  It is shown that the quantum mechanics of a charged particle moving in a
uniform magnetic field in the plane (Landau) or on a planar lattice (Peierls)
is described in all detail by the projective representation theory of the
"euclidean" group of the appropriate configuration space. In the Landau case, a
detailed description of the state space as well as the determination of the
correct Hamiltonian follows from the properties of the real Heisenberg group,
especially the fact that it has an essentially unique irreducible
representation. In the Peierls case, the corresponding groups are infinite
discrete translation groups centrally extended by the circle group. For
irrational flux/plaquette (in units of the flux quantum) these groups are
"almost Heisenberg" in the sense that they have a distinguished irreducible
representation which plays, in the quantum theory, the role of the unique
representation of the real Heisenberg group. The physics is fully determined
by, and is periodic in, the value of the flux/plaquette. The Hamiltonian for
nearest neighbour hopping is the Harper Hamiltonian. Vector potentials are not
introduced.
