Contextuality and noncommutative geometry in quantum mechanics

Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be comeasurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative algebras of observables as being generalisations of classical state spaces to the quantum setting and argue that these generalised geometric spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton-Isham-Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry \`a la Connes et al. To each unital $C^*$-algebra we associate a geometric object acting as a generalised Gel'fand spectrum. We show how any functor $F$ from compact Hausdorff spaces to a suitable target category can be applied directly to these geometric objects to automatically yield an extension $\tilde{F}$ acting on all unital $C^*$-algebras. This procedure is used to give a novel formulation of the operator $K_0$-functor in terms of the topological $K$-functor. We then delineate a $C^*$-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital $C^*$-algebra its lattice of closed, two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.