QED as a many-body theory of worldlines: II. All-order S-matrix formalism

TL;DR

This paper extends a worldline formalism for QED to include real photon emissions and absorptions, providing a unified proof of infrared safety for the S-matrix and offering new insights into IR divergence treatment.

Contribution

It introduces an all-order worldline approach to both virtual and real photon processes in QED, extending previous work and deriving key theorems like Low's and Weinberg's.

Findings

Proved IR safety of the FK S-matrix for real and virtual photons.

Derived a simple formula for N-th rank vacuum polarization tensors.

Connected the worldline approach to Wilsonian IR divergence interpretation.

Abstract

In arXiv:2206.04188, we developed a first-quantized worldline formalism for all-order computations of amplitudes in QED. In particular, we demonstrated in this framework an all-order proof of the infrared safety of the Faddeev-Kulish (FK) S-matrix for virtual exchanges in the scattering of charged fermions. In this work, we extend the worldline formalism for both the Dyson and FK S-matrix to consider further the emission and absorption of arbitrary numbers of photons. We show how Low's theorem follows in this framework and derive Weinberg's theorem for the exponentiation of IR divergences. In particular, we extend our all-order proof of the IR safety of the FK S-matrix to both virtual exchanges and real photon emissions. We argue that the worldline approach leads to a modern Wilsonian interpretation of the IR safety of the FK S-matrix and provides a novel template for the treatment of IR divergences in real-time problems. Using Grassmannian integration methods, we derive a simple and powerful result for N-th rank vacuum polarization tensors. Applications of these methods will be discussed in follow-up work.