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We show in elementary terms the equivalence in a general gauge of aU(1)-gauge theory of a scalar charged particle on a torus T^n = R^n/L to theanalogous theory on R^n constrained by quasiperiodicity under translations inthe 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 theassociated hermitean line bundle E on T^n, since also E admits a globaldescription as a quotient. The components of the covariant derivativescorresponding to a constant (necessarily integral) magnetic field B = dAgenerate a Lie algebra g_Q and together with the periodic functions the algebraof observables O_Q . The non-abelian part of g_Q is a Heisenberg Lie algebrawith the electric charge operator Q as the central generator; the correspondingLie group G_Q acts on the Hilbert space as the translation group up to phasefactors. Also the space of sections of E is mapped into itself by g in G_Q . Weidentify the socalled magnetic translation group as a subgroup of theobservables' group Y_Q . We determine the unitary irreducible representationsof O_Q, Y_Q corresponding to integer charges and for each of them an associatedorthonormal basis explicitly in configuration space. We also clarify how in then = 2m case a holomorphic structure and Theta functions arise on the associatedcomplex torus. These results apply equally well to the physics of chargedscalar particles on R^n and on T^n in the presence of periodic magnetic field Band scalar potential. They are also necessary preliminary steps for theapplication to these theories of the deformation procedure induced by Drinfel'dtwists.

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