Contextuality and noncommutative geometry in quantum mechanics

Thesis

It is argued that the geometric dual of a noncommutative operator algebra represents a notion of quantum state space which differs from existing notions by representing observables as maps from states to outcomes rather than from states to distributions on outcomes. 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 topological tools from their topological prototypes is presented.

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—meant to serve the role of a generalized Gel'fand spectrum. After showing that 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^∼ which acts on all unital C*-algebras, we compare a novel formulation of the operator K₀ functor to the extension K^∼ of the topological K-functor. We then conjecture that the extension of the functor assigning a topological space its topological lattice assigns a unital C*-algebra the topological lattice of its primary ideal spectrum and prove the von Neumann algebraic analogue of this conjecture.

Authors: de Silva, N

• Publication date: 2015
• DOI: 10.5287/ora-n1zn9o4yr
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: operator algebras; k-theory; noncommutative geometry; contextuality; functional analysis; quantum physics; quantum mechanics
• Subjects: Functional analysis (mathematics); 46M40; Quantum theory (mathematics); 46L85; Computer science (mathematics); 46L80; Theoretical physics; 46L10; Analytic Topology or Topology; 58B34