Axioms of Quantum Mechanics in light of Continuous Model Theory

TL;DR

This paper reformulates the informal axioms of quantum mechanics within the framework of Continuous Logic, drawing analogies with algebraic structures to better formalize the theory.

Contribution

It introduces a novel approach by expressing quantum mechanical axioms using Continuous Logic, linking algebraic structures with physical theories.

Findings

Establishes an analogy between cylindric algebras and Hilbert spaces.

Recasts quantum axioms in the language of Continuous Logic.

Provides a foundation for algebraic formalization of quantum mechanics.

Abstract

The aim of this note is to recast somewhat informal axiom system of quantum mechanics used by physicists (Dirac calculus) in the language of Continuous Logic. We note an analogy between Tarski's notion of cylindric algebras, as a tool of algebraisation of first order logic, and Hilbert spaces which can serve the same purpose for continuous logic of physics.