A Topos for Algebraic Quantum Theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos $${\mathcal{T}(A)}$$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra $${\underline{A}}$$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum $${\underline{\Sigma}(\underline{A})}$$ in $${\mathcal{T}(A)}$$ , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on $${\underline{\Sigma}}$$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from $${\underline{\Sigma}}$$ to Scott’s interval domain. Noting that open subsets of $${\underline{\Sigma}(\underline{A})}$$ correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos $${\mathcal{T}(A)}$$.These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.