International Journal of Theoretical Physics

, Volume 52, Issue 3, pp 877–896 | Cite as

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

Article

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 ℝn 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 ℝn 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 gQ and together with the periodic functions the algebra of observables\(\mathcal {O}_{Q}\). The non-abelian part of gQ is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group GQ 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 gGQ. We identify the socalled magnetic translation group as a subgroup of the observables’ groupYQ. 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 ℝn 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.

Keywords

Bloch theory with magnetic field Fiber bundles Gauge symmetry Quantization on manifolds 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Dip. di Matematica e ApplicazioniUniversità “Federico II”NapoliItaly
  2. 2.I.N.F.N.Sez. di Napoli, Complesso MSANapoliItaly

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