On the quantum aspects of the logarithmic corrections to the black hole entropy

Abstract

A new relativity theory, or more concretely an extended relativity theory, actively developed by one of the authors incorporates 3 basic concepts. They are the old idea of Chew about bootstrapping, Nottale's scale relativity, and enlargement of the conventional time-space by inclusion of noncommutative Clifford manifolds where all p-branes are treated on equal footing. The latter allows one to write a master action functional. The resulting functional equation is simplified and applied to the p-loop oscillator. Its solution represents a generalization of the conventional quantum point oscillator. In addition, it exhibits some novel features: an emergence of two explicit scales delineating the asymptotic regimes (Planck scale region and a smooth region of a quantum point oscillator). In the most interesting Planck scale regime, the solution recovers in an elementary fashion some basic relations of string theory (including string tension quantization and string uncertainty relation). It is shown that the degeneracy of the $first$ collective excited state of the p-loop oscillator yields not only the well-known Bekenstein-Hawking area-entropy linear relation but also the logarithmic corrections therein. In addition we obtain for any number of dimensions the Hawking temperature, the Schwarschild radius, and the inequalities governing the area of a black hole formed in a fusion of two black holes. One of the interesting results is a demonstration that the evaporation of a black hole is limited by the upper bound on its temperature, the Planck temperature.