Bloch waves and non-commutative tori of magnetic translations

Abstract

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak's magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant, which is the non-commutative torus of magnetic translations in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. The main focus of our review is the Kähler structure on the Hilbert space of Bloch waves and its inherent quantum toric geometry. We reveal that the metaplectic group Mp(2,R) of the automorphisms of magnetic translation algebras is represented by the quantum optics squeezing operators.