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Completely Discretized, Finite Quantum Mechanics

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Abstract

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism.

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Notes

  1. Discussions of fundamental theories often revolve around approaches to unification and quantum gravity, such as superstring theory. But for the most part all of these approaches live within the standard framework of quantum mechanics; they are specific models under that broad umbrella, rather than alternatives to it.

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Acknowledgements

It is a pleasure to thank Justin Clarke-Doane for helpful comments.

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  1. Departments of Physics & Astronomy and Philosophy, Johns Hopkins University, Baltimore, MD, 21218, USA

    Sean M. Carroll

  2. Santa Fe Institute, Santa Fe, NM, 87501, USA

    Sean M. Carroll

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Carroll, S.M. Completely Discretized, Finite Quantum Mechanics. Found Phys 53, 90 (2023). https://doi.org/10.1007/s10701-023-00726-6

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