The Large Numbers Hypothesis and Quantum Mechanics

In this paper, the suggested similarity between micro and macro-cosmos 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 $\Lambda \sim 10^{38}$, 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 Godel's universe, solution of Einstein equations of gravitation. Besides, we suggest the existence of another, intermediate, scale invariance, with scale factor $\lambda \sim 10^{19}$. 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 $\lambda$-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.