aa r X i v : . [ h e p - t h ] N ov Lie Algebra Quantization by the Star Product
Takao KOIKAWA ∗ ) School of Social Information Studies,Otsuma Women’s UniversityTama 206-0035, Japan
Abstract
We apply the star product quantization to the Lie algebra. The quantization in termsof the star product is well known and the commutation relation in this case is calledthe θ -deformation where the constant θ appears as a parameter. In the applicationto the Lie algebra, we need to change the parameter θ to x -dependent θ ( x ). Thereis no essential difference between the quantization in the quantum mechanics andderiving quantum numbers in the Lie algebra from the viewpoint of the star product.We propose to unify them in higher dimensions, which may be analogous to theKaluza-Klein theory in the classical theory. ∗ ) E-mail: [email protected] introduction The commutation relations(CRs hereafter) of non-commutative coordinates ˆ x i can be classified into several types. Two of the important types of the CRs ofoperators ˆ x i are given by[1] (cid:2) ˆ x i , ˆ x j (cid:3) = i Θ ij , (1) (cid:2) ˆ x i , ˆ x j (cid:3) = iθf ijk ˆ x k , (2)where Θ ij , f ijk and θ are constants. The first CR is the θ -deformation type,and the second CR is the Lie algebra type.The quantization using the star product is mainly applied to the first type.The application of the star product to the second type is less known. It mightbe possible to unify these types as (cid:2) ˆ x i , ˆ x j (cid:3) = i Θ ij (ˆ x ) , (3)where Θ ij (ˆ x ) are functions of ˆ x i s in general. A special case Θ ij (ˆ x ) = Θ ij fallsinto the first type. So far, θ -deformation known as the Moyal quantization hasbeen studied extensively. On the other hand, the space-time operator dependentΘ ij (ˆ x ) case has been less studied from the viewpoint of star product framework.One of the purposes of the present paper is to show the star product re-alization of the Lie algebra type of operator CRs. When f ijk in the CR (2)is the structure constant of su (2), the star product representation of the CRalso yields the algebra in the same way as the operator representation. Thequantization of the Casimir operator is carried out by using functions and starproduct only.When we derive a star product for the Lie algebra type CR, we encounter adifficulty which does not exist in the θ -deformation. In order to clarify thedifficulty we start with reviewing the definition of the star product for the θ -deformation case. We define the star product for functions of x and x , f ( x ) = f ( x , x ) and g ( x ) = g ( x , x ), by f ( x ) ⋆ g ( x )= exp (cid:20) i θ (cid:18) ∂∂x ∂∂x ′ − ∂∂x ∂∂x ′ (cid:19)(cid:21) f ( x ) g ( x ′ ) | x ′ = x , x ′ = x , (4)where θ is a constant. As far as θ is a constant, the differential operators in theexponent never hit θ when the exponential function is expanded. However, when θ is a function of x and x , there occurs a question whether the x -dependent θ is differentiated by the differential operators in the exponent of the exponentialfunction. Though there can be several definitions, we adopt the ansatz that θ ( x ) is not differentiated. In other words, we generalize the above definition tothe following definition for x -dependent θ ( x ) case. We denote the star productby use of the same notation as before f ( x ) ⋆ g ( x ) 2 exp (cid:20) i θ ( x ′′ )2 (cid:18) ∂∂x ∂∂x ′ − ∂∂x ∂∂x ′ (cid:19)(cid:21) f ( x ) g ( x ′ ) | x ′′ = x ′ = x , x ′′ = x ′ = x = ∞ X n =0 n ! (cid:18) i θ ( x ′′ )2 (cid:19) n × (cid:18) ∂∂x ∂∂x ′ − ∂∂x ∂∂x ′ (cid:19) n f ( x ) g ( x ′ ) | x ′′ = x ′ = x , x ′′ = x ′ = x . (5)The last expression is to be read that the Poisson bracket differential opera-tor(see Eq. (6)) acts upon f and g only, and we keep this ansatz whenever wedefine other type of star products throughout this paper.This paper is organized as follows. In the following section, we discuss the x -dependent θ -deformation. We show how such cases appear in curved space-time. In section 3, we review the θ -deformation quantization of one-dimensionalharmonic oscillator which shows one of the most general procedure of Moyalquantization. In section 4, we quantize the Lie algebra su (2) and su (3) usingthe star product in the same way as in the Moyal quantization, after we assignproper components to each element of a matrix (Θ µν ( x )) for the Lie algebra.The quantization is implemented completely in an algebraic way. In section 5,we propose a higher dimensional model where the quantization of a Hamiltonianand the quantization of the Lie algebra in terms of the star product are unifiedin higher dimensions. The last section is devoted to summary and discussion. The procedure of defining the star product for constant θ is a little bit cumber-some as shown in Eq.(4). First we need to distinguish the variables of a function f ( x ) and g ( x ) and then take derivatives by allocating the variables to derivativeoperators. After the derivatives are taken, the variables are set to be equal.When we change the constant θ to a function of x and x , θ ( x ), manipula-tion becomes more complicated. As for a question whether θ ( x ) is differentiatedby the differential operators in the exponent, we adopt the ansatz that the dif-ferential operators operate only on the functions f ( x , x ) and g ( x , x ) andnot on θ ( x ) as in Eq.(5). The merit of this ansatz is that the Poisson bracketoperator defined by ∂∂x ∂∂x − ∂∂x ∂∂x , (6)on the functions always keeps its form because part of the operators neveroperates on θ ( x ), by assumption. We keep this ansatz as a prescription for thegeneralization of the star product in the discussion below. We also comment onthe reason why we need to study x -dependent θ ( x ).In order to generalize the star product, we replace differential operators byvector fields. Then x -dependent coefficient in front of the differential operatorsappear, and we follow the above ansatz in this case. The generalized star product3s explicitly written as f ( x ) ⋆ g ( x ) = µ (cid:26) exp (cid:20) i Θ ab X a O X b (cid:21) f ( x ) O g ( x ) (cid:27) , (7)where the independent vector field X a , ( a = 1 , , · · · , n ) is given by X a = e µa ( x ) ∂ µ , (8)and is assumed to satisfy [ X a , X b ] = 0 . (9)In (7), Θ ab ( a, b = 1 , , · · · , n ) is an antisymmetric constant in indices a and b , and f ( x ) and g ( x ) are functions of n variables x i , ( i = 1 , , · · · , n ). Theexpansion of the generalized star product is written as f ( x ) ⋆ g ( x )= f ( x ) g ( x ) + i ab X a f ( x ) X b g ( x )+ 12! (cid:18) i (cid:19) Θ a b Θ a b ( X a X a f ( x ))( X b X b g ( x ))+ 13! (cid:18) i (cid:19) Θ a b Θ a b Θ a b ( X a X a X a f ( x ))( X b X b X b g ( x )) + · · · = f ( x ) g ( x ) + i µν ∂ µ f ( x ) ∂ ν g ( x )+ 12! (cid:18) i (cid:19) Θ µ ν Θ µ ν ( ∂ µ ∂ µ f ( x ))( ∂ ν ∂ ν g ( x ))+ 13! (cid:18) i (cid:19) Θ µ ν Θ µ ν Θ µ ν ( ∂ µ ∂ µ ∂ µ f ( x ))( ∂ ν ∂ ν ∂ ν g ( x )) + · · · , (10)where the coefficients Θ µν appearing in front of the differential operators arefunctions of x given byΘ µν = Θ µν ( x ) = Θ ab e µa ( x ) e νb ( x ) . (11)Here we have followed the prescription that the differential operators operateonly on the functions f ( x ) and g ( x ), and not on Θ µν ( x ). We thus define thegeneralize star product by f ( x ) ⋆ g ( x )= ∞ X n =0 n ! (cid:18) i Θ µν ( x ′′ )2 (cid:19) n (cid:18) ∂∂x µ ∂∂x ν ′ − ∂∂x ν ∂∂x ν ′ (cid:19) n f ( x ) g ( x ′ ) | x ′′ = x ′ = x . (12)We note that this formula of the generalized star product coincides with theordinary definition of the star product (5) when Θ µν = θǫ µν . We stress thatwe obtain this formula similar to one with a constant coefficient, as the conse-quence of the prescription, otherwise there can appear terms which include thederivatives of vielbein e µa ( x )s. 4e denote the star product CR by[ f ( x ) , g ( x )] ⋆ = f ( x ) ⋆ g ( x ) − g ( x ) ⋆ f ( x ) . (13)We apply the formula to the case f ( x ) = x µ and g ( x ) = x ν to obtain[ x µ , x ν ] ⋆ = i Θ µν ( x ) , (14)where µ, ν = 1 , , · · · , n . In this paper, we do not take up the form of vielbeintoo much. We rather take this equation as a starting point. By assumingpossible forms of Θ µν ( x ), we study what we can obtain from this equation asthe result of possible assumptions. In this section, we quantize the one-dimensional harmonic oscillator by usingthe star product. The energy spectrum is quantized using the stargenvalueequation[2]. The one-dimensional harmonic oscillator falls into an applicationof the generalized star product by inserting Θ µν ( x ) = θǫ µν , ( µ, ν = 1 ,
2) withconstant θ , which is the simplest case of the generalized star product cases,or the ordinary star product used for the θ -deformation. Inserting this intoEq.(14), we obtain [ x , x ] ⋆ = iθ. (15)The star product which we use in this section is obtained from (12) by settingΘ µν ( x ) = θǫ µν . The dimension is two which is the smallest dimension in thephase space of coordinate and momentum. As far as we apply the star productto the θ -deformation to such phase space, the dimension is always even. Thisexample shows contrasting features in comparative with the Lie-algebra typesdiscussed in the following section, which allow for the odd number of dimensions.We introduce new variables a and ¯ a by a = x + ix √ , (16)¯ a = x − ix √ . (17)Then the differentiations with respect to x and x are replaced by those withrespect to a and ¯ a as ∂∂x = 1 √ (cid:18) ∂∂a + ∂∂ ¯ a (cid:19) , (18) ∂∂x = i √ (cid:18) ∂∂a − ∂∂ ¯ a (cid:19) . (19)5nserting these results to the star product, we obtain f ⋆ g = exp (cid:20) θ (cid:18) ∂∂a ∂∂ ¯ a ′ − ∂∂a ∂∂ ¯ a ′ (cid:19)(cid:21) f ( a, ¯ a ) g ( a ′ , ¯ a ′ ) | a ′ = a, ¯ a ′ =¯ a ′ . (20)Then, the CR is written in terms of a and ¯ a as[ a, ¯ a ] ⋆ = θ, (21)suggesting a and ¯ a play roles of an annihilation operator and a creation operatorin the operator formalism, respectively.We next discuss the stargen equation of the one-dimensional harmonic os-cillator Hamiltonian given by H = 12 (( x ) + ( x ) ) = 12 (¯ a ⋆ a + a ⋆ ¯ a ) . (22)We shall study the stargenstates belonging to the stargenvalues of the Hamil-tonian. The stargenstate ψ and stargevalue E correspond to the eigenstate andeigenvalues in the operator formalism, respectively. The stargen-value equationis given by H ⋆ ψ = Eψ. (23)In order to solve this equation, there can be an analytic method and an algebraicmethod. Here we use the algebraic method. We first rearrange the order of a and ¯ a in the Hamiltonian as H = ¯ a ⋆ a + θ a ⋆ ¯ a − θ a ⋆ ¯ a + ¯ a ⋆ a ) . (24)We then define the vacuum state ψ by a ⋆ ψ = 0 . (25)The stargen-value equation is solved by H ⋆ ψ n = E n ψ n , ( n = 0 , , · · · ) (26)where the stargenstate ψ n and the stargen-value E n are given by ψ n = (¯ a⋆ ) n ψ , E n = ( n + 12 ) θ. (27)This is proved by the inductive method. The vacuum state satisfies H ⋆ ψ = (¯ a ⋆ a + θ ⋆ ψ = θ ⋆ ψ , (28)showing that Eq.(27) holds for n = 0. We assume that Eq.(26) holds for n = k .When n = k + 1, the stargen-value equation reads H ⋆ ψ k +1 = H ⋆ ¯ a ⋆ ψ k = (¯ a ⋆ H + θ ¯ a ) ⋆ ψ k = ( k + 1 + 12 ) θψ k +1 , (29)6here we have used [ H, ¯ a ] ⋆ = θ ¯ a. (30)Therefore, Eq.(26) holds also for n = k + 1. We have thus shown that Eq.(23)is solved by Eqs.(26) with Eq.(27) for non-negative integer number n .This is a well-known result in the operator formalism or the Schr¨odingerequation, and we have shown that the same result can be obtained by solvingthe stargen-value equation. We remark here that the energy spectrum is semi-infinite. In this section, examples of x -dependent Θ µν ( x ) in (14) are shown. In the firstsubsection, we show su (2) as an example. In the second subsection, su (3) isexhibited as another example. We present a three dimensional space model in which the coordinates x µ satisfyEq.(14) with a condition n = 3:[ x µ , x ν ] ⋆ = i Θ µν ( x ) , ( µ, ν = 1 , , µν ( x ) is an entity of a 3 × µν ( x )) = 2 θ x − x − x x x − x . (31)Then, Eq.(14) with this assignment yields the su (2) CR in terms of the starproduct [ x µ , x ν ] ⋆ = 2 iθǫ µνλ x λ , (32)where ǫ µνλ are constants antisymmetric in all indices with ǫ = 1.We next discuss the quantization of this algebra. Introducing j ± and j by j ± = x ± ix , (33) j = x , (34)the CRs are rewritten as [ j , j ± ] ⋆ = ± θj ± , (35)[ j + , j − ] ⋆ = 2 θj . (36)7e define j known as a Casimir operator in the operator formalism by j = X i =1 j i , (37)then the CRs with j s become (cid:2) j , j (cid:3) ⋆ = 0 , (38) (cid:2) j ± , j (cid:3) ⋆ = 0 , (39)which show that j is commutable with all j s.In order to obtain stargen-values and stargen functions, we need to solve thestargen-value equations of j and j . Since they are commutable, the stargenfunctions are characterized by two indices which are independent to each other.We denote them by l and m . In the quantization of a harmonic oscillator in theprevious section, we quantized the energy spectrum of the Hamiltonian function.In the present case, we quantize the su (2) algebra in which the stargen functionis that of j and j at the same time. This may be similar to the situation wherea constraint on the eigenstate of a Hamiltonian is imposed.We study the stargen-value equation of j . We define the state belonging tothe stargen-value mθ by j ⋆ f lm = mθf lm , (40)where m is assumed to be an integer or half-integer. As for the range of m , acondition which bounds the range will be imposed later. We also assume j ⋆ f lm = γθ f lm , (41)where γ is assumed to be expressed in terms of l only and its explicit form willalso be computed later.The states f lm ± k with the second quantum number m ± k can be obtainedby operating j ± upon f lm state k times as f lm ± k ∝ ( j ± ⋆ ) k f lm . (42)These can be derived in the following way. By using (35), we obtain j ⋆ ( j ± ⋆ f lm ) = ( j ± ⋆ j ± θj ± ) ⋆ f lm = ( m ± θ ( j ± ⋆ f lm ) . (43)Then we find that j ± ⋆ f lm ∝ f lm ± . (44)By repeating this procedure, Eq.(42) can be proved.As for the range of m , we assume that m ranges from − l to l where weassume that l is integer or half-integer. In order this to be realized, we imposea condition that the value m does not exceed the maximal value l : j + ⋆ f ll = 0 . (45)8y taking a complex conjugate of this equation, we also have f l − l ⋆ j − = 0 . (46)These conditions guarantee the finite range of m ; − l ≤ m ≤ l .Last of all, we determine the explicit form of γ , which is expressed in termsof l . Noting that j = j − ⋆ j + + θj + j , (47)we obtain, by operating j on f ll , j ⋆ f ll = ( j − ⋆ j + + θj + j ⋆ j ) ⋆ f ll = ( l + l ) θ f ll . (48)Then, by operating j − on f ll ( l − m ) times, we obtain j ⋆ f lm = ( l + l ) θ f lm . (49)We have thus derived that γ = l ( l + 1).We have assumed that l takes either the integer value or half-integer value.When l takes integer values, the algebra represents so (3). Setting θ = ¯ h , weobtain the algebra of the angular momentum. When l takes half-integer values,the algebra represents su (2), which can be interpreted as a spin representationor the internal symmetry, isospin. For example, when l is set to be l = 1 /
2, thevalue of m takes either m = 1 / m = − /
2, which correspond to an upstateand a downstate of isospin, respectively.
In this subsection, we discuss su (3) algebra in terms of the star product. Westart with defining the star product. The rank of the root space of su (3) is twoin contrast to one in su (2) case. This makes it complicated to impose constraintsin order to confine the states into the multiplet representation of su (3).We study a eight dimensional space model in which the coordinates x i , ( i =1 , , · · · ,
8) satisfy Eq.(14) with a condition n = 8:[ x µ , x ν ] ⋆ = 2 if µνλ x λ = i Θ µν ( x ) , where f µνλ is the su (3) structure constant and Θ µν ( x ) is given by(Θ µν ( x ))= θ x − x x − x x − x − x x x x − x − x x − x x − x − x x − x − x − x x x x −√ x x − x x − x − x x √ x − x x x − x x − x −√ x x x − x − x − x x √ x √ x −√ x √ x −√ x . x , x ] ⋆ = 2 iθx , [ x , x ] ⋆ = 2 iθx , [ x , x ] ⋆ = 2 iθx , (50)which show that the CRs constitute one of the su (2) subalgebras of su (3).We introduce new variables in order to adapt to the ordinary expressions.We define H i , ( i = 1 ,
2) by H = x , (51) H = 1 √ x , (52)which are commutable with each other[ H , H ] ⋆ = 0 . (53)We further define linear combinations of x i by E ± = ( x ± ix ) / , E ± = ( x ± ix ) / , E ± = ( x ± ix ) / . (54)Some of the CRs are given by[ E , E − ] ⋆ = 2 H , (55)[ E , E − ] ⋆ = 32 H − H , (56)[ E , E − ] ⋆ = 32 H + H . (57)We introduce two component vector ~H by ~H = (cid:18) H H (cid:19) . (58)Then we obtain h ~H, E ± i i ⋆ = ± ~α i E ± i , ( i = 1 , ,
3) (59)where two component vectors ~α i s called root vectors are given by ~α = (cid:18) (cid:19) , ~α = (cid:18) − (cid:19) , ~α = (cid:18) (cid:19) = ~α + ~α . (60)We next study the stargen-value equation of ~H . We denote the stargenfunction of ~H by f I Y ( x ), where I and Y are the stargen-values of H and H ,respectively. The equation reads ~H ⋆ f I Y ( x ) = f I Y ( x ) ⋆ ~H = ~ηf I Y ( x ) , (61)10igure 1: The translations by E , E and E expressed by ~α , ~α and ~α .where ~η = ( I , Y ) t . Though there are left and right operation of ~H in thestargen-value equation, the operation of ~H from left shall be omitted in thefollowing discussion for simplicity, if not necessary.We study the translations of a point specified by the stargen-values ( I , Y )in I Y -plane by the operation of E ± i s. This can be explicitly shown by ~H ⋆ ( E ± i ⋆ f I Y ( x )) = ( ~η ± ~α ± i )( E ± i ⋆ f I Y ( x )) . (62)This shows that the state E ± i ⋆ f I Y ( x ) is the stargen state belonging to thestargen-value ~η ± ~α ± i : E ± i ⋆ f ~η ( x ) ∝ f ~η ± ~α i ( x ) , (63)where f ~η ( x ) = f I Y ( x ). From this result, we find that E ± i translate a state at ~η to one at ~η ± ~α i . The translations by E ± i , ( i = 1 , ,
3) are illustrated in Fig.1.We next show that every state in a multiplet can be visited by these oper-ations by E ± i . As an example, we can visit all the states in the octet startingfrom a state in the octet. Starting from a f ( x ) state, other states can bevisited by successive operations of E ± i s as f ( x ) ∝ E − ⋆ f ( x ) , (64) f − ( x ) ∝ E − ⋆ f ( x ) , (65) f − − ( x ) ∝ E − ⋆ f − ( x ) , (66) f − ( x ) ∝ E ⋆ f − − ( x ) , (67) f ( x ) ∝ E ⋆ f − ( x ) , (68) f − ( x ) ∝ E ⋆ f ( x ) , (69) f ( x ) ∝ E ⋆ f − ( x ) , (70)which is shown in Fig.2. 11igure 2: Traveling in the octet states driven by E ± i starting from f state.When we observe the traveling in the octet in Fig.2, we note that the statedesignated by ( I , Y ) = (0 ,
0) is visited twice. As will be shown later, we have( E − ⋆ ) f ( x ) = 0 , (71)( E ⋆ ) f ( x ) = 0 . (72)Since ( E − ⋆ ) = ( E ⋆ ) , this suggests that the state f ( x ) is doubly degenerate,though it can not be distinguished by the values of I and Y .In the above traveling among the octet states, the path should not exceedthe boundary of the hexagon. These constraints on the path should be imposedso that the traveling is limited inside an octet. After the introduction of theCasimir operator, we shall discuss these constraints.In su (3) algebra, there are two Casimir operators. The stargen-values ofthe Casimir operators should be the same for any state in the octet, becausethey are commutable with all x i s and so E i s. We discuss one of the Casimiroperators. The Casimir operator C ( su (3)) is defined by C ( su (3)) = X i =1 (cid:18) x i (cid:19) . (73)Then, this can be rewritten in terms of H i s and E ± i s as C ( su (3)) = H ( H + 1) + 34 H ( H + 2) + X i =1 E − i ⋆ E i . (74)At each state on the vertices of hexagon representing the octet states, we shouldimpose conditions that forbid the translations outward of the hexagon to theouter states and allow translations inward to the states within the hexagon. Inorder to realize this, we impose the following conditions at ( I , Y ) = (1 , E ∗ f ( x ) = 0 , E − ∗ f ( x ) = 0 , E ∗ f ( x ) = 0 . (75)12t ( I , Y ) = ( , E ∗ f ( x ) = 0 , E ∗ f ( x ) = 0 , E ∗ f ( x ) = 0 , (76)and at ( I , Y ) = ( − , E − ∗ f − ( x ) = 0 , E ∗ f − ( x ) = 0 , E ∗ f − ( x ) = 0 . (77)As for other states at the antipodal vertices, similar conditions with E i s withopposite signed i are obtained by the complex conjugation of the above con-straints. Noting that f ( x ) ∝ E ⋆ f and using the middle equation of (76),we obtain ( E ⋆ ) f = 0 , which is Eq.(72). This shows that the state f is a state of the triplet of su (2).In the similar way, we can show Eq.(71). Since there are two independent su (2)subalgebras of su (3), we have Eqs.(71) and (72) imposed on f independently.We can now evaluate the Casimir operator. As we have mentioned, evalu-ation at any state would be the same. We evaluate it at ( I , Y ) = (1 , C ( su (3)) = H ( H + 2) + 34 H + E − ⋆ E + E ⋆ E − + E − ⋆ E . (78)Then we can evaluate it to obtain C ( su (3)) ⋆ f ( x ) = 3 f ( x ) . (79)The same value can be obtained from other states of the octet by rearrangingthe order of E i and E − i , so that it fits to the conditions at vertex of the hexagon. In section 3, we quantize a Hamiltonian of one-dimensional harmonic oscillatorin the two dimensional phase space in terms of the star product. We applythe star product method also to the Lie algebra. In section 4, we quantizethe Lie algebra su (2) in the three dimensional representation space, and Liealgebra su (3) in the eight dimensional representation space. The quantizationis made by setting CRs among the coordinates in each dimension. Once theCRs among the coordinates are set, or explicit Θ µν ( x ) are determined, thequantization is implemented. The terminology “quantization” is meant by notonly the quantization of the energy levels but also computing quantum numbersin the Lie algebra. As far as quantization by using the star product is concerned,there seems to be no reason to distinguish both quantizations in the phase space13nd representation space. Only difference between them is the choice of entitiesof Θ µν ( x ) in Eq.(14).Then it is natural to unify two quantizations. The unification of the externalspace and the internal space reminds us of the Kaluza-Klein(K-K hereafter)theory[3]. In the K-K theory, the unification of the external space and internalspace in higher dimension together with the dimensional reduction leads to the4-dimensional theory with internal symmetries, like the Einstein gravity with a U (1) gauge field. The K-K theory is a classical theory. In this paper, we claimthat the quantizations are unified in higher dimensions by setting the CRs inhigher dimensions, which brings about the energy levels in the external phasespace and the quantum numbers in the representation space, at the same time.We show this unification idea by a toy model in which the external phasespace is two dimensional phase space and the internal representation space isthree dimensional. Then we discuss the quantization in terms of the star prod-uct in five dimensions. We assume that the internal and external spaces arefactorized. The star product is characterized by Θ µν ( x ) , ( µ, ν = 1 , , · · · , µν ( x )) = θ r − r x − x − x x x − x , (80)where r is some constant. Although we can not determine the magnitude of θ and r separately, we might regard their multiplication θ · r as ¯ h . We factorizethe Wigner function f ( x ) in 2 + 3 dimensions into two parts, one in the externalphase space f ( x , x ) and the other in the internal representation space. Weassume that the Wigner function of five variables is factorized to the Wignerfunction f ( x , x ) of two variables and the Wigner function f ( x , x , x t ) ofthree variables f ( x ) = f ( x , x ) · f ( x , x , x t ) . (81)Then the remaining computation has been already shown in previous sections.When we assume CRs in higher dimensions, the stargen function f ( x , x ) givesrise to the states for the quantized energy spectrum, and the stargen function f ( x , x , x t ) yields the quantum numbers of su(e) like an isospin state. Inthis way, the energy spectrum and the isospin state, which is either upstateor downstate of su (2). These quantum numbers emerge simultaneously by thequantization in higher dimensions. Although this is simply a toy model, we canextend the present model in a more realistic way. In this paper, we have shown the star product construction of CR given by (14),which includes the θ -deformation quantization and the x -dependent Θ µν ( x )14ases like the Lie algebra. In the curved space time, where the coordinatesare not perpendicular to each other and so the vielbein e µi ( x ) is not δ µi , it seemsnatural to use x -dependent Θ µν ( x ). We showed that the CR is reduced to thatof su (2) or su (3) when Θ µν ( x ) has special entities. A special case of constantΘ µν ( x ) is nothing but the θ -deformation. There is no essential difference be-tween the quantization of quantum mechanics and the quantization of the Liealgebra from the viewpoint of the star product quantization. Then, it seems nat-ural to unify the quantizations in the internal space and external phase spaceto higher dimensions. In section 5, we gave a toy model of unified quantiza-tion where one-dimensional harmonic oscillator and su (2) internal symmetryare simultaneously quantized.In section 4 we discuss the quantization of su (2) and su (3). When we com-pare two quantizations, su (2) is much easier because the rank is just one. Inorder to confine states in a multiplet of su (2), we limited outward translationsfrom the multiplet. This limitation corresponds to the vacuum condition in thequantum mechanics like the one-dimensional harmonic oscillator. In su (2), onelimitation brings about another limitation by complex conjugation. Therefore,the number of states is finite. On the other hand, in the θ -deformation thenumber of states is infinite.Though the model in section 5 is just a toy model, we can generalize it tomore realistic model that is comprised of realistic external phase space and real-istic internal symmetry space. Since a Wigner function expresses a probabilityof the corresponding state, it would be possible to compute the possibilities ofvarious dimensional reductions when we obtain all the Wigner functions corre-sponding to those dimensional reductions. For example, it might be possibleto compare the magnitude relation of probabilities of su (3) and su (2) × u (1)by use of the Wigner functions. In the present discussion, we made use of thealgebraic method in finding the energy spectrum and quantum numbers of theLie algebra invariants. But we can also use the analytic method in finding thosequantum numbers. Then we need to solve the stargen-value equation to obtainthe stargen functions in this analytic method. These functions are nothing butthe Wigner functions representing the probabilities. A study investigating inthis direction will be reported in the future.15 eferenceseferences