I mentioned also how multi-p-fractality and the pairing p≈2k, k prime, in p-adic length scale hypothesis can be understood in this scenario. There is however a problem involved with the understanding of the origin of the p-adic length scale hypothesis if the correspondence via common rationals is assumed.
1. How p-adic length scale hypothesis can be understood?
Here the problem and its resolution is discussed (I have discussed this point already earlier but from different view point).
- The mass calculations based on p-adic thermodynamics for Virasoro generator L0 predict that mass squared is proportional to 1/p and Uncertainty Principle implies that Lp is proportional to p1/2 rather than p, which looks more natural if common rationals define the correspondence between real and p-adic physics.
- It would seem that length dp≈ pR, R or order CP2 length, in the induced space-time metric must correspond to a length Lp≈ p1/2R in M4. This could be understood if space-like geodesic lines at real space-time sheet obeying effective p-adic topology are like orbits of a particle performing Brownian motion so that the space-like geodesic connecting points with M4 distance rM has a length rX propto rM2. Geodesic random walk with randomness associated with the motion in CP2 degrees of freedom could be in question. The effective p-adic topology indeed induces a strong local wiggling in CP2 degrees of freedom so that rX increases and can depend non-linearly on rM.
- If the size of the space-time sheet associated with the particle has size dp≈ pR in the induced metric, the corresponding M4 size would be about Lp propto p1/2R and p-adic length scale hypothesis results.
- The strongly non-perturbative and chaotic behavior rX propto rM2 is assumed to continue only up to Lp. At longer length scales the space-time distance dp associated with Lp becomes the unit of space-time distance and geodesic distance rX is in a good approximation given by
rX= (rM/Lp)dp propto p1/2× rM ,
and is thus linear in M4 distance rM.
2. How to understand the smallness of gravitational constant?
The proposed explanation of the p-adic length scale hypothesis allows also to understand the weakness of the gravitational constant as being due to the fact that the space-time distance rX appearing in gravitational force as given by Newtonian approximation is by a factor p1/2 times longer than rM so that the strong gravitational force proportional to Lp2/rX2 scales down by a factor p as rX is expressed in terms of rM. M4 distance rM is indeed the natural variable since distances are measured using space-time sheets as units and their sizes are always measured in M4 metric or almost flat X4 metric.
A more precise argument goes as follows.
- Assume first that the space-time sheet is characterized by single prime p: also multi-p-fractality is possible. The strong gravitational constant Gs characterizes the interactions involving exchanges of string like objects of size scale measured naturally using Lp as a unit. In this case the force is mediated in M4 as an exchange of a particle. By dimensional estimate Gs is proportional to Lp2 and string model picture gives a precise estimate for the numerical factor n in Gs=nLp2.
- The classical gravitational force is mediated via induced metric inside the space-time sheets and in long length scales is proportional to Gs/rX2 propto Lp2/rX2 propto R2/rM2, where R≈ 104G1/2 is CP2 length. Hence the effective gravitational constant is reduced by a factor 1/p and is same for all values of p.
- The value of the gravitational constant is still by a factor of order R2/G≈ 108 too high. A correct value is obtained if multi-p-fractality prevails in such a manner that p1/2 is replaced by n1/2 with n=2× 3× 5...× 23× p. One can visualize the situation as hierarchy of wavelets: to p-adic wavelets very small q=23-adic wawelets are superposed to which in turn q=19-adic ... The earlier estimates for the gravitational constant are consistent with this result and fix the numerical details.
- This approach predicts a p-adic hierarchy of strong gravitons and unstable spin 2 hadrons are excellent candidates for them. It is however not clear whether Newtonian graviton is predicted at all: in other words could the gravitation inside space-time sheets be a purely classical phenomenon? One can certainly imagine the exchange of topologically condensed Newtonian gravitons moving along light-like geodesics along space-time sheets and the lengths of spatial projections of these geodesics would be indeed scaled up by p1/2 factor.