Electroweak origin of cosmological magnetic fields

Magnetic fields may have been generated in the electroweak phase transition through spontaneous symmetry breaking or through the subsequent dynamical evolution of semiclassical field configurations. Here I demonstrate explicitly how magnetic fields emerge spontaneously in the phase transition also when no gradients of the Higgs field are present. Using a simple model, I show that no magnetic fields are generated, at least initially, from classical two-bubble collisions in a first-order phase transition. An improved gauge-invariant definition of the electromagnetic field is advocated which is more appropriate in the sense that it never allows electrically neutral fields to serve as sources for the electromagnetic field. In particular, semiclassical configurations of the $Z$ field alone do not generate magnetic fields. The possible generation of magnetic fields in the decay of unstable $Z$ strings is discussed.