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

TL;DR

This work reframes Dirac–von Neumann quantum mechanics within Continuous Logic, building a CL-based rigged Hilbert space and showing that the canonical infinite model is the CL-ultraproduct of finite lattice-models. It introduces H-structures with two universes, U(∞) and U(n), and uses imaginary elements to access generalized states such as position and momentum eigenvectors. A key result is the ultraproduct identification of the infinite model and the equivalence of local and global Dirac-quantification on Gaussian states, with perturbations extended to anharmonic states via perturbation-theory arguments. Overall, the paper bridges quantum axiomatization and continuous model theory, enabling exact finite-model representations and manifold generalizations of quantum mechanics within CL.

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.