Axioms of Quantum Mechanics in light of Continuous Model Theory

TL;DR

The paper reframes Dirac-style quantum mechanics within the framework of Continuous Logic, proposing an algebraic-logical bridge via cylindric algebras and their continuous-model analogue. It introduces a construction, $\mathfrak{C}(\mathbf{M})$, for a continuous-logic structure $\mathbf{M}$, and proves under natural assumptions that this structure forms a rigged Hilbert space from which $\mathbf{M}$ can be recovered. The work develops the interpretation of QM axioms in continuous model theory, establishes a Riesz-measure-based representation of definable predicates, and provides a main reconstruction result (MainT) linking observables, states, and imaginary sorts to a Gel'fand-triple, with practical consideration given to finite-volume manifolds. Overall, it tightens the conceptual link between quantum axiomatization and logical-algebraic frameworks, enabling a unified continuous-model-theoretic view of quantum structures.

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.