Axiomatic approach to a local gauge quantum field theory and spontaneous symmetry breaking

The covariant (indefinite metric) formulation of quantum electrodynamic is reviewed. The picture of spontaneous symmetry breaking in (abelian and non-abelian) gauge theories is contrasted with the Goldstone theorem is standard (positive metric) quantum field theory. In particular, the photon is viewed as a Goldstone particle for the spontaneously broken local gauge invarience; it belongs to the physical Hilbert space precisely because the corresponding global U(1)-symmetry is exact. A Lorentz and locally gauge invariant nonlocal charged field is constructed first in the framework of classical electrodynamics. Its renormalized quantum version is shown to have infrared finite on shell matrix elements in perturbation theory. It is argued that such a field may give rise to a physical electron state with no need for introducing an additional soft photon cloud that breaks Lorentz invariance in the charged sector