  A short review of recent advances in understanding the dynamics of
relativistic matter in a magnetic field is presented. The emphasis is on the
dynamics related to the generation of the chiral shift parameter in the normal
ground state. We argue that the chiral shift parameter contributes to the axial
current density, but does not modify the conventional axial anomaly relation.
The analysis based on gauge invariant regularization schemes in the
Nambu-Jona-Lasinio model suggests that these findings should be valid also in
gauge theories. It is pointed out that the chiral shift parameter can affect
observable properties of compact stars and modify the key features of the
chiral magnetic effect in heavy ion collisions.
