Noncommutative regime of fundamental physics

TL;DR

This paper develops a noncommutative algebraic model unifying general relativity and quantum mechanics, extending previous work to include noncompact groups and constructing a noncommutative general relativity framework.

Contribution

It introduces a noncommutative algebraic model on a groupoid that unifies general relativity and quantum mechanics, including a noncommutative general relativity and a noncommutative Fock space.

Findings

Model unifies general relativity and quantum mechanics.

Correct correspondence with standard theories established.

Constructs a noncommutative Fock space.

Abstract

We further develop a model unifying general relativity with quantum mechanics proposed in our earlier papers (J. Math. Phys. 38, 5840 (1998); 41, 5168 (2000)). The model is based on a noncommutative algebra $A$ defined on a groupoid $\Gamma = E \times G$ where $E$ is the total space of a fibre bundle over space-time and $G$ a Lie group acting on $E$. In this paper, the algebra $A$ is defined in such a way that the model works also if $G$ is a noncompact group. Differential algebra based on derivations of this algebra is elaborated which allows us to construct a "noncommutative general relativity". The left regular representation of the algebra $A$ in a Hilbert space leads to the quantum sector of our model. Its position and momentum representations are discussed in some detail. It is shown that the model has correct correspondence with the standard theories: with general relativity, by restricting the algebra $A$ to a subset of its center; with quantum mechanics, by changing from the groupoid $\Gamma $ to its algebroid; with classical mechanics, by changing from the groupoid $\Gamma $ to its tangent groupoid. We also construct a noncommutative Fock space based on the proposed model.