Fourier Duality as a Quantization Principle

TL;DR

This paper explores how Fourier duality, via Kac algebras, can serve as a foundational principle for quantization across various phase spaces, extending traditional methods beyond Euclidean settings.

Contribution

It demonstrates the potential of Kac algebras and Fourier duality as a unifying framework for quantization on general locally compact groups and phase spaces.

Findings

Fourier duality guides quantization in non-Euclidean phase spaces

Kac algebras generalize Weyl-Wigner formalism

Modifications are needed for non-trivial phase spaces like the half-plane

Abstract

The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras -- and the duality they incorporate -- are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest non-trivial phase space, the half-plane, it is shown how the structures present in the complete-plane case must be modified. Traces, for example, must be replaced by their noncommutative generalizations - weights - and the correspondence embodied in the Weyl-Wigner formalism is no more complete. Provided the underlying algebraic structure is suitably adapted to each case, Fourier duality is shown to be indeed a very powerful guide to the quantization of general physical systems.