Geometry of Non-Hausdorff Spaces and Its Significance for Physics

TL;DR

This paper explores how analyzing the Hausdorff relation in non-Hausdorff spaces reveals deep geometric and algebraic structures, leading to a natural emergence of quantum-like uncertainty relations without quantization.

Contribution

It introduces a geometric framework using groupoids and operator algebras to analyze non-Hausdorff spaces, connecting topology with quantum-like phenomena.

Findings

Spaces with total Hausdorff relation include Penrose tilings and certain cosmological models.

A groupoid-based algebraic approach allows analysis of otherwise intractable spaces.

Position and momentum operators emerge with a commutation relation resembling Heisenberg's uncertainty principle.

Abstract

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and momentum operators can be defined, and they satisfy a commutation relation which, in the suitable limiting case, reproduces the Heisenberg indeterminacy relation. It should be emphasized that this is neither an additional assumption nor an effect of a quantization process, but simply the consequence of a purely geometric analysis.