# Strict Deformation Quantization via Geometric Quantization in the Bieliavsky Plane

Authors: P. Hurtado; A. Leones; J. B. Moreno
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

Using standard techniques from geometric quantization, we rederive the integral product of functions onℝ2 (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×ℝ¯2, 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 L2ℝ2.


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## Body

## 1. Introduction

LetM 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
(1)Φ:TM→M×M,m,v↦expm−v,expmv,determines a foliationFV on M×M¯ which, if regular, defines a real polarization on the symplectic pair groupoid (cf. [1–3] 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=ℝ2n (actually we here fix n=1 to make matters simpler without any significant loss of generality). Similarly, ℝ2n×ℝ2n can be identified with the cotangent bundle T∗ℝ2n, via the map
(2)Y:x,y↦12x+y,ω^x−y,whereω is the standard symplectic form on ℝ2n and the pullback of the vertical fibration of T∗ℝ2n determine a foliation FV∗. on ℝ2n×ℝ¯2n.The integral version of the Weyl-Moyal product of functions onℝ2n, 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 ℝ2n×ℝ2n¯ one being the polarization FV 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 ℝb2n (ℝb2n is ℝ2n as a (nonmetric) symplectic symmetric space) given explicitly by
(3)g1∗ℏbg2p,q=1ℏ2∫ℝb2×ℝb2cosh2p−p2.cosh2p1−p.cosh2p1−p21/2,(4)g1p1,q1.g2p2,q2.ei/hsinh2p−p1q2+sinh2p2−pq1+sinh2p1−p2qdq1dq2dp1dp2.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 FV∗ is determined by a map of ℝ2n×ℝ2n to T∗ℝ2n given for
(5)Yb:x1,y1;x2,y2↦mx1,y1,x2,y2,ω^x2,y2−x1,y1,where mx1,y1,x2,y2 is the middle point function on ℝb2n (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 ℝ2n). On the other hand, appropriate generalizations of this technique to other noncompact hermitian symmetric spaces can in principle be helpful (for instance, if M=ℍ2 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 ℝb2 (Bieliavsky plane), in this present note. As we shall see below in detail, the geometric quantization pairing of FV∗ and a standard real polarization on ℝ2×ℝ2¯ defines a integral transform from functions on ℝb2 to L2ℝb2.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 the2-d integral transforms given by
(6)Tℏgx1,x2=Cℏcoshx1−x21/2∫ℝgx1+x22,qei/hqsinhx2−x1dq,Sℏfp,q=Cℏ∫ℝfq∗2+p,p−q∗2.ei/hqsinhq∗coshq∗1/2dq∗,where Tℏ is an integral transform defined on the support compact function in ℝb2; x1,x2,p,q∈ℝb2, and Planck’s constant ℏ∈ℝ+ can also be considered a free positive parameter whenever this is convenient. Moreover, for an appropriate choice of constant Cℏ, TℏSℏf=SℏTℏf=f for, f, a support compact function.Thus, in Section2, 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

Letℝb2 the Bieliavsky plane [8], this is ℝ2,ω,s, where ω is the canonical symplectic form on the euclidian plan and s a symmetric on ℝ2, such that, if x1,y1,x2,y2∈ℝ2, then a symmetric is given by the expression:
(7)sx1,y1x2,y2=2x1−x2,2y1cosh2x1−x2−y2.Thus, we have that the middle point function is given by
(8)mx1,y1,x2,y2=x1+x22,y1+y22sechx1+x2,defined by the relation smx1,y1,x2,y2x1,y1=x2,y2.In this case, the mapYb:ℝb2×ℝb2→T∗ℝb2 above, cf. equation (5), is given explicitly by
(9)Ybx1,y1,x2,y2=x1+x22,y1+y22sechx1+x2,y1−y2,x2−x1=p,q;p∗,q∗.Denote byℝb2×ℝ¯b2 the symplectic manifold, such that if ℝb2×ℝ¯b2 has coordinates x1,y1;x2,y2, then the symplectic form is given by
(10)Ω=dx1∧dy1−dx2∧dy2.Moreover, ifT∗ℝb2 has coordinates p,q,p∗,q∗ as above and since Yb is a diffeomorphism with inverse Yb−1 given by
(11)x1=p−q∗2,x2=p+q∗2,y2=qcoshq∗−p∗2,y1=qcoshq∗+p∗2,the symplectic formYb−1∗Ω=Π on T∗ℝb2 is given by
(12)Π=−coshq∗dq∗∧dq+dp∧dp∗.Consider the following respective polarization onT∗ℝb2 and ℝb2×ℝb2,(13)P~=<∂p∗,∂q∗>,F=<∂y1,∂y2>.Thus, from (12), the symplectic potential adapted to the polarizations P~ is given by
(14)ΘP~=−sinhq∗dq−p∗dp,while from (10), the symplectic potential adapted to the polarization F is given by
(15)ΘF=y2dx2−y1dx1.ForF~=Yb∗F, we have from (11) that
(16)ΘF~=qcoshq∗−p∗2.dp+12dq∗−p∗2+qcoshq∗dp−12dq∗−qcoshq∗dp+p∗4dq∗−p∗2dp=−p∗dp+qcoshq∗dq∗,Therefore,(17)ΘP~−ΘF~=dΨ~,where
(18)Ψ~p,q;p∗,q∗=−iℏqsinhq∗,which in terms of the coordinates onℝb2×ℝb2 can be written as
(19)Ψ~∘Yb=Ψx1,y1;x2,y2=−iℏy1+y22sechx1+x2sinhx2−x1.Now, recall that a connection on a hermitian line bundleL associated to the prequantum principal S1-bundle over a symplectic manifold M is given locally by
(20)∇X=X−i/ℏΘX,X∈XM,where Θ is a symplectic potential. Then, consider the polarized section s0 of L over ℝb2×¯ℝb2 adapted to the symplectic potential ΘF and its push-forward s~0 adapted to ΘF~, as well as the polarized section t~0 of L~ over T∗ℝb2 adapted to the symplectic potential ΘP~ and its push-forward t0 adapted to ΘP, where P=Yb−1∗P~, satisfying
(21)∇X~t~0=−i/ℏΘP~t~0;t~0,t~0=1,∇X~s~0=−i/ℏΘF~s~0;s~0,s0~=1,where ·,· is the hermitian product of the line bundle L~ and X~∈XT∗ℝb2, with similar expressions for t0 and s0 in terms of P, F, ·,· on L, and X∈Xℝb2×ℝ¯b2. The polarized sections t~∈ΓP~L~ are given by t~=g¯t~0, with g¯∈C∞T∗ℝb2 satisfying Xg¯=0, for X∈XT∗ℝb2,P~; thus, it follows that g¯ depends only on the variables x,y∈ℝb2 seen as the zero section of T∗ℝ2. Similarly, the polarized sections s∈ΓFL are of the form s=fs0, where f∈C∞ℝb2×ℝb2 and depends only on the variables x1,x2 for x1,y1;x2,y2∈ℝb2×ℝb2.Furthermore, as the prequantum line bundle is a linear bundle, we have thats~0=ϕ~t~0 for a nonvanishing function ϕ~∈C∞T∗ℝb2. Therefore,
(22)∇X~s~0=∇X~ϕ~t~0=X~ϕ~t~0+ϕ~∇X~t~0,whence, we get
(23)dϕ~ϕ~=iℏΘP~−ΘF~=diℏΨ~,cf. (17), (18), and (19), thus ϕ~=CeiΨ~/ℏ. Hence, we have the following.Lemma 1.
Fort∈ΓPL, t~∈ΓP~L~, s∈ΓFL, ands~∈ΓF~L~, the hermitian products of these polarized sections are given modulo multiplicative constants by the formulas:(24)t~,s~=gp,qfx1p,q;p∗,q∗,x2p,q;p∗,q∗eiΨ~p,q;p∗,q∗/ℏ,t,s=gpx1,y1;x2,y2,qx1,y1;x2,y2fx1,x2eiΨx1,y1;x2,y2/ℏ,with px1,y1;x2,y2,qx1,y1;x2,y2 and x1p,q;p∗,q∗,x2p,q;p∗,q∗ obtained from (9) and (11), with Ψ~p,q;p∗,q∗ and Ψx1,y1;x2,y2 given by (18) and (19).Now, as the polarizationP~ is the natural polarization of cotangent bundle, given q∈ℝb2 and m∈π−1q, with π the canonical projection, TqT∗ℝb2/P~m=Tℝb2, thus
(25)Δ−1/2P~m=Δ−1/2TmT∗ℝb2⊗Δ1/2Tp,qℝb2.and so the volume form εΠ of T∗ℝb2 and εω of ℝb2 determine the natural section of Δ−1/2P~m given by
(26)v~=εΠ−1/2.εω1/2.Analogously, for the polarizationF in the pair groupoid ℝb2×ℝ¯b2, we have
(27)Tℝb2×ℝb2/F=Tℝb2,thus, a natural section of Δ−1/2Fm is given by
(28)v=εΩ−1/2.εω1/2,where εΩ is the volume form in ℝb2×ℝb2.SinceYb∗ is an isomorphism, for P=Yb−1∗P~ and F~=Yb∗F, the natural half density in Δ−1/2F~ are given, respectively, by
(29)v~′=Yb∗εΠ−1/2.Yb∗εω1/2v′=εYb−1∗Ω−1/2.εYb−1∗ω1/2.On the other hand, we can see that the polarizationsP~,F, above, satisfy the following property.Lemma 2 (see [17]).
LetP~andFpolarizations above, then ifF~=Yb∗F, we have thatF~andP~are always not transverse, specificallyF~∩P~=<∂p∗>.Now, ifF~∩P~≔D=<∂p∗>, D0≔<∂q,∂p∗,∂q∗> and R≔T∗ℝb2/D, for x∈R and m∈π−1x, with π the canonical projection on T∗ℝb2, we have
(30)TxR=TmMDmeVm=Dm0Dm.Then,(31)Δ1TxR=Δ−1/2F~m⊗Δ−1/2P~m⊗Δ1/2Vm⊗Δ1TmM,with M=T∗ℝb2.Therefore, from definition of pairing and Lemma2, we have that
(32)<t~⊗v~,s~⊗v~′>pr=C∫T∗ℝb2/Dfq∗2+p,p−q∗2.gp,qe−i/hqsinhq∗××coshq∗1/2dpdqdq∗.=<Sℏf,g>L2ℝb2and so T∗ℝb2/D≅ℝ3, for x1=q∗/2+p;x2=p−q∗/2, we obtain
(33)<t~⊗v~,s~⊗v~′>pr=C∫ℝ3fx1,x2.gx1+x22,qei/hqsinhx2−x1××C~x1−x21/2dx1dx2dq=<f,Tℏg>L2ℝb2,with
(34)Tℏ,Sℏ:L2ℝb2→L2ℝb2,given by
(35)Tℏgx1,x2=Ccoshx1−x21/2∫ℝgx1+x22,qei/hqsinhx2−x1dq.and
(36)Sℏfp,q=C∫ℝfq∗2+p,p−q∗2.ei/hqsinhq∗coshq∗1/2dq∗.Now, we have the following result for the integral transform,Tℏ and Sℏ. 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 onℝb2the integral transformTℏabove ((35)) has inverse given bySℏabove ((36)), this is,SℏTℏfp,q=fp,q. Inversely,TℏSℏfp,q=fp,q.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 (ℝb2×ℝ¯b2Ãℝb2)
(37)f⊗gx1,y1;x2,y2=∫ℝb2fx1,y1;x2,y2gx2,y2;x3,y3dx2dy2,is possible to construct a deformed convolution of the functions on ℝb2×ℝ¯b2/F (see appendix) as follows:
(38)f⊗¯gx1,x3=∫ℝfx1,x2gx2,x3dx2,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 onL2ℝb2 as follows:
(39)∗ℏb:L2ℝb2×L2ℝb2→L2ℝb2,g1,g2↦g1∗ℏbg2,where
(40)g1∗ℏbg2=SℏTℏg1⊗¯Tℏg2,which from Proposition 3 and Lemma 4 satisfies the following.Corollary 5.
The product∗ℏbdefined by ((40)) above is associative.Finally, by a straightforward computation, one can easily check the following.Theorem 6.
The formula for the product∗ℏbdefined by ((40)) via ((35)), ((36)), and ((38)) coincides in an obvious with formula ((4)) for the Bieliavsky product.



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