Clifford algebras, Fourier transforms and quantum mechanics

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.

Review paper, 39 pages, to appear in Math. Methods. Appl. Sci

References in corpus (7)

• Directional Uncertainty Principle for Quaternion Fourier Transform
• Spherical harmonics and integration in superspace
• Fourier transform and related integral transforms in superspace
• Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
• Orthogonality of Hermite polynomials in superspace and Mehler type formulae
• Hermite and Gegenbauer polynomials in superspace using Clifford analysis
• Schrödinger equation with delta potential in superspace

Cited by in corpus (4)

• Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)
• $L^p$-$L^q$ boundedness of $(k,a)$-Fourier multipliers with applications to Nonlinear equations
• Algebraic approach to slice monogenic functions
• Solutions for the Lévy-Leblond or parabolic Dirac equation and its generalizations