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 Γ given by the action of a finite group on a space E. We define the algebra A of smooth complex valued functions on Γ, 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.
As well known, the b-boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation ρ, defined on the Cauchy completed total spaceĒ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class [p 0 ] related to the singularity remains in close contact with all other equivalence classes, i.e., if p 0 ∈ cl[p] for every p ∈ E. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra onĒ, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense)
We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in Gen. Rel. Grav. (36, 111-126 (2004)). Generalized symmetries of the model are defined by a groupoid Γ 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 Γ. 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 Γ, 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.
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