Geometry of nonHausdorff spaces and its significance for physics
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 farreaching 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 spacetimes 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 April 2011
 DOI:
 10.1063/1.3574352
 arXiv:
 arXiv:1007.0491
 Bibcode:
 2011JMP....52d3506H
 Keywords:

 algebra;
 cosmology;
 geometry;
 Hilbert spaces;
 quasicrystals;
 spacetime configurations;
 04.62.+v;
 03.65.Aa;
 02.10.v;
 02.40.k;
 98.80.k;
 Quantum field theory in curved spacetime;
 Logic set theory and algebra;
 Geometry differential geometry and topology;
 Cosmology;
 Mathematical Physics
 EPrint:
 13 LaTex pages, no figures