  The properties of the ground state of relativistic matter in a magnetic field
are examined within the framework of a Nambu-Jona-Lasinio model. The main
emphasis of this study is the normal ground state, which is realized at
sufficiently high temperatures and/or sufficiently large chemical potentials.
In contrast to the vacuum state, which is characterized by the magnetic
catalysis of chiral symmetry breaking, the normal state is accompanied by the
dynamical generation of the chiral shift parameter $\Delta$. In the chiral
limit, the value of $\Delta$ determines a relative shift of the longitudinal
momenta (along the direction of the magnetic field) in the dispersion relations
of opposite chirality fermions. We argue that the chirality remains a good
approximate quantum number even for massive fermions in the vicinity of the
Fermi surface and, therefore, the chiral shift is expected to play an important
role in many types of cold dense relativistic matter, relevant for applications
in compact stars. The qualitative implications of the revealed structure of the
normal ground state on the physics of protoneutron stars are discussed. A
noticeable feature of the $\Delta$ parameter is that it is insensitive to
temperature when $T \ll \mu_0$, where $\mu_0$ is the chemical potential, and
{\it increases} with temperature for $T > \mu_0$. The latter implies that the
chiral shift parameter is also generated in the regime relevant for heavy ion
collisions.
