Dirac - von Neumann axioms in the setting of Continuous Model Theory

TL;DR

This paper reformulates the Dirac-von Neumann axioms of quantum mechanics within Continuous Logic, demonstrating that the canonical continuous model can be approximated by finite models and analyzing the Dirac integration quantifier.

Contribution

It introduces a continuous logic framework for quantum axioms and proves the existence of finite approximations, connecting continuous models with ultraproducts of finite models.

Findings

The continuous model is isomorphic to an ultraproduct of finite models.

The Dirac integration quantifier has local and global versions that coincide on Gaussian wave-functions.

Finite models approximate the continuous quantum model effectively.

Abstract

We recast the well-known axiom system of quantum mechanics used by physicists (the Dirac calculus) in the language of Continuous Logic. For the basic version of the axiomatic system we prove that along with the canonical continuous model the axioms have approximate finite models of large sizes, in fact the continuous model is isomorphic to an ultraproduct of finite models. We analyse the continuous logic quantifier corresponding to Dirac integration and show that in finite context it has two versions, local and global, which coincide on Gaussian wave-functions.