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

2025/10/30 by Boris Zilber, Zilber, Boris
Computer Science · Mathematics · Physics and Astronomy · #03C66 #81P10 #Advanced Algebra and Logic #FOS: Mathematics #FOS: Physical sciences #Logic (math.LO) #Mathematical and Theoretical Analysis #Quantum Mechanics and Applications #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2511.01900

openalex publication_date 2025/10/30 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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.

Citations

• Axioms of Quantum Mechanics in light of Continuous Model Theory
• Mathematical ideas and notions of quantum field theory
• An Introduction To Continuous Model Theory
• Ax's theorem with an additive character
• The role of the rigged Hilbert space in quantum mechanics

Related

• Axioms of Quantum Mechanics in light of Continuous Model Theory
• An Introduction To Continuous Model Theory
• On the logical structure of physics
• Continuous model theories for von Neumann algebras
• Continuous theory of operator expansions of finite dimensional Hilbert spaces, continuous structures of quantum circuits and decidability