Groenewold-von Neumann product
via Segal-Bargmann transform

JOHN B. MORENO^*

Universidad del Atlántico, Programa de Matemáticas, Barranquilla, Colombia.

Abstract. 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).

Keywords: Geometric quantization, star product, Segal-Bargmann transform, Fock spaces.
MSC2010: 53D50, 53D55, 30H20.

Producto de Groenewold-von Neumann mediante
una transformada de Segal-Bargmann

Resumen. Usando técnicas de cuantización geométrica, obtenemos el producto de funciones en ℝ², primeramente introducido por von Neumann y posteriormente reintroducido por Groenewold, el cual es la versión integral del producto de Moyal-Weyl. De forma más específica, por el empareamiento de polarizaciones reales en el par grupoide con sus polarizaciones holomorfas estándares, obtenemos una transformada de Segal-Bargamann deformada (por rotación y traslación). Junto con una convolución de funciones en el espacio de Segal-Bargmann, la cual es una deformación natural de la convolución de funciones en el par grupoide, se obtiene el producto de Groenewold-von Neumann en L²(ℝ²).

Palabras clave: Cuantización geométrica, producto estrella, transformada de Segal-Bargmann, espacios Fock.

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^*E-mail: johnmoreno@mail.uniatlantico.edu.co.
Received: 25 July 2015, Accepted: 02 September 2015.
To cite this article: J.B. Moreno, Groenewold-von Neumann product via Segal-Bargmann transform, Rev. Integr. Temas Mat. 33 (2015), No. 2, 135-144.