- p-Adic length scale (PLS) hypothesis states Lp =p1/2R(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).
- The understanding of p-adic length scale evolution is also a problem. The "dark" coupling constant evolution would be αK = gK2/2heff = gK2/2nh0, and the PLS evolution gK2(k)=gK2(max)/k should define independent evolutions since scalings commute with number theory. The total evolution αK= αK(max)/nk would induce also the evolution of other coupling strengths if the coupling strenghts are related to αK by Möbius transformation as suggested.
- The formula heff=nh0 involves the minimal value h0. How could one determine it? p-Adic mass calculations for heff=h lead to the conclusion that the CP2 scale R is roughly 107.5 times longer than Planck length lP. Classical argument however suggests R\simeq lP. If one assumes heff=h0 in the p-adic mass calculations, this is indeed the case for h/h0=(R/lP)2. This ratio follows from number theoretic arguments as h/h0= n0= (7!)2. This gives \alphaK=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.
- Nottale hypothesis predicts gravitational Planck constant ℏgr= GMm/β0 (β0=v0/c is velocity parameter), which has gigantic values. Gravitational fine structure constant is given by αgr= β0/4π. Kepler's law β2=GM/r=rS/2r suggests length scale evolution β2=xrS/2LN = β20,max/N2, where x is proportionality constant, which can be fixed.
Phase transitions changing β0 are possible at LN/agr=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 αK.
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.
For a summary of earlier postings see Latest progress in TGD.