Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

TL;DR

This paper develops a finite noncommutative geometric model unifying general relativity and quantum mechanics, demonstrating how classical spacetime and quantum behavior emerge from the algebra of a finite groupoid.

Contribution

It introduces a finite groupoid-based noncommutative geometry model that unifies gravity and quantum mechanics, with explicit calculations and recovery of classical physics.

Findings

Recovery of standard spacetime geometry through averaging.

Establishment of correspondence with conventional quantum mechanics.

Full computability due to the finite nature of the model.

Abstract

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group G on a space E. We define the algebra of smooth complex valued functions on the groupoid, with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the groupoid algebra, and its correspondence with the standard quantum mechanics is established.