Deformation Quantization by Moyal Star-Product and Stratonovich Chaos

We make a deformation quantization by Moyal star-product on a space of functions endowed with the normalized Wick product and where Stratonovich chaos are well defined.


Introduction
This work follows those of G. Dito [3] (motivated by quantum field theory [4,5,8,27]), of R. Léandre [14,15,16] and G. Dito, R. Léandre [6]. [3] deals with the deformation quantization of Moyal on a Hilbert space: the condition of equivalence of the Moyal deformations is that the chosen perturbation operator is a Hilbert-Schmidt operator but in this case, Moyal and normal products are not equivalents. [15] choose Hida weighted Fock spaces which are very small spaces. This gives spaces of continuous functionals on the path space. The inequivalences of [3] become equivalences. In [6], the Malliavin test algebra is used for the Moyal quantization. A very important remark in [3] and [6] is that the matrix of the associated Poisson structure is still bounded.
Our motivation comes from the fact we deal with deformation quantization on an algebra constituted of Stratonovich chaos. The Connes spaces where our work was possible present some differences with the Hida spaces of [15]. Indeed, Connes spaces are involved with tensor products of Banach spaces and Hida spaces are involved with tensor products of Hilbert spaces [11,17,23]. In infinite dimension analysis, there are two basic objects [21]: • an algebraic model; • a mapping space and a map Ψ defined from algebraic model into the space of functionals on this mapping space.
Let us recall what is the main difference between the point of view of [6] and the point of view of [15,16] and this paper: • The tools of Malliavin calculus on the Wiener space are used in [6]. The Malliavin test algebra is constituted of functionals almost surely defined, because there is no Sobolev imbedding in infinite dimension [19,20,22,25].
• The tools of white noise analysis are used in [15,16] and in this present work. The differential operations and the topological structures are seen at the level of the algebraic model and after they are transported through the map Ψ on a set of functionals which arXiv:1203.3272v2 [math.QA] 16 Mar 2012 are continuous on the Wiener space. [15] and [16] work on the level of the algebraic model. They use a different normalization of annihilation operators which fit with the map Stratonovitch chaos and not with the map Wiener chaos. In the present paper, we transport the algebraic operations of [15,16] on a functional space by using the map Ψ Stratonovitch chaos which was not defined in [15] and [16].
Note that E. Getzler [10] was the first to consider another map than the map Wiener chaos. He used as algebraic space a Connes space and as map Ψ the map of Chen iterated integrals (see [13] for developments). After this quick presentation, we shall define in the second part our Connes functional space on which the multiple Stratonovich integrals or Stratonovich chaos are well defined. We shall finish this second part by a study of the annihilation operator on our functions space. In the third part, the deformation quatization of the Poisson bracket by a Moyal product is defined on the Connes space. The last part is about equivalences of deformations on the Connes space.

Gaussian space
In this section we show the existence of a Gaussian measure on a space of continuous loops which will be our reference measure throughout this work on the algebraic space. We also define Stratonovich chaos and differentiation operators we use in the next section.

Gaussian measure
Let us consider the Hilbert space H := H(S 1 , R d ) such that γ ∈ H verifies: ||γ|| 2 = 1 0 |γ(s)| 2 ds+ 1 0 |γ(s)| 2 ds. We consider B = {B(t) = (B i (t)), t ∈ S 1 } the Wiener process associated to this Hilbert space. We note (·, t) → G(·, t) the symmetric Green kernel. Let h be a continuous function from S 1 with values in R such that G is solution of a second-class linear differential equation and the Green kernel of that equation verifies d 2 ds 2 G(s, 1) − G(s, 1) = 0 but also d ds G(1, 1) − d ds G(0, 1) = 1. We obtain that G(s, 1) = αe −s + βe s where α and β are constants of integration with respect to s. With the conditions in the limits, we find that α = −1 2(1−e −1 ) and β = 1 2(1−e) . Moreover, we know that By the Kolmogorov's criterion of continuity, we deduct that B is Hoelderian.

Connes space
Let us consider the Hilbert space H above and a map γ defined from the circle into R d such that We consider a symmetric tensor product F n where σ is a permutation of the symmetric group of degree n. F i are elements of H and ⊗ denotes the standard tensor product on this Hilbert space. F n can be seen as a function from (S 1 ) n into (R d ) ⊗n . We consider the symmetric Fock space where we consider the Hilbert norm on the symmetric tensor product H⊗ n . We can define the Wiener chaos It is well defined for the symmetric function F n ∈ H ⊗n and the Gaussian process B. The map Wiener chaos realizes as classical an isomorphism between the symmetric Fock space and the L 2 of the Wiener space. The goal of this work is to replace Wiener chaos by Stratonovitch chaos. Let {e i } 1≤i≤d be a canonical basis of R d . We get by Fourier expansion an orthonormal basis of the Hilbert space for some λ > 0 if k ≥ 0 and if k < 0 Consider that (γ i ) i≥1 is an orthonormal basis of H, then an orthonormal basis of H ⊗n takes the form ⊗ denotes the tensor product on R d . σ is a permutation of the symmetric group S N of degree N . For all l = 1, . . . , N j l ∈ J = m p=1 J np and J np = {(i p , k p ), . . . , (i p , k p )} such that |J np | = n p and J n i ∩ J n j = ∅ if i = j and we note by n p for n p ≤ n.
Remark 1. We shall use the orthonormal basis of the symmetric space H ⊗n given by (5) to avoid redundancies in the calculations throughout this work.
Let us consider the space CO k,C of function F given by where F n ∈ H ⊗n is a C ∞ function of n parameters such that for all k, C > 0 where J α = {l 1 , . . . , l α } with J 1 ∪· · ·∪J l = {1, . . . , n} and S Jα = {s l 1 , . . . , s lα }. For α ∈ {1, . . . , l} such that n α ≤ k, we write Definition 1. We call Connes space the set CO ∞− given by for all k, C > 0.
Remark 3. Let us compare with the Hida Fock space. We consider the positive selfadjoint Laplacian ∆ on H and we consider the k th order Sobolev space H k associated with ∆ + 1. It is a Hilbert space. We consider the symmetric tensor product H ⊗n k endowed with its Hilbert norm · n k . Let us consider a sequence F n of H ⊗n Cauchy-Schwarz inequality and the Sobolev imbedding theorem we see that the Connes space is densely continuously imbedded in the Hida Fock space if we consider the standard Hilbert norm · ⊗n k on the symmetric tensor product of H in the definition of the Hida Fock space. Definition 2. Let F = F n and G = G m be two functions of CO ∞− . We define Wick product of F and G by F n ⊗G m is the symmetric tensor product of the functions F n and G m given by with F n and G m verifying both (1) and (2) and where σ is a permutation of the symmetric group S n+m of the space H ⊗(m+n) .
Theorem 1. The Connes space CO ∞− is a topological commutative algebra for the Wick product.
Proof . For the algebraic properties of the Wick product on CO ∞− , see Theorem 2 and Proposition 2. Let us consider By (6) and (7) we have Deformation Quantization by Moyal Star-Product and Stratonovich Chaos 5 Clearly, for all k > 0, there exist k 0 > 0 such that We deduce The theorem is proved.

Multiple Stratonovich integrals
The theory of Stratonovich chaos was initiated in [12,24] but no convenient functional space was defined. Let us consider γ N = γ ⊗n 1 The associated Stratonovitch chaos takes the form We consider multiple Stratonovitch integrals. In Stratonovitch calculus, the Itô formula reduces to the classical one. So (8) gives (9), because in such a case the Stratonovitch-Itô formula is nothing else than the ordinary Fubini theorem. We have where we consider a Stratonovitch differential •dB(s j ). With the integration by parts formula we get and finally These considerations, which take care of the difference of behaviour between Stratonovitch chaos and Wiener chaos into the passage from (8) to (9) allow us to give the next definition and to generalize by linearity I S m (γ N ) in the functions F := F n which are not products. We get Definition 3. The multiple Stratonovich integrals or Stratonovich chaos takes the form and we put Remark 4. For arbitrary small µ > 0, we can choose n µ > 0(n µ depending only on µ) such that for all n > n µ , we have sup s |D Proof . We consider F = F n 1 and G = G n 2 two functions of CO ∞− . Since : F.G : = 1 (n 1 + n 2 )! σ∈S n 1 +n 2 But, by permutating indexes, we have clearly By using Fubini's theorem, we get The theorem is proved.

Dif ferentiation operators
Differentiation operators are annihilation and creation operators. In the case of Hida test algebra (7), these operators are adjoint operators and then their study is simplified. In our case, using Banach spaces to define the Connes space makes that it is difficult to give a definition of an adjoint operator. Then, we just give a description of annihilation operator.
Definition 4. We define annihilation operator on CO ∞− , for all h ∈ H and F = F n ∈ CO ∞− by wheres i means that we make a concatenation at this term.
The annihilation operator is continuous on CO ∞− . Proof . We have just to show that the map F → I S m (F ) is injective. Indeed, considering (10) and the fact that a h is a derivation on the algebraic space , it is clear that if F → I S m (F ) is injective, the proposition is proved.

Poisson space
The theory of deformation quantization was initiated in [1,2]. See [7,18,26] for reviews and [9] for basical background. This section gives some properties of the Poisson structure of the Connes space CO ∞− . We make also the quantization deformation of that Poisson structure in Moyal star-product. We note by K = R or C.
Definition 5. A Poisson structure on CO ∞− is given by a K-bilinear map {·, ·} from CO ∞− × CO ∞− into CO ∞− such that: 1. {·, ·} is antisymmetric, satisfies the Jacobi identity and verifies the Leibniz rule for the Wick product of CO ∞− .
2. For all k, C, there exists K, k 1 , C 1 such that for all F 1 , F 2 ∈ CO ∞− we get We note by CO ∞− [[ ]] the set of formal series with coefficients in the Connes space CO ∞− .

Definition 6. A star-product on CO ∞− [[ ]] is a K[[h]]-bilinear map on CO
For all r ≥ 0, P r is a bilinear map on CO ∞− satisfying: 3. For all r > 0, for all k, C > 0 there are K, k 1 , C 1 > 0 such that for all F 1 , F 2 ∈ CO ∞− , we get We call a deformation of the Poisson bracket on CO ∞− .
We recall that γ i,k are given by (3) and (4). We have where δ k i k j is the Kronecker delta. We note by ω ij = (ω ij ) −1 . The Poisson matrix of Ω is given by Let us note by a γ i,k the annihilation operator associated to γ i,k . Then, for all F, G ∈ CO ∞− we have Proposition 3. {·, ·} defines a Poisson structure on CO ∞− in the sense of Definition 5.

Then we have
Definition 8. The Moyal star-product on CO ∞− is given by The Moyal star-product endowed with the symplectic structure of Ω is well defined on CO ∞− in the sense of Definition 6. Then, we have Proof . Since the Wick product is continuous, for all k, C > 0 there are k 1 , C 1 > 0 such that ||P r (F, G)|| k,C ≤ m,n,k j (Ck 2 j + 1)||a γ i 1 ,k 1 · · · a γ ir ,kr F n || k 1 ,C 1 ||a γ j 1 ,k 1 · · · a γ jr ,kr G m || k 1 ,C 1 .
Using integration by parts, there are k 2 , C 2 > 0 large enough such that Without loss of generality, we get Then It remains to check the algebraic properties. It is enough to prove them if we consider finite sum of γ N because P r apply the product of this space on itself and because by Stone-Weierstrass theorem the set of finite sum of γ N is dense in C0 ∞− . But in [15], these algebraic properties were proved where a completion of Hida type of the set of finite sum (by using an Hida Fock space) was chosen.
The result holds. On I S m F we choose the Banach norm of F .

Equivalence of deformation quantization
Using the model of [3],we show that there are many similarities between the Connes space we use here and the Hida test algebra of [15]. Let us consider the Hilbert space H = H(S 1 , R d ) of functions γ defined from the circle into R d such that Let (e i ) 1≤i≤d be the canonical basis of R d . Considering the Fourier basis of H defined by (3) and (4), we can define on the Hilbert space H ⊕ H * a Poisson structure by Ω(Γ 1 , Γ 2 ) = 1 0 ω(Γ 1 (s), Γ 2 (s))ds + 1 0 ω( d ds Γ 1 (s), d ds Γ 2 (s))ds, where ω = (ω ij ) i,j≥1 is a non-degenerated bilinear and antisymmetric form of R d ⊕ R d such that for all i = j ω ij = 0, and note Γ j∈{1,2} = γ j ⊕ γ * j ∈ H ⊕ H * . We get that Ω acts continuously on CO ∞− × CO ∞− and its Poisson matrix is bounded. Then, the model of [3] holds for the rest of the section.
Definition 9. For all γ ⊕ γ * ∈ H ⊕ H * , we call Wick exponentials, the maps Φ γ,γ * defined by We get a classical result for Hida calculus given by Proof . We shall note by CO W k,C the adherence of Wick exponentials in CO k,C and by CO n k,C the space of the product of n homogeneous polynomials of CO k,C . We are going to use recurrence on the holomorphic function where Γ = γ ⊕ γ * and T = B ⊕ B * . Then, F (λ) can be written under the form and obviously F (λ) ∈ CO W k,C . With Cauchy's differentiation formula It is clear that for all n ≥ 0 Now, it remains just to prove that all products of n homogeneous polynomials are in the adherence. We consider for |z| < 1 the holomorphic function F n+1 is clearly in CO W k,C and by Cauchy's differentiation formula Then F n+1 is also a function of CO W k,C . By computation, we get F n+1 (z) = (n + 1) Then F n+1 (0) = (n + 1) Thus, we proved the recurrence relation in the order (n + 1). We have By the theorem of Stone-Weierstrass, for all k, C > 0 we get that because F is a limit of elements of CO n k,C . We conclude that The proposition is proved.
According to [15], we choose the operator A : γ i,k → α k γ i,k such that |α k | ≤ K|k| µ for some µ > 0. We put Proof . For all k, C > 0, integrating by parts we can find k 2 , C 2 > 0 large enough such that This proves the theorem.
Since T r = T 1 • · · · • T 1 r-times , we get that T r is continuous and as a result T is continuous.
We note by ·, · c : H × H * → R the canonical pairing between H and H * . Then, according to [3], we have the formula Then as in [3], we get This proves the proposition since the Wick exponentials are dense in the Connes space CO ∞− .
Remark 8. In the Connes space CO ∞− endowed with Stratonovich chaos, unlike in [3], Moyal star-product and normal star-product(A = I) are obviously equivalent. We can suppose that equivalences of [15] with the Hida test functional space remain true because the Connes space CO ∞− is very small.