Dirac equation in some homogeneous$sqace-times, separation of variables and exact solutions

TL;DR

This paper generalizes the algebraic separation of variables for the Dirac equation in certain homogeneous space-times, deriving exact solutions for cosmological and gravitational wave backgrounds using hypergeometric functions.

Contribution

It introduces a novel algebraic method to separate variables in the Dirac equation in space-times lacking a complete set of commuting operators, leading to exact solutions.

Findings

Exact solutions for Dirac equation in specific space-times

Reduction to coupled ordinary differential equations

Discussion of Klein-Gordon equation in the same backgrounds

Abstract

In the present article, using a further generalization of the algebraic method of separation of variables, the Dirac equation is separated in a family of space-times where it is not possible to find a complete set of first order commuting differential operators. After separating variables, the Dirac equation is reduced to a set of coupled ordinary differential equations and some exact solutions corresponding to cosmological backgrounds and gravitational waves are computed in terms of hypergeometric functions. By passing, the Klein Gordon equation in this background field is discussed.