Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model

TL;DR

This paper advances a noncommutative geometric model unifying quantum mechanics and gravity, analyzing its observables, especially position and momentum, revealing nonlocal properties and the algebraic structure of these observables.

Contribution

It develops a detailed algebraic and geometric analysis of observables within a noncommutative unification model of quantum mechanics and gravity, highlighting their nonlocal characteristics.

Findings

Position observable is a coderivation of a coalgebra.

Momentum observable is a derivation of an algebra.

The model exhibits nonlocality in its quantum sector.

Abstract

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in {\it Gen. Rel. Grav.} (2004) {\bf 36}, 111-126. Generalized symmetries of the model are defined by a groupoid $\Gamma $ given by the action of a finite group on a space $E$. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on $\Gamma $. We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, "coparallelly" to the well known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on $\Gamma $, the model in its "quantum sector" is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its "gravity sector" such coderivations do exist. They are investigated.