Non-unitary evolution of noncommutative spaces and its application as models of the universe

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Through the recent astrophysical observations, it has become clear that the most part of our universe is composed of the unidentified dark matter and dark energy. Especially the dark energy is a long-standing problem also in elementary particle physics, which is known as the cosmological constant problem or the vacuum energy problem. Probably, the solution requires the development of new quantum theory of gravity, which replaces the general relativity, the classical theory of gravity.General relativity is based on the concept of Riemannian geometry, and its dynamical variable is the metric tensor, which varies continuously on a coordinated spacetime. It is well known that the standard procedure of field quantization cannot be applied to the general relativity, because of the appearance of uncontrollable divergences. Moreover some thought experiments based on the general relativity and the quantum mechanics suggest the existence of the observable minimum length, and do not seem to favor the classical continuous and smooth spacetime. The concept of fuzzy spaces is an interesting candidate for such quantum spacetimes. In this granted project, I have pursued the possibilities to construct quantum gravity based on dynamical theory of filmy spaces. A fuzzy space can be defined by a rank-three tensor which determines the product rules among functions. Thus I have proposed dynamical theory of a rank-three tensor as that of fuzzy spaces. Such a rank-three tensor model is a background-independent theory which describes dynamical spaces of various topologies and dimensions in a unified manner. I have numerically studied the models, and have found that they can produce various physically meaningful spaces as classical solutions, and reproduces general relativity in the classical limit. Thus the rank-three tensor model provides an interesting candidate for quantum gravity well worth further study.