Note on the existence theorem in ``Emergent Quantum Mechanics and Emergent Symmetries''

TL;DR

This paper extends 't Hooft's deterministic models to quantum systems with complete sets of commuting observables, introducing a symmetry of beables and linking quantum numbers to symmetry-breaking conditions, thus advancing the understanding of emergent quantum mechanics.

Contribution

It presents an extension of 't Hooft's theorem to systems with beables, introducing a symmetry of beables and connecting quantum numbers to boundary conditions in deterministic models.

Findings

Introduces symmetry of beables in deterministic models.

Links quantum numbers to symmetry-breaking conditions.

Proposes the Hamiltonian as an emergent beable.

Abstract

Recently 't Hooft demonstrated that ``For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization''. An extension is presented here which covers quantum systems that are characterized by a complete set of mutually commuting Hermitian operators (``beables''). We introduce the symmetry of beables: any complete set of beables is as good as any other one which is obtained through a real general linear group transformation. The quantum numbers of a specific set are related to symmetry breaking initial and boundary conditions in a deterministic model. The Hamiltonian, in particular, can be taken as the emergent beable which provides the best resolution of the evolution of the model universe.