###### Abstract

Starting with the first-order singular Lagrangian, the problem of the quantization of a dynamical system constrained to a submanifold embedded in the higher-dimensional Euclidean space is investigated within the framework of operatorial quantization formalism. Through the projection operator method (POM) with the constraint star-products, it is shown that both of the constraint quantum system with the usual constraint and that with the derivative-type constraint are naturally constructed from one Lagarangian. It is proved that the system with the usual constraint is the sub-system of that with the derivative-type one. Furthermore, the quantization of the dynamical system subject to both of the usual constraint and the derivative-type one is investigated by the POM, and the quantum corrections in the resultant Hamiltonians are discussed.

Unified Description of Quantum Mechanics on a Curved Space

M. Nakamura***E-mail:

Research Institute, Hamamatsu Campus, Tokoha University, Miyakoda-cho 1230, Kita-ku, Hamamastu-shi, Shizuoka 431-2102, Japan

## 1 Introduction

The problem of the quantization of a dynamical system constrained to a submanifold embedded in the higher-dimensional Euclidean space has been extensively investigated as one of the quantum theories on a curved space until now[1, 2, 3, 4]. In order to avoid the unnecessary troublesomeness, the submanifold specified by () in an -dimensional Euclidean space has been considered in many studies, where . As the dynamical system constrained to , then, the dynamical system subject to the constraint and that subject to We shall call the constraint the static constraint, and , the dynamical one. have been presented with the Lagrangians , =, where is the usual second-order nonsingular Lagrangian and , an additional dynamical variable[3, 4]. In this paper, we shall call the system subject to the static constraint system, and the system subject to , the dynamical constraint system.
In the quantization of constraint systems, alternative approach of the first-order singular Lagrangian formalism has been proposed by Faddeev and Jackiw. Starting with the first-order singular Lagrangian containing the term associated to the dynamical constraint, in this paper, we shall accomplish the quantization of the system constrained to within the framework of operatorial quantization formalism. Through the projection operator method(POM) with the constraint star-products, then, it is shown that the resultant constraint quantum systems corresponding to these two constraints, the static constraint quantum system and the dynamical constraint quantum system, are naturally constructed with one Lagrangian, and it is proved, with the alternative approach to Ref., that the static constraint quantum system is the sub-system of the dynamical constraint quantum system . Furthermore, we shall discuss the quantum correction terms due to the projections of operators and the re-ordering of operators in the resultant Hamiltonian.
The quantization of the dynamical sytems subject to both of the static constraint and the dynamical one has been investigated through, for example, the Stückelberg formalism. Starting with the first-order singular Lagrangian containing these two consraint terms, we shall quantize the dynamical system with these two kinds of contraints by the POM, and it is shown that the resultant constraint quantum system is included in the constraint quantum system with the dynamical constraint.
The program of this paper is the following. In Sect.2, we propose the Lagrangian with the dynamical constraint and construct the initial unconstraint quantum system, which we denote . In Sect.3, we construct the the static constraint quantum system, which we denote , the dynamical constraint quantum system, which is denoted by , and the constraint quantum system with the static and dynamical constraints, which is denoted by . Then, it is proved that is the sub-system of , and is included in . Furthermore, the quantum correction terms in the final Hamiltonians are discussed. In Sect.4, the discussion and the some concluding remarks are given.

## 2 Initial Hamiltonian System

Consider the dynamical system described by the first-order singular Lagrangian

 L=L(x,˙x,v,˙v,λ,˙λ)=˙xivi−λ˙G(x)−12vivi−V(x),

where and Here, stands for with .
Following the canonical quantization formulation for constraint systems[8, 9], then, the initial unconstraint quantum system is obtained as follows:
i) Initial canonically conjugate set

 C={(xi,pxi),(vi,piv),(λ,pλ)|i=1,⋯,N},

which obeys the commutator algebra :

 [xi,pxj]=iℏδij,[vi,pjv]=iℏδji,[λ,πλ]=iℏ,(the others)=0,

ii) Initial Hamiltonian

 H={μi(1),ϕ(1)i}+{μi(2),ϕ(2)i}+{μ(3),ϕ(3)}+12vivi+V(x),

where , are the constraint operators corresponding to the primary constraints due to the singularity of the Lagrangian , which are given by

 ϕ{\tiny(1)}i=pxi−vi+λGi(x),ϕ{\tiny(2)}i=piv,ϕ{\tiny(3)}=pλ,

and , and are the Lagrange multiplier operators.
iii) Consistent set of constraints
From the consistency conditions for the time evolusions of constraint operators, the consistent set of constraints, , is given by

 K={ϕ{\tiny(1)}i,ϕ{\tiny(2)}i,ϕ{\tiny(3)},ψ{\tiny(1)}},

where

 ψ{\tiny(1)}=Gi(x)vi,

which is the constraint operator corresponding to the secondary constraint.
Then, the Lagrange multiplier operators are determined as

 μi{\tiny(1)}=vi,μi{\tiny(2)}=−G−1(x)Gi(x)Gkl(x)vkvl−PijVj(x),μ{\tiny(3)}=G−1(x)(Vi(x)Gi(x)−Gij(x)vivj),

where

 G(x)=Gi(x)Gi(x),Pij=δij−G−1(x)Gi(x)Gj(x),

which satisfies

 PijPjk=Pik,PijGj(x)=Gi(x)Pij=0.

iv) Commutator algebra of
The constraint set obeys the commutator algebra :

 A(K):[ϕ{\tiny(1)}i,ϕ{\tiny(2)}j]=−iℏδij,\par[ϕ{\tiny(1)}i,ϕ{\tiny(3)}]=iℏGi(x),\par[ψ{\tiny(1)},ϕ{\tiny(1)}i]=iℏGij(x)vj,\par[ψ{\tiny(1)},ϕ{\tiny(2)}i]=iℏGi(x),(the others)=0.

## 3 The Constraint Quantum System S∗

Starting with the initial system , we shall construct the constraint quantum system , which satisfies , through the star-product quantization formalism of POM.
For this purpose, we first classify into the following two subsets :

 K=K({\tiny A})⊕K{\tiny(B)% }withK({\tiny A})={ϕ{\tiny(% 1)},ϕ{\tiny(2)}},K{\tiny(B)}={ϕ{\tiny(3)},ψ{\tiny(1)}}.

As well as the Dirac bracket formalism, the POM satisfies the iterative property. Terefore, the constraint quantum system can be constructed through the successive projections of .

### 3.1 Successive Projections of S

From the structure of the commutator algebra (2.10), the successive projections of can be uniquely carried out through the following diagram:

 ^P{\tiny(1)}K{\tiny(1)}=0→^P{\tiny(2)}K{\tiny(2)}=0,

where

 K{\tiny(1)}=K{\tiny(A)},K{\tiny(2)}=K{\tiny(B)},

and is the projection operator associated to the subset , that is, . Then, the successive projections of the operators of the system by are carried out through the program designated by the following diagram :

 C^P(1)⟶C(1)^P(2)⟶C(2),

where

 C(n)=^P(n)C(n−1)(n=1,2)

with , which satisfy the projection conditions

 K(n)(C(n))=0(n=1,2).

Then, the ACCS (associated canonically conjugate set)  for the subsets consist of the operators in ,

 Z(n)=Z(n)(C(n−1)).

From (3.7), therefore, the projection operators are also represented as

 ^P(n)=^P(n)(C(n−1))(n=1,2).

Now, The ACCS of the projection operators () are given, respectively, as follows:

The hyper-operators for in the star-product formulation are given as follows:§§§For any operator , and

 ^Ω{\tiny(1)}ηζ=Jαβ^Z{\tiny(1)}(−)α(η)^Z{\tiny(1)}(−)β(ζ)=^ξ{\tiny(1)}(−)i(η)^π{% \tiny(1)}(−)i(ζ)−^π{\tiny(1)}(−)i(η)^ξ{\tiny(1)}(−)i(ζ),^Ω{\tiny(2)}ηζ=Jαβ^Z{% \tiny(2)}(−)α(η)^Z{\tiny(2)}(−)β(ζ)=^ξ{\tiny(2)}(−)(η)^π{\tiny(2)}(−)(ζ)−^π{\tiny(2)}(−)(η)^ξ{\tiny(2)}(−)(ζ).

The operations of and on are presented in Appendix A.

#### 3.1.1 Projected CCS C{\tiny(2)} and commutator algebra of C{\tiny(2)}

Let be . From the projection conditions (3.6), then, becomes as follows:

 C{\tiny(2)}=^PC=C{(^Px,^Ppx),(^Pv,^Ppv),(^Pλ,^Pπλ)}={xi,pxi,vi,λ|i=1,⋯,N},

where the projection conditions are represented as

 ^Pϕ{\tiny(1)}i=pxi−vi+{λ,Gi(x)}=0,^Pϕ{\tiny(2)}i=piv=0,^Pϕ{\tiny(3)}=pλ=0,^Pψ{\tiny(1)}={Gi(x),vi}=0.

Then, the commutator algebra is represented as

#### 3.1.2 Projected Hamiltonian H{\tiny(2)}

According to the projection formulas of symmetrized product, the projected Hamiltonian is obtained in the following way:

 H{\tiny(2)}=^PH=^P{\tiny(2)}^P{\tiny(1)}H=12{vi,vi}+V(x)+V{\tiny(1)}{\tiny C}(x)+V{\tiny(% 2)}{\tiny C}(x),

where is the quantum correction term due to the projection of ,

 V{\tiny(1)}{\tiny C}(x)=ℏ28G−2(x)Gik(x)Gk(x)Gil(x)Gl(x)−ℏ24G−1(x)Gik(x)PklGli,

and , that due to the projection of ,

 V{\tiny(2)}{\tiny C}(x)=ℏ24G−1(x)Gij(x)Gij(x).

Substituting into (3.15), then, one obtains

 H{\tiny(2)}=12{vi,vi}+V(x)+3ℏ28G−2(x)Gik(x)Gk(x)Gil(x)Gl(x).

Thus, we have obtained the final projected quantum system due to the successive projection ,

 S{\tiny(2)}=(C{\tiny(2)},A(C{\tiny(2)}),H{\tiny(2)}).

### 3.2 Constraint Quantum System S∗

From the structure of the commutator algebra , we can consider two constraint quantum systems, one of which is represented in terms of , and the other, in terms of .

#### 3.2.1 Constraint Quantum System S∗{\tiny I}

Let be the CCS in ,

 C∗{\tiny I}={xi,vi|i=1,⋯,N}.

From the projection conditions (3.12), satisfies

 {Pij,pxj}=vi,λ={G−1(x)Gi(x),pi},piv=0,pλ=0.

Let , be , , respectively. Then, the commutator algebra is defined as follows:

 A(C∗{\tiny I}):[xi,xj]=0,[xi,vj]=iℏPij=iℏ(δij−ni(x)nj(x)),[vi,vj]=iℏ{ni;k(x)nj(x)−ni(x)nj;k(x),vk},

which is just equivalent to the commutator algebra of the static constraint quantum system[1, 3, 4]. From the projection conditions (3.20), however, does not reproduce the commutator algebra with respect to and .
The Hamiltonian in is given by the projected Hamiltonian , which is also expressed in the following form:

 H∗{\tiny I}=H{\tiny(2)}=12{vi,vi}+V(x)+332ℏ2G−2(x)G;i(x)G;i(x),

where

 G;i(x)=∂xiG(x).

Thus, the quantum system is defined with

 S∗{\tiny I}=(C∗{\tiny I},A(C∗{\tiny I}),H∗{\tiny I}).

Following , then, it is obvious that , that is,

 ˙G(x)=1iℏ[G(x),H∗{\tiny I}]={PijGj(x),vi}=0.

In the quantum system , thus, is the constant of motion and is conserved through the time evolusion of system under the operatorial formalisn. Because of the projection conditions (3.20), on the other hand, is not complete with respect to . The commutator algerbra of these operators is defined by , which shows that is the subset of ,

 C∗{\tiny I}⊂C{\tiny(2)},

and, therefore, is the sub-system of ,

 S∗{\tiny I}⊂S{\tiny(2)}.

#### 3.2.2 Constraint Quantum System S∗{\tiny II}

The alternative final CCS is given by :

 C∗{\tiny II}={xi,pxi|i=1,⋯,N}.

From , the commutator algebra is defined by the canonically conjugate commutation relations

 A(C∗{\tiny II}):[xi,xj]=0,[xi,pxj]=iℏδij,[pxi,pxj]=0,

which is just identical to the commutator algebara in the dynamical constraint quantum system[3, 4]. According to the projection conditions (3.12), then, , are represented in terms of as follows:

 vi={Pij,pxi},λ={G−1(x)Gi(x),pxi},

and the commutator algebra with respect to and is completely reproduced by . Therefore, is represented as

 C{\tiny(2)}={x,px,v,λ}={x,px,v(x,px),λ(x,px)},

which shows that is just equivalent to ,

 C{\tiny(2)}=C∗{\tiny II}.

From (3.22) and (3.30), the Hamiltonian is represented as

 H∗{\tiny II}=12{{Pik,pxk},{Pil,pxl}}+V(x)+332ℏ2G−2(x)G;i(x)G;i(x),

which contains the products of operators noncommutatable with each other. So, the re-ordering of operators also yields the quantum correction terms in .
We shall rewrite as follows:

 12{{Pik,pxk},{Pil,pxl}}=12pxiPijpxj+Uc(x),

where is the quantum correction term associated to the re-ordering of operators,

 Uc(x)=ℏ24(nk;kl(x)nl(x)+nk;l(x)nl;k(x))+ℏ28(nk;k(x)nl;l(x)+nk(x)ni;k(x)ni;l(x)nl(x)).

Then, is rewritten in the following form:

 H∗{\tiny II}=12pxiPijpxj+V(x)+332ℏ2G−2(x)G;i(x)G;i(x)+Uc(x).

Thus, the quantum system is given by

 S∗{\tiny II}=(C∗{\tiny II},A(S∗{\tiny II}),H∗{\tiny II}),

and it is easily shown that holds in also, that is,

 ˙G(x)=1iℏ[G(x),H∗{\tiny II}]={pxi,PijGj(x)}=0.

### 3.3 Constraint Quantum System S∗{\tiny III}

Consider the dynamical system described by the first-order singular Lagrangian containing terms corresponding to the static and dynamical constraints, ,

 Lsd=L(x,˙x,v,˙v,ν,˙ν,λ,˙λ)=˙xivi−νG(x)−λ˙G(x)−12vivi−V(x).

#### 3.3.1 Initial Unconstraint Quantum System

As well as in the case of the Lagrangian (2.1), the initial unconstraint quantum system is given as follows:
i) Initial canonically conjugate set

 C={(xi,pxi),(vi,piv),(ν,pν),(λ,pλ)|i=1,⋯,N},A(C):[xi,pxj]=iℏδij,[vi,pjv]=iℏδji,[ν,pν]=iℏ,[λ,pλ]=iℏ,(the others)=0.

ii) Consistent set of constraints

 K={ϕ{\tiny(1)}i,ϕ{\tiny(2)}i,ϕ{\tiny(3)},ϕ{\tiny(4)},ψ{\tiny(1)},ψ{\tiny(2)}|i=1,⋯,N},

where

 ϕ{\tiny(1)}i=pxi−vi+λGi(x),ϕ{\tiny(2)}i=piv,ϕ{\tiny(3)}=pν,ϕ{\tiny(4)}=pλ,ψ{\tiny(1)}=G(x),ψ{\tiny(2)}=Gi(x)vi,

which obeys :

 [ϕ{\tiny(1)}i,ϕ{\tiny(2)}% j]=−iℏδij,[ϕ{\tiny(1)}i,ϕ{% \tiny(4)}]=iℏGi(x),[ψ{\tiny(1)},ϕ{\tiny(1)}i]=iℏGi(x)[ψ{\tiny(2)},ϕ{\tiny(1)}i]=iℏGij(x)vj,[ψ{\tiny(2)},ϕ{\tiny(2)}i]=iℏGi(x),(the others)=0.

iii) Initial Hamiltonian

 H={μi(1),ϕ(1)i}+{μi(2),ϕ(2)i}+{μ(3),ϕ(3)}+{μ(4),ϕ(4)}+12vivi+V(x)+νG(x),

where the Lagrange multiplier operators associated with , , and are determined as

 μi{\tiny(1)}=vi,μi{\tiny(2)}=−G−1(x)Gi(x)Gkl(x)vkvl−PijVj(x),μ{\tiny(4)}=ν+G−1(x)(Vi(x)Gi(x)−Gij(x)vivj),

and is indefinite.

#### 3.3.2 Successive Projections of Ssd and Constraint Quantum System S∗{\tiny III}

From the structure of , (3.42), the successive projections of can be carried out through two ways of the projection process designated by the following diagrams, respectively:

 {ϕ{\tiny(1)},ϕ{\tiny(2)}}=0→{ψ{\tiny(1)},ψ{\tiny(2)}}=0→{ϕ{\tiny% (3)},ϕ{\tiny(4)}}=0, (P1)
 {ϕ{\tiny(1)},ϕ{\tiny(2)}}=0→{ϕ{\tiny(4)},ψ{\tiny(2)}}=0→{ϕ{\tiny% (3)},ψ{\tiny(1)}}=0. (P2)

I) Constraint Quantum System 888We shall denote the constraint quantum system by the process (P1) as
In the process (P1), are first class and therefore would be considered as the generators of gauge transformations. Therefore, these are eliminated through imposing the following gauge fixing conditions[10, 11],

 ν=0,λ=0.

Then, is given as follows:

 S∗{\tiny III}(P1)=(C∗,A(C∗),H∗),

where

 C∗={xi,vi|i=1,⋯,N},withpxi=vi,piv=0,A(C∗):[xi,xj]=0,[xi,vj]=iℏPij(x),[vi,vj]=iℏ{G−1(x)(Gik(x)Gj(x)−Gi(x)Gjk(x)),vk},H∗=12{vi,vi}+V(x)+3ℏ28G−2(x)Gik(x)Gk(x)Gil(x)Gl(x).

From (3.46b), thus, it is shown that is equivalent to .
II) Constraint Quantum System
In the process (P2), are first class. As well as in (P1), is eliminated by imposing the gauge fixing condition . On the other hand, remains as the first-class operator.

Then, is constructed in the following way:

 S∗{\tiny III}(P2)=(C∗,A(C∗),H∗),

where

 C∗={xi,pxi|i=1,⋯,N},withvi={Pij(x),pxj},piv=0,ν=pν=0,λ=−{G−1(x)Gi(x),pxi},pλ=0,A(C∗):[xi,pxj]=iℏδij,[xi,xj]=[pxi,pxj]=0,H∗=12{{Pik(x),pxi},{Pjk(x),pxj}}+V(x)+3ℏ28G−2(x)Gik(x)Gk(x)Gil(x)Gl(x).

Then, is the constant of motion, that is,

 ˙G(x)=(1/iℏ)[G(x),H∗]=0.

From (3.47b) and (3.48), thus, it is shown that is equivalent to .

## 4 Discussion and Concluding remarks

We have investigated the quantization of the dynamical system constrained to the submanifold specified by in an -dimensional Euclidean space through the POM with the constraint star-products. Then, we have obtained the following results:
(1) The projected quantum system includes two constraint quantum systems, one of which is with the static constraint, and the other of which, with the dynamical one, where

 S∗{\tiny I}⊂S{\tiny(2)}=S∗{\tiny II}.

Thus, it has been shown that is complete, and , incomplete.
(2) The commutator algebras (3.21) and (3.29) in the constraint quantum systems are identical with the corresponding commutator ones obtained through the Dirac-bracket quantization formulation with the ordinaly second-order Lagrangians.
(3) The Hamiltonians in the constraint quantum systems contain the quantum correction terms due to the sequential projections of and 999In the operatorial approach, Lagrange multipliers are also operators at the first step., which are completely missed in the usual approach with Dirac-bracket quantization. In , further, the Hamiltonian contains the additional correction terms associated to the re-ordering of operators in , which will be obtained in the usual Dirac-bracket quantization procedure.
(4) Under the constraints and , there have been constructed two kinds of constraint quantum systems, and . Then, it has been shown that is equivalent to , and ,
The approach based on first-order singular Lagrangians is considered to be one of the most available procedures for the investigation of noncommutative quantum system. Then, our next task is to extend our approach to noncommutative quantum systems.

Appendix

## Appendix A Operations of ACCS

### a.1 Operations of ^ξ{\tiny(1)}(−), ^π{\tiny(1)}(−) on C{\tiny(0)}

 ^ξ{\tiny(1)}(−)kxi=δki,^π{\tiny(1)}(−)kxi=0,\par^ξ{\tiny(1)}(−)kpxi=−λGki(x),^π{\tiny(1)}(−)kpxi=0,\par^ξ{\tiny(1)}(−)kvi=0,^π{\tiny(1)}(−)kvi=−δki,\par^ξ
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