Abstract
A unique classification of the topological effects associated to quantum mechanics on manifolds is obtained on the basis of the invariance under diffeomorphisms and the realization of the Lie–Rinehart relations between the generators of the diffeomorphism group and the algebra of C ∞ functions on the manifold. This leads to a unique (“Lie–Rinehart”) C *-algebra as observable algebra; its regular representations are shown to be locally Schroedinger and in one to one correspondence with the unitary representations of the fundamental group of the manifold. Therefore, in the absence of spin degrees of freedom and external fields, \( \pi_1{(\mathcal M)}\) appears as the only source of topological effects.
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A preliminary version of this work was presented as a joint work by the first author at the Workshop “Local Quantum Theory”, Vienna September 1997, and made available to the participants as hand written notes.
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Morchio, G., Strocchi, F. Quantum Mechanics on Manifolds and Topological Effects. Lett Math Phys 82, 219–236 (2007). https://doi.org/10.1007/s11005-007-0188-5
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DOI: https://doi.org/10.1007/s11005-007-0188-5