Fundamental solutions for the Dirac equation in curved spacetime and generalized Euler-Poisson-Darboux equation

TL;DR

This paper derives fundamental solutions for the Dirac equation in curved spacetimes with power-type expansion or contraction, utilizing solutions to the generalized Euler-Poisson-Darboux equation via integral transforms.

Contribution

It introduces a method to obtain fundamental solutions for the Dirac equation in specific curved spacetimes using generalized Euler-Poisson-Darboux equations.

Findings

Fundamental solutions for spin-1/2 fields in certain curved spacetimes are explicitly constructed.

The approach links solutions of the Dirac equation to generalized Euler-Poisson-Darboux equations.

The method provides a new tool for analyzing quantum fields in dynamic curved backgrounds.

Abstract

We present the fundamental solutions for the spin-1/2 fields propagating in the spacetimes with power type expansion/contraction and the fundamental solution of the Cauchy problem for the Dirac equation. The derivation of these fundamental solutions is based on formulas for the solutions to the generalized Euler-Poisson-Darboux equation, which are obtained by the integral transform approach.