MORENO, JOHN B. Groenewold-von Neumann product via Segal-Bargmann transform. []. , 33, 2, pp.135-144. ISSN 0120-419X.

Abstract. Using standard techniques from geometric quantization, we rederive the product of functions on ℝ² which was first introduced by von Neumann and later reintroduced by Groenewold and which is the integral version of the Moyal product. More specifically, by pairing the diagonal real polarization on the pair groupoid with its standard holomorphic polarization, we obtain the well-known Segal-Bargmann transform in a rotated and scaled (and half-conjugated) form. Together with a convolution of functions in the Segal-Bargmann space, which is a natural deformation of the usual convolution of functions on the pair groupoid, this defines the Groenewold-von Neumann product on L²(ℝ²)

: Geometric quantization; star product; Segal-Bargmann transform; Fock spaces.

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