Rotations and boosts of Hermite functions

TL;DR

This paper develops transformation matrices for multidimensional Hermite functions under Lorentz transformations, facilitating analysis of spacetime properties, relativistic oscillators, and image processing, while identifying Lie algebra generators.

Contribution

It introduces explicit transformation matrices for Lorentz transformations of Hermite functions in any dimension, linking mathematical tools with physical and computational applications.

Findings

Transformation matrices for Lorentz transformations of Hermite functions.

Identification of Lorentz Lie algebra generators.

Application to basis construction for rotation operators.

Abstract

We provide transformation matrices for arbitrary Lorentz transformations of multidimensional Hermite functions in any dimension. These serve as a valuable tool for analyzing spacetime properties of MHS fields, and aid in the description of the relativistic harmonic oscillator and digital image manipulation. We also focus on finite boosts and rotations around specific axes, enabling us to identify the Lorentz Lie algebra generators. As an application and to establish a contact with the literature we construct a basis in which the two dimensional rotation operator is diagonal. We comment on the use of hypergeometric functions, the Wigner d-functions, Kravchuk polynomials, Jacobi polynomials and generalized associated Legendre functions.