Noncommutative closed Friedman universe

Heller et al. (J Math Phys 46:122501, 2005; Int J Theor Phys 46:2494, 2007) proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra defined on a groupoid having the frame bundle over space–time as its base space. The generalized Einstein equation is assumed in the form of the eigenvalue equation of the Einstein operator on a module of derivations of the algebra . No matter sources are assumed. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, generalized eigenvalues of the Einstein operator reproduce components of the perfect fluid energy-momentum tensor for the usual Friedman model together with the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space–time geometry. These results are briefly discussed.