Abstract
In this contribution certain group-theoretical and topological aspects of quantum theory are reviewed. We consider systems localized on configuration spaces being homogeneous spaces or differentiable manifolds. Our approach to quantum kinematics is based on systems of imprimitivity in the case of homogeneous spaces, and their generalization to manifolds, which we call quantum Borel kinematics. We show that quantum kinematics can be classified in terms of global invariants-quantum numbers of group-theoretical or topological origin. Finally, quantum mechanics of a charged particle in the magnetic field of the Dirac monopole is presented as an example illustrating the interplay of group representation theory and non-trivial topology. The exposition is based on our joint work1,2,3,4 with B. Angermann and P. Štoviček, and Ref. 5.
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Doebner, H.D., Tolar, J. (1986). Symmetry and Topology of the Configuration Space and Quantization. In: Gruber, B., Lenczewski, R. (eds) Symmetries in Science II. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1472-2_10
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DOI: https://doi.org/10.1007/978-1-4757-1472-2_10
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