# Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold

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

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## Abstract

The BRST quantization of particle motion on the hypersurfaceVN−1 embedded in Euclidean space RN is carried out both in Hamiltonian and Lagrangian formalism. Using Batalin-Fradkin-Fradkina-Tyutin (BFFT) method, the second class constraints obtained using Hamiltonian analysis are converted into first class constraints. Then using BFV analysis the BRST symmetry is constructed. We have given a simple example of these kind of system. In the end we have discussed Batalin-Vilkovisky formalism in the context of this (BFFT modified) system.


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## Body

## 1. Introduction

The quantum mechanical analysis of the system in curved space has been examined about the ordering problem for a long time. Primarily, two approaches have been used, canonical quantization and path-integral method [1–15]. Also, the problem of the quantization of a dynamical system constrained to a curved manifold embedded in the higher-dimensional Euclidean space has been extensively investigated as one of the quantum theories on a curved space [16–22]. Here we are taking a non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space [23, 24]. These type of systems and their various properties such as quantization in different approaches and their comparisons has been studied by many authors [25–49]. Here for the first time we have explicitly constructed the BFFT Abelianization and BRST symmetry for the system of a non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space in both Hamiltonian and Lagrangian formalism [50]. The results derived are of highly significant because of nonlinear nature of the constraints of the system. So, for the first BFFT Abelianization and BRST quantization of a system with nonlinear constraints has been constructed explicitly.BRST quantization [51–54] is an important and powerful technique to deal with a field theoretic model with gauge symmetry. It has also been found as symmetry of constrained systems [55–60]. It enlarges the phase space of a gauge theory and restores the symmetry of the gauge fixed action in the extended phase space without changing the physical contents of the theory. BRST symmetry plays a very important role in renormalizing spontaneously broken theories, like standard model and hence it is extremely important to investigate it for different systems. To the best of our knowledge BRST formulation for a non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space which is nonlinear constrained system and is a toy model for a wide class of physical systems has not been developed yet. This motivates us in the study of BRST symmetry for this system. We study non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space following the technique of Dirac constraints analysis. The system is shown to contain second-class constraints which are nonlinear in nature. We will apply Batalin-Fradkin-Fradkina-Tyutin (BFFT) method to convert these second class constraints to first class constraints [61–73]. We will further develop the BFV (Batalin-Fradkin-Vilkovisky) formulation of this BFFT extended theory using the constraints of the theory [74–76] by the Faddeev-Senjanovic technique [77, 78]. The nilpotent BRST charge is constructed in the operator form using the mode expansion of the fields [79, 80]. The result has been verified using simple example of particle on torus [81–84]. At the end we will construct BRST transformation of the system using Batalin-Vilkovisky (BV) quantization method [86–89]. This is the first part of the two part manuscript. In the second part we will discuss BRST quantization of embedding VL in Euclidean space RN where 1≤L<N [90]. This manuscript has been arranged in the following way. In the first section, we have reviewed motion on hypersurface (H.S.) and its equivalence with motion in curved space and also calculated all the possible constraints of the theory using Dirac’s constraints analysis. In the second section, we have reviewed BFFT formalism. In the third section we have constructed first class constraints and Hamiltonian. In the next section we have constructed BRST symmetry for the system based on BFV Formalism. We have also constructed BRST operator. In the fifth section we have given a simple mechanical example of this kind of system. In the sixth section, we have discussed BV quantization of this system based on BFFT formalism. In the last section concluding remark has been made.


## 2. Motion on a Hypersurface: A Review

Consider an N dimensional Euclidean spaceRN, which is specified by a set of Cartesian coordinates x1,x2,⋯,xa,⋯,xN. Further consider in RN the N−1 dimensional H.S., VN−1 subject to the equation fx1,x2,⋯,xN=0. Let us consider the motion of particle on this H.S. with potential Vx [23, 24]. The Lagrangian for this system can be written as,
(1)L0=12·x.ax.a−Vx+λfxhere, δab is the metric, λ is a variable, which is independent of xaa;1,2,⋯,N and the dot denotes the time derivative. The canonical momentum conjugate to xa and λ can be written as
(2)Pa=∂L∂x.a=x.a,Pλ=∂L∂λ.≈0Hamiltonian corresponding to Lagrangian in eqn(1) can be written as,
(3)H0=12·PaPa+Vx−λfx


### 2.1. Equivalence with Particle in Curved Space

Let us consider a general coordinate transformation ofN dimensional Euclidean space [23].
(4)xa⟶qμa:1~N,μ:0~N−1In this coordinate frameqi’s are the coordinates on H.S. and q0 is the coordinate normal to it. In this coordinate frame, constraints will take form
(5)fx↔q0=constantx.a∂afx↔x.a∂aq0=q.a=0Here we take the constantq0 equal to zero. So, the constraints to the particle motion on H.S. are,
(6)q0=0,q.a=0Heredqi is a tangential vector, and dq0 is the normal vector. The metric for dqμ is generally given by,
(7)ds2=∑adxadxa=dxadxa=gμνdqμdqνwhere metric gμν is defined as
(8)gμν=g0000gijHeregi0=g0i=0 which implies that dq0 is normal to the Surface. The inverse of the metric gμν is defined as
(9)gμν=gμν−1=g0000gijHere,(10)g00=1g00,andgij.gjk=δikUnit normal in Cartesian coordinate is defined [23].
(11)na=na=∂fx/∂xa∂bfx.∂bfx1/2Under a general coordinate transformation,na is transformed into nμ as,
(12)nμ=∂qμ∂xa.na,nμ=gμν.nν=∂xa∂qμ.naIn terms of metric, unit normal can be defined as(13)nμ=δμ0g001/2,nμ=δ0μg001/2Using the transformation in general coordinate system discussed above, we will find the equations in modified form as follows:(14)q¨i+Γjkiq.jq.k+gik.∂vq∂qk=0q¨0=0constraints:
(15)q0=q.0=0Here potential is defined as(16)Vx=Vxq=Vq.The equations derived above implies that there is no motion along the normal and the equation of motion is quite the same as Euler-Lagrange equation obtained from LagrangianL=1/2·gijq.iq.j−Vq which implies that classically the equation of motion in a curved space is similar to that of on HS. In the general coordinate system, the form of Hamiltonian derived from Lagrangian is written as:
(17)H=12.gijqpipj+Vq


### 2.2. Hamiltonian Analysis

The primary constraint for this system is(18)Pλ≈0After inclusion of primary constraint our new Hamiltonian has the form(19)HT=12·PaPa+Vx−λfx+uPλwhere u is the Lagrange multiplier. Now, using the Dirac’s technique of constraint analysis [55–60], we will calculate all the constraints of the theory [23].
(20)P.λ=Pλ,HTP=fx≈0P¨λ=fx,HTP=Pa·∂fx∂xa≈0Pλ3=Pa·∂fx∂xa,HTP=PaPb∂a∂bfx−∂bVx−λ∂bfx·∂bfx≈0Pλ4 will vanish and the value of u will be determined from it. All the constraints can be written as,
(21)Φ1=Pλ≈0(22)Φ2=fx≈0(23)Φ3=Dfx≈0(24)Φ4=D2fx−∂bV−λfx·∂bfx=D2fx−∂bΦ·∂bfx≈0where D=Pa∂a and Φ=V−λfx. Now, the Poisson brackets between the constraints have following values,
(25)Φ1,Φ4P=−∂afx·∂afx=−αΦ2,Φ3P=∂afx·∂afx=αΦ2,Φ4P=Pa∂a∂bfx·∂bfx=Pa∂aα=−βΦ3,Φ4P=2∂aDfx·∂aDfx−∂afx.∂aD2fx−∂bΦ·∂bfx=−γThus the matrixΔab between the constraints has the form
(26)Δab=Φa,ΦbP=000−α00α−β0−α0−γαβγ0


## 2.1. Equivalence with Particle in Curved Space

Let us consider a general coordinate transformation ofN dimensional Euclidean space [23].
(4)xa⟶qμa:1~N,μ:0~N−1In this coordinate frameqi’s are the coordinates on H.S. and q0 is the coordinate normal to it. In this coordinate frame, constraints will take form
(5)fx↔q0=constantx.a∂afx↔x.a∂aq0=q.a=0Here we take the constantq0 equal to zero. So, the constraints to the particle motion on H.S. are,
(6)q0=0,q.a=0Heredqi is a tangential vector, and dq0 is the normal vector. The metric for dqμ is generally given by,
(7)ds2=∑adxadxa=dxadxa=gμνdqμdqνwhere metric gμν is defined as
(8)gμν=g0000gijHeregi0=g0i=0 which implies that dq0 is normal to the Surface. The inverse of the metric gμν is defined as
(9)gμν=gμν−1=g0000gijHere,(10)g00=1g00,andgij.gjk=δikUnit normal in Cartesian coordinate is defined [23].
(11)na=na=∂fx/∂xa∂bfx.∂bfx1/2Under a general coordinate transformation,na is transformed into nμ as,
(12)nμ=∂qμ∂xa.na,nμ=gμν.nν=∂xa∂qμ.naIn terms of metric, unit normal can be defined as(13)nμ=δμ0g001/2,nμ=δ0μg001/2Using the transformation in general coordinate system discussed above, we will find the equations in modified form as follows:(14)q¨i+Γjkiq.jq.k+gik.∂vq∂qk=0q¨0=0constraints:
(15)q0=q.0=0Here potential is defined as(16)Vx=Vxq=Vq.The equations derived above implies that there is no motion along the normal and the equation of motion is quite the same as Euler-Lagrange equation obtained from LagrangianL=1/2·gijq.iq.j−Vq which implies that classically the equation of motion in a curved space is similar to that of on HS. In the general coordinate system, the form of Hamiltonian derived from Lagrangian is written as:
(17)H=12.gijqpipj+Vq


## 2.2. Hamiltonian Analysis

The primary constraint for this system is(18)Pλ≈0After inclusion of primary constraint our new Hamiltonian has the form(19)HT=12·PaPa+Vx−λfx+uPλwhere u is the Lagrange multiplier. Now, using the Dirac’s technique of constraint analysis [55–60], we will calculate all the constraints of the theory [23].
(20)P.λ=Pλ,HTP=fx≈0P¨λ=fx,HTP=Pa·∂fx∂xa≈0Pλ3=Pa·∂fx∂xa,HTP=PaPb∂a∂bfx−∂bVx−λ∂bfx·∂bfx≈0Pλ4 will vanish and the value of u will be determined from it. All the constraints can be written as,
(21)Φ1=Pλ≈0(22)Φ2=fx≈0(23)Φ3=Dfx≈0(24)Φ4=D2fx−∂bV−λfx·∂bfx=D2fx−∂bΦ·∂bfx≈0where D=Pa∂a and Φ=V−λfx. Now, the Poisson brackets between the constraints have following values,
(25)Φ1,Φ4P=−∂afx·∂afx=−αΦ2,Φ3P=∂afx·∂afx=αΦ2,Φ4P=Pa∂a∂bfx·∂bfx=Pa∂aα=−βΦ3,Φ4P=2∂aDfx·∂aDfx−∂afx.∂aD2fx−∂bΦ·∂bfx=−γThus the matrixΔab between the constraints has the form
(26)Δab=Φa,ΦbP=000−α00α−β0−α0−γαβγ0


## 3. BFFT Analysis: A Short Review

In this section we will review BFFT technique [61–73], which is used to construct a first class constraint system from a second-class constraint system. We know from the Dirac-Bergmann constraint analysis that second-class constraint of a constrained system satisfy an open algebra. Let us take a system described by a Hamiltonian H0 in a 2 N dimensional phase space. Let us denote the second-class constraints of the system as Ta with a=1,2,⋯,M<2N. These constraints satisfy following algebra
(27)Ta,Tb=Δab,where detΔab≠0. To achieve this goal, we will extend the Hilbert space of the theory by introducing auxiliary fields ηa, one for each second class constraint. This is done to keep the physical degrees of freedom in the extended theory same as in the original theory. These fields satisfy the symplectic algebra,
(28)ηa,ηb=ωabwhere ωab is a constant quantity and detωab≠0. The constraints are now defined in terms of auxiliary field ηa as
(29)T~a=T~aq,p;η=∑n=0∞T~an=∑n=0∞XabnγnηbnγnThe modified constraints satisfy first class constraints algebra. So the Poisson bracket between the constraints are defined as(30)T~a,T~b=0This modified constraint satisfies the boundary condition(31)T~aq,p;0=Taq,p=T~a0,Replacement of eqn(29) into eqn(30) gives recurrence relations, one for each coefficient of ηn.Using this iterative technique we can calculate thenth order correction term T~n. The expression for T~1 is written as
(32)T~a1=Xabq,pηbPutting this expression in [30] and using the boundary condition (31) as well as (27) and (28), we get
(33)Δab+XacωcdXbd=0We notice that this equation does not giveXab univocally, because it also contains the still unknown ωab. We choose ωab in such a way that the new variables are unconstrained. The knowledge of Xab allows us to obtain T~a1. If Ta+T~a1 is strongly involutive then series ends here or we will continue the same process to calculate the higher order terms till we do not get strongly involutive constraints.Another point in the Hamiltonian formalism is that any dynamic functionAq,p (for instance, the Hamiltonian) has also to be properly modified in order to be strongly involutive with the first-class constraints T~a. Denoting the modified quantity by Aq,p;η, we then have [61–66]
(34)T~a,A~=0In addition,A~ has also to satisfy the boundary condition,
(35)A~q,p;0=Aq,p=A~0To obtainA~ an expansion analogous to (29) is considered,
(36)A~=∑n=0∞AnA1 can be written as
(37)A1=−ηaωabXbcq,pTc,A,where ωab and Xab are the inverses of ωab and Xab. It was earlier seen that Ta+Ta1 was strongly involutive if the coefficients Xab do not depend on q,p. However, the same argument does not necessarily apply in this case. Usually we have to calculate other corrections to obtain the final A~.The general expression reads as,(38)An+1=−1n+1ηaωabXbcq,pGcn,where
(39)Gan=∑m=0nTan−m,Amq,p+∑n=0n−2Tan−m,Am+2η+Tan+1,A1ηSimilarly the involutive form of other variables can be obtained using the method described above. Let the initial fields beq and p. Then their involutive form q~ and p~ will follow these relations.
(40)T~,q~=T~,p~=0Now any function ofq~ and p~ will also be strongly involutive, since
(41)T~,F~q~,p~=T~,q~∂F~∂q~+T~,p~∂F~∂p~=0Thus if we take any dynamical variable in the original phase space, its involutive form can be obtained by the replacement(42)Fq,p⟶Fq~,p~=F~q~,p~It is obvious that the initial boundary condition in the BFFT process, namely, the reduction of the involutive function to the original function when the new fields are set to zero, remains preserved.


## 4. Construction of First Class Constraint Theory

As all the constraints of the theory (eqn[21–24]) are second class, we will introduce four possible fields ϑ1,ϑ2,ϑ3,ϑ4 corresponding to each constraint. Relation between these fields will give us possible solution of the eqn(33). Relation between these fields will provide us possible value of ωab. Here we will discuss the solutions (based on author’s recent article on Abelianization of prototypical nonlinear second class system [73]) and will construct the involutive Hamiltonian. Our choice of Poisson Bracket between the fields ϑ1,ϑ2,ϑ3,ϑ4 are
(43)ϑ1,ϑ3=1,ϑ2,ϑ4=1From the above relation, matrixωab between the fields can be written as,
(44)ωab=00100001−10000−100Putting the matrixωab and Δab in the eqn(33), and solving it, one can find many possible values of matrix Xab. Now using the eqn(29) we can calculate the modified constraints as
(45)Φ~1=Pλ−ϑ3(46)Φ~2=fx+ϑ2(47)Φ~3=P¯a−∂af¯xϑ4∂¯afx(48)Φ~4=P¯a−∂af¯xϑ4P¯b−∂bf¯xϑ4∂a∂bf¯x−∂aV¯·∂af¯x+λ∂af¯x∂af¯x+∂af¯x∂af¯xϑ1where all barred quantities are function of xa and ϑ2 and will change to original unbarred quantities in the limit ϑ2⟶0. Here, any field A¯xa,ϑ2 will be written as [73].
(49)A~xa,ϑ2=∑n=0∞Ann!ϑn2also, its partial differentiation wrt. Any field xa can be written as [73].
(50)A~,i=fxiA¯,ϑ4The Poisson bracket between these modified constraints are(51)Φ~i,Φ~j=0where i,j=1,2,3,4. 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. Corrections in Hamiltonian due to different fields ϑ can be calculated as follows. First we will calculate inverse of the matrices ωab and Xab. The matrix ωab is written as
(52)ωab=00−10000−110000100Using the BFFT method discussed in the section III, we will find the involutive Hamiltonian for this system as [73].
(53)H~=12·P¯a−∂af¯xϑ4P¯a−∂af¯xϑ4+V¯x−λ+ϑ1fx+ϑ2As mentioned above, all the barred quantities are functions ofx and ϑ2. It can be easily verified that the Hamiltonian H~ is involutive by computing it’s Poisson bracket with modified constraints of the theory.
(54)H~,Φ~i=0where i=1,2,3,4.


## 5. Hamiltonian BRST Formalism

To construct BRST symmetry for this system, we further extend the theory using Hamiltonian BRST formalism also called BFV formalism [74–76]. Here we will present a simplified form of this formalism.In the BFV formulation associated with a general class of system with first class constraints, we introduce two canonical set of ghost and anti-ghost fieldsCk,P¯k with ghost number 1 and -1, respectively, and Pk,C¯k with ghost number -1 and 1, respectively, with Lagrange multiplier fields Nk,Bk. These ghost-antighost, corresponding momenta and stuckelberg fields satisfy following super algebra,
(55)Ck,P¯l=Pk,C¯l=Nk,Bl=δlkNow, the general expression for nilpotent BRST charge, gauge-fixing fermion and BRST invariant Hamiltonian is given by,(56)Qb=∫dxNCkΦ~k+PkBk(57)Ψ=∫dxNP¯kNk+C¯Kχk(58)HU=HP+HBF−Qb,ΨIn BFV formulation the generating functional is independent of gauge fixing fermion [74–78], hence we have liberty to choose it in the convenient form. It is also worth notice that χk is the Hermitian gauge-fixing function with the identical Grassmann parity as Φ~, and satisfies
(59)det∣χk,Φ~l∣≠0Now, we will apply these general results to the particle on surfaceVN−1 embedded in RN.
(60)Seff=∫dtPax.a+Πϑkϑ.k+BkN.k+P.kC¯k+C.kP¯k−HP−HBF+Qb,ΨIn case of the system described in section IV, the convenient choice isΦ~k calculated in eqn[45–48].Our choice of gauge conditionχk in this case is Φk in eqnand H~=HP+HBF is taken as the BFFT modified Hamiltonian in eqn[53].The canonical brackets for all dynamical variables are written as(61)xa,Pb=δba;ϑi,Πϑj=δij;(62)C¯a,C.b=iδab;Ca,C¯.b=−iδbaNilpotent BRST transformation corresponding to this action is constructed using the relationsbΓ=−Qb,Γ± which is related to infinitesimal BRST transformation as δbΓ=sbΓδΛ. Here δΛ is infinitesimal BRST parameter. Here − sign is for bosonic and + is for fermionic variable. The BRST transformation for the particle on a Riemann surface is,
(63)sbNk=Pk,sbP¯k=Φ~ksbC¯k=Bk,sbCk=sbBk=sbPk=0One can easily verify that these transformations are nilpotent.Using the expressions forQb and Ψ, Effective action (60) is written as
(64)Seff=∫dtPax.a+Πϑkϑ.k+BkN.k+P.kC¯k+C.kP¯k−H~−PkP¯k+NkΦ~k+Bkχk+C¯kCkand the generating functional for this effective theory is represented as
(65)Zψ=∫DϕexpiSeffThe Liouville measureDϕ=∏idξi, where ξi are all dynamical variables Pa,xa,Πϑk,ϑk,Bk,Nk,C¯k,Pk,Ck,P¯k of the theory. Now integrating this generating functional over P and P¯, we get
(66)Zψ=∫Dϕ′expi∫dtPax.a+Πϑkϑ.k+BkN.k+C.kC¯.k−H~+NkΦ~k−CkC¯k−BkχkwhereDϕ′ is the path integral measure for effective theory when integration over fields P and P¯ are carried out. Further integrating over Bk we obtain an effective generating functional as
(67)Zψ=∫Dϕ′′expi∫dtPax.a+Πϑkϑ.k+C.kC¯.k−H~+NkΦ~k−CkC¯k−N.k−χk22where DΦ′′ is the path integral measure corresponding to all the dynamical variables involved in the effective action. The BRST symmetry transformation for this effective theory is written as
(68)sbNk=C.k,sbC¯k=−N.k−χksbPk=sbCk=sbBk=0We know from the literature that BRST charge is nilpotent in nature. Also its operation on the states of Hilbert space will give us the physical subspace of the system.(69)QBRSTphys=0,phys≠QBRST⋯.which can be written as
(70)iCkΦ~kphys=0,iC¯.kNkphys=0This implies that the first class constraints of the system anihilates the physical subspace of the total Hilbert sapce of the system.


### 5.1. Canonical BRST Quantization

The BRST extended action is given by eqn(60).As we know, variation of S will give boundary conditions. To covariantly quantize this system, we will now Fourier decompose the BRST charge [79, 80].
(71)Φ~kx,ϑ,t=12π∑n=0∞Φ~nke−int+Φ~nk†eintPkx,t=12π∑n=0∞Pnke−int+Pnk†eintCkx,t=∑n=0∞Cxnke−int+Cxnk†eintPkx,t=∑n=0∞Pxnae−int+Pxna†eintBkx,t=12π∑n=0∞Bnke−int+Bnk†eintHere the commutation relations between these variables is defined as in eqn(62). Putting these mode expansions in eqn(56) and simplifying, we can easily achieve the operator form of BRST charge.Applying this charge on the states of total Hilbert space will give us physical subspace conditions.


## 5.1. Canonical BRST Quantization

The BRST extended action is given by eqn(60).As we know, variation of S will give boundary conditions. To covariantly quantize this system, we will now Fourier decompose the BRST charge [79, 80].
(71)Φ~kx,ϑ,t=12π∑n=0∞Φ~nke−int+Φ~nk†eintPkx,t=12π∑n=0∞Pnke−int+Pnk†eintCkx,t=∑n=0∞Cxnke−int+Cxnk†eintPkx,t=∑n=0∞Pxnae−int+Pxna†eintBkx,t=12π∑n=0∞Bnke−int+Bnk†eintHere the commutation relations between these variables is defined as in eqn(62). Putting these mode expansions in eqn(56) and simplifying, we can easily achieve the operator form of BRST charge.Applying this charge on the states of total Hilbert space will give us physical subspace conditions.


## 6. Examples ofN−1 Dimensional Embedding in RN

As an example ofN−1 Dimensional Embedding in RN we will discuss particle on torus model.


### 6.1. Particle on Torus

Particle on torus [81–85] is a two dimensional surface embedded in three dimensional space. It has been studied as a toy model for different type of field theories. Here, we will discuss all the important results developed for general system in section IV for this case. Lagrangian for a particle constrained to move on the surface of torus of radius r is
(72)L=12mr.2+12mr2θ.2+12mb+rsinθ2ϕ.2+λr−awhere r,θ,ϕ are toroidal co-ordinates. Their relation with Cartesian coordinates can be written as,
(73)x=b+rsinθcosϕ,y=b+rsinθsinϕ,z=rcosθand λ is the Lagrange multiplier. Here we have considered a torus with axial circle in the x−y plane centered at the origin, of radius b, having a circular cross section of radius r. The angle θ ranges from 0 to 2π, and the angle ϕ from 0 to 2π.The canonical Hamiltonian corresponding to the Lagrangian in eqn(72)is then written as,
(74)H=pr22m+pθ22mr2+pϕ22mb+rsinθ2−λr−awhere pr, pθ,pϕ and pλ are the canonical momenta conjugate to the coordinate r, θϕ and λ, respectively, given by
(75)pr=mr.,pθ=mr2θ.pϕ=mb+rsinθ2ϕ.,pλ≈0Herepλ is the primary constraint of the theory. After inclusion of primary constraint our new Hamiltonian has the form
(76)HT=pr22m+pθ22mr2+pϕ22mb+rsinθ2−λr−a+upλwhere u is a Lagrange multiplier. Now, using Dirac’s method of Hamiltonian analysis, we will calculate all the possible constraints of the theory.
(77)p.λ=pλ,HTP=r−a≈0p¨λ=r−a,HTP=prm≈0pλ3=prm,HTP=1mpθ2mr3+Pphi2sinθmb+rsinθ3+λ≈0Pλ4 will vanish and the value of u will be determined from it. All the constraints can be written as,
(78)Φ1=pλ≈0(79)Φ2=fx=r−a≈0(80)Φ3=Dfx=prm≈0(81)Φ4=D2fx−∂bV−λfx·∂bfx=1mpθ2mr3+Pphi2sinθmb+rsinθ3+λ≈0Now, the Poisson brackets between the constraints have following values,(82)Φ1,Φ4P=−∂afx·∂afx=−1mΦ2,Φ3P=∂afx·∂afx=1mΦ2,Φ4P=Pa∂a∂bfx·∂bfx=Pa∂aα=0Φ3,Φ4P=2∂aDfx·∂aDfx−∂afx·∂aD2fx−∂bΦ·∂bfx=3m3pθ2r4+pϕ2sin2θb+rsinθ4=−γThus the matrixΔab between the constraints has the form
(83)Δab=Φa,ΦbP=000−1m001m00−1m03m3pθ2r4+pϕ2sin2θb+rsinθ41m0−3m3pθ2r4+pϕ2sin2θb+rsinθ40As all the constraints of the theory (81) are second class, we will follow the method of section IV and introduce four possible fields ϑ1,ϑ2,ϑ3,ϑ4 corresponding to each constraint. Relation between these fields will provide us possible value of ωab. Our choice of Poisson bracket between the fields will be same as one taken for the general case. Hence the matrix ωab will have the form of eqn[44].Using the matrixωab and the matrix Δab between the constraints, in the eqn(33), one can find many possible value of matrix Xab.Now, applying the results developed in section IV we can calculate the modified constraints as(84)Φ~1=pλ−ϑ3Φ~2=r−a+ϑ2Φ~3=1mpr−ϑ4Φ~4=1mpθ2mr+ϑ23+pϕ2sinθmb+r+θ2sinθ3+λ−θ1,i=1,2,3,4The 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 section V.The total involutive Hamiltonian for this system will take the form as [73],
(85)H~=pr−ϑ422m+pθ22mr+ϑ22+pϕ22mb+r+ϑ2sinθ2−λ+ϑ1r+ϑ2−aIt can be easily verified that the HamiltonianH~ is involutive by computing it’s Poisson bracket with modified constraints of the theory.
(86)H~,Φ~i=0where i=1,2,3,4.BRST charge for this first class system can be written using above expression, as(87)Qb=iCiΦ~i+iPiBiand corresponding BRST symmetry transformation can be written as
(88)sbNi=Pi,sbP¯i=Φ~isbC¯i=Bi,sbCi=sbPa=sbBi=sbPi=0This shows that result obtained above is true for anyN−1 dimensional surface embedded in RN.


## 6.1. Particle on Torus

Particle on torus [81–85] is a two dimensional surface embedded in three dimensional space. It has been studied as a toy model for different type of field theories. Here, we will discuss all the important results developed for general system in section IV for this case. Lagrangian for a particle constrained to move on the surface of torus of radius r is
(72)L=12mr.2+12mr2θ.2+12mb+rsinθ2ϕ.2+λr−awhere r,θ,ϕ are toroidal co-ordinates. Their relation with Cartesian coordinates can be written as,
(73)x=b+rsinθcosϕ,y=b+rsinθsinϕ,z=rcosθand λ is the Lagrange multiplier. Here we have considered a torus with axial circle in the x−y plane centered at the origin, of radius b, having a circular cross section of radius r. The angle θ ranges from 0 to 2π, and the angle ϕ from 0 to 2π.The canonical Hamiltonian corresponding to the Lagrangian in eqn(72)is then written as,
(74)H=pr22m+pθ22mr2+pϕ22mb+rsinθ2−λr−awhere pr, pθ,pϕ and pλ are the canonical momenta conjugate to the coordinate r, θϕ and λ, respectively, given by
(75)pr=mr.,pθ=mr2θ.pϕ=mb+rsinθ2ϕ.,pλ≈0Herepλ is the primary constraint of the theory. After inclusion of primary constraint our new Hamiltonian has the form
(76)HT=pr22m+pθ22mr2+pϕ22mb+rsinθ2−λr−a+upλwhere u is a Lagrange multiplier. Now, using Dirac’s method of Hamiltonian analysis, we will calculate all the possible constraints of the theory.
(77)p.λ=pλ,HTP=r−a≈0p¨λ=r−a,HTP=prm≈0pλ3=prm,HTP=1mpθ2mr3+Pphi2sinθmb+rsinθ3+λ≈0Pλ4 will vanish and the value of u will be determined from it. All the constraints can be written as,
(78)Φ1=pλ≈0(79)Φ2=fx=r−a≈0(80)Φ3=Dfx=prm≈0(81)Φ4=D2fx−∂bV−λfx·∂bfx=1mpθ2mr3+Pphi2sinθmb+rsinθ3+λ≈0Now, the Poisson brackets between the constraints have following values,(82)Φ1,Φ4P=−∂afx·∂afx=−1mΦ2,Φ3P=∂afx·∂afx=1mΦ2,Φ4P=Pa∂a∂bfx·∂bfx=Pa∂aα=0Φ3,Φ4P=2∂aDfx·∂aDfx−∂afx·∂aD2fx−∂bΦ·∂bfx=3m3pθ2r4+pϕ2sin2θb+rsinθ4=−γThus the matrixΔab between the constraints has the form
(83)Δab=Φa,ΦbP=000−1m001m00−1m03m3pθ2r4+pϕ2sin2θb+rsinθ41m0−3m3pθ2r4+pϕ2sin2θb+rsinθ40As all the constraints of the theory (81) are second class, we will follow the method of section IV and introduce four possible fields ϑ1,ϑ2,ϑ3,ϑ4 corresponding to each constraint. Relation between these fields will provide us possible value of ωab. Our choice of Poisson bracket between the fields will be same as one taken for the general case. Hence the matrix ωab will have the form of eqn[44].Using the matrixωab and the matrix Δab between the constraints, in the eqn(33), one can find many possible value of matrix Xab.Now, applying the results developed in section IV we can calculate the modified constraints as(84)Φ~1=pλ−ϑ3Φ~2=r−a+ϑ2Φ~3=1mpr−ϑ4Φ~4=1mpθ2mr+ϑ23+pϕ2sinθmb+r+θ2sinθ3+λ−θ1,i=1,2,3,4The 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 section V.The total involutive Hamiltonian for this system will take the form as [73],
(85)H~=pr−ϑ422m+pθ22mr+ϑ22+pϕ22mb+r+ϑ2sinθ2−λ+ϑ1r+ϑ2−aIt can be easily verified that the HamiltonianH~ is involutive by computing it’s Poisson bracket with modified constraints of the theory.
(86)H~,Φ~i=0where i=1,2,3,4.BRST charge for this first class system can be written using above expression, as(87)Qb=iCiΦ~i+iPiBiand corresponding BRST symmetry transformation can be written as
(88)sbNi=Pi,sbP¯i=Φ~isbC¯i=Bi,sbCi=sbPa=sbBi=sbPi=0This shows that result obtained above is true for anyN−1 dimensional surface embedded in RN.


## 7. Batalin - Vilkovisky Quantization

We will perform the quantization of system described above along the field-antifield formalism [86–88] for BFFT system discussed in ref [89]. To do so we will introduce antifields ϕA⋆=xμ⋆,ϑν⋆,Nν⋆,Cν⋆ corresponding to the fields ϕA=xμ,ϑν,Nν,Cν. Here, fields xμ,ϑν and Nν are bosonic and have ghost number zero. The ghosts Cν are fermionic and have ghost number one. The corresponding anti-fields have opposite grassmanian parity and ghost number given by minus the ghost number of the corresponding field minus one. Action in terms of fields and antifields is written as
(89)S=S0+∫dtxμ⋆xμ,Φ~νcν+ϑφ⋆ϑφ,Φ~νcν+Nν⋆C.νwhere S0 is defined as
(90)S0=∫dtPμx.μ+Πνϑ.ν−NνΦ~ν−H~HereΦ~ are the modified constraints in eqn[45–48] and H~ is the modified Hamiltonian in eqn(53). Now, this action satisfies the classical master equation
(91)12S,S=0where the antibracket between any two quantities Xϕ,ϕ⋆ and Yϕ,ϕ⋆ is defined as
(92)X,Y=δrXδϕAδlYδϕA⋆−δrXδϕA⋆δlYδϕAHere we assume the de Witt’s notation of sum and integration over intermediary variables, when necessary. In the BV formalism, the BRST differential is introduced using the relation(93)sX=X,Sfor any local functional Xϕ,ϕ⋆. Due to classical master equation and Jacobi identity, s is nilpotent. So, the BV action satisfying the master equation is equivalent to BRST invariance.To fix a gauge, we need to introduce trivial pairsC¯ν,Pν as new fields and the corresponding antifields C¯ν⋆,Pν⋆, as well as a gauge-fixing fermion Ψ. The antifields are eliminated by choosing ϕA⋆=∂Ψ/∂ϕA. We can choose the form of Ψ as
(94)Ψ=C¯νϑνOther possible choices can also be made. It is also necessary to extend the field-antifield action to a nonminimal action,(95)S⟶Snm=S+∫dtPνC¯⋆νin order to implement the gauge fixing introduced by Ψ. The gauge-fixed generating functional is then defined as
(96)ZΨ=∫dϕAdω−1/2df−1/2expiℏSnmϕA,ϕA⋆=∂Ψ∂ϕAIn general, if the classical field-antifield actionS is replaced by some quantum action W expressed as a local functional of fields and antifields and satisfy the so-called quantum master equation
(97)12W,W−iℏΔW=0then the gauge symmetries are not obstructed at quantum level. Here Δ is a potentially singular operator which is defined as
(98)Δ≡δrδϕAδlδϕA⋆and it was assumed that W can be expanded in powers of ℏ as
(99)WϕA,ϕA⋆=SϕA,ϕA⋆+∑p=1∞ℏpMpϕA,ϕA⋆The first two term of the quantum master equation (97) are
(100)S,S=0(101)M1,S=iΔSIfΔS is non-zero and gives a nontrivial result, then there exists some M1 expressed in terms of local fields such that (101) is satisfied. Using cohomological arguments, it can be shown that the quantum master equation, for first order systems with pure second class constraints converted with the use of the BFFT procedure, can always be solved. BRST transformations for the BFFT converted system can be written as
(102)sbNν=C.ν,sbCν=0,sbC¯ν=Pν,sbPν=0sbxμ⋆=−∂S∂xμ,sbϑν⋆=−∂S∂ϑν,sbNν⋆=Φ~ν,sbCν⋆=−xμ⋆xμ,Φ~ν−ϑφ⋆ϑφ,Φ~ν−N.⋆,sbC¯ν⋆=0,sbP¯⋆ν=C¯⋆νThese symmetry transformations are same as the one obtained in [68]. It can be shown, on the basis of argument given in ref [89] that enlarged symmetries due to compensating fields (BFFT variables) are not anomalous. These fields plays non-trivial role at the quantum level because the existence of a counter-term modify expectation values of relevant physical quantities.


## 8. Conclusion

We have for the first time investigated BRST symmetry for a particle moving in a curved spaceVN−1 embedded in a Euclidean space RN in both Hamiltonian and Lagrangian formalism. Using the Dirac’s constraints analysis, we have calculated all the constraints of the system. Using the BFFT technique, second class constraints are converted into first class constraints and corresponding first class Hamiltonian is constructed. In the limit of ϑ⟶0 the constraints and Hamiltonians return to original second class one. Now, using BFV technique we have constructed BRST charge and corresponding BRST invariant action. We have shown that, the action of BRST charge on the state of total Hilbert space will give physical subspace of the system. We have constructed BRST operator using mode expansion technique. This is the first time in the literature that BFFT Abelianization and BRST symmetry is constructed for a nonlinear second class systems explicitly. We have also discussed simple example (particle on torus) of this kind of system. At the end we have discussed Batalin - Vilkovisky quantization of this system based on BFFT formalism. As this model is a toy model for a wide class of physical systems, the results obtained here will be highly useful in studying these systems. The more general case of BRST quantization of embedding VL in Euclidean space RN where 1≤L<N will be discussed in the next part of the paper [90].



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