On quantum mechanics with a magnetic field on R^n and on a torus T^n, and their relation

TL;DR

This paper establishes a gauge-invariant equivalence between scalar charged particles in a magnetic field on R^n and on a torus T^n, providing a global geometric framework and explicit representations relevant for quantum physics and deformation quantization.

Contribution

It introduces a global description of U(1)-gauge theories on T^n via quasiperiodic functions on R^n, characterizes the associated Lie algebra and groups, and constructs explicit irreducible representations, linking geometry with quantum observables.

Findings

Equivalence between theories on R^n and T^n in a magnetic field.

Explicit construction of irreducible representations of observable algebras.

Identification of holomorphic structures and Theta functions on complex tori.

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 T^n = R^n/L to the analogous theory on R^n constrained by quasiperiodicity under translations in the lattice L. The latter theory provides a global description of the former: the quasiperiodic wavefunctions defined on R^n play the role of sections of the associated hermitean line bundle E on 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 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 g in G_Q . We identify the socalled magnetic translation group as a subgroup of the observables' group Y_Q . We determine the unitary irreducible representations of 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 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.