Variational Derivation of Relativistic Fermion-Antifermion Wave Equations in QED

TL;DR

This paper introduces a variational approach to derive relativistic fermion-antifermion wave equations in QED, enabling direct connection between quantum field theory and wave equations, and confirms results with positronium energy levels.

Contribution

It presents a novel variational method that reformulates QED to derive relativistic two-fermion wave equations using a simple Fock-space trial state.

Findings

Energy eigenvalues match known positronium results.

Reformulation simplifies the derivation of relativistic wave equations.

Direct use of photon propagator in Hamiltonian.

Abstract

We present a variational method for deriving relativistic two-fermion wave equations in a Hamiltonian formulation of QED. A reformulation of QED is performed, in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. The reformulation permits one to use a simple Fock-space variational trial state to derive relativistic fermion-antifermion wave equations from the corresponding quantum field theory. We verify that the energy eigenvalues obtained from the wave equation agree with known results for positronium.