To $B$ or not to $B$: Primordial magnetic fields from Weyl anomaly
  The quantum effective action for the electromagnetic field in an expanding
universe has an anomalous dependence on the scale factor of the metric arising
from virtual charged particles in the loops. It has been argued that this Weyl
anomaly of quantum electrodynamics sources cosmological magnetic fields in the
early universe. We examine this long-standing claim by using the effective
action beyond the weak gravitational field limit which has recently been
determined. We introduce a general criteria for assessing the quantumness of
field fluctuations, and show that the Weyl anomaly is not able to convert
vacuum fluctuations of the gauge field into classical fluctuations. We conclude
that there is no production of coherent magnetic fields in the universe from
the Weyl anomaly of quantum electrodynamics, irrespective of the number of
massless charged particles in the theory.
