#### Abstract

Using standard techniques from geometric quantization, we rederive the integral product of functions on (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 , 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 .

#### 1. Introduction

Let be a symplectic symmetric space, its tangent bundle, and let be the symplectic pair groupoid. Because is a symplectic symmetric space, the pushforward of the vertical fibration of under the map

determines a foliation on which, if regular, defines a real polarization on the symplectic pair groupoid (cf. [1–3] for instance). The regularity condition fails if is compact but is satisfied if is noncompact with no compact factors. This short paper considers only the simplest possible case: (actually we here fix to make matters simpler without any significant loss of generality). Similarly, can be identified with the cotangent bundle via the map

where is the standard symplectic form on and the pullback of the vertical fibration of determine a foliation on

The integral version of the Weyl-Moyal product of functions on , 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 one being the polarization 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 ( is 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 is determined by a map of to given for where is the middle point function on (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 ). On the other hand, appropriate generalizations of this technique to other noncompact hermitian symmetric spaces can in principle be helpful (for instance, if 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 (Bieliavsky plane), in this present note. As we shall see below in detail, the geometric quantization pairing of and a standard real polarization on defines a integral transform from functions on to .

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., [12–14]), 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 -d integral transforms given by where is an integral transform defined on the support compact function in ; and Planck’s constant can also be considered a free positive parameter whenever this is convenient. Moreover, for an appropriate choice of constant , for, , a support compact function.

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).

#### 2. The Integral Transform Generated by the Geometric Quantization

Let the Bieliavsky plane [8], this is , where is the canonical symplectic form on the euclidian plan and a symmetric on , such that, if , then a symmetric is given by the expression: Thus, we have that the middle point function is given by defined by the relation

In this case, the map above, cf. equation (5), is given explicitly by

Denote by the symplectic manifold, such that if has coordinates , then the symplectic form is given by

Moreover, if has coordinates as above and since is a diffeomorphism with inverse given by

the symplectic form on is given by

Consider the following respective polarization on and

Thus, from (12), the symplectic potential adapted to the polarizations is given by while from (10), the symplectic potential adapted to the polarization is given by

For , we have from (11) that

Therefore, where

which in terms of the coordinates on can be written as

Now, recall that a connection on a hermitian line bundle associated to the prequantum principal -bundle over a symplectic manifold is given locally by where is a symplectic potential. Then, consider the polarized section of over adapted to the symplectic potential and its push-forward adapted to , as well as the polarized section of over adapted to the symplectic potential and its push-forward adapted to , where , satisfying where is the hermitian product of the line bundle and , with similar expressions for and in terms of , , on , and . The polarized sections are given by , with satisfying , for ; thus, it follows that depends only on the variables seen as the zero section of . Similarly, the polarized sections are of the form , where and depends only on the variables for

Furthermore, as the prequantum line bundle is a linear bundle, we have that for a nonvanishing function . Therefore, whence, we get cf. (17), (18), and (19), thus . Hence, we have the following.

Lemma 1. *For*, , , *and*, *the hermitian products of these polarized sections are given modulo multiplicative constants by the formulas:*with and obtained from (9) and (11), with and given by (18) and (19).

Now, as the polarization is the natural polarization of cotangent bundle, given and with the canonical projection, thus and so the volume form of and of determine the natural section of given by

Analogously, for the polarization in the pair groupoid , we have thus, a natural section of is given by where is the volume form in

Since is an isomorphism, for and , the natural half density in are given, respectively, by

On the other hand, we can see that the polarizations above, satisfy the following property.

Lemma 2 (see [17]). *Let**and**polarizations above, then if**, we have that**and**are always not transverse, specifically*.

Now, if , and , for and , with the canonical projection on , we have

Then, with

Therefore, from definition of pairing and Lemma 2, we have that and so for , we obtain with given by and

Now, we have the following result for the integral transform, and . This result is a direct consequence of the approximation unit theorem and associativity guarantees that define the product to follow.

Proposition 3. *For the support compact function on**the integral transform**above (*(*35*)*) has inverse given by**above (*(*36*)*), this is,**. Inversely,*

The proof of this result is given in an the appendix.

#### 3. 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 (see appendix) as follows: which can be straightforwardly checked to satisfy the following.

Lemma 4. *The deformed convolution defined by (*(*38*)*) above is associative.*

From this, we define a new product on as follows: where which from Proposition 3 and Lemma 4 satisfies the following.

Corollary 5. *The product**defined by (*(*40*)*) above is associative.*

Finally, by a straightforward computation, one can easily check the following.

Theorem 6. *The formula for the product**defined by (*(*40*)*) via (*(*35*)*), (*(*36*)*), and (*(*38*)*) coincides in an obvious with formula (*(*4*)*) for the Bieliavsky product.*

#### Appendix

#### A. Symplectic Groupoids

A general example of the symplectic groupoid is the fundamental groupoid of a symplectic manifold Its elements are homotopy classes of smooth paths , with the usual concatenation product of paths whose endpoints math; reserving the path gives the involution. Here , are the endpoint assignment maps. The manifold embeds en as the submanifold of constant paths, which is Lagragian with respect to the symplectic structure on . When is simply connected, is determined by its endpoints, and the fundamental groupoid may be reexpressed as

We can then write , , and identify with the diagonal submanifold The multiplication and involution are given by

When is simply connected, is determined by its endpoints, and the fundamental groupoid may be reexpressed as

where , , the product and involution is given by

We call this groupoid of symplectic pair groupoid.

Now, if a symplectic manifold simply connected and a polarization of manifold . As is simply connected, we can consider the symplectic groupoid above.

On the other hand, we have the polarization of the symplectic manifold . Without loss of generality, we assume that has dimension , if has coordinates , consider the polarization

Thus, the polarization is generated by ; then, the foliation space has coordinates , as is identified with the diagonal of if , then has coordinates Suppose that are simply connected, we will show that is a groupoid on . In effect, we considered the following graph: where and are the canonical projections.

Then, we get ; analogously, we define . Thus,

given by and . Then, as is the domain of the product in , as they were defined by applications above, we can consider the following set:

Hence, the product in the groupoid is given by define our product in as

where we can see that

On the other hand, know that the involution in is given by . Thus, define the involution on by

where we have and

Therefore, using the definition above, we have that together with , , , and is a groupoid.

Lemma A.1. *Let**a symplectic manifold simply connected,**the symplectic fundamental groupoid on**with the operations**and the polarization**as above (*(*A.4*)*) and (*(*A.5*)*). Then*, *if**is the polarization of**and**the applications given above (*(*A.7*)*), (*(*A.11*)*), and (*(*A.13*)*);**with this applications is a groupoid on**denoted by*

#### B. Inverse of the Integral Transform

In this appendix, I give a proof of Proposition 3. We consider and as (35) and (36), respectively, then

But, for , we have that where we note that the integral (A.17) is not absolutely convergent.

Being as we can not to change the order integration in the usual sense, then introduce the Gaussian factor in the integral above, this is,

Notice that when , the integral (B.3) above is Moreover, it is absolutely convergent and we can take the iterated integral in any order. Thus, where

Then,

for all . If we take the limits when , the left side in (B.6) tends to

Now, making a change of variables in (B.6), we have that

Thus, of the Fourier transform for the Gaussian function in (B.7), we get

which has the following properties: (i) para to do (ii)(iii) se , uniformly in for any

Therefore, is an approximation unit and the inversion formula is a direct consequence of approximation unit theorem, because is an approximation unit when , whence

Analogously, it shows that

As a last remark, we emphasize that the whole treatment presented in this paper generalizes in an obvious way from to for every .

#### Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors would like to express their gratitude to Pedro de Magalhães Rios, Thesis advisor of the Mathematics Department of ICMC-São Paulo University, for his critical reading of the manuscript and many valuable suggestions for improvements.