By Wigner’s theorem on symmetries, the total state space of a quantum system whose symmetries form the group G is the collection of all projective unitary representations of G; these are, in turn, realised as certain unitary representations of the set of all central extensions of G by U(1). Exploiting this relationship, we present in this paper a new approach to the quantum mechanics of an electron in a uniform magnetic field B, in the plane (the Landau electron) and on the 2-torus in the presence of a periodic potential V whose periodicity is that of an N×N lattice (the Peierls electron). For the Landau electron, G is the Euclidean group E(2) whose central extensions arise from the Heisenberg Lie group central extensions, determined by B, of the translation subgroup. The state space is a unitary representation of the direct product of two such groups corresponding to B and -B and the Hamiltonian is a unique element of the universal enveloping algebra of the centrally-extended E(2). The complete quantum theory of the Landau electron follows directly. For the Peierls electron, lattice translation-invariance is possible only if the flux per unit cell Φ takes rational values with denominator N. The state space is a unitary representation of the direct product of a finite Heisenberg group, which is a central extension of the translation group, and a Heisenberg Lie group, both characterised by Φ. The following new results are rigorous consequences. In the empty lattice limit V=0, the energy spectrum is the Landau spectrum with degeneracy equal to the total flux through the sample. As V moves away from zero, every Landau level splits into NΦ discrete sublevels, each of degeneracy N. More generally, for V≠0 of any strength and (periodic) form, and B such that Φ is nonintegral, every point in the spectrum has multiplicity N. The degeneracy is thus proportional to the linear size rather than the area of the sample. Throughout the paper, vector potentials and gauges are dispensed with and many misconceptions thereby removed.
Essential spectrum of a fermionic quantum field model
