Clifford algebras, Fourier transforms and quantum mechanics

TL;DR

This review explores recent generalizations of the Fourier transform linked to Lie algebras and superalgebras, highlighting their mathematical properties and connections to Clifford analysis and special functions.

Contribution

It provides a comprehensive overview of scalar and hypercomplex Fourier transforms derived from Lie algebraic structures, emphasizing their eigenfunctions, spectra, and integral kernels.

Findings

Detailed exposition of various Fourier transforms

Analysis of eigenfunctions and spectral properties

Connections to special functions and Clifford analysis

Abstract

In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel and connection with various special functions.