Axioms of Quantum Mechanics in light of Continuous Model Theory

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 $\mathcal{B}(\mathbb{M}).$