Supervariable and BRST Approaches to a Reparameterization Invariant Non-Relativistic System
aa r X i v : . [ h e p - t h ] D ec Supervariable and BRST Approaches to aReparameterization Invariant Non-Relativistic System
A. K. Rao ( a ) , A. Tripathi ( a ) , R. P. Malik ( a,b )( a ) Physics Department, Institute of Science,Banaras Hindu University, Varanasi - 221 005, (U.P.), India ( b ) DST Centre for Interdisciplinary Mathematical Sciences,Institute of Science, Banaras Hindu University, Varanasi - 221 005, India e-mails: [email protected]; [email protected]; [email protected]
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 absolutelyanticommuting) (anti-)BRST symmetry transformations for the reparameterization invari-ant model of a non-relativistic (NR) free particle whose space ( x ) and time ( t ) variablesare function of an evolution parameter ( τ ). The infinitesimal reparameterization (i.e. 1Ddiffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution pa-rameter ( τ ). We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA)as well as the (anti-)chiral supervariable approach (ACSA) to BRST formalism to discussvarious aspects of our present system. For this purpose, our 1D ordinary theory (parame-terized by τ ) is generalized onto a (1 , Z M = ( τ, θ, ¯ θ ) where a pair of Grassmannian variables satisfythe fermionic relationships: θ = ¯ θ = 0 , θ ¯ θ + ¯ θ θ = 0 and τ is the bosonic evolution param-eter. In the context of ACSA, we take into account only the (1, 1)-dimensional (anti-)chiralsuper sub-manifolds of the general (1, 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 thegauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly 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.PACS numbers: 11.15.-q; 12.20.-m; 11.30.Pb.; 02.20.+b Keywords : Reparameterization invariant non-SUSY and non-relativistic system; modified
BT-supervariable approach; horizontality condition; (anti-)chiral supervariable approach;symmetry invariant restrictions; (anti-)BRST symmetries and (anti-)BRST charges; off-shell nilpotency and absolute anticommutativity properties; CF-type restriction
Introduction
During the last few years, there has been an upsurge of interest in the study of diffeomor-phism invariant theories because one of the key and decisive features of the gravitationaland (super)string theories is the observation that they respect the classical diffeomorphismsymmetry transformations. The latter symmetry transformations can be exploited withinthe framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism [1-4] where the classical diffeomorphism symmetry transformation is elevated to the quantum (anti-)BRST symme-try transformations. In fact, it is the key feature of the BRST formalism that the clas-sical diffeomorphism transformation parameter is traded with the fermionic (anti-)ghostfields/variables at the quantum level. In other words, the (anti-)BRST transformationsare of the supersymmetric (SUSY) kind where the bosonic type of fields/variables trans-form to the fermionic type fields/variables and vice-versa. Two of the key properties of the(anti-)BRST transformations are the on-shell/off-shell nilpotency and absolute anticommu-tativity. 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 inter-pretation. The nilpotency property (i.e. fermionic nature) of the (anti-)BRST symmetries(and their corresponding charges) is also connected with some aspects of the cohomologicalproperties of differential geometry and a few decisive features of supersymmetry.The BRST formalism has been exploited in the covariant canonical quantization of the gauge and diffeomorphism invariant theories in the past. At the classical level, the gaugetheories are characterized by the existence of the first-class constraints [5, 6] on them. Thisfundamental feature is translated, at the quantum level, into the language of the existenceof the Curci-Ferrari (CF)-type restriction(s) when the classical theory is quantized by ex-ploiting the theoretical richness of BRST formalism. Hence, the existence of the CF-typerestriction(s) is the key signature of a BRST quantized version of the gauge and/or dif-feomorphism invariant theory. The CF-type restrictions are (i) deeply connected with thegeometrical objects called gerbes [7, 8], (ii) responsible for the absolute anticommutativityof the quantum (anti-)BRST transformations, and (iii) the root-cause behind the existenceof the coupled (but equivalent) Lagrangians/Lagrangian densities for the (anti-)BRST in-variant quantum theories (corresponding to the classically gauge/diffeomorphism invarianttheories). 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 gaugetheory where the non-trivial CF-condition exists [9].It is the supervariable/superfield approaches [10-21] to BRST formalism that provide thegeometrical 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 isthe horizontality condition (HC) that plays a decisive role as it leads to (i) the derivationof the (anti-)BRST symmetry transformations for only the gauge and anti-)ghost fields, aswell as (ii) the derivation of the CF-type restriction(s). The augmented version of the su-perfield 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 derivationof the (anti-)BRST symmetry transformations for the gauge, (anti-)ghost and matter fields2 ogether in an interacting gauge theory. It has been a challenging problem to incorporatethe diffeomorphism transformation within the ambit of superfield approach to gauge theo-ries (see, e.g. [14-16]) so that one can discuss the gravitational and (super)string theorieswithin the framework of USFA/AVSA. In this direction, a breakthrough has recently beenmade 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 dif-feomorphism invariant theory. This approach has been christened by us as the modified
Bonora-Tonin (BT) superfield approach (MBTSA) to BRST formalism. In a recent coupleof 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 con-text 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 reparame-terization (i.e. 1D diffeomorphism) invariant theory of a massive non-supersymmetric(NSUSY) and non-relativistic (NR) free particle where the standard
NR Lagrangian L ( t )0 ( x, ˙ x ) = m ˙ x (with ˙ x = dx/dt ) is rendered reparameterization invariant by treatingthe “time” variable on a par with x variable [30] that are parameterized by an evolutionparameter τ such that the new Lagrangian L ( τ )0 ( x, ˙ x, t, ˙ t ) = m ˙ x t where ˙ x = dx/dτ and˙ t = dt/dτ . The latter Lagrangian respects the reparameterization symmetry [31, 32] andit has been discussed in different theoretical settings where the noncommutativity of thespacetime appears by the symmetry considerations, constraints analysis, redefinitions ofvariables, etc. This reparameterization invariant model of the free particle ( ˙ p x = ˙ p t = 0)has been discussed by us within the frameworks of BRST formalism as well as quantumgroups [32]. However, in the BRST analysis, we have exploited the gauge symmetry ofthis NSUSY and NR system [32] without discussing anything about the reparameterizationtransformations. In our present investigation, we have applied the beautiful blend of the-oretical ideas from MBTSA and ACSA to derive the proper (anti-)BRST symmetries andCF-type restriction for this NR system. This model is interesting in its own right as it is aNR 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 treatedas 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 ourpresent endeavor. First, so far, we have been able to apply the beautiful blend of theoreticalideas behind MBTSA and ACSA to BRST formalism in the cases of reparameterizationinvariant systems of (i) relativistic non-supersymmetric (NSUSY) scalar free particle, and(ii) supersymmetric (SUSY) (i.e. spinning) relativistic free particle. Thus, it has beena challenging problem for us to apply the same theoretical ideas to discuss the NSUSYand non-relativistic (NR) system of a reparameterization invariant free particle. We haveaccomplished 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 havebeen motivated to see the existence of the same CF-type of restriction in our present caseof reparameterization invariant system of NSUSY and NR free particle. We have been able3o demonstrate that it is the same
CF-type of restriction that exists in the BRST approachto our present NSUSY and NR system. Third, we have found out that the gauge-fixingand Faddeev-Popov (FP) ghost terms for the systems of non-SUSY (i.e. scalar) and SUSY(i.e. spinning) relativistic particles are same within the ambit of BRST formalism. Thus,we have been curious to find out the gauge-fixing and FP-ghost terms for our presentnon-SUSY and NR system. It is surprising that the above terms are same for our presentsystem, too. Finally, our present work is our another modest initial step towards our maingoal of applying the theoretical potential of MBTSA and ACSA to the physical four (3+1)-dimensional (4D) diffeomorphism invariant gravitational and (super)sting theories in thehigher dimensions (i.e.
D >
4) of spacetime.The contents of our present endeavor are organized as follows. In Sec. 2, we recapitu-late the bare essentials of the Lagrangian formulation of our reparameterization invariantnon-relativistic system and discuss the BRST quantization of this model by exploiting its infinitesimal gauge symmetry transformations. Our Sec. 3 is devoted to the applicationof MBTSA for the derivation of (i) the quantum (anti-)BRST symmetry transformationsfor the phase space variables, and (ii) the underlying (anti-)BRST invariant Curci-Ferrari(CF)-type restriction (corresponding to the classical infinitesimal reparameterization sym-metry transformations). The theoretical content of Sec. 4 is related to the derivation of the full set of (anti-)BRST symmetry transformation by requiring the off-shell nilpotency andabsolute anticommutativity properties. We also show the existence of the CF-type restric-tion and deduce the coupled (anti-)BRST invariant Lagrangians for our theory. In Sec. 5,we derive the (anti-)BRST symmetry transformations for the other variables (i.e. differentfrom the phase variables) within the purview of ACSA. Our Sec. 6 deals with the proofof the equivalence of the coupled Lagrangians within the framework of ACSA to BRSTformalism. In Sec. 7, we prove the off-shell nilpotency and absolute anticommutativity ofthe conserved (anti-)BRST charges in the ordinary space and superspace (by exploiting thetheoretical richness of ACSA to BRST formalism). Finally, in Sec. 8, we discuss our keyresults and point out a few future directions for further investigation(s).In our Appendices A, B and C, we perform some explicit computations that supplementas well as corroborate the key claims that have been made in the main body of our text.
Convention and Notations : We adopt the convention of the left-derivative w.r.t. all thefermionic variables of our theory. In the whole body of the text, we denote the fermionic(anti-)BRST transformations by the symbols s ( a ) b and corresponding conserved and nilpo-tent charges carry the notations Q ( a ) b and Q ( ¯ B ) B in different contexts. The general (1 , Z M = ( τ, θ, ¯ θ ) and its chiral and anti-chiral super sub-manifolds 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: θ = 0 , ¯ θ = 0 , θ ¯ θ + ¯ θ θ = 0. Our present section is divided into two parts. In subsection 2.1, we discuss the classical infinitesimal reparameterization and gauge symmetry transformations. Our subsection 2.24eals with the BRST quantization of our system by exploiting the classical gauge symmetrytransformations (which are infinitesimal and continuous).
We begin with the three equivalent reparameterization invariant Lagrangians for the freenon-relativistic and non-SUSY particle as (see e.g. [32] for details) L ( x, ˙ x, t, ˙ t ) = m ˙ x t ,L f ( x, ˙ x, t, ˙ t, p x , p t ) = p x ˙ x + p t ˙ t − E ( p x + 2 m p t ) ,L s ( x, ˙ x, t, ˙ t ) = ˙ x E + m ˙ x t h E m ˙ t − i , (1)where the trajectory of the free non-relativistic particle is embedded in a 2D configurationspace characterized by the coordinates [ x ( τ ) , t ( τ )] and the parameter τ specifies the trajec-tory of the particle as an evolution parameter. The momenta variables ( p x , p t ) are definedby: p x = ( ∂ L/∂ ˙ x ), p t = ( ∂ L/∂ ˙ t ) where ˙ x = ( d x/d τ ), ˙ t = ( d t/d τ ) are the generalized“velocities” w.r.t. the coordinates [ x ( τ ) , t ( τ )] and L stands for any of the three Lagrangiansof Eq. (1). It is self-evident that the 4D phase space, corresponding to the 2D configu-ration space, is characterized by [ x ( τ ) , t ( τ ) , p x ( τ ) , p t ( τ )]. In the above equation (1), themass of the non-relativistic particle is denoted by m and E ( τ ) is the Lagrange multipliervariable that incorporates the constraint p x + 2 m p t ≈ L f and L s . It isstraightforward to note that L and L s contain variables (and their first-order derivative) inthe denominator but L f (i.e. the first-order Lagrangian) does not incorporate any variable(and/or its derivative) in its denominator. Furthermore, the starting Lagrangian ∗ L does not permit the massless ( m = 0) limit but the massless ( m = 0) limit is well-defined for L f and the second-order Lagrangian L s . We would like to stress that the Lagrange multipliervariable behaves like the “gauge” variable due to its transformation property.For our further discussions, we shall concentrate on the first-order Lagrangian L f be-cause it has maximum number of variables (i.e. x, p x , t, p t , E ), allows massless limit andthere 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 1Ddiffeomorphism (i.e. reparameterization) symmetry transformations ( δ r ), namely; δ r x = ǫ ˙ x, δ r p x = ǫ ˙ p x , δ r t = ǫ ˙ t, δ r p t = ǫ ˙ p t ,δ r E = dd τ ( ǫ E ) , δ r L f = dd τ ( ǫ L f ) , (2)the action integral S = R + ∞−∞ d τ L f remains invariant (i.e. δ r S = 0) for the physically well-defined variables in L f and the infinitesimal diffeomorphism transformation parameter ǫ ( τ ) ∗ For a free massive NR particle, the standard action integral is: S t = R + ∞−∞ d t ( m ˙ x ) where ˙ x = ( dx/dt )and time “t” is the evolution parameter. This action has no reparameterization invariance. If the evolutionparameter is τ , then, the action integral is: S τ = ( m/ R + ∞−∞ d τ ( dt/dτ ) ( dx/dτ ) ( dτ /dt ) ( dx/dτ ) ( dτ /dt )which leads [32, 31] to the final action integral as: S τ = ( m/ R + ∞−∞ d τ [( dx/dτ ) ( dτ /dt )] ≡ R + ∞−∞ d τ L .Hence, the starting Lagrangian becomes L ( x, ˙ x, t, ˙ t ) = m ˙ x t where ˙ x = ( dx/dτ ) and ˙ t = ( dt/dτ ). τ −→ τ ′ = g ( τ ) = τ − ǫ ( τ ) where g ( τ ) is a physically well-defined function of τ such thatit is finite at τ = 0 and vanishes off at τ = ±∞ . In fact, the infinitesimal and continuousreparameterization symmetry transformation ( δ r ) is defined as: δ r φ ( τ ) = φ ′ ( τ ) − φ ( τ ) forthe generic variable φ ( τ ) = x ( τ ) , p x ( τ ) , t ( τ ) , p t ( τ ) , E ( τ ) of our present theory.The above infinitesimal and continuous reparameterization symmetry transformations(2) encompass in their folds the gauge symmetry transformations ( δ g ) which are generated(see, e.g. [32] for details) by the first-class constraints: Π E ≈ , ( p x + 2 m p t ) ≈ E is the canonical conjugate momentum corresponding to the variable E ( τ ). Using thefollowing Euler-Lagrange equations of motion (EL-EOMs) from L f , namely;˙ p x = 0 , ˙ p t = 0 , ˙ x = E p x , ˙ t = E m, (3) and identifying the transformation parameters ǫ ( τ ) E ( τ ) = ξ ( τ ), we obtain the infinitesimaland continuous gauge symmetry transformations ( δ g ), from the infinitesimal and continuousreparameterization symmetry transformations (2), as follows: δ g x = ξ p x , δ g t = ξ m, δ g p x = 0 , δ g p t = 0 , δ g E = ˙ ξ. (4)It is elementary now to check that the first-order Lagrangian ( L f ) transforms to a totalderivative under the infinitesimal and continuous gauge transformations ( δ g ), namely; δ g L f = dd τ h ξ p x i , (5)thereby rendering the action integral S = R + ∞−∞ d τ L f invariant (i.e δ g S = 0) under theinfinitesimal and continuous gauge symmetry transformations ( δ g ).We end this subsection with the following decisive comments. First, the 1D diffeomor-phism (i.e. reparameterization) transformations (2) are more general than the infinitesimaland continuous gauge symmetry transformations (4). Second, the Lagrange multiplier vari-able E ( τ ) behaves like a “gauge” variable due to its transformation: δ g E = ˙ ξ in (4). Third,all the three Lagrangians in (1) are equivalent and all of them respect the infinitesimal andcontinuous gauge and reparameterization symmetry transformations [32]. Fourth, all theLagrangians describe the free motion ( ˙ p x = 0 , ˙ p t = 0) of the NR particle. Hence, oursystem is a non-relativistic free ( ˙ p x = ˙ p t = 0) particle. Fifth, the first-order Lagrangian L f is theoretically more interesting to handle because, as pointed out earlier, it incorpo-rates the maximum number of variables. Finally, we can exploit the reparameterization and gauge symmetry transformations (2) and (4) for the BRST quantization. Followingthe usual BRST prescription, we note that the (anti-)BRST symmetry transformations,corresponding to the classical reparameterization symmetry transformation (2), are s ab x = ¯ C ˙ x, s ab p x = ¯ C ˙ p x , s ab t = ¯ C ˙ t, s ab p t = ¯ C ˙ p t , s ab E = dd τ ( ¯ C E ) ,s b x = C ˙ x, s b p x = C ˙ p x , s b t = C ˙ t, s b p t = C ˙ p t , s b E = dd τ ( C E ) , (6)where ( ¯ C ) C are the fermionic ( C = ¯ C = 0 , C ¯ C + ¯ C C = 0) (anti-)ghost variablescorresponding to the classical infinitesimal diffeomorphism transformation parameter ǫ ( τ )6cf. Eq. (2)]. In exactly similar fashion, the (anti-)BRST symmetry transformations,corresponding to the classical gauge symmetry transformations (4), are s ab x = ¯ c p x , s ab p x = 0 , s ab t = ¯ c m, s ab p t = 0 , s ab E = ˙¯ c,s b x = c p x , s ab p x = 0 , s b t = c m, s b p t = 0 , s b E = ˙ c, (7)where the fermionic ( c = ¯ c = 0 , c ¯ c + ¯ c c = 0) variables (¯ c ) c are the FP-(anti-)ghostvariables corresponding to the classical gauge symmetry transformation parameter ξ ( τ ) ofEq. (4). In addition to the (anti-)BRST symmetry transformations in (6) and (7), we havethe following standard (anti-)BRST transformations s ab C = i ¯ B, s ab ¯ B = 0 , s b ¯ C = i B, s b B = 0 ,s ab c = i ¯ b, s ab ¯ b = 0 , s b ¯ c = i b, s b b = 0 , (8)where the pairs ( B, ¯ B ) and ( b, ¯ b ) are the Nakanishi-Lautrup auxiliary variables in the con-texts of the BRST quantization of our reparameterization and gauge invariant system byexploiting the classical reparameterization and gauge transformations, respectively. We have listed the quantum (anti-)BRST symmetries corresponding to the classical gaugesymmetry transformations (4) in our Eqs. (7) and (8). It is elementary to check that these quantum symmetries are off-shell nilpotent ( s a ) b = 0) of order two. The requirement ofthe absolute anticommutativity ( s b s ab + s ab s b = 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: s ab x = ¯ c p x , s ab p x = 0 , s ab t = ¯ c m, s ab p t = 0 ,s ab E = ˙¯ c, s ab ¯ c = 0 , s ab c = − i b, s ab b = 0 ,s b x = c p x , s b p x = 0 , s b t = c m, s b p t = 0 ,s b E = ˙ c, s b c = 0 , s b ¯ c = i b, s b b = 0 . (9)It is straightforward to check that the above (anti-)BRST symmetry transformations areoff-shell nilpotent ( s a ) b = 0) and absolutely anticommuting ( s b s ab + s ab s b = 0) in nature.The (anti-)BRST invariant Lagrangian L b (which is the generalization of the classical L f to its quantum level) can be written as † : L b = L f + s b h − i ¯ c (cid:16) ˙ E + b (cid:17)i ≡ L f + s ab h i c (cid:16) ˙ E + b (cid:17)i , ≡ L f + s b s ab h i E − ¯ c c i ≡ L f − s ab s b h i E − ¯ c c i . (10) † The structure of gauge-fixing and FP-ghost terms is exactly like the 1-form ( A (1) = dx µ A µ ) Abeliangauge theory where we have the BRST-invariant Lagrangian density: L b = − F µν F µν + s b [ − i ¯ c ( ∂ µ A µ + b )] ≡ − F µν F µν + s ab [ − i c ( ∂ µ A µ + b )] ≡ − F µν F µν + s b s ab [ i A µ A µ − ¯ c c ]. Here A µ is the vectorpotential, F µ ν = ∂ µ A ν − ∂ ν A µ is the field strength tensor and rest of the symbols are same as in Eqs.(10) and (11). Note that the 2-form F (2) = d A (1) = ( d x µ ∧ d x ν ) F µν defines the field strength tensor F µν (where d = d x µ ∂ µ in F (2) = d A (1) stands for the exterior derivative of the differential geometry).
7n other words, we have expressed the gauge-fixing and Faddeev-Popov ghost terms in three different ways which, ultimately, lead to the following expression for L b , namely; L b = L f + b ˙ E + b − i ˙¯ c ˙ c, ≡ p x ˙ x + p t ˙ t − E ( p x + 2 m p t ) + b ˙ E + b − i ˙¯ c ˙ c. (11)It should be noted that we have dropped the total derivative terms in obtaining L b from (10).The above equation demonstrates that we have obtained a unique (anti-)BRST invariantLagrangian. This has happened because the CF-type restriction is trivial (i.e. b + ¯ b = 0)in our simple case of NR system. We can explicitly check that: s b L b = dd τ h c p x + b ˙ c i , s ab L b = dd τ h ¯ c p x + b ˙¯ c i , (12)which lead to the derivation of the conserved (anti-)BRST charges [ Q ( a ) b ] as follows: Q ab = b ˙¯ c + 12 ¯ c ( p x + 2 m p t ) ≡ b ˙¯ c − ˙ b ¯ c,Q b = b ˙ c + 12 c ( p x + 2 m p t ) ≡ b ˙ c − ˙ b c. (13)In the last step, we have used ˙ b = − (1 /
2) ( p x + 2 m p t ) which emerges out as the EL-EOMfrom L b w.r.t. the Lagrange multiplier variable E ( τ ) .We close this sub-section with a few crucial and decisive remarks. First, we can checkthat the (anti-)BRST charges are conserved [ ˙ Q ( a ) b = 0] by using the EL-EOMs. Second,the (anti-)BRST charges [ Q ( a ) b ] are off-shell nilpotent [ Q a ) b = 0] of order two due to the direct observations that: s b Q b = s b [ b ˙ c − ˙ b c ] = 0 and s ab Q ab = s ab [ b ˙¯ c − ˙ b ¯ c ] = 0 whichencode in their folds s b Q b = − i { Q b , Q b } = 0 = ⇒ Q b = 0 and s ab Q ab = − i { Q ab , Q ab } =0 = ⇒ Q ab = 0. Third, the above nilpotency is also encoded in: Q b = s b [ b E + i ˙¯ c c ]implying that s b Q b = 0 due to s b = 0 and we also point out that s ab Q ab = 0 due tothe nilpotency ( s ab = 0) of s ab because Q ab = s ab [ b E + i ¯ c ˙ c ]. Fourth, we observe that s ab Q b = i { Q b , Q ab } ≡ − i b ˙ b + i ˙ b b = 0 and s b Q ab = − i { Q ab , Q b } = i b ˙ b − i b ˙ b = 0 whichexplicitly lead to the conclusion that the off-shell nilpotent charges Q ( a ) b are also absolutelyanticommuting ( Q b Q ab + Q ab Q b = 0) in nature. Fifth, the above observation of the absoluteanticommutativity can be also expressed in terms of the nilpotency property because weobserve that Q b = s ab ( − i ˙ c c ) and Q ab = s b ( i ˙¯ c ¯ c ) which imply that s ab Q b = − i { Q b , Q ab } =0 and s b Q ab = − i { Q ab , Q b } = 0 [due to the off-shell nilpotency ( s ab = 0) of the anti-BRSTas well as the off-shell nilpotency ( s b = 0) of the BRST symmetry transformations]. Sixth,it can be seen that the physical space (i.e. | phys > ) of the total Hilbert space of statesis defined by Q b | phys > = 0 which implies that b | phys > ≡ Π E | phys > = 0 and˙ b | phys > ≡ ( p x + 2 m p t ) | phys > = 0. In other words, the Dirac quantization conditions(with the first-class constraints Π E ≈ , p x + 2 m p t ≈
0) are beautifully satisfied. Finally,physicality criterion Q b | phys > = 0 implies that the two physical states | phys ′ > and | phys > belong to the same cohomological class w.r.t. the nilpotent BRST charge Q b ifthey differ by a BRST exact state (i.e. | phys ′ > = | phys > + Q b | χ > for non-null | χ > ).8 Nilpotent and Anticommuting (Anti-)BRST Sym-metries for the Phase Variables: MBTSA
This section is devoted to the derivation of the transformations: s b x = C ˙ x, s b p x = C ˙ p x , s b t = C ˙ t, s b p t = C ˙ p t , s ab x = ¯ C ˙ x, s ab p x = ¯ C ˙ p x , s ab t = ¯ C ˙ t, s ab p t = ¯ C ˙ p t byexploiting the theoretical tricks of MBTSA. Before we set out to perform this exercise, it isessential to pinpoint the off-shell nilpotency and absolute anticommutativity properties ofthe (anti-)BRST symmetry transformations on the phase variables [cf. Eq. (6)]. It can beeasily checked that the off-shell nilpotency requirement (i.e. s a ) b S = 0 , S = x, p x , t, p t )leads to the (anti-)BRST symmetry transformations for the (anti-)ghost variables as: s ab ¯ C = ¯ C ˙¯ C, s b C = C ˙ C. (14)Furthermore, the absolute anticommutativity requirement: { s b , s ab } S = 0 for the genericphase variable S = x, p x , t, p t leads to the following { s b , s ab } S = i [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] ˙ S = ⇒ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 . (15)In other words, the absolute anticommutativity property ( s b s ab + s ab s b = 0) is satisfied ifand only if we invoke the sanctity of the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0.It goes without saying that the above cited requirements of the off-shell nilpotency andabsolute anticommutativity properties are very sacrosanct within the framework of BRSTapproach to gauge and/or reparameterization invariant theories.Against the backdrop of the above discussions, we set out to deduce the (anti-)BRSTsymmetry transformations: s ab S = ¯ C ˙ S, s b S = C ˙ S (with S = x, p x , t, p t ) and the CF-type restrictions: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 within the framework of MBTSA. Towardsthis end in our mind, first of all, we generalize the classical function g ( τ ) [in τ −→ τ ′ = g ( τ ) ≡ τ − ǫ ( τ )] onto a (1 , g ( τ ) −→ ˜ g ( τ, θ, ¯ θ ) = τ − θ ¯ C ( τ ) − ¯ θ C ( τ ) + θ ¯ θ k ( τ ) , (16)where ( ¯ C ) C variables are the (anti-)ghost variables of Eq. (6) and k ( τ ) is a secondaryvariable that has to be determined from the consistency conditions [that include the off-shellnilpotency as well as absolute anticommutativity requirements]. It will be noted that, due tothe mappings: s b ↔ ∂ ¯ θ | θ =0 , s ab ↔ ∂ θ | ¯ θ =0 [14-16], we have taken the coefficients of θ and ¯ θ in Eq. (16) as the (anti-)ghost variables ( ¯ C ) C . This has been done due to our observation inthe infinitesimal reparameterization symmetry transformation ( δ r ) [where δ r τ = − ǫ ( τ )] atthe classical level. Following the basic tenet of BRST formalism, the infinitesimal parameter ǫ ( τ ) has been replaced (in the BRST quantized theory) by the (anti-)ghost variables therebyleading to the (anti-)BRST symmetry transformations: s ab τ = − ¯ C, s b τ = − C .For our present 1D diffeomorphism (i.e. reparameterization) invariant theory, thegeneric variable S ( τ ) 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 Grass-mannian directions of the (1, 2)-dimensional supermanifold, namely;˜ S (cid:2) ˜ g ( τ, θ, ¯ θ ) , θ, ¯ θ ) (cid:3) = S (cid:2) ˜ g ( τ, θ, ¯ θ ) (cid:3) + θ ¯ R (cid:2) ˜ g ( τ, θ, ¯ θ ) (cid:3) + ¯ θ R (cid:2) ˜ g ( τ, θ, ¯ θ ) (cid:3) + θ ¯ θ Q (cid:2) ˜ g ( τ, θ, ¯ θ ) (cid:3) , (17)9here ˜ g ( τ, θ, ¯ θ ) has been given in Eq. (16). It should be noted that all the primary as wellas the secondary supervariables on the r.h.s. of (17) are function of the (1 , all the primary as well as the secondary supervariables as: θ ¯ θ Q ( τ − θ ¯ C − ¯ θ C + θ ¯ θ k ) = θ ¯ θ Q ( τ ) , ¯ θ R ( τ − θ ¯ C − ¯ θ C + θ ¯ θ k ) = ¯ θ R ( τ ) + θ ¯ θ ¯ C ( τ ) ˙ R ( τ ) ,θ ¯ R ( τ − θ ¯ C − ¯ θ C + θ ¯ θ k ) = θ ¯ R ( τ ) − θ ¯ θ C ( τ ) ˙¯ R ( τ ) , S ( τ − θ ¯ C − ¯ θ C + θ ¯ θ k ) = S ( τ ) − θ ¯ C ( τ ) ˙ S ( τ ) − ¯ θ C ( τ ) ˙ S ( τ )+ θ ¯ θ (cid:2) k ( τ ) ˙ S ( τ ) + C ( τ ) ¯ C ( τ ) ¨ S ( τ ) (cid:3) . (18)Collecting all these terms and substituting them into (17), we obtain the following superexpansion for the supervariable on the (1, 2)-dimensional supermanifold, namely;˜ S (cid:2) ˜ g ( τ, θ, ¯ θ ) , θ, ¯ θ (cid:3) = S ( τ ) + θ ( ¯ R − ¯ C ˙ S ) + ¯ θ ( R − C ˙ S )+ θ ¯ θ (cid:2) Q + ¯ C ˙ R + C ˙¯ R + k ˙ S + C ¯ C ¨ S (cid:3) . (19)We now exploit the horizontality condition (HC) which physically implies that all the scalar variables should not transform at all under any kind of spacetime, internal, supersymmetric,etc., transformations. With respect to the 1D space of trajectory of the particle, all thesupervariables on the l.h.s. and r.h.s. of Eq. (18) are scalars . The HC, in our case, is:˜ S (cid:2) ˜ g ( τ, θ, ¯ θ ) , θ, ¯ θ ) (cid:3) = S ( τ ) . (20)The above equality implies that all the coefficients of θ , ¯ θ and θ ¯ θ of Eq. (19) should be setequal to zero. In other words, we have the following: R = C ˙ S, ¯ R = ¯ C ˙ S, Q = C ˙¯ R − ¯ C ˙ R − k ˙ S − C ¯ C ¨ S. (21)Substitutions of the values of R and ¯ R into the expression for Q leads to the following: Q = − ( ¯ C ˙ C + ˙¯ C C ) ˙ S − k ˙ S − ¯ C C ¨ S. (22)As explained before Eq. (20) (i.e. exploiting the key properties of scalars ), it is evidentthat (17) can be finally written as˜ S (cid:2) ˜ g ( τ, θ, ¯ θ ) , θ, ¯ θ ) (cid:3) = S ( τ ) + θ ¯ R ( τ ) + ¯ θ R ( τ ) + θ ¯ θ Q ( τ ) ≡ S ( τ ) + θ ( s ab S ) + ¯ θ ( s b S ) + θ ¯ θ ( s b s ab S ) , (23)where, due to the well known mappings: s b ↔ ∂ ¯ θ | θ =0 , s ab ↔ ∂ θ | ¯ θ =0 [14-16], the coefficientsof θ and ¯ θ are the anti-BRST and BRST symmetry transformations [cf. Eq. (6)]. Wepoint out that the key properties of scalars on the r.h.s. of Eq. (17) implies that we have: S [˜ g ( τ, θ, ¯ θ )] = S ( τ ) , R [˜ g ( τ, θ, ¯ θ )] = R ( τ ) , ¯ R [˜ g ( τ, θ, ¯ θ )] = ¯ R ( τ ) and Q [˜ g ( τ, θ, ¯ θ )] = Q ( τ ).A comparison between (21) and (23) implies that we have already derived the nilpotent(anti-)BRST symmetry transformations: R = s b S = C ˙ S and ¯ R = s ab S = ¯ C ˙ S . In10ther words, we have obtained: s b x = C ˙ x, s b p x = C ˙ p x , s b t = C ˙ t, s b p t = C ˙ p t and s ab x = ¯ C ˙ x, s ab p x = ¯ C ˙ p x , s ab t = ¯ C ˙ t, s ab p t = ¯ C ˙ p t . Furthermore, it is evident that: s b s ab S = Q = − ( ¯ C ˙ C + ˙¯ C C ) ˙ S − k ˙ S − ¯ C C ¨ S. (24)The requirement of the absolute anticommutativity: { s b , s ab } S = 0 implies that s b s ab S = − s ab s b S which, in turn, leads to the following relationships: s b s ab S = s b ¯ R = Q ≡ − ( ¯ C ˙ C + ˙¯ C C ) ˙ S − k ˙ S − ¯ C C ¨ S, − s ab s b S = − s ab R = Q ≡ − ( ¯ C ˙ C + ˙¯ C C ) ˙ S − k ˙ S − ¯ C C ¨ S. (25)The explicit computation of the following, using the (anti-)BRST symmetry transforma-tions of the phase variables in Eqs. (6) and (14), are: s b ¯ R = i B ˙ S − ¯ C ˙ C ˙ S − ¯ C C ¨ S ≡ Q, − s ab R = − i ¯ B ˙ S − ˙¯ C C ˙ S − ¯ C C ¨ S ≡ Q. (26)Equating (25) and (26), we obtain the following interesting relationship: k = − ˙¯ C C − i B ≡ i ¯ B − ¯ C ˙ C = ⇒ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 . (27)In other words, it is the consistency conditions of the BRST formalism that lead to thedetermination of k ( τ ) in Eq. (16) within the ambit of MBTSA. A close look at Eqs. (25),(26) and (27) establishes that a precise determination of Q ( τ ) in (23) leads to (i) thevalidity of the absolute anticommutativity (i.e. { s b , s ab } S = 0) of the off-shell nilpotent(anti-)BRST symmetries, and (ii) the deduction of the (anti-)BRST invariant ‡ CF-typerestriction B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 on our theory.We conclude this section with the following useful and crucial remarks. First, we setout to derive the (anti-)BRST symmetry transformations (corresponding to the classicalreparameterization symmetry transformations) for the phase variables [cf. Eq. (6)]. Wehave accomplished this goal in Eq. (21). Second, we have derived the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 within the purview of MBTSA [cf. Eq. (27)] which is actuallyhidden in the determination of Q ( τ ) in Eq. (23). Third, for the application of the theoreticalpotential of MBTSA, we have taken the full super expansion of the generic supervariable[cf. Eq. (17)] along all the possible Grassmannian directions of the (1, 2)-dimensionalsupermanifold. Fourth, unlike the application of BT-superfield/supervariable approach tothe gauge theories [14-16] where spacetime does not change, in the case of MBTSA, thesuper diffeomorphism transformation (16) has been taken into account in all the basic aswell as secondary supervariables. Fifth, taking into account the inputs from Eqs. (21) and(26), we obtain the following super expansion of the generic variable S ( τ ), namely;˜ S ( h ) ( τ, θ, ¯ θ ) = S ( τ ) + θ ( ¯ C ˙ S ) + ¯ θ ( C ˙ S ) + θ ¯ θ [ i B ˙ S − ¯ C ˙ C ˙ S − ¯ C C ¨ S ] ≡ S ( τ ) + θ ( s ab S ) + ¯ θ ( s b S ) + θ ¯ θ ( s b s ab S ) , (28) ‡ This statement is true only when the whole theory is considered on a submanifold of the Hilbert space ofthe quantum variables where the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 is satisfied. In other words,we explicitly compute s b [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] = ( ddτ ) [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] C − [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] ˙ C and s ab [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] = ( ddτ ) [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] ¯ C − [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] ˙¯ C which implythat s ( a ) b [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] = 0 is true only on the above mentioned submanifold. S = x, p x , t, p t and the superscript ( h ) on the supervariable ˜ S ( τ, θ, ¯ θ ) denotes thatthis supervariable has been obtained after the application of HC. Finally, the standardnilpotent (anti-)BRST symmetry transformations (8) dictate that we can have the following(anti-)chiral super expansions for the supervariables corresponding to ( ¯ C ) C , namely; C ( τ ) −→ F ( c ) ( τ, θ ) = C ( τ ) + θ ( i ¯ B ) ≡ C ( τ ) + θ ( s ab C ) , ¯ C ( τ ) −→ ¯ F ( ac ) ( τ, ¯ θ ) = ¯ C ( τ ) + ¯ θ ( i B ) ≡ ¯ C ( τ ) + ¯ θ ( s b ¯ C ) , (29)where the superscripts ( c ) and ( ac ) denote the chiral and anti-chiral supervariables. Theabove observation gives us a clue that we should exploit the theoretical strength of ACSAto BRST formalism for our further discussions. 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 corre-sponding to the classical infinitesimal and continuous reparameterization symmetry trans-formations (2). We exploit the strength of the sacrosanct requirements of off-shell nilpo-tency and absolute anticommutativity properties. In this context, we point out that wehave already derived s b C = C ˙ C, s ab ¯ C = ¯ C ˙¯ C by invoking the sanctity of the off-shellnilpotency ( s a ) b = 0) property for the phase variables (i.e. s a ) b S = 0 , S = x, p x , t, p t ). Itis interesting to note the following absolute anticommutativity requirements, namely; { s b , s ab } C = 0 = ⇒ s b ¯ B = ˙¯ B C − ¯ B ˙ C, { s b , s ab } ¯ C = 0 = ⇒ s ab B = ˙ B ¯ C − B ˙¯ C, (30)leads to the derivation of the s b ¯ B and s ab B . We can readily check that s b ¯ B = 0 , s ab B = 0are satisfied due to our knowledge of the BRST and anti-BRST symmetry transformations: s b C = C ˙ C, s ab ¯ C = ¯ C ˙¯ C and the fermionic ( C = ¯ C = 0 , C ¯ C + ¯ C C = 0) nature of the(anti-)ghost variables ( ¯ C ) C . We further note that { s b , s ab } B = 0 and { s b , s ab } ¯ B = 0. Therequirement of the absolute anticommutativity on the E ( τ ) variable leads to: { s b , s ab } E ( τ ) = dd τ h i (cid:8) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:9) E ( τ ) i . (31)Thus, we emphasize that the absolute anticommutativity property ( s b s ab + s ab s b = 0) onthe phase variables [cf. Eq. (15)] as well as on the Lagrange multiplier variable [cf. Eq.(31)] are 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 infinitesimalreparameterization symmetry transformations (2)] are as follows: s ab x = ¯ C ˙ x, s ab p x = ¯ C ˙ p x , s ab t = ¯ C ˙ t, s ab p t = ¯ C ˙ p t , s ab E = dd τ ( ¯ C E ) ,s ab C = i ¯ B, s ab ¯ C = ¯ C ˙¯ C, s ab ¯ B = 0 , s ab B = ˙ B ¯ C − B ˙¯ C, (32)12 b x = C ˙ x, s b p x = C ˙ p x , s b t = C ˙ t, s b p t = C ˙ p t , s b E = dd τ ( C E ) ,s b ¯ C = i B, s b C = C ˙ C, s b B = 0 , s b ¯ B = ˙¯ B C − ¯ B ˙ C. (33)The above fermionic symmetry transformations are off-shell nilpotent and absolutely an-ticommuting provided the whole theory is considered on a submanifold of the space ofquantum variables where the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 is satisfied.The existence of the above
CF-type restriction leads to the derivation of the coupled(but equivalent) Lagrangians (i.e. L B and L ¯ B ) as follows: L B = L f + s b s ab h i E − ¯ C C i ,L ¯ B = L f − s ab s b h i E − ¯ C C i . (34)We point out that the terms inside the square brackets are same as in Eq. (10) for the BRSTanalysis of the classical gauge symmetry transformations (4). Furthermore, in contrast tothe unique (anti-)BRST invariant Lagrangian [cf. Eq. (11)] (corresponding to the classical gauge symmetry transformations), we have obtained here a set of coupled (but equivalent)(anti-)BRST invariant Lagrangians in Eq. (34). This has happened because of the factthat the CF-type restriction ( b + ¯ b = 0) is trivial in the case of the former while it is a non-trivial restriction [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0] in the context of the latter .It is straightforward to compute the operation of s ( a ) b on the quantities in the squarebrackets of Eq. (34). In the full blaze of their glory, the coupled (but equivalent) La-grangians L B and L ¯ B are as follows § L B = L f + B h E ˙ E − i (2 ˙¯ C C + ¯ C ˙ C ) i + B − i E ˙ E ˙¯ C C − i E ˙¯ C ˙ C − ˙¯ C ¯ C ˙ C C,L ¯ B = L f − ¯ B h E ˙ E − i (2 ¯ C ˙ C + ˙¯ C C ) i + ¯ B − i E ˙ E ¯ C ˙ C − i E ˙¯ C ˙ C − ˙¯ C ¯ C ˙ C C, (35)where the subscripts B and ¯ B on the Lagrangians are appropriate because L B depends uniquely on the Nakanishi-Lautrup auxiliary variable B (where ¯ B is not present at all).Similarly, the Lagrangian L ¯ B is uniquely dependent on ¯ B . They are coupled because theEL-EOMs with respect to B and ¯ B from L B and L ¯ B , respectively, yield B = − E ˙ E + 2 i ˙¯ C C + i ¯ C ˙ C, ¯ B = E ˙ E − i ¯ C ˙ C − i ˙¯ C C, (36)which lead to the deduction of the CF-type restrictions: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0. Further-more, the condition L B ≡ L ¯ B also demonstrates the existence of the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ 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 termsis same as in the cases of NSUSY (i.e. scalar) and SUSY (i.e. spinning) relativistic particles [23, 24].
13t this stage, we are in the position to study the (anti-)BRST symmetries of the La-grangians L B and L ¯ B . It is straightforward to note that we have the following: s b L B = dd τ h C L f + B C − i B ¯ C ˙ C C + E ˙ E B C + E B ˙ C i , (37) s ab L ¯ B = dd τ h ¯ C L f + ¯ B ¯ C − i ¯ B ˙¯ C ¯ C C − E ˙ E ¯ B ¯ C − E ¯ B ˙¯ C i . (38)The above observations demonstrate that the action integrals S = R ∞−∞ d τ L B and S = R ∞−∞ d τ L ¯ B remain invariant under the SUSY-type (i.e. fermionic) off-shell nilpo-tent, continuous and infinitesimal (anti-)BRST symmetry transformations for the physicalvariables that vanish off at τ = ±∞ . At this crucial juncture, we establish the equivalence of the coupled Lagrangian L B and L ¯ B w.r.t the (anti-)BRST symmetry transformations[ s ( a ) b ]. In this context, we apply s ab on L B and s b on L ¯ B to obtain the following s ab L B = dd τ h ¯ C L f + E ˙ E ( i ˙¯ C ¯ C C + B ¯ C ) + E ( i ˙¯ C ¯ C ˙ C + B ˙¯ C )+ B ¯ C + i (2 B − ¯ B ) ˙¯ C ¯ C C i + (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) (2 i ˙¯ C ¯ C ˙ C − B ˙¯ C − E ˙ E ˙¯ C + i ¨¯ C ¯ C C ) − dd τ (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) (cid:2) B ¯ C + E ˙¯ C (cid:3) , (39) s b L ¯ B = dd τ h C L f + E ˙ E ( i ¯ C C ˙ C − ¯ B C ) + E ( i ˙¯ C C ˙ C − ¯ B ˙ C )+ ¯ B C − i (2 ¯ B − B ) ¯ C C ˙ C i + (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) ( − i ˙¯ C C ˙ C − B ˙ C + E ˙ E ˙ C + i ¯ C ¨ C C )+ dd τ (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) (cid:2) − ¯ B C + E ˙ C (cid:3) , (40)which demonstrate that the coupled Lagrangians L B and L ¯ B (and corresponding actionintegrals) respect both (i.e. BRST and anti-BRST) symmetry transformations together provided the whole theory is considered on a supermanifold in the Hilbert space of quantum variables where the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 is satisfied. It shouldbe recalled that, under the latter restriction, we also have the absolute anticommutativityproperty (i.e. { s b , s ab } = 0 of the (anti-)BRST symmetry transformations.We end this section with the following key comments. First, the properties of theoff-shell nilpotency and absolute anticommutativity are sacrosanct in the realm of BRSTapproach to gauge and/or diffeomorphism invariant theories. Second, physically, the firstproperty (i.e. off-shell nilpotency) implies that these fermionic symmetry transformationsare supersymmetric-type as they transform bosonic variables to fermionic variables andvice-versa. Third, the property of the absolute anticommutativity encodes the linear in-dependence of the BRST and anti-BRST symmetry transformations. Fourth, the absolute14nticommutativity property owes its origin to the existence of the CF-type restrictionswhich are connected with the concepts of gerbes [7, 8]. Fifth, as the classical gauge the-ory is characterized by the first-class constraints, in exactly similar fashion, the quantum gauge and/or diffeomorphism invariant theories are characterized by the existence of theCF-type restrictions within the ambit of BRST formalism. Sixth, the coupled Lagrangians L B and L ¯ B are equivalent because both of them respect BRST and anti-BRST symmetrytransformations as is clear from Eqs. (37)-(40) provided the whole theory is considered onthe submanifold of the total Hilbert space of the quantum variables where the CF-typerestriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 is satisfied.
In this section, we derive the nilpotent (anti-)BRST symmetry transformations [ s ( a ) b ] for all the other variables [cf. Eqs. (32), (33)] besides the phase space variables ( x, p x , t, p t ) whose(anti-)BRST symmetries have already been derived in Sec. 3 by exploiting the theoreticalpotential of MBTSA. To achieve the above goal, we exploit the ideas behind ACSA toBRST formalism [25-29]. In this context, first of all, we focus on the derivation of the BRSTsymmetry transformations: s b B = 0 , s b ¯ B = ˙¯ B C − ¯ B ˙ C, s b C = C ˙ C, s b E = ˙ E C + E ˙ C [cf.Eq. (33)]. For this purpose, we generalize the ordinary variables [ B ( τ ) , ¯ B ( τ ) , C ( τ ) , E ( τ )]onto a (1 , anti-chiral super sub-manifold as follows B ( τ ) −→ B ( τ, ¯ θ ) = B ( τ ) + ¯ θ f ( τ ) , ¯ B ( τ ) −→ ¯ B ( τ, ¯ θ ) = ¯ B ( τ ) + ¯ θ f ( τ ) ,C ( τ ) −→ F ( τ, ¯ θ ) = C ( τ ) + ¯ θ b ( τ ) ,E ( τ ) −→ Σ( τ, ¯ θ ) = E ( τ ) + ¯ θ f ( τ ) , (41)where we note that ( f , f , f ) are the fermionic secondary variables and b ( τ ) is the bosonic secondary variable because of the fermionic (¯ θ = 0) nature of the Grassmannian variable¯ θ which characterizes the anti-chiral super sub-manifold (along with the bosonic evolutionparameter τ ). It is elementary to note that the observation s b B = 0 implies the followingsuper expansion (in view of the fact that ∂ ¯ θ ↔ s b ), namely; B ( b ) ( τ, ¯ θ ) = B ( τ ) + ¯ θ (0) ≡ B ( τ ) + ¯ θ ( s b B ) , (42)where the superscript ( b ) on the anti-chiral supervariable B ( τ, ¯ θ ) denotes that the coefficientof ¯ θ yields the BRST symmetry transformation: s b B = 0 due to the trivial equality: B ( τ, ¯ θ ) = B ( τ ) which emerges from the observation that the Nakanishi-Lautrup auxiliaryvariable B ( τ ) is a BRST invariant quantity [cf. Eq. (33)]. In other words, we have foundout that the secondary variable f ( τ ) = 0 in the super expansions (41).At this stage, we find other non-trivial BRST invariant quantities for the derivation ofthe secondary variables: b , f , f of Eq. (41). We observe that ¶ : s b ( C ˙ x ) = 0 , s b [ ˙¯ B C − ¶ We have specifically taken here s b ( C ˙ x ) = 0 for our purpose. However, one can take the generalexpression: s b S = C ˙ S ( S = x, p x , t, p t ) for the derivation of b ( τ ) = C ˙ C . B ˙ C ] = 0 , s b [ E ˙ C + ˙ E C ] = 0. The basic tenets of the ACSA to BRST formalism requiresthat the quantities in the square brackets have to be independent of the Grassmannianvariables ¯ θ when they are generalized onto a (1, 1)-dimensional super sub-manifold, namely; F ( τ, ¯ θ ) ˙ X ( h, ac ) ( τ, ¯ θ ) = C ( τ ) ˙ x ( τ )˙¯ B ( τ, ¯ θ ) F ( τ, ¯ θ ) − ¯ B ( τ, ¯ θ ) ˙ F ( τ, ¯ θ ) = ˙¯ B ( τ ) C ( τ ) − ¯ B ( τ ) ˙ C ( τ )Σ( τ, ¯ θ ) ˙ F ( τ, ¯ θ ) + ˙Σ( τ, ¯ θ ) F ( τ, ¯ θ ) = E ( τ ) ˙ C ( τ ) + ˙ E ( τ ) C ( τ ) , (43)where X ( h, ac ) is the anti-chiral limit of the full expansion of X ( h ) ( τ, θ, ¯ θ ) obtained after theapplication of HC [cf. Eq. (28)], namely; X ( h ) ( τ, θ, ¯ θ ) = x ( τ ) + θ ( ¯ C ˙ x ) + ¯ θ ( C ˙ x ) + θ ¯ θ [ i B ˙ x − ¯ C ˙ C ˙ x − ¯ C C ¨ x ] , (44)which has been obtained [cf. Eq. (28)] in Sec. 3 using the theoretical strength of MBTSA.In other words, from the top entry of Eq. (43), we have the following restrictions: F ( τ, ¯ θ ) ˙ X ( h, ac ) ( τ, ¯ θ ) = C ( τ ) ˙ x ( τ )= ⇒ (cid:2) C ( τ ) + ¯ θ b ( τ ) (cid:3) (cid:2) ˙ x + ¯ θ ( ˙ C ˙ x + C ¨ x ) (cid:3) = C ( τ ) ˙ x ( τ ) . (45)From the above relationship we obtain b ( τ ) = C ˙ C . Thus, we have the following F ( b ) ( τ, ¯ θ ) = C ( τ ) + ¯ θ ( C ˙ C ) ≡ C ( τ ) + ¯ θ ( s b C ) , (46)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 s b C . We have to use the above superexpansion in the second entry from the top in (43) to obtain the following:˙¯ B ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) − ¯ B ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) = ˙¯ B ( τ ) C ( τ ) − ¯ B ( τ ) ˙ C ( τ ) . (47)In other words, we have the following equality[ ˙¯ B + ¯ θ ˙ f ( τ )] [ C ( τ ) + ¯ θ ( C ˙ C )] − [ ¯ B ( τ ) + ¯ θ f ( τ )] [ ˙ C ( τ ) + ¯ θ ( C ¨ C )]= ˙¯ B ( τ ) C ( τ ) − ¯ B ( τ ) ˙ C ( τ ) , (48)which yields the following condition on the secondary variable f , namely;˙ f C − f ˙ C − ˙¯ B ˙ C C + ¯ B ¨ C C = 0 . (49)It is straightforward to note that f = ˙¯ B C − ¯ B ˙ C satisfies the above condition in a precisemanner. We point out that the last entry (from the top) of Eq. (43) can be re-written, inview of our the super expansion in Eq. (46), as follows:Σ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) + ˙Σ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) = E ( τ ) ˙ C ( τ ) + ˙ E ( τ ) C ( τ ) . (50)The substitutions of expansions from (41) and (46) lead to the following condition on thesecondary variable f ( τ ) [present in the expansion of Σ( τ, ¯ θ )], namely; f ˙ C + ˙ f C − E ¨ C C − ˙ E ˙ C C = 0 , (51)16hich is satisfied by the choice f = E ˙ C + ˙ E C . Hence, we have the following superexpansions (with the BRST symmetry transformations (33) as input), namely;¯ B ( b ) ( τ, ¯ θ ) = ¯ B ( τ ) + ¯ θ ( ˙¯ B C − ¯ B ˙ C ) ≡ ¯ B ( τ ) + ¯ θ ( s b ¯ B ) , Σ ( b ) ( τ, ¯ θ ) = E ( τ ) + ¯ θ ( E ˙ C + ˙ E C ) ≡ E ( τ ) + ¯ θ ( s b E ) , (52)where the coefficients of ¯ θ (in view of ∂ ¯ θ ↔ s b ) are the BRST symmetry transformations(33). For the convenience of the readers, we have performed the explicit computations of f = E ˙ C + ˙ E C and f = ˙¯ B C − ¯ B ˙ C in our Appendix B. It is clear that we have alreadycomputed the BRST transformations s b B = 0 , s b C = C ˙ C, s b ¯ B = ˙¯ B C − ¯ B ˙ C, s b E = E ˙ C + ˙ E C by exploiting the virtues of ACSA in Eqs. (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 Sec. 3, we have already computed s ab S = ¯ C ˙ S ( S = x, p x , t, p t ) and s ab C = i ¯ B byexploiting MBTSA to BRST formalism. Our objective in the present part of our sectionis to derive: s ab ¯ B = 0 , s ab ¯ C = ¯ C ˙¯ C, s ab B = ˙ B ¯ C − B ˙¯ C, s ab E = E ˙¯ C + ˙ E ¯ C by exploitingACSA to BRST formalism. In this context, first of all, we generalize the ordinary variablesonto a (1 , chiral super sub-manifold as¯ B ( τ ) −→ ¯ B ( τ, θ ) = ¯ B ( τ ) + θ ¯ f ( τ ) ,B ( τ ) −→ B ( τ, θ ) = B ( τ ) + θ ¯ f ( τ ) , ¯ C ( τ ) −→ ¯ F ( τ, θ ) = ¯ C ( τ ) + θ ¯ b ( τ ) ,E ( τ ) −→ Σ( τ, θ ) = E ( τ ) + θ ¯ f ( τ ) , (53)where ( ¯ f , ¯ f , ¯ f ) are the fermionic secondary variables, ¯ b ( τ ) is a bosonic secondary variableand the above (1 , chiral super sub-manifold is parameterized by ( τ, θ ). It isstraightforward to note that s ab ¯ B = 0 implies that: ¯ B ( τ, θ ) = B ( τ ) and, as a consequence,we have ¯ f ( τ ) = 0 which leads to¯ B ( ab ) ( τ, θ ) = ¯ B ( τ ) + θ (0) ≡ ¯ B ( τ ) + θ ( s ab ¯ B ) , (54)where the superscript ( ab ) on the chiral supervariable denotes that we have obtained s ab ¯ B =0 as the coefficient of θ . The other useful and interesting anti-BRST invariant quantitiesof our interest [cf. Eq. (32)] are: s ab [ ˙ B ¯ C − B ˙¯ C ] = 0 , s ab [ E ˙¯ C + ˙ E ¯ C ] = 0 , s ab [ ¯ C ˙ x ] = 0 . (55)The quantities in the square brackets can be generalized onto the (1 , chiral super sub-manifold. Following the fundamental requirement(s) of ACSA to BRST formal-ism, these quantities must be independent of the Grassmannian variable θ . In other words,we have the following restrictions on the chiral supervariables˙ B ( τ, θ ) ¯ F ( τ, θ ) − B ( τ, θ ) ˙¯ F ( τ, θ ) = ˙ B ( τ ) ¯ C ( τ ) − B ( τ ) ˙¯ C ( τ ) , Σ( τ, θ ) ˙¯ F ( τ, θ ) + ˙Σ( τ, θ ) ¯ F ( τ, θ ) = E ( τ ) ˙¯ C ( τ ) + ˙ E ( τ ) ¯ C ( τ ) , ¯ F ( τ, θ ) ˙ X ( h,c ) ( τ, θ ) = ¯ C ( τ ) ˙ x ( τ ) , (56)17here X ( h,c ) ( τ, θ ) is the chiral limit of the super expansion in Eq. (44). In other words, wehave the following explicit expression for the supervariable X ( h,c ) ( τ, θ ), namely; X ( h,c ) ( τ, θ ) = x ( τ ) + θ ( ¯ C ˙ x ) . (57)Taking the expansions from (53) and (57), we find that the last entry of Eq. (56) yields:¯ b ( τ ) = ¯ C ˙¯ C . Hence, we have obtained the following super expansion¯ F ( ab ) ( τ, θ ) = ¯ C ( τ ) + θ ( ¯ C ˙¯ C ) ≡ ¯ C ( τ ) + θ ( s ab ¯ C ) , (58)where the superscript ( ab ) on the chiral supervariable on the l.h.s. denotes that it has beenderived after the application of the anti-BRST invariant restriction in (56). The coefficient θ is nothing but the anti-BRST symmetry transformation: s ab ¯ C = ¯ C ˙¯ C . This equationalso shows that ∂ θ ↔ s ab and it leads to the anti-BRST symmetry for ¯ C .We utilize now the two top entries of (56) where we use the explicit expansion for¯ F ( ab ) ( τ, θ ) of (58) in the following restrictions on the supervariables, namely;˙ B ( τ, θ ) ¯ F ( ab ) ( τ, θ ) − B ( τ, θ ) ˙¯ F ( ab ) ( τ, θ ) = ˙ B ( τ ) ¯ C ( τ ) − B ( τ ) ˙¯ C ( τ ) , Σ( τ, θ ) ˙¯ F ( ab ) ( τ, θ ) + ˙Σ( τ, θ ) ¯ F ( ab ) ( τ, θ ) = E ( τ ) ˙¯ C ( τ ) + ˙ E ( τ ) ¯ C ( τ ) . (59)The substitutions from (53) and (58) lead to:˙¯ f ¯ C − ¯ f ˙¯ C − ˙ B ˙¯ C ¯ C + B ¨¯ C ¯ C = 0 , ¯ f ˙¯ C + ˙¯ f ¯ C − E ¨¯ C ¯ C − E ˙¯ C ¯ C = 0 . (60)It is straightforward, following the theoretical tricks of Appendix B, to find out the solutionsfor the secondary variables ¯ f ( τ ) and ¯ f ( τ ) which are as follows:¯ f ( τ ) = ˙ B ¯ C − B ˙¯ C, ¯ f = E ˙¯ C + ˙ E ¯ C. (61)Substitutions of these secondary variables into the super expansions (53) leads to the de-termination of the anti-BRST symmetry transformations for the variables B ( τ ) and E ( τ )as the coefficients of θ in the followingΣ ( ab ) ( τ, θ ) = E ( τ ) + θ ( E ˙¯ C + ˙ E ¯ C ) ≡ E ( τ ) + θ ( s ab E ( τ )) , B ( ab ) ( τ, θ ) = B ( τ ) + θ ( ˙ B ¯ C − B ˙¯ C ) ≡ B ( τ ) + θ ( s ab B ( τ )) , (62)where the superscript ( ab ) on the chiral supervariable denotes that these supervariableshave been obtained after the applications of the anti-BRST invariant restrictions (56).Moreover, the above observation establishes that: s ab ⇔ ∂ θ which implies that the nilpo-tency ( s ab = 0 , ∂ θ = 0) properties of s ab and ∂ θ are connected with each-other. Thus, wehave obtained all the anti-BRST symmetry transformations (besides the phase variables)in our Eqs. (54), (58) and (62). This completes our discussion on the derivation of theoff-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations(32) and (33) within the ambit of ACSA to BRST formalism.18 Symmetry Invariance of the Lagrangians: ACSA
In this section, we establish the equivalence of the coupled Lagrangian L B and L ¯ B as far asthe (anti-)BRST symmetry invariance (within the purview of ACSA to BRST formalism)is concerned. We accomplish this objective by generalizing the ordinary Lagrangians totheir counterpart super
Lagrangians as L ¯ B → ˜ L ( c )¯ B ( τ, θ ) = ˜ L ( c ) f ( τ, θ ) − ¯ B ( ab ) ( τ, θ ) h Σ ( ab ) ( τ, θ ) ˙Σ ( ab ) ( τ, θ ) − i (cid:8) F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ )+ ˙¯ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) (cid:9)i + 12 ¯ B ( ab ) ( τ, θ ) ¯ B ( ab ) ( τ, θ ) − i Σ ( ab ) ( τ, θ ) Σ ( ab ) ( τ, θ ) ˙¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) − i Σ ( ab ) ( τ, θ ) ˙Σ ( ab ) ( τ, θ ) ¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) − ˙¯ F ( ab ) ( τ, θ ) ¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) , (63) L B → ˜ L ( ac ) B ( τ, ¯ θ ) = ˜ L ( ac ) f ( τ, ¯ θ ) + B ( b ) ( τ, ¯ θ ) h Σ ( b ) ( τ, ¯ θ ) ˙Σ ( b ) ( τ, ¯ θ ) − i (cid:8) F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ )+ ¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) (cid:9)i + 12 B ( b ) ( τ, ¯ θ ) B ( b ) ( τ, ¯ θ ) − i Σ ( b ) ( τ, ¯ θ ) Σ ( b ) ( τ, ¯ θ ) ˙¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) − i Σ ( b ) ( τ, ¯ θ ) ˙Σ ( b ) ( τ, ¯ θ ) ˙¯ F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) − ˙¯ F ( b ) ( τ, ¯ θ ) ¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) , (64)where ˜ L ( c ) f and ˜ L ( ac ) f are the generalizations of the first-order Lagrangian ( L f ) to its coun-terpart chiral and anti-chiral super Lagrangians as˜ L ( c ) f ( τ, θ ) = P ( h,c ) x ( τ, θ ) ˙ X ( h,c ) ( τ, θ ) + P ( h,c ) t ( τ, θ ) ˙ T ( h,c ) ( τ, θ ) − Σ ( ab ) ( τ, θ )2 h P ( h,c ) x ( τ, θ ) P ( h,c ) x ( τ, θ ) + 2 m P ( h,c ) t ( τ, θ ) i , ˜ L ( ac ) f ( τ, ¯ θ ) = P ( h,ac ) x ( τ, ¯ θ ) ˙ X ( h,ac ) ( τ, ¯ θ ) + P ( h,ac ) t ( τ, ¯ θ ) ˙ T ( h,ac ) ( τ, ¯ θ ) − Σ ( b ) ( τ, ¯ θ )2 h P ( h,ac ) x ( τ, ¯ θ ) P ( h,ac ) x ( τ, ¯ θ ) + 2 m P ( h,ac ) t ( τ, ¯ θ ) i , (65)where the superscripts ( c ) and ( ac ) denote the chiral and anti-chiral generalizations andthe rest of the supervariables with superscripts ( b ) and ( ab ) have already been explainedearlier in Sec. 5. The supervariables with superscripts ( h, c ) and ( h, ac ) are the chiral and anti-chiral limits of the super phase variables ( X ( h ) , P ( h ) x , T ( h ) , P ( h ) t ) that have beenobtained after the application of HC. Thus, these are the counterparts of the ordinaryphase variables ( x, p x , t, p t ) and they have been explained in Sec. 3. In the above equation(65), the super phase variables with superscript ( h, c ) and ( h, ac ) can be expressed in termsof the generic supervariable as follows S ( τ ) → S ( h,c ) ( τ, θ ) = S ( τ ) + θ [ ¯ C ˙ S ( τ )] ,S ( τ ) → S ( h,ac ) ( τ, ¯ θ ) = S ( τ ) + ¯ θ [ C ˙ S ( τ )] , (66)19here the (anti-)chiral supervariables on the l.h.s. stand for the super phase variables( X, P x , T, P t ) with the proper chiral and anti-chiral superspace coordinates ( τ, θ ) and ( τ, ¯ θ )as their arguments. The set of supervariables ( X, P x , T, P t ) are the generalizations of the ordinary phase variables ( x, p x , t, p t ) to their (anti-)chiral counterparts onto the (1 , general (1 , ∂∂ θ ˜ L ( c ) f ( τ, θ ) = dd τ [ ¯ C L f ] ⇐⇒ s ab L f = dd τ [ ¯ C L f ] ,∂∂ ¯ θ ˜ L ( ac ) f ( τ, ¯ θ ) = dd τ [ C L f ] ⇐⇒ s b L f = dd τ [ C L f ] . (67)In other words, we have captured the (anti-)BRST invariance of the first-order Lagrangian( L f ) in view of the mappings: s b ↔ ∂ ¯ θ , s ab ↔ ∂ θ . Since in the ordinary space, the (anti-)BRST symmetry transformations acting on L f produce the total derivatives [cf. Eq. (67)],the action integral S = R + ∞−∞ d τ L f remains invariant under the transformations s ( a ) b .At this stage, we focus on the (anti-)BRST invariance of the coupled Lagrangians L B and L ¯ B [cf. Eqs. (38), (37)]. We can express these invariances within the ambit of ACSA(in view of the mappings: s b ↔ ∂ ¯ θ , s ab ↔ ∂ θ ), namely; ∂∂ θ ˜ L ( c )¯ B ( τ, θ ) = dd τ h ¯ C L f − e ˙ e ¯ B ¯ C − e ¯ B ˙¯ C + ¯ B ¯ C − i ¯ B ˙¯ C ¯ C C i = s ab L ¯ B , (68) ∂∂ ¯ θ ˜ L ( ac ) B ( τ, ¯ θ ) = dd τ h C L f + e ˙ e B C + e B ˙ C + B C − i B ¯ C ˙ C C i = s b L B , (69)where the super Lagrangians ˜ L ( c )¯ B ( τ, θ ) and ˜ L ( ac ) B ( τ, ¯ θ ) have been already quoted in Eqs. (63)and (64). It is interesting to note that the r.h.s. of (68) and (69) are same as we have foundin the ordinary space [cf. Eq. (38), (37)]. To prove the equivalence of the Lagrangians L B and L ¯ B w.r.t. the (anti-)BRST symmetry transformations [cf. Eqs. (32), (33)] within thepurview of ACSA, we generalize the ordinary Lagrangians L B and L ¯ B as follows L B → ˜ L ( c ) B ( τ, θ ) = ˜ L ( c ) f ( τ, θ ) + B ( ab ) ( τ, θ ) h Σ ( ab ) ( τ, θ ) ˙Σ ( ab ) ( τ, θ ) − i { F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ )+ ¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) } i + 12 B ( ab ) ( τ, θ ) B ( ab ) ( τ, θ ) − i Σ ( ab ) ( τ, θ ) Σ ( ab ) ( τ, θ ) ˙¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) − i Σ ( ab ) ( τ, θ ) ˙Σ ( ab ) ( τ, θ ) ˙¯ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) − ˙¯ F ( ab ) ( τ, θ ) ¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) , (70) L ¯ B → ˜ L ( ac )¯ B ( τ, ¯ θ ) = ˜ L ( ac ) f ( τ, ¯ θ ) − ¯ B ( b ) ( τ, ¯ θ ) h Σ ( b ) ( τ, ¯ θ ) ˙Σ ( b ) ( τ, ¯ θ ) − i (cid:8) F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ )+ ˙¯ F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) (cid:9)i + 12 ¯ B ( b ) ( τ, ¯ θ ) ¯ B ( b ) ( τ, ¯ θ ) − i Σ ( b ) ( τ, ¯ θ ) Σ ( b ) ( τ, ¯ θ ) ˙¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) − i Σ ( b ) ( τ, ¯ θ ) ˙Σ ( b ) ( τ, ¯ θ ) ¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) − ˙¯ F ( b ) ( τ, ¯ θ ) ¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) , (71)20here all the notations and symbols have already been explained earlier. To find out theresult of the operations of s b on L ¯ B and s ab on L B , we observe the following (in view of themappings: s b ↔ ∂ ¯ θ , s ab ↔ ∂ θ ), namely; ∂∂ θ ˜ L ( c ) B ( τ, θ ) = dd τ h ¯ C L f + E ˙ E ( i ˙¯ C ¯ C C + B ¯ C ) + E ( i ˙¯ C ¯ C ˙ C + B ˙¯ C )+ B ¯ C + i (2 B − ¯ B ) ˙¯ C ¯ C C i + (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) (2 i ˙¯ C ¯ C ˙ C − E ˙ E ˙¯ C − B ˙¯ C + i ¨¯ C ¯ C C ) − dd τ (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) ( B ¯ C + E ˙¯ C ) ≡ s ab L B , (72) ∂∂ ¯ θ ˜ L ( ac )¯ B ( τ, ¯ θ ) = dd τ h C L f − E ˙ E ( i ¯ C ˙ C C + ¯
B C ) − E ( i ˙¯ C ˙ C C + ¯ B ˙ C )+ ¯ B C + i (2 ¯ B − B ) ¯ C ˙ C C i + (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) (cid:2) i ¯ C ¨ C C + 2 i ˙¯ C ˙ C C − B ˙ C + E ˙ E ˙ C (cid:3) + dd τ (cid:2) i ( ¯ C ˙ C − ˙¯ C C ) + B + ¯ B (cid:3) ( E ˙ C − ¯ B C ) ≡ s b L ¯ B , (73)within the framework of ACSA. It is self-evident, from the r.h.s. of (72) and (73), that wehave the BRST invariance of L ¯ B and anti-BRST invariance of L B if and only if our wholetheory is considered on the submanifold of the Hilbert space of quantum variables wherethe CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 is satisfied.We end this section with the following crucial remarks. First of all, we have capturedthe BRST and anti-BRST invariance of L B and L ¯ B , respectively, in the terminology ofACSA on the (anti-)chiral super submanifolds [cf. Eqs. (68), (69)]. Second, we have also demonstrated the anti-BRST invariance of L B and BRST invariance of L ¯ B in the super-space formalism [cf. Eqs. (72), (73)] where the theoretical techniques of ACSA have playedvery important roles. Third, we have also expressed the (anti-)BRST invariance of thefirst-order Lagrangian L f in Eq. (67). Finally, we have proven the equivalence of L B and L ¯ B within the framework of ACSA in the Eqs. (68), (69), (72) and (73). Our present section is divided into two subsections. In subsection 7.1, we discuss the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges inthe ordinary space. Our subsection 7.2 deals with the above properties within the realm ofACSA to BRST formalism. In other words, we capture the off-shell nilpotency and absoluteanticommutativity of the conserved fermionic (anti-)BRST charges in the superspace bytaking the theoretical inputs from ACSA. 21 .1 Nilpotency and Anticommutativity: Ordinary Space
The perfect symmetry invariance of L ¯ B under the anti-BRST symmetry transformations[cf. Eq. (38)] and L B under the BRST symmetry transformations [cf. Eq. (37)] allow usto compute the Noether conserved charges by using the standard techniques of Noether’stheorem (applied to the Lagrangians L ¯ B and L B ) as Q ¯ B = ¯ C E p x + 2 m p t ) + ¯ B ¯ C − i ¯ B ˙¯ C ¯ C C − ¯ B E ˙ E ¯ C − ¯ B E ˙¯ C,Q B = C E p x + 2 m p t ) + B E ˙ C − i B ¯ C ˙ C C + B E ˙ E C + B C, (74)where conserved ( ˙ Q ¯ B = 0 , ˙ Q B = 0) (anti-)BRST charges are denoted by Q ( ¯ B ) B . Theconservation law ( ˙ Q ¯ B = 0 , ˙ Q B = 0) can be proven by using the EL-EOMs derived fromthe coupled Lagrangians L ¯ B and L B . For readers’ convenience, we prove the conservation( ˙ Q B = 0) of the BRST charge by using the EL-EOMs derived from L B in our Appendix C.First of all, we concentrate on the proof of the off-shell nilpotency properties of the(anti-)BRST charges Q ( ¯ B ) B . In this context, we note that the following EL-EOMs w.r.t.the variable E from L ¯ B and L B , respectively, yield the following:˙¯ B E − i E ˙¯ C ˙ C + i E ¯ C ¨ C −
12 ( p x + 2 m p t ) = 0 , ˙ B E + i E ˙¯ C ˙ C − i E ¨¯ C C + 12 ( p x + 2 m p t ) = 0 . (75)The above equations can be used to recast Q ¯ B and Q B as follows: Q (1)¯ B = E ( ˙¯ B ¯ C − ¯ B ˙¯ C + i ˙¯ C ¯ C ˙ C ) − i ¯ B ˙¯ C ¯ C C − ¯ B E ˙ E ¯ C + ¯ B ¯ C,Q (1) B = E ( B ˙ C − ˙ B C − i ˙¯ C ˙ C C ) − i B ¯ C ˙ C C + B E ˙ E C + B C, (76)Using the following EL-EOMs w.r.t. the variables C and ¯ B , respectively, from L ¯ B , namely; i ¯ B ˙¯ C + 2 i ˙¯ B ¯ C − i E ˙ E ˙¯ C − i E ¨¯ C − i ˙ E ¯ C − i E ¨ E ¯ C + ¨¯ C ¯ C C + 2 ˙¯ C ¯ C ˙ C = 0 , ¯ B = E ˙ E − i (2 ¯ C ˙ C + ˙¯ C C ) , (77)we obtain the following exact and interesting expression for the anti-BRST charge: Q (1)¯ B −→ Q (2)¯ B = E ( ˙¯ B ¯ C − ¯ B ˙¯ C + i ˙¯ C ¯ C ˙ C ) + i E ¨¯ C ¯ C C + 2 i E ˙ E ˙¯ C ¯ C C ≡ s ab [ i E ( ¯ C ˙ C − ˙¯ C C )] . (78)At this juncture, we apply the basic principle behind the relationship between the contin-uous symmetry transformations (e.g. s ab ) and its generator [ Q (2)¯ B ] which implies that: s ab Q (2)¯ B = − i { Q (2)¯ B , Q (2)¯ B } = 0 ⇒ [ Q (2)¯ B ] = 0 ⇔ s ab = 0 . (79)Thus, we observe that the off-shell nilpotency ([ Q (2)¯ B ] = 0) of the anti-BRST charge Q (2)¯ B and the anti-BRST symmetry transformations ( s ab ) are inter-related . Thus, we have proven22he off-shell nilpotency of the anti-BRST charge Q (2)¯ B . In exactly similar fashion, we exploitthe following EL-EOMs w.r.t. the variables ¯ C and B from the Lagrangian L B i B ˙ C + 2 i ˙ B C + 3 i E ˙ E ˙ C + i E ¨ C + i ˙ E C + i E ¨ E C + ¯ C ¨ C C + 2 ˙¯ C ˙ C C = 0 ,B = − E ˙ E + i (2 ˙¯ C C + ¯ C ˙ C ) , (80)to recast the BRST charge Q (1) B into another interesting form (i.e. Q (2) B ) as Q (1) B → Q (2) B = E ( B ˙ C − i ˙¯ C ˙ C C − ˙ B C ) − i E ˙ E ¯ C ˙ C C − i E ¯ C ¨ C C ≡ s b [ i E ( ˙¯ C C − ¯ C ˙ C )] (81)which turns out to be an exact quantity w.r.t. s b . Thus, we find that we have the following: s b Q (2) B = − i { Q (2) B , Q (2) B } = 0 ⇒ [ Q (2) B ] = 0 ⇔ s b = 0 . (82)In other words, we have proven the off-shell nilpotency ([ Q (2) B ] = 0) of the BRST charge Q (2) B . Once again, we find that off-shell nilpotency ( s b = 0) of the BRST symmetry trans-formations and the off-shell nilpotency ([ Q (2) B ] = 0) are intertwined an intimate manner.We now focus on the proof of the absolute anticommutativity property of the BRSTcharge with the anti-BRST charge and vice-versa. First of all, let us focus on the BRSTcharge Q (2) B [cf. Eq. (81)]. Using the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0, wecan easily check the following transformation: Q (2) B −→ Q (3) B = E ( ˙¯ B C − i ˙¯ C ˙ C C − ¯ B ˙ C ) − i E ˙ E ¯ C ˙ C C ≡ s ab [ i E ˙ C C ] . (83)In other words we have been able to express the above BRST charge as an exact formw.r.t. the anti-BRST symmetry transformations ( s ab ). This is an interesting observationbecause using the relationship between the continuous symmetry transformations and theirgenerators, we can obtain the following from (83), namely; s ab Q (3) B = − i { Q (3) B , Q (3)¯ B } = 0 ⇔ s ab = 0 . (84)Thus we have been able to demonstrate that the absolute anticommutativity of the BRSTcharge with the anti-BRST charge is connected with the off-shell nilpotency ( s ab = 0) ofthe anti-BRST symmetry transformation ( s ab ). In exactly similar fashion, we can have adifferent form of the anti-BRST charge Q (2)¯ B [cf. Eq. (78)] by using the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0. In other words, we have the following interesting transformation: Q (3)¯ B −→ Q (3)¯ B = E ( B ˙¯ C + 2 i ˙¯ C ¯ C ˙ C − ˙ B ¯ C ) + 2 i E ˙ E ˙¯ C ¯ C C ≡ s b [ i E ˙¯ C ¯ C ] . (85)It is straightforward to note that we have the following relationship: s b Q (3)¯ B = − i { Q (3)¯ B , Q (3) B } = 0 ⇔ s b = 0 . (86)23n other words, we note that the absolute anticommutativity of the anti-BRST charge with the BRST charge is intimately connected with the off-shell nilpotency ( s b = 0) ofthe BRST symmetry transformations ( s b ). This completes our discussions on the off-shellnilpotency 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 ofthe CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 on our theory. The key observations of subsection 7 . superspace by using thebasic terminology of ACSA. Keeping in our mind the mappings ∂ ¯ θ ↔ s b , ∂ θ ↔ s ab , we notethat the (anti-)BRST charges Q ( ¯ B ) B [cf. Eqs. (78), (81)] can be expressed as: Q ¯ B = ∂∂ θ h i E ( ab ) ( τ, θ ) E ( ab ) ( τ, θ ) (cid:8) ¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) − ˙¯ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) (cid:9)i = Z d θ h i E ( ab ) ( τ, θ ) E ( ab ) ( τ, θ ) (cid:8) ¯ F ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) − ˙¯ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) (cid:9)i (87) Q B = ∂∂ ¯ θ h i E ( b ) ( τ, ¯ θ ) E ( b ) ( τ, ¯ θ ) (cid:8) ˙¯ F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) − ¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) (cid:9)i = Z d ¯ θ h i E ( b ) ( τ, ¯ θ ) E ( b ) ( τ, ¯ θ ) (cid:8) ˙¯ F ( b ) ( τ, ¯ θ ) F ( b ) ( τ, ¯ θ ) − ¯ F ( b ) ( τ, ¯ θ ) ˙ F ( b ) ( τ, ¯ θ ) (cid:9)i . (88)It is straightforward to observe that we have the following: ∂∂ θ Q ¯ B = 0 ⇔ s ab Q ¯ B = 0 ⇔ Q B = 0 ⇔ ∂ θ = 0 ,∂∂ ¯ θ Q B = 0 ⇔ s b Q B = 0 ⇔ Q B = 0 ⇔ ∂ θ = 0 . (89)Thus, the off-shell nilpotency of the (anti-)BRST charges is connected with the nilpotency( ∂ θ = 0 , ∂ θ = 0) of the translational generators ( ∂ θ , ∂ ¯ θ ) along the Grassmannian direc-tions of the chiral and anti-chiral (1 , ordinary space if weremember the mappings: s b ↔ ∂ ¯ θ , s ab ↔ ∂ θ [14-16].As far as the absolute anticommutativity property is concerned, we note that the ex-pressions of the (anti-)BRST charges in (85) and (83) can be translated into the superspace where we can exploit the theoretical tools of ACSA. To accomplish this goal, we keep inour knowledge the mappings: s b ↔ ∂ ¯ θ , s ab ↔ ∂ θ to recast the expressions (85) and (83) as: Q (3)¯ B = ∂∂ ¯ θ h i E ( b ) ( τ, ¯ θ ) E ( b ) ( τ, ¯ θ ) ˙¯ F ( b ) ( τ, ¯ θ ) ¯ F ( b ) ( τ, ¯ θ ) i ≡ Z d ¯ θ h i E ( b ) ( τ, ¯ θ ) E ( b ) ( τ, ¯ θ ) ˙¯ F ( b ) ( τ, ¯ θ ) ¯ F ( b ) ( τ, ¯ θ ) i ,Q (3) B = ∂∂ θ h − i E ( ab ) ( τ, θ ) E ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) i ≡ Z d θ h − i E ( ab ) ( τ, θ ) E ( ab ) ( τ, θ ) ˙ F ( ab ) ( τ, θ ) F ( ab ) ( τ, θ ) i . (90)24t is now straightforward to check that the following are true, namely; ∂ ¯ θ Q (3)¯ B = 0 ⇔ s b Q (3)¯ B = 0 ⇔ { Q (3)¯ B , Q (3) B } = 0 ⇔ ∂ θ = 0 ,∂ θ Q (3) B = 0 ⇔ s ab Q (3) B = 0 ⇔ { Q (3) B , Q (3)¯ B } = 0 ⇔ ∂ θ = 0 , (91)which establishes the fact that the ACSA to BRST formalism distinguishes between the two types of absolute anticommutativity properties. In other words, we note that theabsolute anticommutativity of the BRST charge with the anti-BRST charge is connectedwith the nilpotency ( ∂ θ = 0) of the translational generator ( ∂ θ ) along the Grassmanniandirection of the (1, 1)-dimensional chiral super sub-manifold. On the contrary, the absoluteanticommutativity of the anti-BRST charge with the BRST charge is connected with thenilpotency ( ∂ θ = 0) of the translational generator ( ∂ ¯ θ ) along the Grassmannian directionof the (1, 1)-dimensional anti-chiral super sub-manifold. In our present endeavor, we have purposely taken a reparameterization invariant NR andNSUSY system so that we could discuss theoretical aspects that are different from ourearlier works on the NSUSY relativistic scalar and SUSY relativistic spinning particles[23, 24]. However, we have demonstrated in our present investigation that ( i ) the CF-type restriction, and ( ii ) the sum of gauge-fixing and Faddeev-Popov ghost terms are same for our present NR and NSUSY system as have been shown by us for the relativisticparticles (in our earlier works [23, 24]). The above observations are interesting resultsof our present investigation which establish the universality of the (anti-)BRST invariantCF-type restriction for the 1D diffeomorphism invariant (i.e. reparameterization) theories.The CF-type restriction(s) are the hallmark of a quantum theory that is BRST quantized . In fact, for a D-dimensional diffeomorphism invariant theory, it has beenshown [22, 33] that the universal CF-type restriction for a BRST quantized theory is: B µ + ¯ B µ + i ( ¯ C ρ ∂ ρ C µ + C ρ ∂ ρ ¯ C µ ) = 0 where µ, ρ = 0 , , , ...D − B µ and ¯ B µ arethe Nakanishi-Lautrup auxilary fields and the (anti-)ghost fields ( ¯ C µ ) C µ correspond tothe D-dimensional diffeomorphism parameter ǫ µ ( x ) in the infinitesimal trnasformation: x µ → x ′ µ = x µ − ǫ µ ( x ). The universality of the above CF-type restriction implies that,for our 1D diffeomorphism (i.e. reparameterization) invariant theory, the CF-type restric-tion is: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0. This is what we have obtained from various theoreticaltricks in our present endeavor. The existance of the CF-type restriction is very funda-mental to a BRST-quantized theory as it is connected with the geometrical object calledgerbes [7, 8]. Physically, the existence of the CF-type restriction leads to the independentnature of the BRST and anti-BRST symmetries (and corresponding conserved charges) atthe 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 (su-per)string and gravitational theories which are also diffeomorphism invariant. In fact, inour earlier work on a bosonic string theory [34], we have shown the existence of the CF-typerestriction in the context of BRST quantization and it has turned out to be the 2D version25f the universal
CF-type restriction for the D-dimensional diffeomorphism invariant theory.This has happened because the bosonic string theory has the 2D diffeomorphism invarianceon the 2D world-sheet. We plan to apply the beautiful blend of MBTSA and ACSA toderive all the (anti-)BRST symmetries as well as the 2D version of the CF-type restrictionin the case of a bosonic string theory of our interest [35].
Acknowledgments
Two of us (AKR and AT) acknowledge the financial support from the BHU-fellowshipprogram of the Banaras Hindu University (BHU)-Varanasi under which the present inves-tigation has been carried out.
Appendix A: The CF-Type Restriction from L B ≡ L ¯ B In this Appendix, we provide the step-by-step derivation of the CF-type restriction byrequiring the equivalence of the coupled Lagrangian L B and L ¯ B [cf. Eq. (35)]. A closelook at them demonstrates that if we demand L B ≡ L ¯ B , the terms that are common wouldcancel out. For instance, we have cancellations of terms L f , − i E ˙¯ C ˙ C and − ˙¯ C ¯ C ˙ C C thatare present both in L B and L ¯ B . Thus, we are left with the following equality: B B [ E ˙ E − i (2 ˙¯ C C + ¯ C ˙ C )] − i E ˙ E ˙¯ C C ≡ ¯ B B [ E ˙ E − i (2 ¯ C ˙ C + ˙¯ C C )] − i E ˙ E ¯ C ˙ C. ( A. E ˙ E [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] , ( A. all the terms from the r.h.s. to the l.h.s.. At this stage,excluding (A.2), the left-over terms on the l.h.s. and the r.h.s. are B − ¯ B − i B ˙¯ C C − i B ¯ C ˙ C − i ¯ B ¯ C ˙ C − i ¯ B ˙¯ C C = 0 ( A. B − ¯ B − i ( B + ¯ B ) ˙¯ C C − i ( B + ¯ B ) ¯ C ˙ C − i B ˙¯ C C − i ¯ B ¯ C ˙ C = 0 . ( A. B − ¯ B − i [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] ˙¯
C C − i [ B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C )] ¯ C ˙ C i B ˙¯ C C − i ¯ B ¯ C ˙ C = 0 . ( A. A. (cid:2) E ˙ E − i ˙¯ C C − i ¯ C ˙ C (cid:3) (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) + B − ¯ B − i B ˙¯ C C − i ¯ B ¯ C ˙ C = 0 . ( A. − i B ˙¯ C C = − i B ˙¯ C C − i B ˙¯ C C and − i ¯ B ¯ C ˙ C = − i ¯ B ¯ C ˙ C − i ¯ B ¯ C ˙ C and re-arranging the terms, we end up with the following final result: (cid:2) B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) (cid:3) (cid:2) E ˙ E + 12 { B − ¯ B − i ( ˙¯ C C + ¯ C ˙ C ) } (cid:3) = 0 . ( A. B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 on our theory due to the equivalence L B ≡ L ¯ B . This is due to the fact that, in no way, we can state that the other combination: E ˙ E + { B − ¯ B − i ( ˙¯ C C + ¯ C ˙ C ) } = 0.On the contrary, the CF-type restriction: B + ¯ B + i ( ¯ C ˙ C − ˙¯ C C ) = 0 has been proven fromvarious angles (including Eq. (36)).We end this Appendix with the concluding remark that we have derived the CF-typerestriction on our theory from theoretical requirements related with the symmetries of thecoupled (but equivalent) Lagrangians and the absolute anticommutativity properties. How-ever, our present derivation of the CF-type restriction is more direct as well as transparent . Appendix B: On the derivation of f = E ˙ C + ˙ E C and f = ˙¯ B C − ¯ B ˙ C The theoretical content of this Appendix is, first of all, devoted to the explicit derivationof f ( τ ) in the expansion of Σ( τ, ¯ θ ) in the anti-chiral super expansions (41). Towards thisobjective in mind, we focus on Eq. (51) where the first-order differential equation w.r.t.the evolution parameter τ for f has been expressed. We can re-write it as f ˙ C + ˙ f C − ˙ E ˙ C C − E ¨ C C = 0 ⇒ dd τ [ f C ] − dd τ [( ˙ E C + E ˙ C ) C ] = 0 , ( B. C = 0) of the ghost variable ( C ). The aboveequation can be re-expressed in a different but useful form as the total derivative w.r.t. τ : dd τ [ { f − ( E ˙ C + ˙ E C ) } C ] = 0 . ( B. τ = −∞ to τ = + ∞ (which are the limiting cases for τ in our theory), we obtain the following relationship:[ f − ( E ˙ C + ˙ E C )] C = 0 . ( B. all the physical variables of the Lagrangian L B and the secondary variable f ( τ ) vanish off at τ = ± ∞ . For C = 0, we obtain the desired result: f = E ˙ C + ˙ E C . We have taken C = 0because the whole set of BRST symmetry transformations in Eq. (33) is true only whenthe ghost variable C ( τ ) has the non-trivial and non-zero value.We now concentrate on the precise determination of f ( τ ) of the super-expansion (41).In other words, we wish to show that f = ˙¯ B C − ¯ B ˙ C . For this purpose, we note that wehave a first-order differential equation w.r.t. the evolution parameter τ for the secondaryvariable f ( τ ) in Eq. (49). This can be re-expressed as follows f ˙ C + ˙ f C − f ˙ C − B ˙ C C + ˙¯ B ˙ C C + ¯ B ¨ C C = 0 ( B. f ˙ C and ˙¯ B ˙ C C . The above equation implies that wehave now its modified form (with total derivatives) as: dd τ [ f C ] − f ˙ C + ˙¯ B ˙ C C ) + dd τ [ ¯ B ˙ C C ] = 0 ⇒ dd τ h f C + ¯ B ˙ C C i − h ( f − ˙¯ B C ) ˙ C i = 0 . ( B. C = ˙ C = 0) property of the ghost variables C and ˙ C , we can recastthe above equation in the following interesting form where [ f − ( ˙¯ B C − ¯ B ˙ C )] appears verynicely in the individual terms of the following difference, namely; dd τ h { f − ( ˙¯ B C − ¯ B ˙ C ) } C i − h { f − ( ˙¯ B C − ¯ B ˙ C ) } ˙ C i = 0 . ( B. total derivative in the first term to obtain: dd τ h { f − ( ˙¯ B C − ¯ B ˙ C ) } C i − h { f − ( ˙¯ B C − ¯ B ˙ C ) } ˙ C i = 0 . ( B. f − ( ˙¯ B C − ¯ B ˙ C ) = χ leads us to the following (cid:16) dd τ χ (cid:17) C − χ ˙ C = 0 ⇒ ˙ χ C = χ ˙ C. ( B. C and taking into account the fermionic (i.e. C = 0) natureof the ghost variable C , we obtain the following0 = χ ˙ C C ⇒ χ = 0 (cid:2) for C ˙ C = 0 (cid:3) . ( B. s b C = C ˙ C [cf. Eq. (33)]. As a consequence, the combination of the variables C ˙ C = 0. Ifthe symmetry of a theory is the guiding principle behind its beauty, it is physically correctto assume that s b C = C ˙ C = 0. Hence, our conclusion in (B.9), is correct which leads tothe derivation of f = ˙¯ B C − ¯ B ˙ C from χ = 0.28 ppendix C: On the Proof of the Conservation Law We take up here the expression for the BRST charge Q B [cf. Eq. (74)] that has beenderived using the Noether theorem [cf. Sec. 7]. We exploit the EL-EOM derived from theLagrangian L B to recast the expression for ˙ Q B , namely;˙ Q B = ˙ C E p x + 2 m p t ) + C ˙ E p x + 2 m p t ) + C E ( p x ˙ p x + m p t )+2 E ˙ E B ˙ C + E B ˙ C + E B ¨ C + ˙ E B C + E ¨ E B C + E ˙ E ˙ B C + E ˙ E B ˙ C + 2 B ˙ B C − i ˙ B ¯ C ˙ C C − i B ˙¯ C ˙ C C − i B ¯ C ¨ C C, ( C. Q B = − i E ¨¯ C ˙ C C − i E ˙ E ˙¯ C ˙ C C + 3 E ˙ E B ˙ C + E B ¨ C + ˙ E B C + E ¨ E B C + 2 B ˙ B C + B ˙ C − i ˙ B ¯ C ˙ C C − i B ˙¯ C ˙ C C − i B ¯ C ¨ C C, ( C. L B as:˙ p x = 0 , ˙ p t = 0 ,
12 ( p x + 2 m p t ) = − ˙ B E + i E ¨¯ C C − i E ˙¯ C ˙ C. ( C. reduced form as follows˙ Q B = i B ˙¯ C ˙ C C − i ˙ B ¯ C ˙ C C − i E ˙ E ˙¯ C ˙ C C − i E ¨¯ C ˙ C C, ( C. L B w.r.t. the variable ¯ C as2 ˙ B C + B ˙ C + 3 E ˙ E ˙ C + E ¨ C + ˙ E C + E ¨ E C − i ¯ C ¨ C C − i ˙¯ C ˙ C C = 0 . ( C. zero by using the following EL-EOMfrom L B w.r.t. the variable C , namely; i ˙ B ¯ C − i B ˙¯ C + i E ˙ E ˙¯ C + i E ¨¯ C − ¨¯ C ¯ C C − C ¯ C ˙ C = 0 . ( C. Q ¯ B = 0) of the anti-BRST charge [cf. Eq. (74)] which has been derivedby exploiting the theoretical tricks of the Noether theorem. References [1] C. Becchi, A. Rouet, R. Stora,
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