Completely Discretized, Finite Quantum Mechanics

TL;DR

This paper introduces a finite, discrete version of quantum mechanics based on a finite-dimensional Hilbert space, with potential implications for cosmology and philosophical questions about mathematical realism.

Contribution

It proposes a model of quantum mechanics with a finite number of states and discrete time evolution, differing from standard continuous models.

Findings

Schrödinger evolution becomes periodic under certain conditions.

System visits only a finite set of states due to discretization.

Potential cosmological and philosophical implications.

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, Schr\"odinger 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.