Causal perturbative QED and infra-red states

Abstract

At the classical level the electromagnetic field can be well identified at the spatial infinity. Staruszkiewicz pointed out that the quantization of the electromagnetic field at spatial infinity is essentially unique and follows from the two fundamental principles: 1) gauge invariance and 2) canonical commutation relations for canonically conjugated generalized coordinates, and constructed a simple and mathematically transparent quantum theory of the Coulomb field, predicting (among other things) a relation between the theory of unitary representations of the $SL(2, \mathbb{C})$ group and the fine structure constant. Until now this theory has stayed outside the main stream of the perturbative development in QED, mainly due to the unsolved infra-red-type (IR) problems in the perturbative approach. Recently however there has been performed a more careful analysis of free fields including the mass less free gauge fields, such as the electromagnetic potential field, their Wick and chronological products, which revealed the need for a more careful and white noise construction of these fields, and which opened a way to resolve IR problems (at least those which shows up at each order separately). Comparison of the perturbatively constructed field at spatial infinity with the quantum phase field of the Staruszkiewicz theory leads to the proof of universality of the unit of charge.