Questions related to coupling constant evolution

There are several open questions related to the hierarchy of Planck constants and p-adic coupling constant evolution in the TGD framework.

1. p-Adic length scale (PLS) hypothesis states L_p =p^1/2R(CP₂), Is this hypothesis correct in this recent form and can one deduce this hypothesis or its generalization from the basic physics of TGD defined by Kähler function of the "world of classical worlds" (WCW)? The fact, that the scaling of the roots of polynomial does not affect the algebraic properties of the extension forcesn to conlude that p-adic prime does not depend on purely algebraic properties of EQ. In particular, the proposed identification of p as a ramified prime of EQ must be given up.

   Number theoretical universality suggests the formula exp(Δ K)= pⁿ, where Δ K is the contribution to Kähler function of WCW for a given space-time surface inside causal diamond (CD).

2. The understanding of p-adic length scale evolution is also a problem. The "dark" coupling constant evolution would be α_K = g_K²/2h_eff = g_K²/2nh₀, and the PLS evolution g_K²(k)=g_K²(max)/k should define independent evolutions since scalings commute with number theory. The total evolution α_K= α_K(max)/nk would induce also the evolution of other coupling strengths if the coupling strenghts are related to α_K by Möbius transformation as suggested.
3. The formula h_eff=nh₀ involves the minimal value h₀. How could one determine it? p-Adic mass calculations for h_eff=h lead to the conclusion that the CP₂ scale R is roughly 10^7.5 times longer than Planck length l_P. Classical argument however suggests R\simeq l_P. If one assumes h_eff=h₀ in the p-adic mass calculations, this is indeed the case for h/h₀=(R/l_P)². This ratio follows from number theoretic arguments as h/h₀= n₀= (7!)². This gives \alpha_K=n₀/kn, and perturbation theory can converge even for n=1 for sufficiently long p-adic length scales. Gauge coupling strengths are predicted to be practically zero at gravitational flux tubes so that only gravitational interaction is effectively present. This conforms with the view about dark matter.
4. Nottale hypothesis predicts gravitational Planck constant ℏ_gr= GMm/β₀ (β₀=v₀/c is velocity parameter), which has gigantic values. Gravitational fine structure constant is given by α_gr= β₀/4π. Kepler's law β²=GM/r=r_S/2r suggests length scale evolution β²=xr_S/2L_N = β²_0,max/N², where x is proportionality constant, which can be fixed.

   Phase transitions changing β₀ are possible at L_N/a_gr=N² and these scales correspond to radii for the gravitational analogs of the Bohr orbits of hydrogen. p-Adic length scale hierarchy is replaced by that for the radii of Bohr orbits. The simplest option is that β₀ obeys a coupling constant evolution induced by α_K.

   This picture conforms with the existing applications and makes it possible to understand the value of β₀ for the solar system, and is consistent with the application to the superfluid fountain effect.

See the article Questions about coupling constant evolution or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

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