Dequantization of Noncommutative Spaces and Dynamical Noncommutative Geometry

TL;DR

This paper explores a mathematical framework for relating noncommutative spaces to their commutative counterparts through sheaf theory, revealing new structures like strings and branes and proposing a dynamic notion of space.

Contribution

It introduces a sheafification approach for noncommutative spaces, identifying stalks with those on commutative moment spaces, and develops a dynamical noncommutative geometry framework.

Findings

Noncommutative stalks can be identified with sheaves on moment spaces.

Spectral families and observables are definable as filtrations on noncommutative topologies.

Proposes a new dynamical system perspective of space with variable noncommutative structures.

Abstract

The purely mathematical root of the dequantization constructions is the quest for a sheafification needed for presheaves on a noncommutative space. The moment space is constructed as a commutative space, approximating the noncommutative space appearing as a dynamical space, via a stringwise construction. The main result phrased is purely mathematical, i.e. the noncommutative stalks of some sheaf on the noncommutative space can be identified to stalks of some sheaf associated to it on the commutative geometry (topology) of the moment space. This may be seen as a (partial) inverse to the deformation--quantization idea, but in fact with a much more precise behaviour of stalks of sheaves. The method, based on minimal axiomatics necessary to rephrase continuity principles in terms of partial order (noncommutative topology) exclusively, leads to the appearance of objects like strings and (M--)branes. Also, spectral families and observables may be defined and studied as separated filtrations on the noncommutative topology! We highlight the relation with pseudo--places and generalized valuation theory. Finally, we hint at a new notion of ``space'' as a dynamical system of noncommutative topologies with the same commutative shadow (4 dimensional space--time if you wish) and variable (but isomorphic) moment spaces, which for special choices may be thought of as a higher dimensional brane--space.