The large numbers hypothesis and quantum mechanics

In this paper, the suggested similarity between micro and macrocosmos is extended to quantum behavior, postulating that quantum mechanics, like general relativity and classical electrodynamics, is invariant under discrete scale transformations. This hypothesis leads to a large scale quantization of angular momenta. Using the scale factor Λ ~ 10³⁸, the corresponding quantum of action, obtained by scaling the Planck constant, is close to the Kerr limit for the spin of the universe - when this is considered as a huge rotating black hole - and to the spin of Gödel’s universe, solution of Einstein equations of gravitation. Besides, we suggest the existence of another, intermediate, scale invariance, with scale factor λ ~ 10¹⁹. With this factor we obtain, from Fermi’s scale, the values for the gravitational radius and for the collapse proper time of a typical black hole, besides the Kerr limit value for its spin. It is shown that the mass-spin relations implied by the two referred scale transformations are in accordance with Muradian’s Regge-like relations for galaxy clusters and stars. Impressive results are derived when we use a λ-scaled quantum approach to calculate the mean radii of planetary orbits in solar system. Finally, a possible explanation for the observed quantization of galactic redshifts is suggested, based on the large scale quantization conjecture.