  We investigate the strong coupling limit of lattice QCD in the Hamiltonian
formulation for systems with non-zero baryon density. In leading order the
Hamiltonian looks like an antiferromagnet that is invariant under global
U(N_f)xU(N_f) and local SU(N_c). Physically it describes meson dynamics with a
fixed background of baryon density. We study this Hamiltonian with several
baryon number distributions, and concentrate on the global symmetries of the
ground state and on the properties of low lying excitations. In particular, for
uniform non-zero baryon density we write the partition function as a path
integral that is tractable in the limit of large N_c. We find that the ground
state spontaneously breaks chiral symmetry as well as discrete lattice
rotations in a way that depends on N_f and the density. The low energy
excitations include type I and type II Goldstone bosons. The energies of the
latter are of order 1/N_c, and are quadratic in momentum. Bosons of either type
can develop anisotropic dispersion relations.
