Infrared Divergence in QED and the Fluctuation of Electromagnetic Fields

Abstract

We establish a no-go result for the infrared sector of quantum electrodynamics. Using the standard Fock-space formulation, we show that gauge invariance enforces coherent soft-photon phases that guarantee the Bloch--Nordsieck/Kinoshita--Lee--Nauenberg cancellation for all inclusive observables. The infrared divergences of perturbative amplitudes therefore do not signal any physical instability of the theory, but reflect the universal quantum dressing cloud inseparably accompanying charged particles. We further demonstrate that the stochastic interpretation suggested by the Schwinger--Keldysh effective action does not apply to four-dimensional Maxwell theory. Although infrared-sensitive imaginary terms appear in the SK effective action and can be rewritten via a Hubbard--Stratonovich transformation, we prove that conformal invariance forbids any infrared growth of these terms. As a consequence, the associated auxiliary field cannot be interpreted as a Langevin force, even in de~Sitter spacetime. These results exclude infrared-induced classical stochastic dynamics for gauge fields and clarify the physical distinction between QED and nearly massless scalar fields in de~Sitter space.