On geometric probability, holography, Shilov boundaries and the four physical coupling constants of nature, Prog

By recurring to Geometric Probability methods, it is shown that the coupling constants, αEM;αW;αC associated with Electromagnetism, Weak and the Strong (color) force are given by the ratios of the ratios of the measures of the Shilov boundaries Q2 = S1×RP 1; Q3 = S2×RP 1; S5, respectively, with respect to the ratios of the measures µ[Q5]/µN [Q5] associated with the 5D conformally compactified real Minkowski spacetime M̄5 that has the same topology as the Shilov boundary Q5 of the 5 complex-dimensional poly-disc D5. The homogeneous symmetric complex domain D5 = SO(5, 2)/SO(5) × SO(2) corresponds to the conformal relativistic curved 10 real-dimensional phase space H10 associated with a particle moving in the 5D Anti de Sitter space AdS5. The geometric coupling constant associated to the gravitational force can also be obtained from the ratios of the measures involving Shilov boundaries. We also review our derivation of the observed vacuum energy density based on the geometry of de Sitter ( Anti de Sitter) spaces. 1. The Fine Structure Constant and Geometric Probability Geometric Probability [21] is the study of the probabilities involved in geometric problems, e.g., the distributions of length, area, volume, etc. for geometric objects under stated conditions. One of the most famous problem is the Buffon’s Needle Problem of finding the probability that a needle of length l will land on a line, given a floor with equally spaced parallel lines a distance d apart. The problem was first posed by the French naturalist Buffon in 1733. For l < d the probability is P