Monday, January 30, 2006

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≈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).

  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 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.

  2. 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.

  3. 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.

  4. 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.

  1. 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.

  2. 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.

  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 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.

  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 p1/2 factor.

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

Matti Pitkanen

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