, Volume 52, Issue 3, pp 877-896
Date: 13 Nov 2012

On Quantum Mechanics with a Magnetic Field on ℝ n and on a Torus \(\mathbb{T}^{n}\) , and Their Relation

Abstract

We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus \(\mathbb{T}^{n}=\mathbb{R}^{n}/\varLambda\) to the analogous theory on ℝ n constrained by quasiperiodicity under translations in the lattice Λ. The latter theory provides a global description of the former: the quasiperiodic wavefunctions ψ defined on ℝ n play the role of sections of the associated hermitean line bundle E on \(\mathbb{T}^{n}\) , since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B=dA generate a Lie algebra g Q and together with the periodic functions the algebra of observables \(\mathcal {O}_{Q}\) . 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 gG Q . We identify the socalled magnetic translation group as a subgroup of the observables’ group Y Q . We determine the unitary irreducible representations of \(\mathcal{O}_{Q},Y_{Q}\) corresponding to integer charges and for each of them an associated orthonormal basis explicitly in configuration space. 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 on ℝ n and on \(\mathbb{T}^{n}\) in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel’d twists.