Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 33 No. 2 (2015): Revista Integración
Research and Innovation Articles

Groenewold-von Neumann product via Segal-Bargmann transform

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  • John B. Moreno
John B. Moreno
Universidad del Atlántico
DOI: https://doi.org/10.18273/revint.v33n2-2015004

Published 2015-12-04

Keywords

  • Geometric quantization,
  • star product,
  • Segal-Bargmann transform,
  • Fock spaces

How to Cite

Moreno, J. B. (2015). Groenewold-von Neumann product via Segal-Bargmann transform. Revista Integración, Temas De matemáticas, 33(2), 135–144. https://doi.org/10.18273/revint.v33n2-2015004

Abstract

Using standard techniques from geometric quantization, we re-derive 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 R2 × R2 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).

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.



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