PROGRESS IN PHYSICS On the Coupling Constants, Geometric Probability and Complex Domains∗

By recurring to Geometric Probability methods it is shown that the coupling constants, αEM , αW , αC , associated with the electromagnetic, weak and strong (color) force are given by the ratios of measures of the sphere S and the Shilov boundaries Q3 =S 2 ×RP , squashed S, respectively, with respect to the Wyler measure ΩWyler[Q4] of the Shilov boundary Q4 =S ×RP 1 of the poly-disc D4 (8 real dimensions). The latter measure ΩWyler[Q4] is linked to the geometric coupling strength αG associated to the gravitational force. In the conclusion we discuss briefly other approaches to the determination of the physical constants, in particular, a program based on the Mersenne primes p-adic hierarchy. The most important conclusion of this work is the role played by higher dimensions in the determination of the coupling constants from pure geometry and topology alone and which does not require to invoke the anthropic principle.