On twisted symmetries and quantum mechanics with a magnetic field on noncommutative tori

Abstract: We study the twist-induced deformation procedure of a torus T^n and of quantum mechanics of a scalar charged quantum particle on T^n in the presence of a magnetic field B. We first summarize our recent results regarding the equivalence of the undeformed theory on T^n to the analogous one on R^n subject to a quasiperiodicity constraint: we describe the sections of the associated hermitean line bundle on T^n as wavefunctions ψ∈C^∞(R^n) periodic up to a suitable phase factor V depending on B and require the covariant derivative components ∇_a to map the space X^V of such ψ’s into itself. The ∇_a corresponding to a constant B generate a Lie algebra g_Q and together with the periodic functions the algebra O_Q of observables. 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. The unitary irreducible representations of O_Q ,YQ corresponding to integer charges are parametrized by a point in the reciprocal torus. We then apply the ⋆-deformation procedure induced by a Drinfel’d twist F ∈ Ug_Q ⊗Ug_Q, sticking for simplicity to abelian twists, to the symmetry Hopf algebra Ug_Q , to the algebra X of functions on T^n and to O_Q in a gauge-independent way, to X^V and to the action of O_Q on the latter in a specific gauge. X^V , O_Q are ‘rigid’, i.e. isomorphic to X_⋆^V , O_Q⋆ , although X and X_⋆ are not isomorphic and therefore X_⋆^V as a X_⋆-bimodule is not isomorphic to the X-bimodule X^V.