The Feynman Variational Principle in the Worldline Representation of Field Theory

TL;DR

This paper applies Feynman's variational principle within the worldline representation to relativistic field theories, providing non-perturbative results for scalar and fermionic systems, including QED.

Contribution

It introduces a variational approach using a quadratic trial action in the worldline formalism for relativistic field theories, extending previous non-relativistic polaron techniques.

Findings

Non-perturbative covariant results for vertex functions

Application to scattering processes in scalar theories

Progress in incorporating spin in fermionic systems

Abstract

Following Feynman's treatment of the non-relativistic polaron problem, similar techniques are used to study relativistic field theories: after integrating out the bosonic degrees of freedom the resulting effective action is formulated in terms of particle trajectories (worldlines) instead of field operators. The Green functions of the theory are then approximated variationally on the pole of the external particles by using a retarded quadratic trial action. Application to a scalar theory gives non-perturbative, covariant results for vertex functions and scattering processes. Recent progress in dealing with the spin degrees of freedom in fermionic systems, in particular Quantum Electrodynamics, is discussed. We evaluate the averages needed in the Feynman variational principle for a general quadratic trial action and study the structure of the dressed fermion propagator.