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This paper reinterprets the foundational axioms of quantum mechanics through the lens of continuous model theory, establishing a rigorous structural analogy between Hilbert spaces and cylindric algebras. By reformulating the Dirac–von Neumann formalism in a logical setting, it bridges abstract algebraic logic with quantum physics without introducing new physical predictions or experimental claims. The work is primarily of theoretical interest to researchers in mathematical foundations and quantum information theory rather than immediate product or policy implications.
Axioms of Quantum Mechanics in light of Continuous Model Theory
arXiv:2506.02029v3 Announce Type: replace-cross Abstract: We revisit Dirac's axiomatization of quantum mechanics and show that it can essentially be reformulated in a more familiar logical setting - continuous model theory. Our aim is twofold: (i) to present the Dirac--von Neumann formalism in a genuinely axiomatic manner suitable for logicians, and (ii) to exhibit a structural analogy between Hilbert spaces and Tarski's cylindric algebras, which were introduced in the program of algebraisation of first-order logic. Recall that the cylindric algebra $\mathfrak{C}(\mathbb{A})$ of a first order structure $\mathbb{A}$ allows to recover $\mathbb{A}$ up to elementary equivalence. For a general continuous structure $\mathbb{M}$, we introduce an analogue $\mathcal{B}(\mathbb{M})$ of the cylindric algebra of a first--order structure. Under natural tameness assumptions, $\mathcal{B}(\mathbb{M})$ takes the form of a (rigged) Hilbert space with operators, and $\mathbb{M}$ can be recovered from $\mathcal{B}(\mathbb{M}).$
Abstract:We revisit Dirac's axiomatization of quantum mechanics and show that it can essentially be reformulated in a more familiar logical setting - continuous model theory. Our aim is twofold: (i) to present the Dirac--von Neumann formalism in a genuinely axiomatic manner suitable for logicians, and (ii) to exhibit a structural analogy between Hilbert spaces and Tarski's cylindric algebras, which were introduced in the program of algebraisation of first-order logic. Recall that the cylindric algebra $\mathfrak{C}(\mathbb{A})$ of a first order structure $\mathbb{A}$ allows to recover $\mathbb{A}$ up to elementary equivalence. For a general continuous structure $\mathbb{M}$, we introduce an analogue $\mathcal{B}(\mathbb{M})$ of the cylindric algebra of a first--order structure. Under natural tameness assumptions, $\mathcal{B}(\mathbb{M})$ takes the form of a (rigged) Hilbert space with operators, and $\mathbb{M}$ can be recovered from $\mathcal{B}(\mathbb{M}).$
Logic (math.LO); Quantum Physics (quant-ph)
arXiv:2506.02029 [math.LO]
(or arXiv:2506.02029v3 [math.LO] for this version)
https://doi.org/10.48550/arXiv.2506.02029 Focus to learn more arXiv-issued DOI via DataCite
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