Cotangent bundle quantization: entangling of metric and magnetic field

Abstract

For manifolds M of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space L 2 (T * M) and construct an irreducible representation of this algebra in L 2 (M). This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over T * M is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in M. The quantization of δ-functions induces a family of symplectic reflections in T * M and generates a magneto-geodesic connection Γ on T * M. This symplectic connection entangles, on the phase space level, the original affine structure on M and the magnetic field. In the classical approximation, the 2 -part of the quantum product contains the Ricci curvature of Γ and a magneto-geodesic coupling tensor.

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

“…The quantum dynamics of charged particles in external gauge fields on flat and curved manifolds is discussed within the star-product formalism in a gauge-invariant manner in Refs. [53,54].…”

Deformation quantization on the cotangent bundle of a Lie group

“…The algebra F (M ) will play the role of the space of test functions. The following construction is based on [13,33]. We will assume that the group G is compact.…”

“…In the literature can be found many works on the topic of deformation quantization on the cotangent bundle of a Riemannian manifold [7][8][9][10][11][12][13], however most of these works do not provide a complete theory of quantization. In [9,10] authors constructed a quantization scheme with the use of a symbol calculus for pseudodifferential operators on a Riemannian manifold.…”

“…Such case was studied in [28] from the perspective of strict deformation quantization. Some ideas of our theory are based on the work [13] where authors consider non-formal deformation quantization on the cotangent bundle of a Riemannian manifold Q. In this paper the exponential map exp q on Q is required to be a diffeomorphism on the whole tangent space T q Q onto the whole manifold Q.…”

The Kirillov picture for the Wigner particle

“…Since the r and p-variables are conjugate, by lemma 4, in the rest of this section we shall write p b rather than p b for the components of p, as is customary 16. The simpler ordinary 'magnetic' Poisson bracket has been the object of several studies leading up to an (already rather inexplicit) magnetic Weyl-Moyal product[31,32] 17. From this (or a similar) formula it should be clear that under quantization one expects functional-analytic complications in the ket space-of the kind discussed in[35].J M Gracia-Bondía et al…”