Star product and star function

Abstract

We give a brief review on star products and star functions [8, 9]. We introduce a star product on polynomials. Extending the product to functions on complex space, we introduce exponential element in the star product algebra. By means of the star exponential functions we can define several functions called star functions in the algebra.We show certain examples.

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1. Star product on polynomials 1.1. Moyal product The Moyal product is a well-known example of star product [2, 3]. For polynomials f; g of the variables (u1; : : : ; um; v1; : : : ; vm) , the Moyal product f O g is given by the power series of the bidifferential operators @v !@ u @u !@ v such that f O g = f exp i~ 2 @v !@ u @u !@ v - g = f X1 k=0 1 k! i~ 2 k @v !@ u @u !@ v -k g = fg + i~ 2 f @v !@ u @u !@ v - g + 1 2! i~ 2 2 f @v !@ u @u !@ v -2 g + + 1 k! i~ 2 k f @v !@ u @u !@ v -k g + (1.1.1) where ~ is a positive number and the overleft arrow @ means that the partial derivative is acting on the polynomial on the left and the overright arrow similar, for example f @v !@ u @u !@ v - g = Xm j=1 @vjf @ujg @ujf @vjg : Although the Moyal product is defined as a formal power series of bidifferential operators, this becomes a finite sum on polynomials. P r o p o s i t i o n 1.1.1. The Moyal product is well-defined on polynomials, and associative. Other typical star products are normal product N , anti-normal product A given similarly by f N g = f exp i~ @v !@ u - g; f A g = f exp n i~ @u !@ v -o g: These are also well-defined on polynomials and associative. By direct calculation we see easily P r o p o s i t i o n 1.1.2. (i) For these star products, the generators (u1; : : : ; um; v1; : : : ; vm) satisfy the canonical commutation relations [uk; vl] L = i~ kl; [uk; ul] L = [vk; vl] L = 0; (k; l = 1; 2; : : : ;m) where L stands for O , N and A . (ii) Then the algebras (C[u; v]; L) (L = O;N;A) are mutually isomorphic and isomorphic to the Weyl algebra. STAR PRODUCT AND STAR FUNCTION 283 Actually the algebra isomorphism IN O : (C[u; v]; O) ! (C[u; v]; N) is given explicitly by the power series of the differential operator such as IO N (f) = exp i~ 2 @u@v (f) = X1 l=0 1 l! i~ 2 l (@u@v)l (f): (1.1.2) And other isomorphisms are given in the similar form. R e m a r k 1.1.1. We remark here that these facts are well-known as ordering problem in physics [1]. 1.2. Star product Now we define a star product on complex domain by generalizing the previous products. Let be an arbitrary n - n complex matrix. We consider a bidifferential operator @w ! @w = ( @w1 ; ; @wn) ( ! @w1 ; ; ! @wn) = Xn k;l=1 kl @wk ! @wl (1.2.3) where (w1; ;wn) is a generators of polynomials. Now we define a star product similar to (1) by D e f i n i t i o n 1.2.1. f g = f exp @w ! @w - g: (1.2.4) R e m a r k 1.2.1. [9] (i) The star product is a generalization of the previous products. Actually if we put = 0 1m 1m 0 then we have the Moyal product if = 2 0 0 1m 0 , then we have the normal product if = 2 0 1m 0 0 then we have the anti-normal product (ii) If is a symmetric matrix, the star product is commutative. Then similarly as before we see easily Theorem 1.2.1. For an arbitrary , the star product is well-defined on polynomials, and associative. 284 A. Yoshioka 1.3. Equivalence and geometric picture of Weyl algebra In this section, we take as a special class of matrices in order to representWeyl algebra, cf. [4, 7]. We consider the following complex matrices : = J + K; where K is an arbitrary 2m - 2m complex symmetric matrix and J = 0 1m 1m 0 : Since is determined by K , we denote the star product by K instead of . We consider polynomials in variables (w1; ;w2m) = (u1; ; um; v1; ; vm) . By a easy calculation one obtains P r o p o s i t i o n 1.3.1. (i) For a star product K , the generators (u1; : : : ; um; v1; : : : ; vm) satisfy the canonical commutation relations [uk; vl] K = i~ kl; [uk; ul] K = [vk; vl] K = 0; (k; l = 1; 2; : : : ;m): (ii) Then the algebra (C[u; v]; K) is isomorphic to the Weyl algebra, and the algebra is regarded as a polynomial representation of the Weyl algebra. Equialence As in the case of typical star products, we have algebra isomorphisms as follows. P r o p o s i t i o n 1.3.2. For arbitrary star product algebras (C[u; v]; K1 ) and (C[u; v]; K2 ) we have an algebra isomorphism IK2 K1 : (C[u; v]; K1 ) ! (C[u; v]; K2 ) given by the power series of the differential operator @w(K2 K1)@w such that IK2 K1 (f) = exp i~ 4 @w(K2 K1)@w (f); where @w(K2 K1)@w = P kl(K2 K1)kl@wk@wl . R e m a r k 1.3.1. 1. By the previous proposition we see the algebras (C[u; v]; K) are mutually isomorphic and isomorphic to the Weyl algebra. Hence we have a family of star product algebras f(C[u; v]; K)gK where each element is regarded as a polynomial representation of the Weyl algebra. 2. The above equivalences are also possible to make for star products for arbitrary ’s with a common skew symmetric part. STAR PRODUCT AND STAR FUNCTION 285 By a direct calculation we have Theorem 1.3.1. Isomorphisms satisfy the following chain rule: 1. IK1 K3 IK3 K2 IK2 K1 = Id , 8K1;K2;K3 2. IK2 K1 1 = IK1 K2 , 8K1;K2 According to the previous theorem, we introduce an infinite dimensional bundle and a connection over it and using parallel sections of this bundle we have a geometric picture for the family of the star product algebras f(C[u; v]; K)gK . Algebra bundle We set S = fKg the space of all 2m-2m symmetric complex matrices. We consider a trivial bundle over S whose fibers are the star product algebras : E = Y K2S (C[u; v]; K) ! S; 1(K) = (C[u; v]; K): Then the previous proposition shows that fibers (C[u; v]; K) are mutually isomorphic and are isomorphic to the Weyl algebra, and the isomorphisms IK2 K1 give an isomorphism between fibers. Connection and parallel sections For a curve C : K = K(t) in the base space S , starting from K(0) = K , we define a parallel translation of a polynomial f 2 (C[u; v]; K) by f(t) = exp i~ 4 @w(K(t) K)@w(f): It is easy to see f(0) = f . By differentiating the parallel translation we have a connection of this bundle such that rXf(K) = d dt f(t)jt=0(K) = i~ 4 @wX@w f(K)jt=0; X = _K (t)jt=0; where f(K) is a smooth section of the bundle E . We set P the space of all parallel sections of this bundle. Since IK2 K1 are algebra isomorphisms IK2 K1 (f(K1) K1 g(K1)) = IK2 K1 (f(K1) K2 IK2 K1 (g(K1) ; we have a star product on the space of parallel sections f; g 2 P by f g(K) = f(K) K g(K): Then we have 286 A. Yoshioka Theorem 1.3.2. (i) The space of the parallel sections P consists of the sections such that rXf = i~ 4 @wX@w f = 0; 8X: (ii) The space P is canonically equipped with the star product , and the associative algebra (P; ) is isomorphic to the Weyl algebra. R e m a r k 1.3.2. The algebra (P; ) is regarded as a geometric realization of the Weyl algebra. 2. Extension to functions We want to extend the star products for an arbitrary complex matrix from polynomials to functions, cf. [6]. 2.1. Star product on certain holomorphic function space We want to transfer the star products from polynomials to functions. However, the product is not necessarily convergent for ordinary smooth functions, hence we need to restrict the product to certain subset of smooth functions. There may be many such spaces. In this note we consider the following space of certain entire functions. Semi-norm Let f(w) be a holomorphic function on Cn . For a positive number p , we consider a family of semi-norms fj jp;sgs>0 given by jfjp;s = sup w2Cn jf(w)j exp( sjwjp); jwj = p jw1j2 + + jwnj2: Space We put Ep = ff : entire j jfjp;s < 1; 8s > 0g: With the semi-norms the space Ep becomes a Fr´echet space. As to the star products, we have for any matrix . Theorem 2.1.1. (i) For 0 < p 6 2 , (Ep; ) is a Frech´et algebra. That is, the product converges for any elements, and the product is continuous with respect to this topology. (ii) Moreover, for any 0 with the common skew symmetric part with , the map I 0 = exp i~ 4 @w( 0 )@w is a well-defined algebra isomorphism from (Ep; ) to (Ep; 0 ) . That is, the expansion convergies for every element, and the operator is continuous with respect to this topology. (iii) For p > 2 , the multiplication : Ep - Eq ! Ep is a well-defined for q such that (1=p) + (1=q) = 2 , and (Ep; ) is a Eq -bimodule. STAR PRODUCT AND STAR FUNCTION 287 3. Star exponentials Since we have a complete topological algebra, we can consider exponential elements in the star product algebra (Ep; ) , cf. [9]. 3.1. Definition For a polynomial H , we want to define a star exponential exp (tH =i~) . However, except special cases, the expansion X n tn n! H i~ n is not convergent, so we define a star exponential by means of a differential equation. D e f i n i t i o n 3.1.1. The star exponential exp (tH =i~) is given as a solution of the following differential equation d dt Ft = H Ft; F0 = 1: (3.1.1) 3.2. Examples We are interested in the star exponentials of linear, and quadratic polynomials. For these, we can solve the differential equation and obtain explicit form. For simplicity, we take as above: = K + J where K is a complex symmetric matrix. First we remark the following. For a linear polynomial l = P2m j=1 ajwj , we see directly that an ordinary exponential function el satisfies el =2 E1; 2 E1+ ; 8 > 0: Then put a Fr´echet space Ep+ = \\q>pEq: Linear case P r o p o s i t i o n 3.2.1. For l = P j ajwj =< a;w >, aj 2 C, we have exp t l i~ = exp t2aKa 4i~ exp t l i~ 2 E1+: Quadratic case P r o p o s i t i o n 3.2.2. For Q = hwA;wi where A is a 2m-2m complex symmetric matrix, exp t(Q =i~) = 2m p detM exp 1 i~ w J e 2t J M w ; where M = I KJ + e 2t (I + KJ) and = AJ . 288 A. Yoshioka R e m a r k 3.2.1. The star exponentials of linear functions are belonging to E1+ then the star products are convergent and continuous. But it is easy to see exp t(Q =i~) 2 E2+; =2 E2 and hence star exponentials exp t(Q =i~) are difficult to treat. Some anomalous phenomena happen, cf. [5]. 4. Star functions There are many applications of star exponential functions, cf. [8]. In this note we show examples using a linear star exponentials. In what follows, we consider the star product for the simple case where = 0 0 0 ; 2 C: Then we see easily that the star product is commutative and explicitly given by p1 p2 = p1 exp i~ 2 @w1 ! @w1 p2: This means that the algebra is essentially reduced to the space of functions of one variable w1 . Thus, we consider functions f(w) , g(w) of one variable w 2 C and we consider a commutative star product with complex parameter such that f(w) g(w) = f(w) exp n 2 @ w !@ w o g(w): 4.1. Star Hermite function Recall the identity exp p 2tw 1 2 t2 = X1 n=0 Hn(w) tn n! ; where Hn(w) is an Hermite polynomial. We remark here that exp p 2tw 1 2 t2 = exp ( p 2tw ) = 1: Since exp ( p 2tw ) = P1 n=0( p 2tw )n tn n! we have Hn(w) = ( p 2tw )n = 1: We define -Hermite function by Hn(w; ) = ( p 2tw )n; (n = 0; 1; 2; ); STAR PRODUCT AND STAR FUNCTION 289 with respect to product. Then we have exp ( p 2tw ) = X1 n=0 Hn(w; ) tn n! : Trivial identity d dt exp ( p 2tw ) = p 2w exp ( p 2tw ) yields at every 2 C the identity p 2 H0 n(w; ) + p 2wHn(w; ) = Hn+1(w; ); (n = 0; 1; 2; ): The exponential law exp ( p 2sw ) exp ( p 2tw ) = exp ( p 2(s + t)w ) yields at every 2 C the identity X k+l=n n! k! l! Hk(w; ) Hl(w; ) = Hn(w; ): 4.2. Star theta function In this note we consider the Jacobi’s theta functions by using star exponentials as an application. A direct calculation gives exp i tw = exp(i tw ( =4)t2): Hence for Re > 0 , the star exponential exp ni w = exp(ni w ( =4)n2) is rapidly decreasing with respect to integer n and then we can consider summations for satisfying Re > 0 X1 n= 1 exp 2ni w = X1 n= 1 exp 2ni w n2 = X1 n= 1 qn2 e2ni w; (q = e ): This is Jacobi’s theta function 3(w; ) . Then we have expression of theta functions as 1 (w) = 1 i X1 n= 1 ( 1)n exp (2n + 1)i w; 2 (w) = X1 n= 1 exp (2n + 1)i w; 3 (w) = X1 n= 1 exp 2ni w; 4 (w) = X1 n= 1 ( 1)n exp 2ni w: 290 A. Yoshioka Remark that k (w) is the Jacobi’s theta function k(w; ) , k = 1; 2; 3; 4 , respectively. It is obvious by the exponential law exp 2i w k (w) = k (w) (k = 2; 3); exp 2i w k (w) = k (w) (k = 1; 4): Then using exp 2i w = e e2i w and the product formula directly we have e2i w k (w + i ) = k (w) (k = 2; 3); e2i w k (w + i ) = k (w) (k = 1; 4): 4.3. -delta functions Since the -exponential exp (itw ) = exp(itw ( =4)t2) is rapidly decreasing with respect to t when Re > 0 , then the integral of -exponential Z 1 1 exp (it(w a) ) dt = Z 1 1 exp (it(w a) )dt = Z 1 1 exp(it(w a) ( =4)t2)dt converges for any a 2 C. We put a star -function (w a) = Z 1 1 exp (it(w a) )dt; which has a meaning at with Re > 0 . It is easy to see that for any element p (w) 2 P (C) , we have p (w) (w a) = p(a) (w a); w (w) = 0: Using the Fourier transform we have P r o p o s i t i o n 4.3.1. 1 (w) = 1 2 X1 n= 1 ( 1)n (w + 2 + n ) 2 (w) = 1 2 X1 n= 1 ( 1)n (w + n ) 3 (w) = 1 2 X1 n= 1 (w + n ) 4 (w) = 1 2 X1 n= 1 (w + 2 + n ): Now, we consider the with the condition Re > 0 . Then we calculate the integral and obtain (w a) = 2 p p exp 1 (w a)2 : STAR PRODUCT AND STAR FUNCTION 291 Then we have 3(w; ) = 1 2 X1 n= 1 (w + n ) = X1 n= 1 p p exp 1 (w + n )2 = p p exp 1 X1 n= 1 exp 2n 1 w 1 n2 2 = p p exp 1 3 ( 2 w i ; 2 ): We also have similar identities for other -theta functions by the similar way. The author is grateful to V. F. Molchanov and S. Berceanu for valuable discussions and also grateful to the organizers for warm hospitality.