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Quantum Mechanics on the Torus, Klein Bottle and Projective Sphere

In: Symmetries in Science IX

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  • Christoph Schulte

    (TU Clausthal, Institut für Theoretische Physik A)

Abstract

The Borel quantization shows that there is a topological dependence of the “free” dynamics on the configuration space M,on which a quantum mechanical system is localized. Unitarily inequivalent quantization mappings are classified by elements (α, D) in π l* (M) × R. In the frame work of Borel quantization the quantization parameter D gives rise to a non-linear Schrödinger equation [1, 2], which reduces to a linear one for D = 0. Our procedure is motivated by the isomorphism of elements in π 1*(M) in the set of equivalence classes of complex line bundles with flat connection [3]. Using these flat connections we will construct a Laplacian on the complex line bundle.

Suggested Citation

  • Christoph Schulte, 1997. "Quantum Mechanics on the Torus, Klein Bottle and Projective Sphere," Springer Books, in: Bruno Gruber & Michael Ramek (ed.), Symmetries in Science IX, pages 313-323, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-5921-4_23
    DOI: 10.1007/978-1-4615-5921-4_23
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