A finite-dimensional model of quantum mechanics

Stanley Gudder, Vladimir Naroditsky

R. Jagannathan, Thalanayar S. Santhanam, Ramabhadra Vasudevan

R. Jagannathan, Thalanayar S. Santhanam

Velimir Jurdjevic

Tom Banks

We investigate modifications of quantum mechanics (QM) that replace the unitary group in a finite dimensional Hilbert space with a finite group and determine the minimal sequence of subgroups necessary to approximate QM arbitrarily closely for general choices of Hamiltonian. This mathematical study reveals novel insights about 't Hooft's Ontological Quantum Mechanics, and the derivation of statistical mechanics from quantum mechanics. We show that Kornyak's proposal to understand QM as classical dynamics on a Hilbert space of one dimension higher than that describing the universe, supplemented by a choice of the value of a naturally conserved quantum operator in that classical evolution can probably be a model of the world we observe.

Denys I. Bondar, Renán Cabrera, HERSCHEL A. RABITZ

Utilizing operational dynamic modeling [D. I. Bondar et al., Phys. Rev. Lett. 109, 190403 (2012)], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechanics is a straightforward discretization of the corresponding infinite-dimensional formulation.