Conceptual inconsistencies in finite-dimensional quantum and classical mechanics

TL;DR

This paper shows that finite-dimensional models of quantum and classical mechanics violate fundamental principles like Ehrenfest's theorems and face other conceptual issues, challenging the idea that discretization preserves the original theories.

Contribution

It reveals fundamental inconsistencies in finite-dimensional representations of quantum and classical dynamics, including violations of Ehrenfest's theorems and issues with defining key physical concepts.

Findings

Finite-dimensional models violate Ehrenfest theorems.

Problems with defining free particles and potential forces.

Non-Hermitian mechanics share similar issues.

Abstract

Utilizing operational dynamic modeling [Phys. Rev. Lett. 109, 190403 (2012); arXiv:1105.4014], 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.