Although a few new results are presented, this is mainly a review article on the relationship between finite-dimensional quantum mechanics and finite groups. The main motivation for this discussion is the hidden subgroup problem of quantum computation theory. A unifying role is played by a mathematical structure that we call a Hilbert *-algebra. After reviewing material on unitary representations of finite groups we discuss a generalized quantum Fourier transform. We close with a presentation concerning position-momentum measurements in this framework.
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Gudder, S. Quantum Mechanics on Finite Groups. Found Phys 36, 1160–1192 (2006). https://doi.org/10.1007/s10701-006-9060-1
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DOI: https://doi.org/10.1007/s10701-006-9060-1