Strict deformation quantization via geometric quantization in the Bieliavsky plane
Summary: Using standard techniques from geometric quantization, we rederive the integral product of functions on \(\mathbb{R}^2\) (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 \(\mathbb{R}^2\times \overline{\mathbb{R}}^2\), 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 \(L^2\left( \mathbb{R}^2\right)\).
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