On the relation between quantum mechanics with a magnetic field on R^n and on a torus T^n

Details On the relation between quantum mechanics with a magnetic field on R^n and on a torus T^n On the relation between quantum mechanics with a magnetic field on R^n and on a torus T^n

2011-02-28

arxiv.org/abs/1103.0034v2

Physics

Mathematical Physics

Latex2e file, 20 pages. Some improvement and references added

Scientific paper

We consider a scalar charged quantum particle on R^n subject to a background U(1) gauge potential A. We clarify in the general setting how the requirement that the wavefunctions be `quasiperiodic' under translations in a lattice L leads to a periodic field strength B=dA with integral fluxes and to the analogous theory on the torus T^n=R^n/L: the wavefunctions defined on R^n play the role of sections of the associated hermitean line bundle E on T^n, since E can be described as a quotient, beside by local trivializations. The covariant derivatives corresponding to a constant B generate a Lie algebra g_Q and together with the periodic functions the `algebra of observables' O_Q. The non-abelian part of g_Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G_Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by a g in G_Q. We identify the socalled magnetic translation group as a subgroup of the observables' group Y_Q and determine the unitary irreducible representations of O_Q,Y_Q corresponding to integer charges. We also clarify how in the n=2m case a holomorphic structure and Theta functions arise on the associated complex torus. These results apply equally well to the physics of charged scalar particles both on T^n and on R^n in the presence of periodic magnetic field B and scalar potential.

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