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    Magnetic quantization over Riemannian manifolds

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    Karasev_Magnetic_quantization.pdf (101.1Kb)
    Date
    2006-07-31
    Author
    Karasev, MV
    Osborn, TA
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    Abstract
    We demonstrate that Weyl's pioneering idea (1918) to intertwine metric and magnetic fields into a single joint connection can be naturally realized, on the phase space level, by the gauge-invariant quantization of the cotangent bundle with magnetic symplectic form. Quantization, for systems over a noncompact Riemannian configuration manifold, may be achieved by the introduction of a magneto-metric analog of the Stratonovich quantizer - a family of invertible, selfadjoint operators representing quantum delta functions. Based on the quantizer, we construct a generalized Wigner transform that maps Hilbert-Schmidt operators into L-2 phase-space functions. The algebraic properties of the quantizer allow one to extract a family of symplectic reflections, which are then used to (i) derive a simple, explicit, and geometrically invariant formula for the noncommutative product of functions on phase space, and (ii) construct a magneto-metric connection on phase space. The classical limit of this product is given by the usual multiplication of functions (zeroth-order term), the magnetic Poisson bracket (first-order term), and by the magneto-metric connection (second-order term).
    URI
    http://hdl.handle.net/1993/2933
    DOI
    10.1139/p06-027
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