Commun. Math. Phys. 365, 375–429 (2019) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-018-3222-9 Mathematical Physics

Contextuality and Noncommutative Geometry in Quantum Mechanics

Nadish de Silva1 , Rui Soares Barbosa2 1 Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK. E-mail: [email protected]; URL: http://nadi.sh 2 Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. E-mail: [email protected]; URL: http://www.cs.ox.ac.uk/people/rui.soaresbarbosa/

Received: 20 October 2016 / Accepted: 8 June 2018 Published online: 7 January 2019 – © The Author(s) 2019

Abstract: Observable properties of a classical physical system can be modelled deter- ministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. 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 jointly measurable, 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 operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these gener- alised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quan- tum 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 à la Connes et al. To each unital C∗-algebra A we associate a geometric object—a diagram of topolog- ical spaces collating quotient spaces of the noncommutative space underlying A—that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F˜ acting on all unital C∗-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a fini- ˜ ˜ tary variant K f of the extension K 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 the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture. 376 N. de Silva, R. S. Barbosa

1. Introduction

The mathematical description of classical physical systems exhibits an elegant interplay between algebraic aspects of observables and geometric aspects of states. A system can be described in two (dually) equivalent ways depending on whether one takes states or observables as primary. Adopting a realist or ontological perspective, one starts with a space of states and constructs observables as (continuous) functions from states to scalar values. Conversely, adopting an operational or epistemic perspective, one starts with an algebra of observables and constructs states as (homomorphic) functions from observables to scalar values. Such state-observable dualities are manifestations of the duality between geometry and algebra that is a common thread running throughout mathematics. We are particularly interested in the interplay described by the Gel'fand– Na˘ımark duality [44] between the categories of unital commutative C∗-algebras (of observables) and compact Hausdorff spaces (of states). Quantum systems are described by their C∗-algebra of obse