Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess