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

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.Comment: 20 LaTex pages, General Relativity and Gravitation, in pres

Geometry and General Relativity in the Groupoid Model with a Finite Structure Group

In a series of papers we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra ${\cal A}_{\Gamma}$ defined on a transformation groupoid $\Gamma$ determined by the action of the Lorentz group on the frame bundle $(E, \pi_M, M)$ over space-time $M$. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group $G$ and the frame bundle is trivial $E=M\times G$. The model is fully computable. We define the Einstein-Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space-time (giving the "general relativistic limit"), the extra terms that appear due to our generalization can be interpreted as "matter terms", as in Kaluza-Klein-type models. To illustrate this effect we further simplify the metric matrix to a block diagonal form, compute for it the generalized Einstein equations and find two of their "Friedmann-like" solutions for the special case when $G =\mathbb{Z}_2$. One of them gives the flat Minkowski space-time (which, however, is not static), another, a hyperbolic, linearly expanding universe.Comment: 32 page