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 T n = R n /Λ 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 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 ∈ 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.

Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

“…Magnetic translations on T d have been studied in e.g. [Fio13,DGTS20] (for constant magnetic fields B), but our treatment is more general and also fits in with expectations from string theory.…”

Modules Over the Noncommutative Torus and Elliptic Curves

“…In the next proposition, we study a transform very close to the Weil-Brezin-Zak transform used in solid state physics [9, Section 1.10] (see also the end of section 2 in [8]). In the following, [n] := n mod p.…”

“…One might then use a suitable Drinfeld twist [6] based on U(h 3 )[[ ]], with h 3 the Lie algebra of H 3 , to deform vector bundles over T 2 into modules for the noncommutative torus. A detailed study of Heisenberg twists and their use to study the dynamics of charged particles on the noncommutative torus, along the lines of [7,8], is postponed to future works.…”

On twisted symmetries and QM with a magnetic field on NC tori

“…It is a ψ 0,k = 0. Up to a gaussian factor, the ψ n,k are Jacobi Theta functions or their derivatives and are analytic in z = x 1 + ix 2 [11].…”

“…Therefore we cannot apply the standard ⋆deformation eX n e ⋆ X n ⋆ choosing H = U g 0 . The way out is based on our recent results [11], which we summarize in section 2. Describing Γ(T n ,E) as a subspace X V of C ∞ (R n ) whose elements are periodic up to a suitable phase factor V , we have shown that Γ(T n ,E) is a module of a central extension of G 0 that we call the projective translation group G Q ; the central generator in the Lie algebra g Q is the electric charge operator Q.…”