General Relativity on Random Operators

TL;DR

This paper develops a mathematical framework that unifies general relativity and quantum mechanics using noncommutative algebra and random operators, enabling a generalized form of Einstein's equations.

Contribution

It introduces a novel algebraic structure linking differential geometry of space-time with quantum operator algebras, advancing the unification of gravity and quantum physics.

Findings

Existence of a dense subalgebra isomorphic to the original algebra ${f A}$.

Transfer of geometric constructions to the space of random operators.

Identification of mathematical obstacles to full unification of gravity and quantum theory.

Abstract

We 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 $\Gamma $ 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.