Strict Deformation Quantization via Geometric Quantization in the Bieliavsky Plane

Using standard techniques from geometric quantization, we rederive the integral product of functions on R2 (non-Euclidian) which was introduced by Pierre Bieliavsky as a contribution to the area of strict quantization. More specifically, by pairing the nontransverse real polarization on the pair groupoid R2 × R2, we obtain the well-defined integral transform. Together with a convolution of functions, which is a natural deformation of the usual convolution of functions on the pair groupoid, this readily defines the Bieliavsky product on a subset of L2ðR2Þ.


Introduction
Let M be a symplectic symmetric space, TM its tangent bundle, and let M × M be the symplectic pair groupoid. Because M is a symplectic symmetric space, the pushforward of the vertical fibration of TM under the map determines a foliation F V on M × M which, if regular, defines a real polarization on the symplectic pair groupoid (cf. [1][2][3] for instance). The regularity condition fails if M is compact but is satisfied if M is noncompact with no compact factors. This short paper considers only the simplest possible case: M = ℝ 2n (actually we here fix n = 1 to make matters simpler without any significant loss of generality). Similarly, ℝ 2n × ℝ 2n can be identified with the cotangent bundle T * ðℝ 2n Þ, via the map where ω is the standard symplectic form on ℝ 2n and the pullback of the vertical fibration of T * ðℝ 2n Þ determine a foliation F V * : on ℝ 2n × ℝ 2n : The integral version of the Weyl-Moyal product of functions on ℝ 2n , also known as the Groenewold-von Neumann product, has been obtained and reobtained in various ways since the original work of von Neumann [4]. But in [5], Gracia-Bondia and Varilly rederived this product via geometric quantization (see [6] for example), using the pairing of two nontransversal real polarizations on the pair groupoid ℝ 2n × ℝ 2n one being the polarization F V described above. Now, Pierre Bieliavsky gets more recently (see [7,8]), as a contribution to the area of Strict Quantization, the integral product of functions on ℝ 2n b (ℝ 2n b is ℝ 2n as a (nonmetric) symplectic symmetric space) given explicitly by This type of product was initially considered for Weinstein and Zakrzewski in the so-called WKB-quantization program (see [9]). Here, we will rederive this product, again via geometric quantization and again using pairing of polarizations, but now pairing the real polarization F V * is determined by a map of ℝ 2n × ℝ 2n to T * ðℝ 2n Þ given for where m½ðx 1 , y 1 Þ, ðx 2 , y 2 Þ is the middle point function on ℝ 2n b (see [10]). Although our derivation presented below could be considered a simple exercise in geometric quantization, we have not yet found it explicitly done in detail, in the literature (the method originally developed in [11] is totally different, using a map to the Weyl product) and it also allows us to obtain the associativity of this product as a direct corollary of our construction. In fact, the main idea for this derivation is already found in the aforementioned paper by Gracia-Bondia and Varilly ( [5], for Euclidian case ℝ 2n ). On the other hand, appropriate generalizations of this technique to other noncompact hermitian symmetric spaces can in principle be helpful (for instance, if M = ℍ 2 is the hyperbolic plane). This fact shall be thoroughly explored in subsequent papers and constitutes the main motivation for our working out this technique in detail for the case of ℝ 2 b (Bieliavsky plane), in this present note. As we shall see below in detail, the geometric quantization pairing of F V * and a standard real polarization on ℝ 2 × ℝ 2 defines a integral transform from functions on ℝ 2 b to L 2 ðℝ 2 b Þ. It is well known that geometric quantization can be used to construct the integral transforms, such as Laplace transform, Fourier transform, Segal-Bargmann transform (cf. e.g., [12][13][14]), and the generalized Segal-Bargmann transform for Lie groups of compact type can also be developed using geometric quantization (cf. [15,16]).
In this short note, again via geometric quantization, we shall obtain the 2-d integral transforms given by where T ℏ is an integral transform defined on the support compact function in ℝ 2 b ; ðx 1 , x 2 Þ, ðp, qÞ ∈ ℝ 2 b , and Planck's constant ℏ ∈ ℝ + can also be considered a free positive parameter whenever this is convenient. Moreover, for an appropri- Thus, in Section 2, we present our detailed derivation of this transform, (cf. (35), (36)), which immediately generalizes to all even dimensional cases. Finally, in Section 3, combining this transform with a natural deformation of the usual convolution of functions on the pair groupoid, we obtain the integral formulation of the Bieliavsky product, which is given by (4).

The Integral Transform Generated by the Geometric Quantization
Let ℝ 2 b the Bieliavsky plane [8], this is ðℝ 2 , ω, sÞ, where ω is the canonical symplectic form on the euclidian plan and s a symmetric on ℝ 2 , such that, if ðx 1 , y 1 Þ, ðx 2 , y 2 Þ ∈ ℝ 2 , then a symmetric is given by the expression: Thus, we have that the middle point function is given by defined by the relation s m½ðx 1 ,y 1 Þ,ðx 2 ,y 2 Þ ðx 1 , y 1 Þ = ðx 2 , y 2 Þ: (5), is given explicitly by Denote by has coordinates ðx 1 , y 1 ; x 2 , y 2 Þ, then the symplectic form is given by Moreover, if T * ℝ 2 b has coordinates ðp, q, p * , q * Þ as above and since Y b is a diffeomorphism with inverse Y −1 b given by the symplectic form ðY −1 b Þ * ðΩÞ = Π on T * ℝ 2 b is given by Consider the following respective polarization on Thus, from (12), the symplectic potential adapted to the polarizationsP is given by 2 Abstract and Applied Analysis while from (10), the symplectic potential adapted to the polarization F is given by ForF = ðY b Þ * ðFÞ, we have from (11) that Therefore, which in terms of the coordinates on ℝ 2 b × ℝ 2 b can be written as Now, recall that a connection on a hermitian line bundle L associated to the prequantum principal S 1 -bundle over a symplectic manifold M is given locally by where Θ is a symplectic potential. Then, consider the polarized section s 0 of L over ℝ 2 b ×ℝ 2 b adapted to the symplectic potential Θ F and its push-forwards 0 adapted to ΘF, as well as the polarized sectiont 0 ofL over T * ℝ 2 b adapted to the symplectic potential ΘP and its push-forward t 0 adapted to Θ P , where P = ðY −1 b Þ * P , satisfying where ð·, · Þ is the hermitian product of the line bundleL and X ∈ XðT * ℝ 2 b Þ, with similar expressions for t 0 and s 0 in terms of P, F, ð·, · Þ on L, and X ∈ Xðℝ 2 b × ℝ 2 b Þ. The polarized sectionst ∈ ΓPL are given byt = gt 0 , with g ∈ C ∞ ðT * ℝ 2 b Þ satisfying X g = 0, for X ∈ XðT * ℝ 2 b ,PÞ; thus, it follows that g depends only on the variables ðx, yÞ ∈ ℝ 2 b seen as the zero section of T * ℝ 2 . Similarly, the polarized sections s ∈ Γ F L are of the form s = f s 0 , where f ∈ C ∞ ðℝ 2 b × ℝ 2 b Þ and depends only on the variables ðx 1 , x 2 Þ for ðx 1 , y 1 ; x 2 , y 2 Þ ∈ ℝ 2 b × ℝ 2 b : Furthermore, as the prequantum line bundle is a linear bundle, we have thats 0 = e ϕt 0 for a nonvanishing function e ϕ ∈ C ∞ ðT * ℝ 2 b Þ. Therefore, whence, we get cf. (17), (18), and (19), thus e ϕ = Ce iΨ/ℏ . Hence, we have the following.
Now, as the polarizationP is the natural polarization of cotangent bundle, given q ∈ ℝ 2 b and m ∈ π −1 ðqÞ, with π the canonical projection, T q ðT * ℝ 2 b Þ/P m = Tℝ 2 b , thus and so the volume form ε Π of T * ℝ 2 b and ε ω of ℝ 2 b determine the natural section of Analogously, for the polarization F in the pair groupoid thus, a natural section of Δ −1/2 ðF m Þ is given by where ε Ω is the volume form in ℝ 2 b × ℝ 2 b : Since ðY b Þ * is an isomorphism, for P = ðY −1 b Þ * ðPÞ and F = ðY b Þ * ðFÞ, the natural half density in Δ −1/2 ðFÞ are given, respectively, by 3 Abstract and Applied Analysis On the other hand, we can see that the polarizationsP, F, above, satisfy the following property.
x ∈ R and m ∈ π −1 ðxÞ, with π the canonical projection on T * ℝ 2 b , we have Then, with M = T * ℝ 2 b : Therefore, from definition of pairing and Lemma 2, we have that and so T * ðℝ 2 b Þ/D ≅ ℝ 3 , for x 1 = ðq * /2Þ + p ; x 2 = p − ðq * /2Þ, we obtain <t ⊗ṽ,s ⊗ṽ′> pr = C with T ℏ , S ℏ : given by and Now, we have the following result for the integral transform, T ℏ and S ℏ . This result is a direct consequence of the approximation unit theorem and associativity guarantees that define the product to follow.

Rederiving the Bieliavsky Product
Starting from the usual convolution of functions on the symplectic pair groupoid ( is possible to construct a deformed convolution of the functions on ðℝ 2 b × ℝ 2 b Þ/F (see appendix) as follows: which can be straightforwardly checked to satisfy the following.
From this, we define a new product on L 2 ðℝ 2 b Þ as follows: * b ℏ : where which from Proposition 3 and Lemma 4 satisfies the following.
Finally, by a straightforward computation, one can easily check the following.