QUANTUM-THEORY OF LANDAU AND PEIERLS ELECTRONS FROM THE CENTRAL EXTENSIONS OF THEIR SYMMETRY GROUPS

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 proj ective unitary representations of G; these are, in turn, realised as c ertain unitary representations of the set of all central extensions of G by U(1). Exploiting this relationship, we present in this paper a n ew approach to the quantum mechanics of an electron in a uniform magne tic field B, in the plane (the Landau electron) and on the a-torus in the presence of a periodic potential V whose periodicity is that of an N x N lattice (the Peierls electron). For the Landau electron, G is t he Euclidean group E(2) whose central extensions arise from the Heisen berg Lie group central extensions, determined by B, of the translation subgroup. The state space is a unitary representation of the direct p roduct of two such groups corresponding to B and -B and the Hamiltonia n is a unique element of the universal enveloping algebra of the centr ally-extended E(2). The complete quantum theory of the Landau electron follows directly, For the Peierls electron, lattice translation-invar iance is possible only if the flux per unit cell Phi takes rational va lues with denominator N. The state space is a unitary representation o f the direct product of a finite Heisenberg group, which is a central extension of the translation group, and a Heisenberg Lie group, both c haracterised by Phi. The following new results are rigorous consequenc es. In the empty lattice limit V = 0, the energy spectrum is the Landa u spectrum with degeneracy equal to the total flux through the sample. As V moves away from zero, every Landau level splits into N Phi discr ete sublevels, each of degeneracy N. More generally, for V not equal 0 of any strength and (periodic) form, and B such that Phi, is noninteg ral, every point in the spectrum has multiplicity N. The degeneracy is thus proportional to the linear size rather than the area of the samp le. Throughout the paper, vector potentials and gauges are dispensed w ith and many misconceptions thereby removed.