Geometry of spin-field coupling on the worldline

TL;DR

This paper develops a geometric framework within the worldline formalism to describe spin-field interactions, combining string-inspired and loop-space methods, and applies it to quantum electrodynamics to derive effective actions.

Contribution

It introduces a novel geometric representation of spin couplings in the worldline approach, unifying various methods and providing explicit formulas for effective actions in QED.

Findings

Rederived the Heisenberg-Euler effective action at one-loop.

Provided a geometric interpretation of the Pauli term.

Presented closed-form worldline representations for all-loop order effective actions.

Abstract

We derive a geometric representation of couplings between spin degrees of freedom and gauge fields within the worldline approach to quantum field theory. We combine the string-inspired methods of the worldline formalism with elements of the loop-space approach to gauge theory. In particular, we employ the loop (or area) derivative operator on the space of all holonomies which can immediately be applied to the worldline representation of the effective action. This results in a spin factor that associates the information about spin with "zigzag" motion of the fluctuating field. Concentrating on the case of quantum electrodynamics in external fields, we obtain a purely geometric representation of the Pauli term. To one-loop order, we confirm our formalism by rederiving the Heisenberg-Euler effective action. Furthermore, we give closed-form worldline representations for the all-loop order effective action to lowest nontrivial order in a small-N_f expansion.