p-Adic length scale hypothesis and the weakness of the gravitational constant

In the posting Reduction of long length scale real physics to short length scale p-adic physics and vice versa I discussed the idea that long range p-adically fractal physics can be reduced to local p-adic physics with p-adic continuity and smoothness alone implying the universal characteristics of long scale physics. Real field equations would determine local real physics and their p-adic counterparts the global real physics.

I mentioned also how multi-p-fractality and the pairing p≈2^k, 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).

1. 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 L_p is proportional to p^1/2 rather than p, which looks more natural if common rationals define the correspondence between real and p-adic physics.

2. It would seem that length d_p≈ pR, R or order CP₂ length, in the induced space-time metric must correspond to a length L_p≈ p^1/2R 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 r_M has a length r_X propto r_M². 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 r_X increases and can depend non-linearly on r_M.

3. If the size of the space-time sheet associated with the particle has size d_p≈ pR in the induced metric, the corresponding M⁴ size would be about L_p propto p^1/2R and p-adic length scale hypothesis results.

4. The strongly non-perturbative and chaotic behavior r_X propto r_M² is assumed to continue only up to L_p. At longer length scales the space-time distance d_p associated with L_p becomes the unit of space-time distance and geodesic distance r_X is in a good approximation given by

   r_X= (r_M/L_p)d_p propto p^1/2× r_M ,

   and is thus linear in M⁴ distance r_M.

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 r_X appearing in gravitational force as given by Newtonian approximation is by a factor p^1/2 times longer than r_M so that the strong gravitational force proportional to L_p²/r_X² scales down by a factor p as r_X is expressed in terms of r_M. M⁴ distance r_M 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.

1. Assume first that the space-time sheet is characterized by single prime p: also multi-p-fractality is possible. The strong gravitational constant G_s characterizes the interactions involving exchanges of string like objects of size scale measured naturally using L_p as a unit. In this case the force is mediated in M⁴ as an exchange of a particle. By dimensional estimate G_s is proportional to L_p² and string model picture gives a precise estimate for the numerical factor n in G_s=nL_p².

2. The classical gravitational force is mediated via induced metric inside the space-time sheets and in long length scales is proportional to G_s/r_X² propto L_p²/r_X² propto R²/r_M², where R≈ 10⁴G^1/2 is CP₂ length. Hence the effective gravitational constant is reduced by a factor 1/p and is same for all values of p.

3. 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 p^1/2 is replaced by n^1/2 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.

4. 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 p^1/2 factor.

For more details see the chapter TGD as a Generalized Number Theory I: p-Adicization Program of TGD.

Matti Pitkanen