dorsal/arxiv
View SchemaCotangent bundle quantization: Entangling of metric and magnetic field
| Authors | M. V. Karasev, T. A. Osborn |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0505144 |
| URL | https://arxiv.org/abs/quant-ph/0505144 |
| DOI | 10.1088/0305-4470/38/40/006 |
| Journal | J.Phys.A: Math.Gen., 2005, v.38, 8549-8578 |
Abstract
For manifolds $\cal M$ 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 $L^2(T^*\cal M)$ and construct an irreducible representation of this algebra in $L^2(\cal M)$. 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 $T^*\cal M$ 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 $\cal M$. The quantization of $\delta$-functions induces a family of symplectic reflections in $T^*\cal M$ and generates a magneto-geodesic connection $\Gamma$ on $T^*\cal M$. This symplectic connection entangles, on the phase space level, the original affine structure on $\cal M$ and the magnetic field. In the classical approximation, the $\hbar^2$-part of the quantum product contains the Ricci curvature of $\Gamma$ and a magneto-geodesic coupling tensor.
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"abstract": "For manifolds $\\cal M$ of noncompact type endowed with an affine connection\n(for example, the Levi-Civita connection) and a closed 2-form (magnetic field)\nwe define a Hilbert algebra structure in the space $L^2(T^*\\cal M)$ and\nconstruct an irreducible representation of this algebra in $L^2(\\cal M)$. This\nalgebra is automatically extended to polynomial in momenta functions and\ndistributions. Under some natural conditions this algebra is unique. The\nnon-commutative product over $T^*\\cal M$ is given by an explicit integral\nformula. This product is exact (not formal) and is expressed in invariant\ngeometrical terms. Our analysis reveals this product has a front, which is\ndescribed in terms of geodesic triangles in $\\cal M$. The quantization of\n$\\delta$-functions induces a family of symplectic reflections in $T^*\\cal M$\nand generates a magneto-geodesic connection $\\Gamma$ on $T^*\\cal M$. This\nsymplectic connection entangles, on the phase space level, the original affine\nstructure on $\\cal M$ and the magnetic field. In the classical approximation,\nthe $\\hbar^2$-part of the quantum product contains the Ricci curvature of\n$\\Gamma$ and a magneto-geodesic coupling tensor.",
"arxiv_id": "quant-ph/0505144",
"authors": [
"M. V. Karasev",
"T. A. Osborn"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/38/40/006",
"journal_ref": "J.Phys.A: Math.Gen., 2005, v.38, 8549-8578",
"title": "Cotangent bundle quantization: Entangling of metric and magnetic field",
"url": "https://arxiv.org/abs/quant-ph/0505144"
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