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 Gamma given by the action of a finite group on a space E. We define the algebra A of smooth complex valued functions on 0, 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 algebra A, and its correspondence with the standard quantum mechanics is established.