On the four dimensional Conformal Anomaly, Fractal Spacetime and the Fine Structure Constant

Abstract

Antoniadis, Mazur and Mottola (AMM) two years ago computed the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D Gravity. The fractal dimension was determined by the coefficient of the Gauss-Bonnet topological term associated with the conformal gravitational anomaly and was found to be greater than 4. It is explicitly shown how one can relate the value of the Hausdorff dimension computed by AMM to the universal dimensional fluctuation of spacetime $\epsilon $ given by $\phi^3/2$, where $\phi$ is the Golden Mean $0.618..$. Based on the infrared scaling limit of the theory and using recent Renormalization Group arguments by El Naschie, we conjecture that the unknown coefficient $Q^2$, associated with the four dimensional gravitational conformal anomaly, could be precisely equal to the inverse fine structure constant values ranging between $137.036 $ and 137.081. Our results generate decimal digits up to any $arbitrary$ number.