Star Product and Star Exponential Akira Yoshioka, Dept. of Math. Sci., Tokyo University of Science. June. 05 2009. Varna START �
This talk is based on the joint work with H. Omori, Y. Maeda and N. Myazaki. Abstract We extend star products by means of complex symmet- ric matrices. We obtain a family of star products. We consider star exponentials with respect to these star products, and we obtain several interesting identities. Plan ➀ First we explain general setting; introducing the con- cept of q -number functions. ➁ Then we consider the example of star exponential and its application. 1( � ) �⊞
§ 1. A family of star products § 1.1. Moyal product, normal and anti-normal prod- ucts It is well known that the star products such as the Moyal product, normal product and the anti-normal product are obtained by fixing the orderings in the Weyl algebra. These are products on polynimals and the obtained al- gebras are all isomorphic to the Weyl algebra. § 1.2, Extension We extend these products and we obtain a family star products parametrized by the space of all complex sym- metric matrices. The intertwiners are also extended to these star prod- ucts, and then all star product algebras are also mutually isomorphic and isomorophic to the Weyl algebra. 2( � ) �⊞
§ 1.3. Definition of star product For simplicity, we consider star products of 2 variables ( u 1 , u 2 ). The general case for ( u 1 , u 2 , · · · , u 2 m ) is similar. 1. First we consider biderivation � � λ 11 λ 12 For a complex matrix Λ = ∈ M 2 ( C ), we λ 21 λ 22 consider a bi-derivation acting on complex polynoimals p 1 ( u 1 , u 2 ) , p 2 ( u 1 , u 2 ) ∈ P ( C 2 ) such that 2 � ← ∂ Λ − − → � ← − → − � p 2 p 2 = p 1 p 1 ∂ λ kl ∂ u k ∂ u l k,l =1 2 � = λ kl ∂ u k p 1 ∂ u l p 2 (1) k,l =1 3( � ) �⊞
2. Star product We fix the skew symmetric matrix � � 0 1 J = (2) − 1 0 For an arbitrary complex symmetric matrix K ∈ S C (2) we put Λ = J + K and we define a product ∗ K on the space of complex poly- nomials p 1 ( u 1 , u 2 ) , p 2 ( u 1 , u 2 ) ∈ P ( C 2 ); � i � � ← − ∂ Λ − → p 1 ∗ K p 2 = p 1 exp ∂ p 2 2 = p 1 p 2 + i � � ← − ∂ Λ − → � 2 p 1 ∂ p 2 � i � � n + · · · + 1 � ← − ∂ Λ − → � n p 1 ∂ p 2 + · · · (3) n ! 2 4( � ) �⊞
3. Associativity We have Proposition 1 For an arbitrary complex symmetric matrix K ∈ S C (2) the product ∗ K is associtaive on the space of all complex polynomials P ( C 2 ) . 4. Isomorphic to the Weyl algebra CCR For an artibrary K ∈ S C (2), the product ∗ K satisfies the canonical commutation relations [ u k , u l ] ∗ K = u k ∗ K u l − u l ∗ K u k = i � δ kl , k, l = 1 , 2 . (4) and hence it follows that all algebras ( P ( C 2 ) , ∗ K ) are iso- morphic to the Weyl algebra W 2 of two generators u 1 , u 2 . 5( � ) �⊞
Intertwiners The algebra isomorphis (intertwiners) I K 2 K 1 : ( P ( C 2 ) , ∗ K 1 ) → ( P ( C 2 ) , ∗ K 2 ) (5) are explicitly given by � i � � I K 2 4 ( K 2 − K 1 ) ∂ 2 K 1 ( p ) = exp p (6) where 2 � ( K 2 − K 1 ) ∂ 2 = ( K 2 − K 1 ) kl ∂ u k ∂ u l (7) kl =1 We have the relations Proposition 2 (i) I K 3 K 2 I K 2 K 1 = I K 3 K 1 (ii) ( I K 2 K 1 ) − 1 = I K 1 K 2 6( � ) �⊞
Infinitesimal intertwiner By differentiating the intertwiner with respect to K , we obtain the infinitesimal intertwiner at K dt I K + tκ ∇ κ ( p ) = d ( p ) | t =0 = i � 4 κ∂ 2 p (8) K Then the infinitesimal intertwiner satisfies ∇ κ ( p 1 ∗ K p 2 ) = ∇ κ ( p 1 ) ∗ K p 2 + p 1 ∗ K ∇ κ ( p 2 ) (9) for any p 1 ( u 1 , u 2 ) , p 2 ( u 1 , u 2 ) ∈ P ( C 2 ). 7( � ) �⊞
§ 1.4. q -number polynomials � � ( P ( C 2 ) , ∗ K ) In the star product algebras K ∈S C (2) , the al- gebras ( P ( C 2 ) , ∗ K 1 ) and ( P ( C 2 ) , ∗ K 2 ) are mutually isomor- phic by the intertwiner I K 2 K 1 and the elements p 1 ∈ ( P ( C 2 ) , ∗ K 1 ) and p 2 ∈ ( P ( C 2 ) , ∗ K 2 ) are identified when p 2 = I K 2 K 1 ( p 1 ) (10) In order to give a geometric picture to the family of star � � ( P ( C 2 ) , ∗ K ) product algebras K ∈S C (2) , we introduce an algebra bundle over S C (2) whose fibres consisit of the Weyl algebra in the following way. 8( � ) �⊞
1. Algebra bundle We consider the the trivial bundle π : P = P ( C 2 ) × S C (2) → S C (2) (11) whose fibre over K ∈ S C (2) consists of the star product algebra π − 1 ( K ) = ( P ( C 2 ) , ∗ K ) (12) 2. Flat connection and parallel translation On this bundle, we regard the infinitesimal intertwiner ∇ as a flat connection and the intertwiner I K 2 K 1 as its parallel translation. We consider a section ˜ p ∈ Γ( P ) of this bundle satisfying p ( K 2 ) = I K 2 ˜ K 1 (˜ p ( K 1 )) (13) This means that ˜ p is parallel ∇ κ ˜ p ( K ) = 0 (14) 9( � ) �⊞
3. q -number polynomial We denote by P ( P ) the space of all parallel sections, and call an element ˜ p ∈ P ( P ) q -number polynomial . Due to the identies I K 3 K 2 I K 2 K 1 = I K 3 K 1 and ( I K 2 K 1 ) − 1 = I K 1 K 2 the intertwiners naturally induce the product ∗ on P ( P ). Then the algebra ( P ( P ) , ∗ ) is regarded as a geometric realization of the Weyl algebra. 10( � ) �⊞
§ 2. q -number functions We introduce a locally convex topology into the family of star product algebras by means of a system of semi-norms. We take the completion of the algebras and then we can consider star exponentials. 1. Topology We introduce a topology into P ( C 2 ) by a system of semi- norms in the following way. Let ρ be a positive number. For every s > 0 we define a semi-norm for polynomials by u ∈ C 2 | p ( u 1 , u 2 ) | exp ( − s | u | ρ ) | p | s = sup (15) Then the system of semi-norms {| · | s } s> 0 defines a locally convex topology T ρ on P ( C 2 ). 11( � ) �⊞
echet space E ρ ( C 2 ) 2. Fr´ We take the completion of P ( C 2 ) with re- Definition echet space E ρ ( C 2 ). spect to the topology T ρ , we obtain a Fr´ Proposition 3 For a positive number ρ , the Fr´ echet space E ρ consists of entire functions on the complex plane C 2 with finite semi-norm for every s > 0 , namely, � � E ρ ( C 2 ) = f ∈ H ( C 2 ) | | f | s < + ∞ , ∀ s > 0 (16) Continuity for the case 0 < ρ ≤ 2 As to the continuitiy of star products and intertwiners, the space E ρ ( C 2 ), 0 < ρ ≤ 2 is very good, namely, we have the following Theorem 1 On E ρ ( C 2 ) , 0 < ρ ≤ 2 every product ∗ K is continuous, and every intertwiner I K 2 K 1 : ( E ρ ( C 2 ) , ∗ K 1 ) → ( E ρ ( C 2 ) , ∗ K 2 ) is continuous. 12( � ) �⊞
Continuity as a bimodule for the case ρ > 2 As to the spaces E ρ ( C 2 ) for ρ > 2, the situation is no so good, but still we have the following. Theorem 2 For ρ > 2 , take ρ ′ > 0 such that ρ ′ + 1 1 ρ = 1 then every star product ∗ K defines a continuous bilinear product ∗ K : E ρ ( C 2 ) × E ρ ′ ( C 2 ) → E ρ ( C 2 ) , E ρ ′ ( C 2 ) × E ρ ( C 2 ) → E ρ ( C 2 ) This means that ( E ρ ( C 2 ) , ∗ K ) is a continuous E ρ ′ ( C 2 ) - bimodule. 13( � ) �⊞
3. q -number functions The case 0 < ρ ≤ 2 Due to the previous theorem, we can introduce a similar concept as q -number polynomials into the Fr´ echet spaces. Namely, the star product ∗ K is well defined on E ρ ( C 2 ) and then we consider the trivial bundle π : E ρ = E ρ ( C 2 ) × S C (2) → S C (2) (17) with fibre over the point K ∈ S C (2) consists of π − 1 ( K ) = ( E ρ ( C 2 ) , ∗ K ) (18) The intertwiners I K 2 K 1 are well defined for any K 1 , K 2 ∈ S C (2) and then the bundle E ρ has a flat connection ∇ and the parallel translation is the intertwiner. The space of flat sections of the bundle denoted by F ρ naturally has the product ∗ and can be regarded as a com- pletion of the Weyl algebra W 2 . 14( � ) �⊞
4. Remark to the case ρ > 2 For the case ρ > 2, at present it is not clear whether the intertwiners are well-defined and whether the product ∗ K are well defined. However the flat connection ∇ is still well defined on π : E ρ = E ρ ( C 2 ) × S C (2) → S C (2), so we can define a space F ρ of all parallel sections of this bundle even for ρ > 2. For ρ > 2, we are trying to extend the product ∗ K and also the intertwiners I K 2 K 1 by means of some regularizations, for example, Borel-Laplace transform, or finite part regular- ization. I hope to construct such a concept in near future. 15( � ) �⊞
5. Star expoenential The space of q -number functions F ρ is a complete topo- logical algebra for 0 < ρ ≤ 2 (even ρ > 2 for future under some regularization). We can consider exponential element ∞ � H � t n H i � ∗ · · · ∗ H � exp ∗ t = (19) i � n ! i � n =0 � �� � n in this algebra. For a q -number polynomial H ∈ P ( P ), we define the star exponenial exp ∗ t ( H/i � ) by the differential equation � H � � H � � H � d = H dt exp ∗ t i � ∗ exp ∗ t , exp ∗ t | t =0 = 1 (20) i � i � i � 16( � ) �⊞
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