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.
Texto Completo disponible en PDF
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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.