# Exact relativistic Green's functions for the time-independent potentials

## TL;DR

This paper introduces an exact method for constructing relativistic Green's functions in systems with time-independent interactions, simplifying calculations by relating them to nonrelativistic Green's functions, with applications in QED and QCD.

## Contribution

The paper presents a novel, exact approach to relativistic Green's functions for time-independent potentials, connecting them to nonrelativistic counterparts and validating with specific physical examples.

## Key findings

- Exact agreement with Schwinger relativistic form in magnetic field problem
- Validation of the formalism with Coulomb potential
- Simplification of relativistic Green's function calculations

## Abstract

Relativistic formalism of Green's functions is dicussed in QCD and QED,where the relativistic Green's functions are constructed using the Schwinger proper time formalism and the Fock-Feynman-Schwinger method.As a result a simple and exact method is found for the relativistic systems,where the interaction can be written in a time-independent form.In this case one can write the relativistic Green's function as a one-dimensional integral of the corresponding nonrelativistic Green's function.The explicit example for the problem of a charge in the constant magnetic field is discussed in detail,and the exact agreement with the Schwinger relativistic form is demonstrated. A similar analysis is performed in the relativistiv Coulomb probem,supporting the accuracy of the proposed relativistic formalism.