Vacuum polarisation without infinities

TL;DR

This paper provides a mathematically rigorous computation of electric vacuum polarisation, demonstrating rigorous regularisation schemes and deriving the second order perturbation expression without infinities.

Contribution

It introduces mathematically rigorous versions of dimensional and Pauli-Villars regularisation schemes for vacuum polarisation calculations.

Findings

Rigorous derivation of vacuum polarisation current

First rigorous implementation of combined regularisation schemes

Expression determined up to a real degree of freedom

Abstract

In honour of Detlef D\"urr, we report on a mathematical rigorous computation of the electric vacuum polarisation current and extract the well-known expression for the second order perturbation. Intermediate steps in the presented calculation demonstrate, to the knowledge of the authors for the first time, mathematical rigorous versions of the combined dimensional and Pauli-Villars regularisation schemes. These are employed as computational tools to infer convenient integral representations during the computation. The said second order expression is determined up to a remaining degree of freedom of a real number -- without ill-defined terms from start to end.