QMATH/GAMP Lecture: Contextuality and Noncommuative Geometry in Quantum Mechanics – University of Copenhagen

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QMATH/GAMP Lecture: Contextuality and Noncommuative Geometry in Quantum Mechanics

Speaker: Nadish de Silva, University College London 

Title: Contextuality and noncommutative geometry in quantum mechanics

The geometric dual of a noncommutative operator algebra represents a notion of quantum state space.  This notion differs from the standard one by representing observables as maps from states to deterministic outcomes rather than from states to probabilistic distributions on outcomes.  We present a program of solving for an explicitly geometric manifestation of quantum state space by adapting the spectral presheaf, a construction meant to analyze contextuality in quantum mechanics, to derive simple reconstructions of noncommutative geometric tools from their topological prototypes.  In this way, we directly relate the topos-theoretic approach to quantum physics with the program of noncommutative operator geometry.

We associate to each unital C*-algebra A a geometric object---a diagram of topological spaces representing quotient spaces of the noncommutative space underlying A---that plays 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.  We use this to give a novel formulation of the operator K_0 functor via the extension K' of the topological K-functor.

We then prove a correspondence between ideals of a von Neumann algebra and clopen sets of its associated spatial diagram; further, we 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 primary ideal spectrum.

Relevant arXiv links: