# Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

Authors: Vipul Kumar Pandey
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

We have previously developed the BRST quantization on the hypersurfaceVN−1 embedded in N-dimensional Euclidean space RN in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of L-dimensional manifold VL embedded in RN with 1≤L<N. The result is essentially the same as the previous one. We have also verified the results obtained here using a simple example of particle motion on a torus knot.


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## Body

## 1. Introduction

In the previous work [1] we have considered the BRST quantization of the motion on a hypersurface VN embedded in the N-dimensional Euclidean space RN based on Batalin-Fradkin-Fradkina-Tyutin (BFFT) Abelianized Batalin-Fradkin-Vilkovisky (BFV) and Batalin-Vilkovisky (BV) formalisms. It is a well-known result in differential geometry that any N-dimensional Riemann manifold can be locally embedded in the NN+1/2-dimensional Euclidean space but cannot be embedded in N+1 dimension generally [2]. In this sense, the considerations in the previous manuscript [1] should be extended to general Riemannian manifolds. This is the motivation of the present manuscript.It is well known in the literature that the quantization of the system in curved space has been extensively studied about the ordering problem using two different approaches, canonical and path integral [3–17]. At the same time, the quantization of dynamical systems constrained to curved manifolds embedded in the higher-dimensional Euclidean space has been extensively investigated as one of the quantum theories [18–24]. Here, a nonrelativistic particle constrained to move on a curved surface embedded in the higher dimensional Euclidean space [25, 26] has been taken. These systems and the various properties they possess have been investigated by many authors [27–51]. This has motivated us to extend our previous work [1] to more general class of systems discussed in [26].Becchi-Rout-Stora-Tyutin (BRST) quantization [52–55] is on of the most significant technique to deal with a system with constraints. It has also been found to be symmetry of general class of constrained systems [56–61]. In this quantization method, we enlarge the total Hilbert space of the gauge system under study and bring back the gauge symmetry of the gauge fixed action in the extended phase space, keeping the physical contents of the theory unchanged. BRST symmetry plays very significant role in renormalization of spontaneously broken gauge theories like standard model and hence is of very high significance for different kind of systems. To the best of our knowledge, there is no literature available which studied the BRST symmetry for a particle moving on a hypersurface VL1≤L<N embedded in the Euclidean space RN. This motivates us in the study of BRST symmetry for this system. We will do the constraint analysis of this system using the Dirac’s technique. The system is shown to contain second-class constraints. The BFFT method will be used to convert the second class constraints to first class one [62–76]. Then, the BRST charge and symmetries will be constructed for this BFFT Abelianized system using BFV method [77–79] with the help of Faddeev-Senjanovic technique [80, 81]. In the limit L⟶N−1, the system will return to system in [1]. The results developed in the manuscript have been verified using a model of particle on torus knot [82–86]. At the end, BV-BRST quantization of the BFFT Abelianized system will be investigated [87–90]. Recently, Lagrangian Abelianization procedure for constrained systems has also been developed [91, 92].This is the second and final part of the work. In this part, we will discuss BRST quantization of embeddingVL in Euclidean space RN, where 1≤L<N. The paper has been organized in the following way. In the second section, we have reviewed motion in curved space and also calculated all the possible constraints of the theory using Dirac’s constraint analysis method. In the third section, we have reviewed BFFT formalism. In Section 4, we have constructed first-class constraints and Hamiltonians for the general class of systems under study. In the next section, we will construct BRST symmetry for this system based on BFV formalism. In Section 6, we have shown consistency of our results in the limit L⟶N−1. In Section 7, we have given a simple example of this kind of systems. In Section 8, we have discussed BV quantization of this system based on BFFT formalism. In the next section, concluding remarks have been made. In the end, we have discussed some important calculations in the appendix.


## 2. Classical Mechanics onVL in RN

Let us consider anN-dimensional Euclidean space RN, a point in which is specified by a set of Cartesian coordinates
(1)XA:x1,x2,⋯,xN.Further consider inRN an L-dimensional Riemann subspace, VL1≤L<N, a point of which is specified by a set of coordinates qk [26],
(2)qk:q1,q2,⋯,qL.The metric of this system is defined asgijqk. We can construct in RN a set of curvilinear coordinates including q1,q2,⋯,qL(3)qμ:q1,q2,⋯,qL,qL+1,⋯,qN.Let us assume thatqL+aa:1~N−L are the intrinsic coordinates normal to VL [26]. We can also use the notation
(4)Qa≡qL+a,a:1~N−L.Then, the subspaceVL can be defined as
(5)Qa=qL+a=0.The metric for the curvilinear coordinatesqμ in RN is defined as
(6)g~μν=g~ijNibNjaG~ab,where Nia and G~ab are defined as
(7)Nia≡g~μ=i,ν=L+a,G~ab≡g~μ=L+a,ν=L+b.It is worth noting that the metricgij=g~ijqk,Qa=0 is induced metric on VL and the metric Gab=G~abqk,Qa=0 can be defined as some function on VL. Using this assumption, we will get N=0, when Q=0. So, the metric on VL have form
(8)gμν=g~ij00G~ab,and the inverse matrix is defined as
(9)gμν=gij00Gab,which implies that
(10)gμζ.gζν=δμν,which can further be written as
(11)gij·gjk=δki,Gab·Gbc=δac.We know thatg~μν can also be written as
(12)g~μν=∂x∂qμ·∂x∂qν.From here, we can obtain following relations [26]:
(13)0=∂x∂qk·∂x∂QaQ=0=∑AekA·haA,gijqk=∂x∂qi·∂x∂qjQ=0=∑AeiA·ejA,Gabqk=∂x∂Qa·∂x∂QbQ=0=∑AhaA·hbA,where eiA and haA are defined as
(14)eiAqk≡∂xA∂qiQ=0,haAqk≡∂xA∂QaQ=0.Here,eiA are called the natural frame and gives the induced metric on VL. The inverse metric of g~μν is given by
(15)g~μν=∇qμ·∇qν,where ∇≡∂/∂x. From this, we obtain
(16)gμν=g~μνQ=0=∇qμ·∇qνQ=0.The Lagrangian for the particle motion onVL is defined as [26]
(17)L=12·ẋAẋA−Vx+λaQax.Here,``A" varies between 1 to N and ``a" from 1 to N−L. The metric for the coordinate xA is δAB. λa′S are variables independent of xA, and the dot denotes the time derivative. The canonical momentum conjugate to xA and λa can be written as
(18)PA≡∂L∂ẋA=ẋA,Πa≡∂L∂λ̇a≈0.Hamiltonian corresponding to Lagrangian in Equation (17) can be written as
(19)H0=12·PAPA+Vx−λaQax.


### 2.1. Hamiltonian Analysis

The primary constraint for the system under study is defined as(20)Πa≈0.After including the primary constraint, the new Hamiltonian is written as(21)HT=12·PAPA+Vx−λaQax+uaΠa,where ua′S are a set of Lagrange multipliers for the system. Now, we will perform the constraint analysis of the given system using the Dirac’s technique of constraint analysis [56–61]. All the constraints of the theory can be calculated in the following manner [26]:
(22)Π̇a=Πa,HTP=Qa,Π¨a=Qa,HTP=PA·∂AQa,Πa3=DQa,HTP=D2Qa−∇Qa·∇V−λdQd.The constraintΠa4=0 determines the ua′S, and the procedure is over. So, the explicit form of the constraints are
(23)Φ1a=Πa≈0,Φ2a=Qa≈0,Φ3a=DQa≈0,Φ4a=PAPB∂A∂BQa−∇Qa·∇V−λdQd=D2Qa−∇Qa·∇Φ≈0.Here,D and Φ are defined as D=PA∂A, Φ=V−λaQa. Also, the product of partial derivatives is defined as ∇f·∇g≡∑A∂Af·∂Ag.The Poisson brackets between the constraints are defined as [26]
(24)Φ1a,Φ4bP=−∇Qa·∇Qb≡−αab,Φ2a,Φ3bP=∇Qa·∇Qb≡αab,Φ2a,Φ4bP=2∇Qa·∇DQb≡−βab,Φ3a,Φ4bP=2∇DQa·∇DQb−∇Qa·∇Φ4b≡−γab,Φ3a,Φ3bP=∇DQa·∇Qb−∇Qa·∇DQb≡ρab,Φ4a,Φ4bP=2∇Φ4a·∇DQa−∇Φ4b·∇DQa≡εab.Other Poisson brackets vanish. It is worth noting that(25)αab=αba,symmetric,ρab=−ρba,antisymmetric,εab=−εba,antisymmetric.Thus, the matrixΔijab between the constraints has the form
(26)Δijab≡Φia,ΦjbP=000−αab00αab−βab0−αabρab−γabαabβbaγbaεab.It is worth noting here thata,b=1,⋯,N−1. Hence, each element of the matrix Δijab is a N−1×N−1 matrix. Thus, the matrix Δijab is a 4N−1×4N−1 matrix.


## 2.1. Hamiltonian Analysis

The primary constraint for the system under study is defined as(20)Πa≈0.After including the primary constraint, the new Hamiltonian is written as(21)HT=12·PAPA+Vx−λaQax+uaΠa,where ua′S are a set of Lagrange multipliers for the system. Now, we will perform the constraint analysis of the given system using the Dirac’s technique of constraint analysis [56–61]. All the constraints of the theory can be calculated in the following manner [26]:
(22)Π̇a=Πa,HTP=Qa,Π¨a=Qa,HTP=PA·∂AQa,Πa3=DQa,HTP=D2Qa−∇Qa·∇V−λdQd.The constraintΠa4=0 determines the ua′S, and the procedure is over. So, the explicit form of the constraints are
(23)Φ1a=Πa≈0,Φ2a=Qa≈0,Φ3a=DQa≈0,Φ4a=PAPB∂A∂BQa−∇Qa·∇V−λdQd=D2Qa−∇Qa·∇Φ≈0.Here,D and Φ are defined as D=PA∂A, Φ=V−λaQa. Also, the product of partial derivatives is defined as ∇f·∇g≡∑A∂Af·∂Ag.The Poisson brackets between the constraints are defined as [26]
(24)Φ1a,Φ4bP=−∇Qa·∇Qb≡−αab,Φ2a,Φ3bP=∇Qa·∇Qb≡αab,Φ2a,Φ4bP=2∇Qa·∇DQb≡−βab,Φ3a,Φ4bP=2∇DQa·∇DQb−∇Qa·∇Φ4b≡−γab,Φ3a,Φ3bP=∇DQa·∇Qb−∇Qa·∇DQb≡ρab,Φ4a,Φ4bP=2∇Φ4a·∇DQa−∇Φ4b·∇DQa≡εab.Other Poisson brackets vanish. It is worth noting that(25)αab=αba,symmetric,ρab=−ρba,antisymmetric,εab=−εba,antisymmetric.Thus, the matrixΔijab between the constraints has the form
(26)Δijab≡Φia,ΦjbP=000−αab00αab−βab0−αabρab−γabαabβbaγbaεab.It is worth noting here thata,b=1,⋯,N−1. Hence, each element of the matrix Δijab is a N−1×N−1 matrix. Thus, the matrix Δijab is a 4N−1×4N−1 matrix.


## 3. BFFT Formalism

In this section, we will discuss the main results of BFFT technique, which is used to Abelianize the second class constraint systems. The basic idea behind the scheme is to introduce additional phase space variablesΘmn, besides the existing physical degrees of freedom q,p of the system such that all the constraints in the extended space of the system are first class. This means that the original constraints and Hamiltonian have to be modified accordingly by putting BFFT-extension terms in them. To achieve this, we will use the results discussed in [66, 68, 69]. Let us consider a set of constraints Φnm,Λj and a Hamiltonian operator H. We know from the Dirac’s constraint analysis that second-class constraints of a constrained system satisfy an open algebra. These constraints and Hamiltonian satisfy following algebra:
(27)Φnmq,p,Φrsq,p≈Δnrmsq,p≠0,Φnmq,p,Λrjq,p≈0,Λjq,p,Λtq,p≈0,Λjq,p,Hq,p≈0.``≈" means that the equality holds on the constraint surface. The additional fields satisfy the symplectic algebra:
(28)Θmn,Θsr=ωmsnr,where ωmsnr is a constant quantity and detωmsnr≠0. The constraints are now defined in terms of auxiliary field Θmn as
(29)Φ~nm=Φ~nmq,p;Θmn,This modified constraint satisfies the boundary condition(30)Φ~nmq,p;0=Φnmq,p.These modified constraints should satisfy first-class constraints algebra. So, the Poisson bracket between the constraints are defined as(31)Φ~nm,Φ~rs=0.The solution of Equation (31) can be achieved by considering an expansion of Φnm, as
(32)Φ~nm=∑k=0∞Φ~nmk,where Φ~mk≈OΘk. The first-order correction in the field is [66, 68, 69]
(33)Φ~nm1=Xnrmsq,pΘsr.Putting the expression of Equation (33) in Equation (31) and using the boundary condition given in Equations (30) and (27) as well as Equation (28), we get
(34)Δnrms+XncmdωdfceXresf=0.We notice that Equation (34) does not give a single solution for Xijab, because there is still unknown matrix ωabij. We can make choices for ωabij in such a way that the newly defined variables are unconstrained in nature. Using the value of the matrix ωabij, we can calculate the possible solutions of Xijab from Equation (34). Using the value of Xijab, we can obtain Θ~nm1. If Θnm+Θ~nm1 is strongly involutive in nature, then series will end; otherwise, it will continue in the same way till we do not get strongly involutive constraints. The explicit expression of higher order corrections in the field Φ is
(35)Φ~nmk+1=−1k+2ΘbcXcebdωdfegBgnfmk,k≥1,where Bmnba is defined as
(36)Brsbak=∑l=0kΦ~rbk−l,Φ~salq,p+∑l=0k−2Φ~rbk−l,Φ~sal+2Θ,k≥2,Brsba1=Φ~rb0,Φ~sa1q,p−Φ~ra0,Φ~sb1q,p.In the above expressions, we have defined(37)XmnabXbcnr=ωmnabωbcnr=δmrδca.Another important part of the BFFT formalism is that any dynamical variablefq,p has also to be modified in the same way as discussed above in order to be strongly involutive with the modified constraints Φ~nm. Denoting the modified quantity by fq,p;Θ, we then have
(38)Φ~nm,f~=0.Apart from that, modified variablef~ must also satisfy the boundary condition given below:
(39)f~q,p;0=fq,p.To obtainf~ as an analogous expansion to Equation (32), we will consider
(40)f~=∑k=0∞fk,where f~k is also a term which is of the order n in Θ′s. The expression in Equation (38) above gives us f~1(41)f~1=−ΘnaωabnoXombcq,pΦ~cm,f,where ωabmn and Xmnab are the inverses of ωmnab and Xabmn.The corrections in the physical variablef can be written in the more general form as
(42)f~k+1=−1k+1ΘnaωabnoXombcq,pGfcmk,where
(43)Gabk=∑l=0kΦ~rbk−l,flq,p+∑l=0k−2Φ~rbk−l,fl+2Θ+Φ~rbk+1,f1Θ.In the similar way, we can find the involutive form of other variables using the BFFT method described above.Let us take the initial fields asq and p. Then, the involutive form of these fields (q~ and p~) will satisfy the following relations.
(44)Φ~,q~=Φ~,p~=0.Similarly, any function of the physical variablesq~ and p~ will also satisfy the strong involution relation, since
(45)Θ~,F~q~,p~=Θ~,q~∂F~∂q~+Θ~,p~∂F~∂p~=0.So, if we are taking any dynamical variable in the original phase space, it can be written in involutive form as(46)Fq,p⟶Fq~,p~=F~q~,p~.It is very much obvious that the initial boundary condition in the BFFT formalism, namely, the reduction of the involutive physical variables to the original physical variables, when the new fields are set to zero, remains preserved.


## 4. Construction of the First-Class Constraint Theory

We can easily observe that all the constraints of the theory in Equation (23) are of second class in nature. To change them in the first-class constraints, we will introduce 4L set of possible BFFT fields Θa1,Θa2,Θa3,Θa4. Here, each set of newly introduced fields will correspond to a set of constraints. We will define some relation between these BFFT fields which will help us in the Abelianization of the constraints of the theory. Using the relation between newly introduced fields, we will define ω which will give us the possible solution of Equation (34). Here, we will discuss the Abelianiaztion of constraints and Hamiltonian for the Particle motion on the surface VL1≤L<N in the Riemann manifold RN (based on [75]).Our choice of Poisson Bracket between the fieldsΘa1,Θa2,Θa3,Θa4,a=1,⋯,N−1 are
(47)Θa1,Θb3=Iab,Θa2,Θb4=Iab,where Iab is a N−1×N−1 unitary matrix.From the relation between the fieldsΘ, we can find matrix ωabij as
(48)ωaibj=00Iab0000Iab−Iab0000−Iab00.Using the matrixωaibj defined above and the matrix Δijab between the constraints in Equation (26), we can calculate the possible value of matrix Xijab. Now, using the matrix Xijab, we can write the modified constraints as
(49)Φ~1a=Πa−Θa3,Φ~2a=Qa+Θa2,Φ~3a=PA−∂AQ¯bΘb4∂AQ¯a,Φ~4a=PA−∂AQ¯bΘb4PB−∂BQ¯cΘc4∂A∂BQ¯aΘb1−∂AV¯·∂AQ¯a+λd∂AQ¯a∂AQ¯d+∂AQ¯a∂AQ¯eΘe1.It is worth to mention here that all the barred quantities defined in Equation (49) are function of coordinates xk and fields Θa2 and will take the form of the original unbarred quantities in the limit Θa2⟶0. Here, any field f¯xk,Θa2 will be written as [75]
(50)f¯xk,Θa2=∑n=0∞fann!Θna2.Also, the partial differentiation of fieldf¯ wrt. any field xk can be written as [75].
(51)f¯,i=Qaif¯,Θa4.Then, the Poisson bracket between these modified set of constraints is(52)Φ~ia,Φ~ja=0,where i,j=1,2,3,4, and a,b=1,2,⋯,N−1. We can conclude from Equation (52) that the modified constraints are involutive in nature. Hence, we have successfully converted the second class constraints of the theory into first-class constraints.Now, we will construct the involutive Hamiltonian for this system.Corrections in the Hamiltonian due to different fieldsΘ can be calculated as follows. We will start it by calculating the inverse of the matrices ωaibj and Xijab. The inverse of the matrix ωaibj can be easily written as
(53)ωijab=00−Iab0000−IabIab0000Iab00.The total Hamiltonian with corrections due to BFFT field can be written as(54)H~=12·PA−∂AQ¯bΘb4PA−∂AQ¯cΘc4+V¯x−λa+Θa1Qax+Θa2.Here,α,β,γ,ρ,ε are N−1×N−1 matrices, and Θaii=1,2,3,4 takes 4N−1 possible values.Now, by calculating the Poisson bracket between the modified constraints and the HamiltonianH~, it can be easily verified that modified Hamiltonian is involutive in nature.
(55)H~,Φ~ia=0,where i=1,2,3,4 and a=1,2,⋯,N−1.


## 5. Hamiltonian BRST Quantization

### 5.1. Charge and Symmetry

In this section, we will construct BRST symmetry for the particle motion on the surfaceVL1≤L<N in the Riemann manifold RN. To construct that, we will use the Hamiltonian BRST formalism, also called BFV-BRST formalism [60, 61, 77–79].In the BFV-BRST technique associated to a general class of system with first-class constraints, we introduce two canonical set of ghost and anti-ghost fieldsC,P¯ with ghost number 1 and -1, respectively, and P,C¯ with ghost number -1 and 1, respectively, with Lagrange multiplier fields N,B for each set of constraints. As there are 4N−1 set of constraints, we will introduce two 4N−1 sets of canonical ghost and anti-ghost fields Cka,P¯ka, Pka,C¯kb and Lagrange multiplier fields Nka,Bka. These fields and corresponding momenta satisfy the following superalgebra:
(56)Cka,P¯lb=Pka,C¯lb=Nka,Blb=δlkIab.Now, using the BFV formalism, we can write the general expression for nilpotent BRST charge, gauge-fixing fermion, and BRST invariant Hamiltonian for the system under study as(57)QBRST=∫dxNCakΩ~ka+PakBka,(58)Ψ=∫dxNP¯kaNak+C¯akχka,(59)HU=HP+HBF−QBRST,Ψ.In the BFV-BRST formulation, the generating functional does not depend on gauge fixing fermion [77–81]; hence, one has freedom to choose it in the convenient form. It is also worth noting here that gauge-fixing fermion χka is Hermitian in nature and of the similar Grassmann parity as of Φ~a. They also satisfy
(60)detχka,Φ~bl≠0.We will apply the general results obtained above to the particle on surfaceVL embedded in RN.
(61)Seff=∫dtPAẋA+ΠΘakΘ̇ka+BkaṄak+ṖakC¯ka+ĊakP¯ka−HP−HBF+QBRST,Ψ.For the system under study, the modified constraintsΦ~ka have been calculated in Equation (49).Our choice of gauge conditionχka for the motion on the surface VL in the Riemann manifold RN is Φka in Equation (23).Here,H~=HP+HBF is taken as the BFFT-modified Hamiltonian for this system obtained in Equation (54).We will define the canonical brackets for all dynamical variables as(62)xA,PB=δAB;Θia,ΠΘjb=λia,Πbj=δijIab;C¯ka,Ċbl=iδklIab;Cka,C¯̇bl=−iδklIab.Nilpotent BRST transformations corresponding the action in Equation (61) can be easily constructed using the relation SBRSTΓ=−QBRST,Γ±. Its relation with infinitesimal BRST transformation SBRST can be established as δBRST=SBRSTΓδΛ. δΛ defined here is an infinitesimal BRST parameter. Here, “−” and “+” signs are defined for bosonic and fermionic variables, respectively. The BRST transformation for the particle motion on general class of Riemann surfaces embedded in higher dimensional Euclidean spaces is written as
(63)SBRSTNak=Pak,SBRSTP¯ak=Φ~ak,SBRSTC¯ak=−Bak,SBRSTPA=SBRSTCak=SBRSTΠak=SBRSTPak=0.It can be easily verified that these transformations are nilpotent in nature.Using the expressions for BRST chargeQBRST and gauge-fixing fermion Ψ, effective action in Equation (61) can be written as
(64)Seff=∫dtPAẋA+ΠΘakΘ̇ka+BkaṄak+ṖakC¯ka+ĊakP¯ka−H~−PakP¯ka+NkaΦ~ak+Bkaχak+C¯kaCak.The generating functional for this effective action can be written as(65)ZΨ=∫DφeiSeff.The Liouville measureDφ for the generating functional is defined as
(66)Dφ=∏idΞi.Here,Ξi are all dynamical variables PA,xA,ΠΘak,Θka,Nak,Bka,C¯ak,Pka,Caa,P¯ka of the theory. Now, performing the integration over ghost and antighost momenta Pka and P¯ka, we will get
(67)Zψ=∫Dφ′expi∫dtPAẋA+ΠΘakΘ̇ka+BkaṄak+ĊkaC¯̇ak−H~+NkaΦ~ak−CakC¯ka−Bkaχak.Here,Dφ′ is the new path integral measure for the effective action after the performance of integral over the fields P and P¯. Further performance of integral over fields Bak will give us the new effective generating functional as
(68)Zψ=∫Dϕ′′expi∫dtPAẋA+ΠΘakΘ̇ka+ĊkaC¯̇ak−H~+NkaΦ~ak−CakC¯ka−Ṅka−χka22,where Dφ′′ is the new path integral measure which corresponds to all the dynamical variables left out after integration. Now, the new BRST transformation for the modified action in Equation (68) is written as
(69)SBRSTNak=Ċak,SBRSTC¯ak=−Ṅak−χak,SBRSTPA=SBRSTCak=SBRSTBak=SBRSTPak=0.It is well known in the literature that BRST charges are nilpotent in nature. We also know that the operation of these charges on the states of total Hilbert space gives us the physical subspace of the system.(70)QBRSTphys=0,phys≠QBRST⋯,which can be further written in more explicit form for this system as
(71)iCkaΦ~akphys=0,iC¯̇kaNakphys=0.The result of Equation (71) implies that the first-class constraints of the system under study will annihilate the physical subspace of the total Hilbert space of the system.


## 5.1. Charge and Symmetry

In this section, we will construct BRST symmetry for the particle motion on the surfaceVL1≤L<N in the Riemann manifold RN. To construct that, we will use the Hamiltonian BRST formalism, also called BFV-BRST formalism [60, 61, 77–79].In the BFV-BRST technique associated to a general class of system with first-class constraints, we introduce two canonical set of ghost and anti-ghost fieldsC,P¯ with ghost number 1 and -1, respectively, and P,C¯ with ghost number -1 and 1, respectively, with Lagrange multiplier fields N,B for each set of constraints. As there are 4N−1 set of constraints, we will introduce two 4N−1 sets of canonical ghost and anti-ghost fields Cka,P¯ka, Pka,C¯kb and Lagrange multiplier fields Nka,Bka. These fields and corresponding momenta satisfy the following superalgebra:
(56)Cka,P¯lb=Pka,C¯lb=Nka,Blb=δlkIab.Now, using the BFV formalism, we can write the general expression for nilpotent BRST charge, gauge-fixing fermion, and BRST invariant Hamiltonian for the system under study as(57)QBRST=∫dxNCakΩ~ka+PakBka,(58)Ψ=∫dxNP¯kaNak+C¯akχka,(59)HU=HP+HBF−QBRST,Ψ.In the BFV-BRST formulation, the generating functional does not depend on gauge fixing fermion [77–81]; hence, one has freedom to choose it in the convenient form. It is also worth noting here that gauge-fixing fermion χka is Hermitian in nature and of the similar Grassmann parity as of Φ~a. They also satisfy
(60)detχka,Φ~bl≠0.We will apply the general results obtained above to the particle on surfaceVL embedded in RN.
(61)Seff=∫dtPAẋA+ΠΘakΘ̇ka+BkaṄak+ṖakC¯ka+ĊakP¯ka−HP−HBF+QBRST,Ψ.For the system under study, the modified constraintsΦ~ka have been calculated in Equation (49).Our choice of gauge conditionχka for the motion on the surface VL in the Riemann manifold RN is Φka in Equation (23).Here,H~=HP+HBF is taken as the BFFT-modified Hamiltonian for this system obtained in Equation (54).We will define the canonical brackets for all dynamical variables as(62)xA,PB=δAB;Θia,ΠΘjb=λia,Πbj=δijIab;C¯ka,Ċbl=iδklIab;Cka,C¯̇bl=−iδklIab.Nilpotent BRST transformations corresponding the action in Equation (61) can be easily constructed using the relation SBRSTΓ=−QBRST,Γ±. Its relation with infinitesimal BRST transformation SBRST can be established as δBRST=SBRSTΓδΛ. δΛ defined here is an infinitesimal BRST parameter. Here, “−” and “+” signs are defined for bosonic and fermionic variables, respectively. The BRST transformation for the particle motion on general class of Riemann surfaces embedded in higher dimensional Euclidean spaces is written as
(63)SBRSTNak=Pak,SBRSTP¯ak=Φ~ak,SBRSTC¯ak=−Bak,SBRSTPA=SBRSTCak=SBRSTΠak=SBRSTPak=0.It can be easily verified that these transformations are nilpotent in nature.Using the expressions for BRST chargeQBRST and gauge-fixing fermion Ψ, effective action in Equation (61) can be written as
(64)Seff=∫dtPAẋA+ΠΘakΘ̇ka+BkaṄak+ṖakC¯ka+ĊakP¯ka−H~−PakP¯ka+NkaΦ~ak+Bkaχak+C¯kaCak.The generating functional for this effective action can be written as(65)ZΨ=∫DφeiSeff.The Liouville measureDφ for the generating functional is defined as
(66)Dφ=∏idΞi.Here,Ξi are all dynamical variables PA,xA,ΠΘak,Θka,Nak,Bka,C¯ak,Pka,Caa,P¯ka of the theory. Now, performing the integration over ghost and antighost momenta Pka and P¯ka, we will get
(67)Zψ=∫Dφ′expi∫dtPAẋA+ΠΘakΘ̇ka+BkaṄak+ĊkaC¯̇ak−H~+NkaΦ~ak−CakC¯ka−Bkaχak.Here,Dφ′ is the new path integral measure for the effective action after the performance of integral over the fields P and P¯. Further performance of integral over fields Bak will give us the new effective generating functional as
(68)Zψ=∫Dϕ′′expi∫dtPAẋA+ΠΘakΘ̇ka+ĊkaC¯̇ak−H~+NkaΦ~ak−CakC¯ka−Ṅka−χka22,where Dφ′′ is the new path integral measure which corresponds to all the dynamical variables left out after integration. Now, the new BRST transformation for the modified action in Equation (68) is written as
(69)SBRSTNak=Ċak,SBRSTC¯ak=−Ṅak−χak,SBRSTPA=SBRSTCak=SBRSTBak=SBRSTPak=0.It is well known in the literature that BRST charges are nilpotent in nature. We also know that the operation of these charges on the states of total Hilbert space gives us the physical subspace of the system.(70)QBRSTphys=0,phys≠QBRST⋯,which can be further written in more explicit form for this system as
(71)iCkaΦ~akphys=0,iC¯̇kaNakphys=0.The result of Equation (71) implies that the first-class constraints of the system under study will annihilate the physical subspace of the total Hilbert space of the system.


## 6. Consistency with Previous Result

We can check the consistency of the results of this system in the limit ofN−L=1 [1]. In this limit, elements of the matrix will transform as
(72)αab⟶α,βab⟶β,γab⟶γ.From the antisymmetry (25),
(73)ρab⟶0,εab⟶0.The inverse matrix elements will transform under this limit as(74)A⟶−γα2,B⟶βα2,C⟶γα2,D⟶−βα2,E,F⟶0.Here,E,F vanishes due to asymmetry.Also, under the transformationQa⟶fx, the modified constraints in Equation (49) takes the form of modified constraints of L=N−1 case:
(75)Φ~1=Π−Θ3,Φ~2=fx+Θ2,Φ~3=P¯k−∂kf¯xΘ4∂¯kfx,Φ~4=P¯k−∂kf¯xΘ4P¯l−∂lf¯xΘ4∂k∂lf¯x−∂kV¯·∂kf¯x+λ∂kf¯x∂kf¯x+∂kf¯x∂kf¯xΘ1.The form of modified Hamiltonian in Equation (54) under these transformations is
(76)H~=12·P¯k−∂kf¯xΘ4P¯k−∂kf¯xΘ4+V¯x−λ+Θ1fx+Θ2.Similary, BRST charge and BRST symmetry can be written in this limit as(77)QBRST=CkΦ~k+PkBk,SBRSTNk=Ċk,SBRSTC¯k=−Ṅk−χk,SBRSTPa=SBRSTCk=SBRSTBk=SBRSTPk=0.These are the same constraints, Hamiltonian, BRST charge, and symmetry which we have obtained in our previous work [1]. This shows that all the results obtained here, (49), (54), (63), and (69), using BFFT formalism are consistent with the previous results [1] in the limit L⟶N−1.


## 7. Examples ofL1≤L<N-Dimensional Embedding in RN

As an example ofL-dimensional embedding in RN, we will discuss particle on torus knot [82–86]. We will discuss all the important results developed for general system in this case.


### 7.1. Particle on Torus Knot

Particle on torus knot is a one-dimensional surface embedded in three-dimensional space. It is a special kind of knot that lies on the surface of unknotted torus inR3. It is specified by a set of co-prime integers p and q. A torus knot of type p,q winds p times around the rotational symmetry axis of the torus and q times around a circle in the interior of the torus. The toroidal coordinate system is a suitable choice to study this system. Toroidal coordinates are related to Cartesian coordinates (x1,x2,x3) in following ways:
(78)x1=asinhηcosϕcoshη−cosθ,x2=asinhηsinϕcoshη−cosθ,x3=asinθcoshη−cosθ,where 0≤η≤∞, −π≤θ≤π, and 0≤ϕ≤2π. A toroidal surface is represnted by some specific value of η (say η0). Parameters a and η0 are written as a2=R2−d2 and coshη0=R/D, where R and D are major and minor radius of torus, respectively.Lagrangian for a particle constrained to move on the surface of torus knot is(79)L=12ma2η̇2+θ̇2+sinh2ηϕ̇2coshη−cosθ2+λpθ+qϕ,where r,θ,ϕ are toroidal coordinates for toric geometry and λ is the Lagrange multiplier. The canonical Hamiltonian corresponding to the Lagrangian in Equation (79) is then written as
(80)H=coshη−cosθ22ma2pη2+pθ2+pϕ2sinh2η−λpθ+qϕ,where pη, pθ,pϕ, and pλ are the canonical momenta conjugate to the coordinate η, θ, ϕ, and λ, respectively, defined as
(81)pη=ma2η̇coshη−cosθ2,pθ=ma2θ̇coshη−cosθ2,pϕ=ma2sinh2ηϕ̇coshη−cosθ2,pλ≈0.Thepλ is the primary constraint of the theory.After inclusion of primary constraint, our new Hamiltonian has the form(82)HT=coshη−cosθ22ma2pη2+pθ2+pϕ2sinh2η−λpθ+qϕ+upλ.Now, using Dirac’s method of Hamiltonian analysis, we will calculate all the possible constraints of the theory as(83)ṗλ=pλ,HTP=pθ+qϕ≈0,(84)p¨λ=pθ+qϕ,HTP=coshη−cosθ2ma2ppθ+qpϕsinh2η≈0,(85)pλ3=coshη−cosθ2ma2ppθ+qpϕsinh2η,HTP=coshη−cosθ2ma22ppθsinhη+2qpϕsinhη−2qpϕcoshηcoshη−cosθsinh3ηpηcoshη−cosθma2+2sinθppθ+qpϕsinh2ηpθcoshη−cosθma2−pcoshη−cosθma2sinθpη2+pθ2+pϕ2sinh2η−λp+λq2sinh2η≈0.pλ4 will vanish, and the value of u will be determined from it. All the constraints can be written as
(86)Φ1a=pλ,Φ2a=Qa=pθ+qϕ,Φ3a=DQa=coshη−cosθ2ma2ppθ+qpϕsinh2η,Φ4a=PAPB∂A∂BQa−∇Qa·∇V−λdQd=D2Qa−∇Qa·∇Φ=coshη−cosθ2ma22ppθsinhη+2qpϕsinhη−2qpϕcoshηcoshη−cosθsinh3ηpηcoshη−cosθma2+2sinθppθ+qpϕsinh2ηpθcoshη−cosθma2−pcoshη−cosθma2sinθpη2+pθ2+pϕ2sinh2η−λp+λq2sinh2η.Now, the Poisson brackets between the constraints have following values:(87)Φ1a,Φ4bP=−∇Qa·∇Qb=−coshη−cosθ2ma2p2+q2sinh2η≡−αab,(88)Φ2a,Φ3bP=∇Qa·∇Qb=coshη−cosθ2ma2p2+q2sinh2η≡αab,(89)Φ2a,Φ4bP=2∇Qa·∇DQb=2coshη−cosθ3m2a4pppηsinhη+sinθppθ+qpϕsinh2η+qqpηsinhη1−coshηcoshη−cosθsinh2η+sinθsinh2ηqpθ−ppϕ≡−βab.Similarly, the Poisson bracket between other constraintsΦ3a,Φ4b (A.1), Φ3a,Φ3b (A.2), and Φ4a,Φ4b (A.3) has been explicitly calculated in the appendix. All these brackets will be nonzero and will be equal to γab,ρab, and εab. Thus, the matrix between the constraints will take exactly the same form of matrix Δijab in Equation (26).As all the constraints of the theory (86) are second class, we will follow the method of Section 4 and introduce four possible fields Θa1,Θa2,Θa3,Θa4 corresponding to each constraint. Relation between these fields will provide us possible value of ωabij. Our choice of Poisson bracket between the fields will be same as one taken for the general case. Hence, the matrix ωabij will have the form of Equation (48).Using the matrixωabij and the matrix Δabij, in Equation (33), one can find many possible value of matrix Xabij.Now, applying the results developed in Section4, we can calculate the modified constraints as
(90)Φ~1a=pλ−Θa3,Φ~2a=pθ+qϕ+Θa2,Φ~3a=coshη−cosθ−Θa2/2p2ma2pθ−pΘ4ap+pϕ−qΘ4aqsinh2η,Φ~4a=coshη−cosθ−Θa2/2p2ma22ppθ−pΘ4asinhη+2qpϕ−qΘ4asinhη−2qpϕ−qΘ4acoshηcoshη−cosθ−Θa2/2psinh3ηpηcoshη−cosθ−Θa2/2pma2+2sinθ−Θa22pppθ−pΘ4a+qpϕ−qΘ4asinh2ηpθ−pΘ4acoshη−cosθ−Θa2/2pma2−pcoshη−cosθ−Θa2/2pma2sinθ−Θa22ppη2+pθ−pΘ4a2+pϕ−qΘ4a2sinh2η−λp+λq2sinh2η+coshη−cosθ−Θa2/2pma2p2+q2sinh2ηΘa1.The Poisson bracket between these modified constraints vanishes which shows that modified constraints are involutive. Hence, we have converted the second class constraints of the theory into first class.Now, we will construct first-class Hamiltonian for this system using the results in Section5.The total involutive Hamiltonian for this system will take the form as [75]
(91)H~=coshη−cosθ−Θa2/2p22ma2pη2+pθ−pΘ4a2+pϕ−qΘ4a2sinh2η−λ+Θa1pθ+qϕ+Θa2.It can be easily verified that the HamiltonianH~ is involutive by computing its Poisson bracket with modified constraints of the theory.
(92)H~,Φ~ia=0,where i=1,2,3,4.BRST charge for this first-class system can be written using above expression, as(93)QBRST=iCaiΦ~ia+iPaiBia,and corresponding BRST symmetry transformation can be written as
(94)SBRSTNak=Pak,SBRSTP¯ak=Φ~ak,SBRSTC¯ak=−Bak,SBRSTPA=SBRSTCak=SBRSTΠak=SBRSTPak=0.This shows that the result obtained in Section4 is true for any L1≤L<N-dimensional surface embedded in RN.


## 7.1. Particle on Torus Knot

Particle on torus knot is a one-dimensional surface embedded in three-dimensional space. It is a special kind of knot that lies on the surface of unknotted torus inR3. It is specified by a set of co-prime integers p and q. A torus knot of type p,q winds p times around the rotational symmetry axis of the torus and q times around a circle in the interior of the torus. The toroidal coordinate system is a suitable choice to study this system. Toroidal coordinates are related to Cartesian coordinates (x1,x2,x3) in following ways:
(78)x1=asinhηcosϕcoshη−cosθ,x2=asinhηsinϕcoshη−cosθ,x3=asinθcoshη−cosθ,where 0≤η≤∞, −π≤θ≤π, and 0≤ϕ≤2π. A toroidal surface is represnted by some specific value of η (say η0). Parameters a and η0 are written as a2=R2−d2 and coshη0=R/D, where R and D are major and minor radius of torus, respectively.Lagrangian for a particle constrained to move on the surface of torus knot is(79)L=12ma2η̇2+θ̇2+sinh2ηϕ̇2coshη−cosθ2+λpθ+qϕ,where r,θ,ϕ are toroidal coordinates for toric geometry and λ is the Lagrange multiplier. The canonical Hamiltonian corresponding to the Lagrangian in Equation (79) is then written as
(80)H=coshη−cosθ22ma2pη2+pθ2+pϕ2sinh2η−λpθ+qϕ,where pη, pθ,pϕ, and pλ are the canonical momenta conjugate to the coordinate η, θ, ϕ, and λ, respectively, defined as
(81)pη=ma2η̇coshη−cosθ2,pθ=ma2θ̇coshη−cosθ2,pϕ=ma2sinh2ηϕ̇coshη−cosθ2,pλ≈0.Thepλ is the primary constraint of the theory.After inclusion of primary constraint, our new Hamiltonian has the form(82)HT=coshη−cosθ22ma2pη2+pθ2+pϕ2sinh2η−λpθ+qϕ+upλ.Now, using Dirac’s method of Hamiltonian analysis, we will calculate all the possible constraints of the theory as(83)ṗλ=pλ,HTP=pθ+qϕ≈0,(84)p¨λ=pθ+qϕ,HTP=coshη−cosθ2ma2ppθ+qpϕsinh2η≈0,(85)pλ3=coshη−cosθ2ma2ppθ+qpϕsinh2η,HTP=coshη−cosθ2ma22ppθsinhη+2qpϕsinhη−2qpϕcoshηcoshη−cosθsinh3ηpηcoshη−cosθma2+2sinθppθ+qpϕsinh2ηpθcoshη−cosθma2−pcoshη−cosθma2sinθpη2+pθ2+pϕ2sinh2η−λp+λq2sinh2η≈0.pλ4 will vanish, and the value of u will be determined from it. All the constraints can be written as
(86)Φ1a=pλ,Φ2a=Qa=pθ+qϕ,Φ3a=DQa=coshη−cosθ2ma2ppθ+qpϕsinh2η,Φ4a=PAPB∂A∂BQa−∇Qa·∇V−λdQd=D2Qa−∇Qa·∇Φ=coshη−cosθ2ma22ppθsinhη+2qpϕsinhη−2qpϕcoshηcoshη−cosθsinh3ηpηcoshη−cosθma2+2sinθppθ+qpϕsinh2ηpθcoshη−cosθma2−pcoshη−cosθma2sinθpη2+pθ2+pϕ2sinh2η−λp+λq2sinh2η.Now, the Poisson brackets between the constraints have following values:(87)Φ1a,Φ4bP=−∇Qa·∇Qb=−coshη−cosθ2ma2p2+q2sinh2η≡−αab,(88)Φ2a,Φ3bP=∇Qa·∇Qb=coshη−cosθ2ma2p2+q2sinh2η≡αab,(89)Φ2a,Φ4bP=2∇Qa·∇DQb=2coshη−cosθ3m2a4pppηsinhη+sinθppθ+qpϕsinh2η+qqpηsinhη1−coshηcoshη−cosθsinh2η+sinθsinh2ηqpθ−ppϕ≡−βab.Similarly, the Poisson bracket between other constraintsΦ3a,Φ4b (A.1), Φ3a,Φ3b (A.2), and Φ4a,Φ4b (A.3) has been explicitly calculated in the appendix. All these brackets will be nonzero and will be equal to γab,ρab, and εab. Thus, the matrix between the constraints will take exactly the same form of matrix Δijab in Equation (26).As all the constraints of the theory (86) are second class, we will follow the method of Section 4 and introduce four possible fields Θa1,Θa2,Θa3,Θa4 corresponding to each constraint. Relation between these fields will provide us possible value of ωabij. Our choice of Poisson bracket between the fields will be same as one taken for the general case. Hence, the matrix ωabij will have the form of Equation (48).Using the matrixωabij and the matrix Δabij, in Equation (33), one can find many possible value of matrix Xabij.Now, applying the results developed in Section4, we can calculate the modified constraints as
(90)Φ~1a=pλ−Θa3,Φ~2a=pθ+qϕ+Θa2,Φ~3a=coshη−cosθ−Θa2/2p2ma2pθ−pΘ4ap+pϕ−qΘ4aqsinh2η,Φ~4a=coshη−cosθ−Θa2/2p2ma22ppθ−pΘ4asinhη+2qpϕ−qΘ4asinhη−2qpϕ−qΘ4acoshηcoshη−cosθ−Θa2/2psinh3ηpηcoshη−cosθ−Θa2/2pma2+2sinθ−Θa22pppθ−pΘ4a+qpϕ−qΘ4asinh2ηpθ−pΘ4acoshη−cosθ−Θa2/2pma2−pcoshη−cosθ−Θa2/2pma2sinθ−Θa22ppη2+pθ−pΘ4a2+pϕ−qΘ4a2sinh2η−λp+λq2sinh2η+coshη−cosθ−Θa2/2pma2p2+q2sinh2ηΘa1.The Poisson bracket between these modified constraints vanishes which shows that modified constraints are involutive. Hence, we have converted the second class constraints of the theory into first class.Now, we will construct first-class Hamiltonian for this system using the results in Section5.The total involutive Hamiltonian for this system will take the form as [75]
(91)H~=coshη−cosθ−Θa2/2p22ma2pη2+pθ−pΘ4a2+pϕ−qΘ4a2sinh2η−λ+Θa1pθ+qϕ+Θa2.It can be easily verified that the HamiltonianH~ is involutive by computing its Poisson bracket with modified constraints of the theory.
(92)H~,Φ~ia=0,where i=1,2,3,4.BRST charge for this first-class system can be written using above expression, as(93)QBRST=iCaiΦ~ia+iPaiBia,and corresponding BRST symmetry transformation can be written as
(94)SBRSTNak=Pak,SBRSTP¯ak=Φ~ak,SBRSTC¯ak=−Bak,SBRSTPA=SBRSTCak=SBRSTΠak=SBRSTPak=0.This shows that the result obtained in Section4 is true for any L1≤L<N-dimensional surface embedded in RN.


## 8. Batalin-Vilkovisky Quantization

In the current section of the manuscript, we are going to discuss the quantization of the Particle motion on the surfaceVL1≤L<N in the Riemann manifold RN using the field-antifield formalism [87–89] developed for BFFT systems in [90]. We will start by introducing 4N−1 set of antifields ϖμk⋆=xA⋆,Θak⋆,λak⋆,Cak⋆ corresponding to the fields ϖkμ=xA,Θka,λka,Cka. Here, fields xA,Θak, and λak are bosonic in nature and have ghost number zero, whereas the ghost fields Cak are fermionic in nature and have ghost number one. Antifields corresponding to these fields have opposite Grassmann parity, and their ghost numbers are given by minus the ghost number of the corresponding fields minus one.BV-action for this system in terms of fields and antifields is written as(95)S=S0+∫dtxA⋆xA,Φ~akCka+Θbk⋆Θbk,Φ~alCla+λak⋆Ċka,where action S0 is defined as
(96)S0=∫dtPAẋA+ΠakΘ̇ka−λkaΦ~ak−H~.HereΦ~ia and H~ are modified constraints and modified Hamiltonian in Equation (49) and Equation (54), respectively. The BV-action defined in Equation (95) satisfies the classical master equation
(97)12S,S=0,where the antibracket between any two dynamical variables Xϖ,ϖ⋆ and Yϖ,ϖ⋆ is defined as
(98)X,Y=δrXδϖμδlYδϖμ⋆−δrXδϖμ⋆δlYδϖμ.Here, de Witt’s notation of sum and integration over intermediary variables will be assumed, whenever necessary. The BRST differential in the BV formalism can be introduced using the relationsX=X,S for any local functional Xϖ,ϖ⋆ of fields. Nilpotentcy of the BRST operator s can be proved using classical master equation and Jacobi identity. So, the BV action satisfying the master equation is equivalent to its BRST invariance.To fix gauges, we will extend the Hilbert space to introduce4N−1 pairs of ghost-antighost fields and corresponding momenta C¯ak,Pak, C¯ak⋆,Pak⋆, as well as gauge-fixing fermions Ψ. These antifields can be easily eliminated by choosing ϖμ⋆=∂Ψ/∂ϖμ. One of the possible forms of Ψ we can choose for the given system is
(99)Ψ=C¯akΘka.We have liberty to make other possible choices also. Now, we will extend the BV action defined above to a nonminimal action(100)S⟶Snm=S+∫dtPakC¯k⋆a,in order to implement the many set of gauge fixing conditions introduced by Ψ. Now, the generating functional for gauge-fixed action is defined as
(101)Zψ=∫dϖμdω−1/2df−1/2expiℏSnmϖμ,ϖμ⋆=∂Ψ∂ϖμ.Now, we will replace the original classical field–antifield actionS by some quantum action Σ which is expressed as a local functional of fields and antifields and also satisfy a new equation called quantum master equation defined as
(102)12Σ,Σ−iℏΔΣ=0.Then, the gauge symmetries of the extended action are not obstructed at quantum level. Here,Δ acts as an operator and is defined as
(103)Δ≡δrδϖμδlδϖμ⋆.It is also assumed here that the quantum actionΣ can be expanded in powers of ℏ in the following manner:
(104)Σϖμ,ϖμ⋆=Sϖμ,ϖμ⋆+∑p=1∞ℏpMpϖμ,ϖμ⋆.The first two term of the quantum master Equation (102) reads as
(105)S,S=0,M1,S=iΔS.It can be easily observed that ifΔS is nonzero and gives a nontrivial result, then there exists some M1 which can be expressed in terms of local fields such that Equation (105) is satisfied. Using the cohomological arguments, it can be easily shown that the quantum master equation for the first order systems with pure second-class constraints converted to first class by the use of the BFFT procedure can always be solved. BRST transformations for the fields and antifields for the BFFT Abelianized system can be written as follows:
(106)SBRSTNkν=Ċkν,SBRSTCkν=0,SBRSTC¯kν=Pkν,SBRSTPkν=0,SBRSTxA⋆=−∂S∂xA,SBRSTΘνk⋆=−∂kS∂Θν,SBRSTNνk⋆=Φ~kν,SBRSTP¯k⋆ν=C¯k⋆ν,SBRSTCνk⋆=−xA⋆xA,Φ~νk−Θli⋆Θil,Φ~νk−Ṅk⋆,SBRSTC¯νk⋆=0.The symmetry transformations obtained in Equation (106) are identical to the one obtained in Equation (63) using BFV formalism. Also, we can easily show on the basis of argument given in [90] that the enlarged symmetries due to the compensating fields (BFFT variables) are nonanomalous in nature. These BFFT fields also plays very significant role at the quantum level because of the existence of a counterterm, by modifying the expectation values of the relevant physical quantities.


## 9. Result and Discussion

The BRST symmetry for a particle moving in a curved spaceVL1≤L<N embedded in a Euclidean space RN is investigated in both Hamiltonian and Lagrangian formalism. All the constraints of the system have been calculated using Dirac’s Hamiltonian analysis. Using the algebra of constraints, we have found that all the constraints of the system are second class. To construct a gauge invariant theory, we have used the BFFT technique. Using this technique, all the second class constraints of the system are converted into first-class constraints, and corresponding Hamiltonian is also constructed explicitly. Using the involution of Hamiltonian with first-class constraints, this Hamiltonian is shown to be first class. In the limit of Θ⟶0, the constraints and Hamiltonian returns to original second-class constraints and Hamiltonian. Now, using this gauge invariant system, we have constructed BRST charge, symmetries, and the BRST invariant action. For constructing BRST symmetry from the first-class constraint system, BFV formalism is used. These BRST charges acting on the states of the total Hilbert space will annihilate the physical subspace of it. From it, we have deduced that first-class constraints operating on total Hilbert space of the system will annihilate its physical subspace which can be used as a physicality criteria for the BRST invariant system. We have shown that the general results derived here for any surface embedded in RN is consistent with the results of previous work [1]. We have also discussed particle motion on the torus knot surface as an example of this kind of system. In this example, we have explicitly calculated all the constraints of this system and converted them to first-class constraints. It has been found that diagonal elements ρ and ε of matrix (of the Poisson’s bracket between the constraints) are nonzero if we take two different torus knot systems. In the limit of a⟶b, the ρ,ε will vanish, and we will achieve the commutative case of [1]. We have also constructed the first-class Hamiltonian, BRST charge, and symmetry for this system. It has been shown that all the results deduced for the general system are consistent with this system. At the end, we have discussed Batalin-Vilkovisky quantization of this system based on BFFT formalism. Here, also, we have explicitly calculated the BRST symmetry for the general system which is consistent with the symmetry derived from Hamiltonian formalism. This again proves the equivalence between the Hamiltonian and Lagrangian formalism. Recently, a more general technique of Lagrangian Abelianization has been developed [91, 92]. It will be interesting to apply this technique to the motion in Riemann manifold.



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