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. Moti-vated 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 T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum (A) in T (A), which in our approach plays the role of the quantum phase space of the sys-tem. 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 , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from to Scott’s interval domain. Noting that open subsets of (A) correspond to proposition