Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

Pandey, Vipul Kumar

doi:https://doi.org/10.1155/2022/6410245

Advances in High Energy Physics

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Research Article | Open Access

Volume 2022 | Article ID 6410245 | https://doi.org/10.1155/2022/6410245

Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

Vipul Kumar Pandey¹

Academic Editor: Torsten Asselmeyer-Maluga

Received14 Mar 2022

Accepted20 Jul 2022

Published23 Aug 2022

Abstract

We have previously developed the BRST quantization on the hypersurface embedded in -dimensional Euclidean space in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of -dimensional manifold embedded in with . 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.

1. Introduction

In the previous work [1] we have considered the BRST quantization of the motion on a hypersurface embedded in the -dimensional Euclidean space 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 -dimensional Riemann manifold can be locally embedded in the -dimensional Euclidean space but cannot be embedded in 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 embedded in the Euclidean space . 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 , 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 embedding in Euclidean space , where . 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 . 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 on in

Let us consider an -dimensional Euclidean space , a point in which is specified by a set of Cartesian coordinates

Further consider in an -dimensional Riemann subspace, , a point of which is specified by a set of coordinates [26],

The metric of this system is defined as . We can construct in a set of curvilinear coordinates including

Let us assume that are the intrinsic coordinates normal to [26]. We can also use the notation

Then, the subspace can be defined as

The metric for the curvilinear coordinates in is defined as where and are defined as

It is worth noting that the metric is induced metric on and the metric can be defined as some function on . Using this assumption, we will get , when . So, the metric on have form and the inverse matrix is defined as which implies that which can further be written as

We know that can also be written as

From here, we can obtain following relations [26]: where and are defined as

Here, are called the natural frame and gives the induced metric on . The inverse metric of is given by where . From this, we obtain

The Lagrangian for the particle motion on is defined as [26]

Here, varies between 1 to and from 1 to . The metric for the coordinate is . are variables independent of , and the dot denotes the time derivative. The canonical momentum conjugate to and can be written as

Hamiltonian corresponding to Lagrangian in Equation (17) can be written as

2.1. Hamiltonian Analysis

The primary constraint for the system under study is defined as

After including the primary constraint, the new Hamiltonian is written as where 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]:

The constraint determines the , and the procedure is over. So, the explicit form of the constraints are

Here, and are defined as , . Also, the product of partial derivatives is defined as .

The Poisson brackets between the constraints are defined as [26]

Other Poisson brackets vanish. It is worth noting that

Thus, the matrix between the constraints has the form

It is worth noting here that . Hence, each element of the matrix is a matrix. Thus, the matrix is a 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 variables , besides the existing physical degrees of freedom 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 and a Hamiltonian operator . 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:

means that the equality holds on the constraint surface. The additional fields satisfy the symplectic algebra: where is a constant quantity and . The constraints are now defined in terms of auxiliary field as

This modified constraint satisfies the boundary condition

These modified constraints should satisfy first-class constraints algebra. So, the Poisson bracket between the constraints are defined as

The solution of Equation (31) can be achieved by considering an expansion of , as where . The first-order correction in the field is [66, 68, 69]

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

We notice that Equation (34) does not give a single solution for , because there is still unknown matrix . We can make choices for in such a way that the newly defined variables are unconstrained in nature. Using the value of the matrix , we can calculate the possible solutions of from Equation (34). Using the value of , we can obtain . If 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 is where is defined as

In the above expressions, we have defined

Another important part of the BFFT formalism is that any dynamical variable has also to be modified in the same way as discussed above in order to be strongly involutive with the modified constraints . Denoting the modified quantity by , we then have

Apart from that, modified variable must also satisfy the boundary condition given below:

To obtain as an analogous expansion to Equation (32), we will consider where is also a term which is of the order n in . The expression in Equation (38) above gives us where and are the inverses of and .

The corrections in the physical variable can be written in the more general form as where

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 as and . Then, the involutive form of these fields ( and ) will satisfy the following relations.

Similarly, any function of the physical variables and will also satisfy the strong involution relation, since

So, if we are taking any dynamical variable in the original phase space, it can be written in involutive form as

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 set of possible BFFT fields . 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 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 in the Riemann manifold (based on [75]).

Our choice of Poisson Bracket between the fields are where is a unitary matrix.

From the relation between the fields , we can find matrix as

Using the matrix defined above and the matrix between the constraints in Equation (26), we can calculate the possible value of matrix . Now, using the matrix , we can write the modified constraints as

It is worth to mention here that all the barred quantities defined in Equation (49) are function of coordinates and fields and will take the form of the original unbarred quantities in the limit . Here, any field will be written as [75]

Also, the partial differentiation of field wrt. any field can be written as [75].

Then, the Poisson bracket between these modified set of constraints is where and . 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 fields can be calculated as follows. We will start it by calculating the inverse of the matrices and . The inverse of the matrix can be easily written as

The total Hamiltonian with corrections due to BFFT field can be written as

Here, are matrices, and takes possible values.

Now, by calculating the Poisson bracket between the modified constraints and the Hamiltonian , it can be easily verified that modified Hamiltonian is involutive in nature. where and .

5. Hamiltonian BRST Quantization

5.1. Charge and Symmetry

In this section, we will construct BRST symmetry for the particle motion on the surface in the Riemann manifold . 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 fields with ghost number 1 and -1, respectively, and with ghost number -1 and 1, respectively, with Lagrange multiplier fields for each set of constraints. As there are set of constraints, we will introduce two sets of canonical ghost and anti-ghost fields , and Lagrange multiplier fields . These fields and corresponding momenta satisfy the following superalgebra:

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

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 is Hermitian in nature and of the similar Grassmann parity as of . They also satisfy

We will apply the general results obtained above to the particle on surface embedded in .

For the system under study, the modified constraints have been calculated in Equation (49).

Our choice of gauge condition for the motion on the surface in the Riemann manifold is in Equation (23).

Here, 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

Nilpotent BRST transformations corresponding the action in Equation (61) can be easily constructed using the relation . Its relation with infinitesimal BRST transformation can be established as . 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

It can be easily verified that these transformations are nilpotent in nature.

Using the expressions for BRST charge and gauge-fixing fermion , effective action in Equation (61) can be written as

The generating functional for this effective action can be written as

The Liouville measure for the generating functional is defined as

Here, are all dynamical variables of the theory. Now, performing the integration over ghost and antighost momenta and , we will get

Here, is the new path integral measure for the effective action after the performance of integral over the fields and . Further performance of integral over fields will give us the new effective generating functional as where 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

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. which can be further written in more explicit form for this system as

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 of [1]. In this limit, elements of the matrix will transform as

From the antisymmetry (25),

The inverse matrix elements will transform under this limit as

Here, vanishes due to asymmetry.

Also, under the transformation , the modified constraints in Equation (49) takes the form of modified constraints of case:

The form of modified Hamiltonian in Equation (54) under these transformations is

Similary, BRST charge and BRST symmetry can be written in this limit as

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 .

7. Examples of -Dimensional Embedding in

As an example of -dimensional embedding in 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 in . It is specified by a set of co-prime integers and . A torus knot of type winds times around the rotational symmetry axis of the torus and 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 () in following ways: where , , and . A toroidal surface is represnted by some specific value of (say ). Parameters and are written as and , where and are major and minor radius of torus, respectively.

Lagrangian for a particle constrained to move on the surface of torus knot is where are toroidal coordinates for toric geometry and is the Lagrange multiplier. The canonical Hamiltonian corresponding to the Lagrangian in Equation (79) is then written as where , , and are the canonical momenta conjugate to the coordinate , , , and , respectively, defined as

The is the primary constraint of the theory.

After inclusion of primary constraint, our new Hamiltonian has the form

Now, using Dirac’s method of Hamiltonian analysis, we will calculate all the possible constraints of the theory as

will vanish, and the value of will be determined from it. All the constraints can be written as

Now, the Poisson brackets between the constraints have following values:

Similarly, the Poisson bracket between other constraints (A.1), (A.2), and (A.3) has been explicitly calculated in the appendix. All these brackets will be nonzero and will be equal to , and . Thus, the matrix between the constraints will take exactly the same form of matrix 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 corresponding to each constraint. Relation between these fields will provide us possible value of . Our choice of Poisson bracket between the fields will be same as one taken for the general case. Hence, the matrix will have the form of Equation (48).

Using the matrix and the matrix , in Equation (33), one can find many possible value of matrix .

Now, applying the results developed in Section 4, we can calculate the modified constraints as

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 Section 5.

The total involutive Hamiltonian for this system will take the form as [75]

It can be easily verified that the Hamiltonian is involutive by computing its Poisson bracket with modified constraints of the theory. where .

BRST charge for this first-class system can be written using above expression, as and corresponding BRST symmetry transformation can be written as

This shows that the result obtained in Section 4 is true for any -dimensional surface embedded in .

8. Batalin-Vilkovisky Quantization

In the current section of the manuscript, we are going to discuss the quantization of the Particle motion on the surface in the Riemann manifold using the field-antifield formalism [87–89] developed for BFFT systems in [90]. We will start by introducing set of antifields corresponding to the fields . Here, fields , and are bosonic in nature and have ghost number zero, whereas the ghost fields 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 where action is defined as

Here and 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 where the antibracket between any two dynamical variables and is defined as

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 relation for any local functional of fields. Nilpotentcy of the BRST operator 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 introduce pairs of ghost-antighost fields and corresponding momenta , , as well as gauge-fixing fermions . These antifields can be easily eliminated by choosing . One of the possible forms of we can choose for the given system is

We have liberty to make other possible choices also. Now, we will extend the BV action defined above to a nonminimal action in order to implement the many set of gauge fixing conditions introduced by . Now, the generating functional for gauge-fixed action is defined as

Now, we will replace the original classical field–antifield action by some quantum action which is expressed as a local functional of fields and antifields and also satisfy a new equation called quantum master equation defined as

Then, the gauge symmetries of the extended action are not obstructed at quantum level. Here, acts as an operator and is defined as

It is also assumed here that the quantum action can be expanded in powers of in the following manner:

The first two term of the quantum master Equation (102) reads as

It can be easily observed that if is nonzero and gives a nontrivial result, then there exists some 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:

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 space embedded in a Euclidean space 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 , 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 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 and of matrix (of the Poisson’s bracket between the constraints) are nonzero if we take two different torus knot systems. In the limit of , the 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.

Appendix

Poisson Bracket between the Constraints of the Particle on Torus Knot System

The calculation of Poisson bracket between some of the constraints for particle on the torus knot model is straightforward but cumbersome process. We have calculated these brackets explicitly here.

The Poisson bracket between constraints is written as

Poisson bracket is written as

Similarly, can be written as

Data Availability

No data has been used to support the study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Copyright © 2022 Vipul Kumar Pandey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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