On the relativistic two-point Green's function

TL;DR

This paper introduces a simplified, computationally efficient method for deriving the Green's function of the relativistic Dirac equation with spherical symmetry, linking it clearly to the nonrelativistic case.

Contribution

It presents a new approach for solving the Dirac equation that simplifies the Green's function representation and enhances computational efficiency.

Findings

Simplified expressions for Dirac-Oscillator and Dirac-Coulomb Green's functions.

Quadratic number of terms in Whittaker functions, reducing complexity.

Off-diagonal elements expressed as sums of relativistic terms.

Abstract

Using a recently developed approach for solving the three dimensional Dirac equation with spherical symmetry, we obtain simple representations for the Green's function of the Dirac-Oscillator and Dirac-Coulomb problems. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a map of the latter into the former. An advantage of this representation is in the computational economy as depicted by the number of terms, which are quadratic in the Whittaker functions. These are single terms except for the off-diagonal elements of the Dirac-Coulomb Green's function which, nonetheless, simplifies into the sum of a first order and third order relativistic terms.