General Relativity on Random Operators

Simple item page
dc.creatorHeller, Michael
dc.creatorPysiak, Leszek
dc.creatorSasin, Wieslaw
dc.date2008-10-14
dc.date.accessioned2026-07-07T10:09:55Z
dc.date.available2026-07-07T10:09:55Z
dc.descriptionWe present a mathematical structure which unifies mathematical structures of general relativity and quantum mechanics. It consists of the noncommutative algebra of compactly supported, complex valued functions ${\mathcal A}$, with convolution as multiplication, on a groupoid $Γ$ the base of which is the total space $E$ of the frame bundle over space-time $M$. A differential geometry based on derivations of ${\mathcal A}$ suitably generalizes the standard differential geometry of space-time, and the algebra ${\mathcal A}$, when represented in a bundle of Hilbert spaces, defines a von Neumann algebra ${\mathcal M}$ of random operators that generalizes the usual quantum mechanics. The main result of the present paper is that there exists a space ${\mathcal M_0}$, dense in ${\mathcal M}$, that is isomorphic with the algebra ${\mathcal A}$. This isomorphism allows us to transfer all differentially geometric constructions, generalized Einstein's equations including, made with the help of ${\mathcal A}$ (and its derivations) to the space ${\mathcal M_0}$. In this way, we obtain a generalization of general relativity in terms of random operators on a bundle of Hilbert spaces. However, this generalization cannot be extended to the whole of ${\mathcal M}$, and this is the main mathematical obstacle, at least in this approach, to fully unify theory of gravity with physics of quanta.
dc.description17 LaTex pages
dc.identifierhttps://arxiv.org/abs/0810.2404
dc.identifierhttp://arxiv.org/abs/0810.2404
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/171470
dc.subjectGeneral Relativity and Quantum Cosmology
dc.titleGeneral Relativity on Random Operators
dc.typetext

Files

Collections

arXiv