Finite Deformations of Quantum Mechanics

Abstract: We investigate modifications of quantum mechanics (QM) that replace theunitary group in a finite dimensional Hilbert space with a finite group anddetermine the minimal sequence of subgroups necessary to approximate QMarbitrarily closely for general choices of Hamiltonian. This mathematical studyreveals novel insights about ’t Hooft’s Ontological Quantum Mechanics, and thederivation of statistical mechanics from quantum mechanics. We show thatKornyak’s proposal to understand QM as classical dynamics on a Hilbert space ofone dimension higher than that describing the universe, supplemented by achoice of the value of a naturally conserved quantum operator in that classicalevolution can probably be a model of the world we observe.

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