Quantum Electrodynamics with Anharmonic Waves

TL;DR

This paper extends quantum electrodynamics by incorporating anharmonic plane waves as a basis, enabling non-perturbative analysis and generalizing Feynman rules to include harmonic generation, with potential applications to complex quantum phenomena.

Contribution

It develops a framework for using anharmonic plane waves in QED, generalizes Feynman rules, and proposes a non-perturbative approach via Dyson-Schwinger equations.

Findings

Generalized Feynman rules are equivalent to standard rules after summing over harmonics.

Harmonic generation diagrams are of order alpha^2 for photons and alpha for electrons.

A non-perturbative method using anharmonic operators in Dyson-Schwinger equations is proposed.

Abstract

This is the second step of a program to use anharmonic plane waves as basis set in non-perturbative quantum field theory. The general framework developed previously is applied to quantum electrodynamics. To test the compatibility with standard quantum electrodynamics, the Feynman rules are generalized to anharmonic waves by expanding the field operators into anharmonic plane waves. A sum rule for the Fourier coefficients of anharmonic waves ensures that the generalized Feynman rules are equivalent to the standard rules after summing over all harmonics. It is possible to construct diagrams for the generation of harmonics. They are of O(alpha^2) for photons and of O(alpha) for electrons. To tackle intrinsically non-perturbative phenomena it is proposed to insert anharmonic field operators into the Dyson-Schwinger equations while retaining only the lowest harmonics.