Strict deformation quantization for a particle in a magnetic field

Abstract

Recently, we introduced a mathematical framework for the quantization of a particle in a variable magnetic field. It consists in a modified form of the Weyl pseudodifferential calculus and a C * -algebraic setting, these two points of view being isomorphic in a suitable sense. In the present paper we leave Planck's constant vary, showing that one gets a strict deformation quantization in the sense of Rieffel. In the limit → 0 one recovers a Poisson algebra induced by a symplectic form defined in terms of the magnetic field.

On the Lipschitz Continuity of Spectral Bands of Harper-Like and Magnetic Schrödinger Operators

“…These phases appear very naturally in the continuous case, see [7,8,19,22,23,24,25,26], where it is shown that if a(x) is the transverse gauge generated by a globally bounded magnetic field |b(x)| ≤ 1, then φ(x, x ′ ) can be chosen to be the path integral of a(x) on the segment linking x ′ with x. This is the same as the magnetic flux of b through the triangle generated by x, x ′ and the origin.…”

Graphene and non-Abelian quantization

“…The non-commutativity in the momenta algebra we are interested in can be related to the deformation quantization of Poissonian structures developed in [28] and considered as a kind of magnetic quantization [29,30].…”

Magnetic twisted actions on general abelian $C^*$-algebras

“…[8,11]) on spaces of functions that are smooth under the action θ × τ * of Ξ on Ω × X * . This Poisson algebras can be represented by families of subalgebras of BC ∞ (Ξ), indexed essentially by the orbits of Ω, each one endowed with the Poisson structure induced by a magnetic symplectic form [13]. For simplicity, we shall concentrate on a Poisson subalgebra consisting of functions which have Schwartz-type behavior in the variable ξ ∈ X * .…”

“…In this way one gets a multiplication on S(X * ; C ∞ 0 (Ω)) which generalizes the magnetic Weyl composition of symbols of [12,13,4] (and to which it reduces, actually, if Ω is just a compactification of the configuration space X ). Together with complex conjugation, they endow S(X * ; C ∞ 0 (Ω)) with the structure of a * -algebra.…”

“…The latter no longer commute amongst each other due to the presence of the magnetic field. It was shown in [13] that the family of twisted crossed products indexed by ∈ (0, 1] can be understood as a strict deformation quantization (in the sense of Marc Rieffel) of a natural Poisson algebra defined by a symplectic form which is the sum of the canonical symplectic form in Ξ and a magnetic contribution.…”