Universality of the Weyl-Heisenberg symmetry and its covariant quantisations

TL;DR

This paper reviews the universality of Weyl-Heisenberg symmetry across different phase space manifolds and its applications in signal analysis and quantum formalism, highlighting their foundational role in various mathematical and physical contexts.

Contribution

It provides an elementary overview of how Weyl-Heisenberg symmetry arises and is utilized in both signal processing and quantum mechanics, including an application to Majorana stellar constellations.

Findings

Weyl-Heisenberg symmetry is fundamental across multiple phase space manifolds.

Displacement operators and Fourier analysis underpin this symmetry.

Application to Majorana stellar constellation demonstrates practical relevance.

Abstract

The Weyl-Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalisations. In this review we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time-frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.