=−sabR=Q≡−C¯Ċ+C¯̇CṠ−kṠ−C¯CS¨.The explicit computations of the following, using the (anti-)BRST symmetry transformations of the phase variables in Equations (6) and (14), are
(26)sbR¯=iBṠ−C¯ĊṠ−C¯CS¨≡Q,−sabR=−iB¯Ṡ−C¯̇CṠ−C¯CS¨≡Q.Equating (25) and (26), we obtain the following interesting relationship:
(27)k=−C¯̇C−iB≡iB¯−C¯Ċ⇒B+B¯+iC¯Ċ−C¯̇C=0.In other words, it is the consistency conditions of the BRST formalism that lead to the determination ofkτ in Equation (16) within the ambit of MBTSA. A close look at Equations (25)–(27) establishes that a precise determination of Qτ in (23) leads to (i) the validity of the absolute anticommutativity (i.e., sb,sabS=0) of the off-shell nilpotent (anti-)BRST symmetries and (ii) the deduction of the (anti-)BRST invariant (This statement is true only when the whole theory is considered on a submanifold of the Hilbert space of the quantum variables where the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 is satisfied. In other words, we explicitly compute sbB+B¯+iC¯Ċ−C¯̇C=d/dτB+B¯+iC¯Ċ−C¯̇CC−B+B¯+iC¯Ċ−C¯̇CĊ and sabB+B¯+iC¯Ċ−C¯̇C=d/dτB+B¯+iC¯Ċ−C¯̇CC¯−B+B¯+iC¯Ċ−C¯̇CC¯̇ which imply that sabB+B¯+iC¯Ċ−C¯̇C=0 is true only on the abovementioned submanifold.) CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 on our theory. This statement (i.e., the (anti-)BRST invariance) is true only when the whole theory is considered on a submanifold of the Hilbert space of the quantum variables where the CF-type restriction B+B¯+iC¯Ċ−ĊC=0 is satisfied. In other words, we explicitly compute sbB+B¯+iC¯Ċ−C¯̇C=d/dτB+B¯+iC¯Ċ−C¯̇CC−B+B¯+iC¯Ċ−C¯̇CĊ and sbB+B¯+iC¯Ċ−C¯̇C=d/dτB+B¯+iC¯Ċ−C¯̇CC¯−B+B¯+iC¯Ċ−C¯̇CC¯̇ which imply that sabB+B¯+iC¯Ċ−C¯̇C=0 is true only on the abovementioned submanifold.We conclude this section with the following useful and crucial remarks. First, we set out to derive the (anti-)BRST symmetry transformations (corresponding to the classical reparameterization symmetry transformations) for the phase variables (cf. Equation (6)). We have accomplished this goal in Equation (21). Second, we have derived the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 within the purview of MBTSA (cf. Equation (27)) which is actually hidden in the determination of Qτ in Equation (23). Third, for the application of the theoretical potential of MBTSA, we have taken the full super expansion of the generic supervariable (cf. Equation (17)) along all the possible Grassmannian directions of the 1,2-dimensional supermanifold. Fourth, unlike the application of the BT superfield/supervariable approach to the gauge theories [14–16] where spacetime does not change, in the case of MBTSA, the super diffeomorphism transformation (16) has been taken into account in all the basic as well as secondary supervariables. Fifth, taking into account the inputs from Equations (21) and (26), we obtain the following super expansion of the generic variable Sτ, namely,
(28)S~hτ,θ,θ¯=Sτ+θC¯Ṡ+θ¯CṠ+θθ¯iBṠ−C¯ĊṠ−C¯CS¨≡Sτ+θsabS+θ¯sbS+θθ¯sbsabS,where S=x,px,t,pt and the superscript h on the supervariable S~τ,θ,θ¯ denotes that this supervariable has been obtained after the application of HC. Finally, the standard nilpotent (anti-)BRST symmetry transformations (8) dictate that we can have the following (anti)chiral super expansions for the supervariables corresponding to C¯C, namely,
(29)Cτ⟶Fcτ,θ=Cτ+θiB¯≡Cτ+θsabC,C¯τ⟶F¯acτ,θ¯=C¯τ+θ¯iB≡C¯τ+θ¯sbC¯,where the superscripts c and ac denote the chiral and antichiral supervariables. The above observation gives us a clue that we should exploit the theoretical strength of ACSA to BRST formalism for our further discussions.


## 4. Coupled Lagrangians and Quantum (Anti-)BRST Symmetries Corresponding to the Classical Reparameterization Symmetry Transformations

In addition to the quantum (anti-)BRST symmetries in (6), (8), and (14), we derive all the other off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries corresponding to the classical infinitesimal and continuous reparameterization symmetry transformations (2). We exploit the strength of the sacrosanct requirements of off-shell nilpotency and absolute anticommutativity properties. In this context, we point out that we have already derived sbC=CĊ and sabC¯=C¯C¯̇ by invoking the sanctity of the off-shell nilpotency sab2=0 property for the phase variables (i.e., sab2S=0,S=x,px,t,pt). It is interesting to note that the following absolute anticommutativity requirements, namely,
(30)sb,sabC=0⇒sbB¯=B¯̇C−B¯Ċ,sb,sabC¯=0⇒sabB=ḂC¯−BC¯̇,lead to the derivation of the sbB¯ and sabB. We can readily check that sb2B¯=0 and sab2B=0 are satisfied due to our knowledge of the BRST and anti-BRST symmetry transformations sbC=CĊ and sabC¯=C¯C¯̇and the fermionic C2=C¯2=0,CC¯+C¯C=0 nature of the (anti)ghost variables C¯C. We further note that sb,sabB=0 and sb,sabB¯=0. The requirement of the absolute anticommutativity on the Eτ variable leads to
(31)sb,sabEτ=ddτiB+B¯+iC¯Ċ−C¯̇CEτ.Thus, we emphasize that the absolute anticommutativity propertysbsab+sabsb=0 on the phase variables (cf. Equation (15)) as well as on the Lagrange multiplier variable (cf. Equation (31)) is satisfied if and only if the CF-type restriction is invoked. In the full blaze of glory, the quantum (anti-)BRST symmetry transformations (corresponding to the infinitesimal reparameterization symmetry transformations (2)) are as follows:
(32)sabx=C¯ẋ,sabpx=C¯ṗx,sabt=C¯ṫ,sabpt=C¯ṗt,sabE=ddτC¯E,sabC=iB¯,sabC¯=C¯C¯̇,sabB¯=0,sabB=ḂC¯−BC¯̇,(33)sbx=Cẋ,sbpx=Cṗx,sbt=Cṫ,sbpt=Cṗt,sbE=ddτCE,sbC¯=iB,sbC=CĊ,sbB=0,sbB¯=B¯̇C−B¯Ċ.The abovefermionic symmetry transformations are off-shell nilpotent and absolutely anticommuting provided that the whole theory is considered on a submanifold of the space of quantum variables where the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 is satisfied.The existence of theabove CF-type restriction leads to the derivation of the coupled (but equivalent) Lagrangians (i.e., LB and LB¯) as follows:
(34)LB=Lf+sbsabiE22−C¯C2,LB¯=Lf−sabsbiE22−C¯C2.We point out that the terms inside the square brackets are thesame as in Equation (10) for the BRST analysis of the classical gauge symmetry transformations (4). Furthermore, in contrast to the unique (anti-)BRST invariant Lagrangian (cf. Equation (11)) (corresponding to the classical gauge symmetry transformations), we have obtained here a set of coupled (but equivalent) (anti-)BRST invariant Lagrangians in Equation (34). This has happened because of the fact that the CF-type restriction b+b¯=0 is trivial in the case of the former while it is a nontrivial restriction B+B¯+iC¯Ċ−C¯̇C=0 in the context of the latter.One can readily compute the operation ofsab on the quantities in the square brackets of Equation (34). In the full blaze of their glory, the coupled (but equivalent) Lagrangians LB and LB¯ are as follows (It will be worthwhile to mention here that the form of the gauge-fixing and Faddeev-Popov ghost terms is the same as in the cases of NSUSY (i.e., scalar) and SUSY (i.e., spinning) relativistic particles [23, 24].):
(35)LB=Lf+BEĖ−i2C¯̇C+C¯Ċ+B22−iEĖC¯̇C−iE2C¯̇Ċ−C¯̇C¯ĊC,LB¯=Lf−B¯EĖ−i2C¯Ċ+C¯̇C+B¯22−iEĖC¯Ċ−iE2C¯̇Ċ−C¯̇C¯ĊC,where the subscripts B and B¯ on the Lagrangians are appropriate because LB depends uniquely on the Nakanishi-Lautrup auxiliary variable B (where B¯ is not present at all). Similarly, the Lagrangian LB¯ is uniquely dependent on B¯. They are coupled because the EL-EOMs with respect to B and B¯ from LB and LB¯, respectively, yield
(36)B=−EĖ+2iC¯̇C+iC¯Ċ,B¯=EĖ−2iC¯Ċ−iC¯̇C,which lead to the deduction of the CF-type restrictions B+B¯+iC¯Ċ−C¯̇C=0. Furthermore, the condition LB≡LB¯ also demonstrates the existence of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 on our theory (cf. Appendix A below). It will be worthwhile to mention here that the form of the gauge-fixing and Faddeev-Popov ghost terms in Equation (35) is the same as in the cases of NSUSY (i.e., scalar) and SUSY (i.e., spinning) relativistic particles [23, 24].At this stage, we are in the position to study the (anti-)BRST symmetries of the LagrangiansLB and LB¯. It is straightforward to note that we have the following:
(37)sbLB=ddτCLf+B2C−iBC¯ĊC+EĖBC+E2BĊ,(38)sabLB¯=ddτC¯Lf+B¯2C¯−iB¯C¯̇C¯C−EĖB¯C¯−E2B¯C¯̇.The above observations demonstrate that the action integralsS1=∫−∞∞dτLB and S2=∫−∞∞dτLB¯ remain invariant under the SUSY-type (i.e., fermionic) off-shell nilpotent, continuous, and infinitesimal (anti-)BRST symmetry transformations for the physical variables that vanish off at τ=±∞. At this crucial juncture, we establish the equivalence of the coupled Lagrangians LB and LB¯ w.r.t. the (anti-)BRST symmetry transformations sab. In this context, we apply sab on LB and sb on LB¯ to obtain the following:
(39)sabLB=ddτC¯Lf+EĖiC¯̇C¯C+BC¯+E2iC¯̇C¯Ċ+BC¯̇+B2C¯+i2B−B¯C¯̇C¯C+B+B¯+iC¯Ċ−C¯̇C2iC¯̇C¯Ċ−2BC¯̇−EĖC¯̇+iC¯¨C¯C−ddτB+B¯+iC¯Ċ−C¯̇CBC¯+E2C¯̇,(40)sbLB¯=ddτCLf+EĖiC¯CĊ−B¯C+E2iC¯̇CĊ−B¯Ċ+B¯2C−i2B¯−BC¯CĊ+B+B¯+iC¯Ċ−C¯̇C−2iC¯̇CĊ−2B¯Ċ+EĖĊ+iC¯C¨C+ddτB+B¯+iC¯Ċ−C¯̇C+E2Ċ−B¯C,which demonstrate that the coupled Lagrangians LB and LB¯ (and corresponding action integrals) respect both (i.e., BRST and anti-BRST) symmetry transformations together provided that the whole theory is considered on a supermanifold in the Hilbert space of quantum variables where the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 is satisfied. It should be recalled that, under the latter restriction, we also have the absolute anticommutativity property (i.e., sb,sab=0) of the (anti-)BRST symmetry transformations.We end this section with the following key comments. First, the properties of the off-shell nilpotency and absolute anticommutativity aresacrosanct in the realm of the BRST approach to gauge and/or diffeomorphism invariant theories. Second, physically, the first property (i.e., off-shell nilpotency) implies that these fermionic symmetry transformations are of supersymmetric type as they transform bosonic variables to fermionic variables and vice versa. Third, the property of the absolute anticommutativity encodes the linear independence of the BRST and anti-BRST symmetry transformations. Fourth, the absolute anticommutativity property owes its origin to the existence of the CF-type restrictions which are connected with the concepts of gerbes [7, 8]. Fifth, as the classical gauge theory is characterized by the first-class constraints, in an exactly similar fashion, the quantum gauge and/or diffeomorphism (i.e., (anti-)BRST) invariant theories are characterized by the existence of the CF-type restrictions within the ambit of BRST formalism. Sixth, the coupled Lagrangians LB and LB¯ are equivalent because both of them respect BRST and anti-BRST symmetry transformations as is clear from Equations (37)–(40) provided that the whole theory is considered on the submanifold of the total Hilbert space of the quantum variables where the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 is satisfied.


## 5. Quantum Off-Shell Nilpotent (Anti-)BRST Symmetries of the Other Variables: ACSA

In this section, we derive the nilpotent (anti-)BRST symmetry transformationssab for all the other variables (cf. Equations (32) and (33)) besides the phase space variables (x, px, t, and pt)whose (anti-)BRST symmetries have already been derived in Section 3 by exploiting the theoretical potential of MBTSA. To achieve the above goal, we exploit the ideas behind ACSA to BRST formalism [25–29]. In this context, first of all, we focus on the derivation of the BRST symmetry transformations sbB=0, sbB¯=B¯̇C−B¯Ċ, sbC=CĊ, and sbE=ĖC+EĊ (cf. Equation (33)). For this purpose, we generalize the ordinary variables (Bτ, B¯τ, Cτ, and Eτ) onto a 1,1-dimensional antichiral super submanifold as follows:
(41)Bτ⟶Bτ,θ¯=Bτ+θ¯f1τ,B¯τ⟶B¯τ,θ¯=B¯τ+θ¯f2τ,Cτ⟶Fτ,θ¯=Cτ+θ¯b1τ,Eτ⟶Στ,θ¯=Eτ+θ¯f3τ,where we note that f1, f2, and f3 are the fermionic secondary variables and b1τ is the bosonic secondary variable because of the fermionic θ¯2=0 nature of the Grassmannian variable θ¯ which characterizes the antichiral super submanifold (along with the bosonic evolution parameter τ). It is elementary to note that the observation sbB=0 implies the following super expansion (in view of the fact that ∂θ¯↔sb), namely,
(42)Bbτ,θ¯=Bτ+θ¯0≡Bτ+θ¯sbB,where the superscript b on the antichiral supervariable Bτ,θ¯ denotes that the coefficient of θ¯ yields the BRST symmetry transformation sbB=0 due to the trivial equality Bτ,θ¯=Bτ which emerges from the observation that the Nakanishi-Lautrup auxiliary variable Bτ is a BRST invariant quantity (cf. Equation (33)). In other words, we have found out that the secondary variable f1τ=0 in the super expansions (41).At this stage, we findother nontrivial BRST invariant quantities for the derivation of the secondary variables b1, f2, and f3 of Equation (41). We observe that (We have specifically taken here sbCẋ=0 for our purpose. However, one can take the general expression sbS=CṠS=x,px,t,pt for the derivation of b1τ=CĊ.) sbCẋ=0, sbB¯̇C−B¯Ċ=0, and sbEĊ+ĖC=0. We have specifically taken here sbCẋ=0 for our purpose. However, one can take the general expression sbS=CṠS=x,px,t,pt for the derivation of b1τ=CĊ. The basic tenet of the ACSA to BRST formalism requires that the quantities in the square brackets have to be independent of the Grassmannian variable θ¯ when they are generalized onto a 1,1-dimensional super submanifold, namely,
(43)Fτ,θ¯Ẋh,acτ,θ¯=Cτẋτ,Bτ,θ¯Fτ,θ¯−Bτ,θ¯Ḟτ,θ¯=B¯̇τCτ−B¯τĊτ,Στ,θ¯Ḟτ,θ¯+Σ̇τ,θ¯Fτ,θ¯=EτĊτ+ĖτCτ,where Xh,ac is the antichiral limit of the full expansion of Xhτ,θ,θ¯ obtained after the application of HC (cf. Equation (28)), namely,
(44)Xhτ,θ,θ¯=xτ+θC¯ẋ+θ¯Cẋ+θθ¯iBẋ−C¯Ċẋ−C¯Cx¨,which has been obtained (cf. Equation (28)) in Section 3 using the theoretical strength of MBTSA. In other words, from the top entry of Equation (43), we have the following restriction:
(45)Fτ,θ¯Ẋh,acτ,θ¯=Cτẋτ⇒Cτ+θ¯b1τẋ+θ¯Ċẋ+Cx¨=Cτẋτ.From the above relationship, we obtainb1τ=CĊ. Thus, we have the following:
(46)Fbτ,θ¯=Cτ+θ¯CĊ≡Cτ+θ¯sbC,where the superscript b on the l.h.s. of the supervariable denotes that the coefficient of θ¯ is nothing but the BRST symmetry transformation sbC. We have to use the above super expansion in the second entry from the top in (43) to obtain the following:
(47)Bτ,θ¯Fbτ,θ¯−Bτ,θ¯Ḟbτ,θ¯=B¯̇τCτ−B¯τĊτ.In other words, we have the following equality:(48)B¯̇+θ¯ḟ2τCτ+θ¯CĊ−B¯τ+θ¯f2τĊτ+θ¯CC¨=B¯̇τCτ−B¯τĊτ,which yields the following condition on the secondary variable f2, namely,
(49)ḟ2C−f2Ċ−B¯̇ĊC+B¯C¨C=0.It is straightforward to note thatf2=B¯̇C−B¯Ċ satisfies the above condition in a precise manner. We point out that the last entry (from the top) of Equation (43) can be rewritten, in view of our the super expansion in Equation (46), as follows:
(50)Στ,θ¯Ḟbτ,θ¯+Σ̇τ,θ¯Fbτ,θ¯=EτĊτ+ĖτCτ.The substitutions of expansions from (41) and (46) lead to the following condition on the secondary variable f3τ (present in the expansion of Στ,θ¯), namely,
(51)f3Ċ+ḟ3C−EC¨C−ĖĊC=0,which is satisfied by the choice f3=EĊ+ĖC. Hence, we have the following super expansions (with the BRST symmetry transformations (33) as input), namely,
(52)Bbτ,θ¯=B¯τ+θ¯B¯̇C−B¯Ċ≡B¯τ+θ¯sbB¯,Σbτ,θ¯=Eτ+θ¯EĊ+ĖC≡Eτ+θ¯sbE,where the coefficients of θ¯ (in view of ∂θ¯↔sb) are the BRST symmetry transformations (33). For the convenience of the readers, we have performed the explicit computations of f3=EĊ+ĖC and f2=B¯̇C−B¯Ċ in Appendix B. It is clear that we have already computed the BRST transformations sbB=0, sbC=CĊ, sbB¯=B¯̇C−B¯Ċ, and sbE=EĊ+ĖC by exploiting the virtues of ACSA in Equations (42), (46), and (52).We concentrate now on the derivation of the anti-BRST symmetry transformations (32) by exploiting the theoretical strength of ACSA to BRST formalism. It is obvious that, in Section 3, we have already computed sabS=C¯ṠS=x,px,t,pt and sabC=iB¯ by exploiting MBTSA to BRST formalism. Our objective in the present part of our section is to derive sabB¯=0, sabC¯=C¯C¯̇, sabB=ḂC¯−BC¯̇, and sabE=EC¯̇+ĖC¯ by exploiting ACSA to BRST formalism. In this context, first of all, we generalize the ordinary variables onto a 1,1-dimensional chiral super submanifold as
(53)B¯τ⟶Bτ,θ=B¯τ+θf¯1τ,Bτ⟶Bτ,θ=Bτ+θf¯2τ,C¯τ⟶F¯τ,θ=C¯τ+θb¯1τ,Eτ⟶Στ,θ=Eτ+θf¯3τ,where f1, f2, and f3 are the fermionic secondary variables, b¯1τ is a bosonic secondary variable, and the above 1,1-dimensional chiral super submanifold is parameterized by τ,θ. It is straightforward to note that sabB¯=0 implies that Bτ,θ=Bτ, and as a consequence, we have f¯1τ=0 which leads to
(54)Babτ,θ=B¯τ+θ0≡B¯τ+θsabB¯,where the superscript ab on the chiral supervariable denotes that we have obtained sabB¯=0 as the coefficient of θ. The other useful and interesting anti-BRST invariant quantities of our interest (cf. Equation (32)) are
(55)sabḂC¯−BC¯̇=0,sabEC¯̇+ĖC¯=0,sabC¯ẋ=0.The quantities in the square brackets can be generalized onto the1,1-dimensional chiral super submanifold. Following the fundamental requirement(s) of ACSA to BRST formalism, these quantities must be independent of the Grassmannian variable θ. In other words, we have the following restrictions on the chiral supervariables:
(56)Bτ,θF¯τ,θ−Bτ,θF¯̇τ,θ=ḂτC¯τ−BτC¯̇τ,Στ,θF¯̇τ,θ+Σ̇τ,θF¯τ,θ=EτC¯̇τ+ĖτC¯τ,F¯τ,θẊh,cτ,θ=C¯τẋτ,where Xh,cτ,θ is the chiral limit of the super expansion in Equation (44). In other words, we have the following explicit expression for the supervariable Xh,cτ,θ, namely,
(57)Xh,cτ,θ=xτ+θC¯ẋ.Taking the expansions from (53) and (57), we find that the last entry of Equation (56) yields b¯1τ=C¯C¯̇. Hence, we have obtained the following super expansion:
(58)F¯abτ,θ=C¯τ+θC¯C¯̇≡C¯τ+θsabC¯,where the superscript ab on the chiral supervariable on the l.h.s. denotes that it has been derived after the application of the anti-BRST invariant restriction in (56). The coefficient of θ is nothing but the anti-BRST symmetry transformation sabC¯=C¯C¯̇. This equation also shows that ∂θ↔sab and it leads to the anti-BRST symmetry for C¯.We utilize now thetwo top entries of (56) where we use the explicit expansion for F¯abτ,θ of (58) in the following restrictions on the supervariables, namely,
(59)Bτ,θF¯abτ,θ−Bτ,θF¯̇abτ,θ=ḂτC¯τ−BτC¯̇τ,Στ,θF¯̇abτ,θ+Σ̇τ,θF¯abτ,θ=EτC¯̇τ+ĖτC¯τ.The substitutions from (53) and (58) lead to
(60)f¯̇2C¯−f¯2C¯̇−ḂC¯̇C¯+BC¯¨C¯=0,f¯3C¯̇+f¯̇3C¯−EC¯¨C¯−EC¯̇C¯=0.It is straightforward, following the theoretical tricks of AppendixB, to find out the solutions for the secondary variables f¯2τ and f¯3τ which are as follows:
(61)f¯2τ=ḂC¯−BC¯̇,f¯3=EC¯̇+ĖC¯.Substitutions of these secondary variables into the super expansions (53) lead to the determination of the anti-BRST symmetry transformations for the variables Bτ and Eτ as the coefficients of θ in the following:
(62)Σabτ,θ=Eτ+θEC¯̇+ĖC¯≡Eτ+θsabEτ,Babτ,θ=Bτ+θḂC¯−BC¯̇≡Bτ+θsabBτ,where the superscript ab on the chiral supervariable denotes that these supervariables have been obtained after the applications of the anti-BRST invariant restrictions (56). Moreover, the above observation establishes that sab⇔∂θ which implies that the nilpotency sab2=0,∂θ2=0 properties of sab and ∂θ are connected with each other. Thus, we have obtained all the anti-BRST symmetry transformations (besides the phase variables) in our Equations (54), (58), and (62). This completes our discussion on the derivation of the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations (32) and (33) within the ambit of ACSA to BRST formalism.


## 6. Symmetry Invariance of the Lagrangians: ACSA

In this section, we establish theequivalence of the coupled Lagrangians LB and LB¯ as far as the (anti-)BRST symmetry invariance (within the purview of ACSA to BRST formalism) is concerned. We accomplish this objective by generalizing the ordinary Lagrangians to their counterpart super Lagrangians as
(63)LB¯⟶L~B¯cτ,θ=L~fcτ,θ−Babτ,θΣabτ,θΣ̇abτ,θ−i2F¯abτ,θḞabτ,θ+F¯̇abτ,θFabτ,θ+12Babτ,θBabτ,θ−iΣabτ,θΣabτ,θF¯̇abτ,θḞabτ,θ−iΣabτ,θΣ̇abτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θF¯abτ,θḞabτ,θFabτ,θ,(64)LB⟶L~Bacτ,θ¯=L~facτ,θ¯+Bbτ,θ¯Σbτ,θ¯Σ̇bτ,θ¯−i2F¯̇bτ,θ¯Fbτ,θ¯+F¯bτ,θ¯Ḟbτ,θ¯+12Bbτ,θ¯Bbτ,θ¯−iΣbτ,θ¯Σbτ,θ¯F¯̇bτ,θ¯Ḟbτ,θ¯−iΣbτ,θ¯Σ̇bτ,θ¯F¯̇bτ,θ¯Fbτ,θ¯−F¯̇bτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯Fbτ,θ¯,where L~fc and L~fac are the generalizations of the first-order Lagrangian Lf to its counterpart chiral and antichiral super Lagrangians as
(65)L~fcτ,θ=Pxh,cτ,θẊh,cτ,θ+Pth,cτ,θṪh,cτ,θ−Σabτ,θ2Pxh,cτ,θPxh,cτ,θ+2mPth,cτ,θ,L~facτ,θ¯=Pxh,acτ,θ¯Ẋh,acτ,θ¯+Pth,acτ,θ¯Ṫh,acτ,θ¯−Σbτ,θ¯2Pxh,acτ,θ¯Pxh,acτ,θ¯+2mPth,acτ,θ¯,where the superscripts c and ac denote the chiral and antichiral generalizations and the rest of the supervariables with superscripts b and ab have already been explained earlier in Section 5. The supervariables with superscripts h,c and h,ac are the chiral and antichiral limits of the super phase variables (Xh, Pxh, Th, and Pth) that have been obtained after the application of HC. Thus, these are the counterparts of the ordinary phase variables (x, px, t, and pt) and they have been explained in Section 3. In the above Equation (65), the super phase variables with superscript h,c and h,ac can be expressed in terms of the generic supervariable as follows:
(66)Sτ⟶Sh,cτ,θ=Sτ+θC¯Ṡτ,Sτ⟶Sh,acτ,θ¯=Sτ+θ¯CṠτ,where the (anti)chiral supervariables on the l.h.s. stand for the super phase variables (X, Px, T, and Pt) with the proper chiral and antichiral superspace coordinates τ,θ and τ,θ¯ as their arguments. The set of supervariables (X, Px, T, and Pt) is the generalization of the ordinary phase variables (x, px, t, and pt) to their (anti)chiral counterparts onto the 1,1-dimensional (anti)chiral super submanifolds of the general1,2-dimensional supermanifold. It is straightforward to check that the following is true, namely,
(67)∂∂θL~fcτ,θ=ddτC¯Lf⇔sabLf=ddτC¯Lf,∂∂θ¯L~facτ,θ¯=ddτCLf⇔sbLf=ddτCLf.In other words, we have captured the (anti-)BRST invariance of the first-order LagrangianLf in view of the mappings sb↔∂θ¯ and sab↔∂θ. Since in the ordinary space the (anti-)BRST symmetry transformations acting on Lf produce the total derivatives (cf. Equation (67)), the action integral S=∫−∞+∞dτLf remains invariant under the transformations sab.At this stage, we focus on the (anti-)BRST invariance of the coupled LagrangiansLB and LB¯ (cf. Equations (37) and (38)). We can express these invariances within the ambit of ACSA (in view of the mappings sb↔∂θ¯ and sab↔∂θ), namely,
(68)∂∂θL~B¯cτ,θ=ddτC¯Lf−eėB¯C¯−e2B¯C¯̇+B¯2C¯−iB¯C¯̇C¯C=sabLB¯,(69)∂∂θ¯L~Bacτ,θ¯=ddτCLf+eėBC+e2BĊ+B2C−iBC¯ĊC=sbLB,where the super Lagrangians L~B¯cτ,θ and L~Bacτ,θ¯ have been already quoted in Equations (63) and (64). It is interesting to note that the r.h.s. of (68) and (69) are the same as we have found in the ordinary space (cf. Equations (37) and (38)). To prove the equivalence of the Lagrangians LB and LB¯ w.r.t. the (anti-)BRST symmetry transformations (cf. Equations (32) and (33)) within the purview of ACSA, we generalize the ordinary Lagrangians LB and LB¯ as follows:
(70)LB⟶L~Bcτ,θ=L~fcτ,θ+Babτ,θΣabτ,θΣ̇abτ,θ−i2F¯̇abτ,θFabτ,θ+F¯abτ,θḞabτ,θ+12Babτ,θBabτ,θ−iΣabτ,θΣabτ,θF¯̇abτ,θḞabτ,θ−iΣabτ,θΣ̇abτ,θF¯̇abτ,θFabτ,θ−F¯̇abτ,θF¯abτ,θḞabτ,θFabτ,θ,(71)LB¯⟶L~B¯acτ,θ¯=L~facτ,θ¯−Bbτ,θ¯Σbτ,θ¯Σ̇bτ,θ¯−i2F¯bτ,θ¯Ḟbτ,θ¯+F¯̇bτ,θ¯Fbτ,θ¯+12Bbτ,θ¯Bbτ,θ¯−iΣbτ,θ¯Σbτ,θ¯F¯̇bτ,θ¯Ḟbτ,θ¯−iΣbτ,θ¯Σ̇bτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯−F¯̇bτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯Fbτ,θ¯,where all the notations and symbols have already been explained earlier. To find out the result of the operations of sb on LB¯ and sab on LB, we observe the following (in view of the mappings sb↔∂θ¯ and sab↔∂θ), namely,
(72)∂∂θL~Bcτ,θ=ddτC¯Lf+EĖiC¯̇C¯C+BC¯+E2iC¯̇C¯Ċ+BC¯̇+B2C¯+i2B−B¯C¯̇C¯C+B+B¯+iC¯Ċ−C¯̇C2iC¯̇C¯Ċ−EĖC¯̇−2BC¯̇+iC¯¨C¯C−ddτB+B¯+iC¯Ċ−C¯̇CBC¯+E2C¯̇≡sabLB,(73)∂∂θ¯L~B¯acτ,θ¯=ddτCLf−EĖiC¯ĊC+B¯C−E2iC¯̇ĊC+B¯Ċ+B¯2C+i2B¯−BC¯ĊC+B+B¯+iC¯Ċ−C¯̇CiC¯C¨C+2iC¯̇ĊC−2B¯Ċ+EĖĊ+ddτiC¯Ċ−C¯̇C+B+B¯E2Ċ−B¯C≡sbLB¯,within the framework of ACSA. It is self-evident, from the r.h.s. of (72) and (73), that we have the BRST invariance of LB¯ and anti-BRST invariance of LB if and only if our whole theory is considered on the submanifold of the Hilbert space of quantum variables where the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 is satisfied.We end this section with the following crucial remarks. First of all, we have captured the BRST and anti-BRST invariances ofLB and LB¯, respectively, in the terminology of ACSA on the (anti)chiral super submanifolds (cf. Equations (68) and (69)). Second, we have also demonstrated the anti-BRST invariance of LBand BRST invariance of LB¯ in the superspace formalism (cf. Equations (72) and (73)) where the theoretical techniques of ACSA have played very important roles. Third, we have also expressed the (anti-)BRST invariance of the first-order Lagrangian Lf in Equation (67). Finally, we have proven the equivalence of LB and LB¯ within the framework of ACSA in Equations (68), (69), (72), and (73).


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

Our present section is divided into two sections. In Section7.1, we discuss the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges in the ordinary space. Section 7.2 deals with the above properties within the realm of ACSA to BRST formalism. In other words, we capture the off-shell nilpotency and absolute anticommutativity of the conserved fermionic (anti-)BRST charges in the superspace by taking the theoretical inputs from ACSA.


### 7.1. Nilpotency and Anticommutativity: Ordinary Space

The perfect symmetry invariances ofLB¯ under the anti-BRST symmetry transformations (cf. Equation (38)) andLB under the BRST symmetry transformations (cf. Equation (37)) allow us to compute the Noether conserved charges by using the standard techniques of the Noether theorem (applied to the action integrals corresponding to the Lagrangians LB¯ and LB) as
(74)QB¯=C¯E2px2+2mpt+B¯2C¯−iB¯C¯̇C¯C−B¯EĖC¯−B¯E2C¯̇,QB=CE2px2+2mpt+BE2Ċ−iBC¯ĊC+BEĖC+B2C,where conserved Q̇B¯=0,Q̇B=0 (anti-)BRST charges are denoted by QB¯B. The conservation law Q̇B¯=0,Q̇B=0 can be proven by using the EL-EOMs derived from the coupled Lagrangians LB¯ and LB. For readers’ convenience, we prove the conservation Q̇B=0 of the BRST charge by using the EL-EOMs derived from LB in Appendix C.First of all, we concentrate on the proof of the off-shell nilpotency properties of the (anti-)BRST chargesQB¯B. In this context, we note that the following EL-EOMs w.r.t. the variable E from LB¯ and LB, respectively, yield the following:
(75)B¯̇E−iEC¯̇Ċ+iEC¯C¨−12px2+2mpt=0,ḂE+iEC¯̇Ċ−iEC¯¨C+12px2+2mpt=0.The above equations can be used to recastQB¯ and QB as follows:
(76)QB¯1=E2B¯̇C¯−B¯C¯̇+iC¯̇C¯Ċ−iB¯C¯̇C¯C−B¯EĖC¯+B¯2C¯,QB1=E2BĊ−ḂC−iC¯̇ĊC−iBC¯ĊC+BEĖC+B2C.Using the following EL-EOMs w.r.t. the variablesC and B¯, respectively, from LB¯, namely,
(77)iB¯C¯̇+2iB¯̇C¯−3iEĖC¯̇−iE2C¯¨−iĖ2C¯−iEE¨C¯+C¯¨C¯C+2C¯̇C¯Ċ=0,B¯=EĖ−i2C¯Ċ+C¯̇C,we obtain the following exact and interesting expression for the anti-BRST charge:
(78)QB¯1⟶QB¯2=E2B¯̇C¯−B¯C¯̇+iC¯̇C¯Ċ+iE2C¯¨C¯C+2iEĖC¯̇C¯C≡sabiE2C¯Ċ−C¯̇C.At this juncture, we apply the basic principle behind the relationship between the continuous symmetry transformations (e.g.,sab) and its generator QB¯2 which implies that
(79)sabQB¯2=−iQB¯2,QB¯2=0⇒QB¯22=0⇔sab2=0.Thus, we observe that the off-shell nilpotencyQB¯22=0 of the anti-BRST charge QB¯2 and the anti-BRST symmetry transformations sab are interrelated. Thus, we have proven the off-shell nilpotency of the anti-BRST charge QB¯2. In an exactly similar fashion, we exploit the following EL-EOMs w.r.t. the variables C¯ and B from the Lagrangian LB:
(80)iBĊ+2iḂC+3iEĖĊ+iE2C¨+iĖ2C+iEE¨C+C¯C¨C+2C¯̇ĊC=0,B=−EĖ+i2C¯̇C+C¯Ċ,to recast the BRST charge QB1 into another interesting form (i.e., QB2) as
(81)QB1⟶QB2=E2BĊ−iC¯̇ĊC−ḂC−2iEĖC¯ĊC−iE2C¯C¨C≡sbiE2C¯̇C−C¯Ċ,which turns out to be an exact quantity w.r.t. sb. Thus, we find that we have the following:
(82)sbQB2=−iQB2,QB2=0⇒QB22=0⇔sb2=0.In other words, we have proven the off-shell nilpotencyQB22=0 of the BRST charge QB2. Once again, we find that the off-shell nilpotency sb2=0 of the BRST symmetry transformations and the off-shell nilpotency QB22=0 are intertwined in an intimate manner.We now focus on the proof of the absolute anticommutativity property of the BRST charge with the anti-BRST charge and vice versa. First of all, let us focus on the BRST chargeQB2 (cf. Equation (81)). Using the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0, we can easily check the following transformation:
(83)QB2⟶QB3=E2B¯̇C−2iC¯̇ĊC−B¯Ċ−2iEĖC¯ĊC≡sabiE2ĊC.In other words, we have been able to express the above BRST charge as anexact form w.r.t. the anti-BRST symmetry transformations sab. This is an interesting observation because using the relationship between the continuous symmetry transformations and their generators, we can obtain the following from (83), namely,
(84)sabQB3=−iQB3,QB¯3=0⇔sab2=0.Thus, we have been able to demonstrate that the absolute anticommutativity of the BRST chargewith the anti-BRST charge is connected with the off-shell nilpotency sab2=0 of the anti-BRST symmetry transformation sab. In an exactly similar fashion, we can have a different form of the anti-BRST charge QB¯2 (cf. Equation (78)) by using the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0. In other words, we have the following interesting transformation:
(85)QB¯3⟶QB¯3=E2BC¯̇+2iC¯̇C¯Ċ−ḂC¯+2iEĖC¯̇C¯C≡sbiE2C¯̇C¯.It is straightforward to note that we have the following relationship:(86)sbQB¯3=−iQB¯3,QB3=0⇔sb2=0.In other words, we point out that the absolute anticommutativity of the anti-BRST chargewith the BRST charge is intimately connected with the off-shell nilpotency sb2=0 of the BRST symmetry transformations sb. This completes our discussions on the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges in the ordinary space. In a subtle manner, the observations in (83) and (85) prove the validity of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0on our theory.


### 7.2. Nilpotency and Anticommutativity: ACSA

The key observations of Section7.1 can be translated into the superspace by using the basic terminology of ACSA. Keeping in our mind the mappings ∂θ¯↔sb and ∂θ↔sab, we note that the (anti-)BRST charges QB¯B (cf. Equations (78) and (81)) can be expressed as
(87)QB¯=∂∂θiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ=∫dθiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ,(88)QB=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯Fbτ,θ¯−F¯bτ,θ¯Ḟbτ,θ¯=∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯Fbτ,θ¯−F¯bτ,θ¯Ḟbτ,θ¯.It is straightforward to observe that we have the following:(89)∂∂θQB¯=0⇔sabQB¯=0⇔QB¯2=0⇔∂θ2=0,∂∂θ¯QB=0⇔sbQB=0⇔QB2=0⇔∂θ¯2=0.Thus, the off-shell nilpotency of the (anti-)BRST charges is connected with the nilpotency∂θ2=0,∂θ¯2=0 of the translational generators ∂θ,∂θ¯ along the Grassmannian directions of the chiral and antichiral1,1-dimensional super submanifolds. This observation is consistent with our discussion of the nilpotency property in the ordinary space if we remember the mappings sb↔∂θ¯ and sab↔∂θ [14–16].As far as the absolute anticommutativity property is concerned, we note that the expressions of the (anti-)BRST charges in (83) and (85) can be translated into the superspace where we can exploit the theoretical tools of ACSA. To accomplish this goal, we keep in our knowledge the mappings sb↔∂θ¯ and sab↔∂θ to recast the expressions (83) and (85) as
(90)QB¯3=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯≡∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯,QB3=∂∂θ−iEabτ,θEabτ,θḞabτ,θFabτ,θ≡∫dθ−iEabτ,θEabτ,θḞabτ,θFabτ,θ.It is now straightforward to check that the following are true, namely,(91)∂θ¯QB¯3=0⇔sbQB¯3=0⇔QB¯3,QB3=0⇔∂θ¯2=0,∂θQB3=0⇔sabQB3=0⇔QB3,QB¯3=0⇔∂θ2=0,which establishes the fact that the ACSA to BRST formalism distinguishes between the two types of absolute anticommutativity properties. In other words, we observe that the absolute anticommutativity of the BRST charge with the anti-BRST charge is connected with the nilpotency ∂θ2=0 of the translational generator ∂θ along the Grassmannian direction of the 1,1-dimensional chiral super submanifold. On the contrary, the absolute anticommutativity of the anti-BRST charge with the BRST charge is connected with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ along the Grassmannian direction of the 1,1-dimensional antichiral super submanifold.


## 7.1. Nilpotency and Anticommutativity: Ordinary Space

The perfect symmetry invariances ofLB¯ under the anti-BRST symmetry transformations (cf. Equation (38)) andLB under the BRST symmetry transformations (cf. Equation (37)) allow us to compute the Noether conserved charges by using the standard techniques of the Noether theorem (applied to the action integrals corresponding to the Lagrangians LB¯ and LB) as
(74)QB¯=C¯E2px2+2mpt+B¯2C¯−iB¯C¯̇C¯C−B¯EĖC¯−B¯E2C¯̇,QB=CE2px2+2mpt+BE2Ċ−iBC¯ĊC+BEĖC+B2C,where conserved Q̇B¯=0,Q̇B=0 (anti-)BRST charges are denoted by QB¯B. The conservation law Q̇B¯=0,Q̇B=0 can be proven by using the EL-EOMs derived from the coupled Lagrangians LB¯ and LB. For readers’ convenience, we prove the conservation Q̇B=0 of the BRST charge by using the EL-EOMs derived from LB in Appendix C.First of all, we concentrate on the proof of the off-shell nilpotency properties of the (anti-)BRST chargesQB¯B. In this context, we note that the following EL-EOMs w.r.t. the variable E from LB¯ and LB, respectively, yield the following:
(75)B¯̇E−iEC¯̇Ċ+iEC¯C¨−12px2+2mpt=0,ḂE+iEC¯̇Ċ−iEC¯¨C+12px2+2mpt=0.The above equations can be used to recastQB¯ and QB as follows:
(76)QB¯1=E2B¯̇C¯−B¯C¯̇+iC¯̇C¯Ċ−iB¯C¯̇C¯C−B¯EĖC¯+B¯2C¯,QB1=E2BĊ−ḂC−iC¯̇ĊC−iBC¯ĊC+BEĖC+B2C.Using the following EL-EOMs w.r.t. the variablesC and B¯, respectively, from LB¯, namely,
(77)iB¯C¯̇+2iB¯̇C¯−3iEĖC¯̇−iE2C¯¨−iĖ2C¯−iEE¨C¯+C¯¨C¯C+2C¯̇C¯Ċ=0,B¯=EĖ−i2C¯Ċ+C¯̇C,we obtain the following exact and interesting expression for the anti-BRST charge:
(78)QB¯1⟶QB¯2=E2B¯̇C¯−B¯C¯̇+iC¯̇C¯Ċ+iE2C¯¨C¯C+2iEĖC¯̇C¯C≡sabiE2C¯Ċ−C¯̇C.At this juncture, we apply the basic principle behind the relationship between the continuous symmetry transformations (e.g.,sab) and its generator QB¯2 which implies that
(79)sabQB¯2=−iQB¯2,QB¯2=0⇒QB¯22=0⇔sab2=0.Thus, we observe that the off-shell nilpotencyQB¯22=0 of the anti-BRST charge QB¯2 and the anti-BRST symmetry transformations sab are interrelated. Thus, we have proven the off-shell nilpotency of the anti-BRST charge QB¯2. In an exactly similar fashion, we exploit the following EL-EOMs w.r.t. the variables C¯ and B from the Lagrangian LB:
(80)iBĊ+2iḂC+3iEĖĊ+iE2C¨+iĖ2C+iEE¨C+C¯C¨C+2C¯̇ĊC=0,B=−EĖ+i2C¯̇C+C¯Ċ,to recast the BRST charge QB1 into another interesting form (i.e., QB2) as
(81)QB1⟶QB2=E2BĊ−iC¯̇ĊC−ḂC−2iEĖC¯ĊC−iE2C¯C¨C≡sbiE2C¯̇C−C¯Ċ,which turns out to be an exact quantity w.r.t. sb. Thus, we find that we have the following:
(82)sbQB2=−iQB2,QB2=0⇒QB22=0⇔sb2=0.In other words, we have proven the off-shell nilpotencyQB22=0 of the BRST charge QB2. Once again, we find that the off-shell nilpotency sb2=0 of the BRST symmetry transformations and the off-shell nilpotency QB22=0 are intertwined in an intimate manner.We now focus on the proof of the absolute anticommutativity property of the BRST charge with the anti-BRST charge and vice versa. First of all, let us focus on the BRST chargeQB2 (cf. Equation (81)). Using the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0, we can easily check the following transformation:
(83)QB2⟶QB3=E2B¯̇C−2iC¯̇ĊC−B¯Ċ−2iEĖC¯ĊC≡sabiE2ĊC.In other words, we have been able to express the above BRST charge as anexact form w.r.t. the anti-BRST symmetry transformations sab. This is an interesting observation because using the relationship between the continuous symmetry transformations and their generators, we can obtain the following from (83), namely,
(84)sabQB3=−iQB3,QB¯3=0⇔sab2=0.Thus, we have been able to demonstrate that the absolute anticommutativity of the BRST chargewith the anti-BRST charge is connected with the off-shell nilpotency sab2=0 of the anti-BRST symmetry transformation sab. In an exactly similar fashion, we can have a different form of the anti-BRST charge QB¯2 (cf. Equation (78)) by using the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0. In other words, we have the following interesting transformation:
(85)QB¯3⟶QB¯3=E2BC¯̇+2iC¯̇C¯Ċ−ḂC¯+2iEĖC¯̇C¯C≡sbiE2C¯̇C¯.It is straightforward to note that we have the following relationship:(86)sbQB¯3=−iQB¯3,QB3=0⇔sb2=0.In other words, we point out that the absolute anticommutativity of the anti-BRST chargewith the BRST charge is intimately connected with the off-shell nilpotency sb2=0 of the BRST symmetry transformations sb. This completes our discussions on the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges in the ordinary space. In a subtle manner, the observations in (83) and (85) prove the validity of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0on our theory.


## 7.2. Nilpotency and Anticommutativity: ACSA

The key observations of Section7.1 can be translated into the superspace by using the basic terminology of ACSA. Keeping in our mind the mappings ∂θ¯↔sb and ∂θ↔sab, we note that the (anti-)BRST charges QB¯B (cf. Equations (78) and (81)) can be expressed as
(87)QB¯=∂∂θiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ=∫dθiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ,(88)QB=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯Fbτ,θ¯−F¯bτ,θ¯Ḟbτ,θ¯=∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯Fbτ,θ¯−F¯bτ,θ¯Ḟbτ,θ¯.It is straightforward to observe that we have the following:(89)∂∂θQB¯=0⇔sabQB¯=0⇔QB¯2=0⇔∂θ2=0,∂∂θ¯QB=0⇔sbQB=0⇔QB2=0⇔∂θ¯2=0.Thus, the off-shell nilpotency of the (anti-)BRST charges is connected with the nilpotency∂θ2=0,∂θ¯2=0 of the translational generators ∂θ,∂θ¯ along the Grassmannian directions of the chiral and antichiral1,1-dimensional super submanifolds. This observation is consistent with our discussion of the nilpotency property in the ordinary space if we remember the mappings sb↔∂θ¯ and sab↔∂θ [14–16].As far as the absolute anticommutativity property is concerned, we note that the expressions of the (anti-)BRST charges in (83) and (85) can be translated into the superspace where we can exploit the theoretical tools of ACSA. To accomplish this goal, we keep in our knowledge the mappings sb↔∂θ¯ and sab↔∂θ to recast the expressions (83) and (85) as
(90)QB¯3=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯≡∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯,QB3=∂∂θ−iEabτ,θEabτ,θḞabτ,θFabτ,θ≡∫dθ−iEabτ,θEabτ,θḞabτ,θFabτ,θ.It is now straightforward to check that the following are true, namely,(91)∂θ¯QB¯3=0⇔sbQB¯3=0⇔QB¯3,QB3=0⇔∂θ¯2=0,∂θQB3=0⇔sabQB3=0⇔QB3,QB¯3=0⇔∂θ2=0,which establishes the fact that the ACSA to BRST formalism distinguishes between the two types of absolute anticommutativity properties. In other words, we observe that the absolute anticommutativity of the BRST charge with the anti-BRST charge is connected with the nilpotency ∂θ2=0 of the translational generator ∂θ along the Grassmannian direction of the 1,1-dimensional chiral super submanifold. On the contrary, the absolute anticommutativity of the anti-BRST charge with the BRST charge is connected with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ along the Grassmannian direction of the 1,1-dimensional antichiral super submanifold.


## 8. Conclusions

In our present endeavor, we have purposely taken a reparameterization invariant NR and NSUSY system so that we could discuss theoretical aspects that are different from our earlier works on the NSUSY relativisticscalar and SUSY relativistic spinning particles [23, 24]. We have demonstrated, however, in our present investigation that (i) the CF-type restriction and (ii) the sum of gauge-fixing and Faddeev-Popov ghost terms are the same for our present NR and NSUSY system as have been shown by us for the relativistic particles (in our earlier works [23, 24]). The above observations are interesting results of our present investigation which establish the universality of the (anti-)BRST invariant CF-type restriction for the 1D diffeomorphism invariant (i.e., reparameterization) theories.The CF-type restriction(s) are the hallmark of aquantum theory that is BRST-quantized. In fact, for a D-dimensional diffeomorphism invariant theory, it has been shown [22, 33] that the universal CF-type restriction for a BRST-quantized theory is Bμ+B¯μ+iC¯ρ∂ρCμ+Cρ∂ρC¯μ=0, where μ,ρ=0,1,2,⋯,D−1, Bμ and B¯μ are the Nakanishi-Lautrup auxiliary fields, and the (anti)ghost fields C¯μCμ correspond to the D-dimensional diffeomorphism parameter εμx in the infinitesimal transformation xμ⟶xμ′=xμ−εμx. The universality of the above CF-type restriction implies that, for our 1D diffeomorphism (i.e., reparameterization) invariant theory, the CF-type restriction is B+B¯+iC¯Ċ−C¯̇C=0. This is what we have obtained from various theoretical tricks in our present endeavor. The existence of the CF-type restriction is very fundamental to a BRST-quantized theory as it is connected with the geometrical objects called gerbes [7, 8]. Physically, the existence of the CF-type restriction leads to the independent nature of the BRST and anti-BRST symmetries (and corresponding conserved charges) at the quantum level (that are connected with a given classical local symmetry).Our present work (and earlier works [23, 24]) can be generalized to the cases of (super)string and gravitational theories which are also diffeomorphism invariant. In fact, in our earlier work on a bosonic string theory [34], we have shown the existence of the CF-type restriction in the context of its BRST quantization and it has turned out to be the 2D version of the universal CF-type restriction for the D-dimensional diffeomorphism invariant theory. It is gratifying to pinpoint the fact that we have derived the CF-type restrictions Ba+B¯a+iC¯m∂mCa+Cm∂mC¯a=0 (with a,m=0,1) for a model of bosonic string theory [34]. This has happened because the bosonic string theory has the 2D diffeomorphism invariance on the 2D worldsheet. We have applied the beautiful blend of MBTSA and ACSA to derive all the (anti-)BRST symmetries as well as the 2D version of the CF-type restriction in the case of a bosonic string theory of our interest [35]. In our present investigation, we have utilized only one and/or two Grassmannian variables because there are only two nilpotent symmetries in the theory. If a theory is endowed with the nilpotent (anti-)BRST as well as (anti-)co-BRST symmetries, then we have to invoke four numbers of the Grassmannian variables. We are currently exploring such kinds of possibilities.



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*Source: 5593434-2021-07-23.xml*