Relativistic Spherical Functions on the Lorentz Group

TL;DR

This paper derives explicit formulas for Lorentz group matrix elements using complex angular momentum, Fuchsian differential equations, and hypergeometric functions, advancing the mathematical understanding of relativistic spherical functions.

Contribution

It provides new explicit expressions for Lorentz group matrix elements in terms of hypergeometric functions, connecting representation theory with special functions.

Findings

Derived matrix elements using complex angular momentum

Connected Laplace-Beltrami operators to Fuchsian differential equations

Expressed matrix elements via hypergeometric functions for various representations

Abstract

Matrix elements of irreducible representations of the Lorentz group are calculated on the basis of complex angular momentum. It is shown that Laplace-Beltrami operators, defined in this basis, give rise to Fuchsian differential equations. An explicit form of the matrix elements of the Lorentz group has been found via the addition theorem for generalized spherical functions. Different expressions of the matrix elements are given in terms of hypergeometric functions both for finite-dimensional and unitary representations of the principal and supplementary series of the Lorentz group.