Cotangent bundle quantization: Entangling of metric and magnetic field

Abstract

For manifolds $\mathcal{M}$ 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 $L^2(T^{\ast }\mathcal{M})$ and construct an irreducible representation of this algebra in $L^2(\mathcal{M})$. 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 $T^{\ast }\mathcal{M}$ 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 $\mathcal{M}$. The quantization of $\delta$-functions induces a family of symplectic reflections in $T^{\ast }\mathcal{M}$ and generates a magneto-geodesic connection $\Gamma$ on $T^{\ast }\mathcal{M}$. This symplectic connection entangles, on the phase space level, the original affine structure on $\mathcal{M}$ and the magnetic field. In the classical approximation, the ${\hslash }^2$-part of the quantum product contains the Ricci curvature of $\Gamma$ and a magneto-geodesic coupling tensor.

Latex, 38 pages, 5 figures, minor corrections

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Cited by in corpus (6)

• Two-parameter Asymptotics in Magnetic Weyl Calculus
• Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics
• The Kirillov picture for the Wigner particle
• Gradient expansion of the non-Abelian gauge-covariant Moyal star-product
• The spectral density function of the renormalized Bochner Laplacian on a symplectic manifold
• Deformation quantization on the cotangent bundle of a Lie group