Tuesday, May 16, 2006

Large values of Planck constant and coupling constant evolution

There has been intensive evolution of ideas induced by the understanding of large values of Planck constants. This motivated a separate chapter which I christened as "Does TGD Predict the Spectrum of Planck Constants?". I have commented earlier about various ideas related to this topic and comment here only the newest outcomes.

1. hbargr as CP2 Planck constant

What gravitational Planck constant means has been somewhat unclear. It turned out that hbargr can be interpreted as Planck constant associated with CP2 degrees of freedom and its huge value implies that also the von Neumann inclusions associated with M4 degrees of freedom meaning that dark matter cosmology has quantal lattice like structure with lattice cell given by Ha/G, Ha the a=constant hyperboloid of M4+ and G subgroup of SL(2,C). The quantization of cosmic redshifts provides support for this prediction.

2. Is Kähler coupling strength invariant under p-adic coupling constant evolution

Kähler coupling constant is the only coupling parameter in TGD. The original great vision is that Kähler coupling constant is analogous to critical temperature and thus uniquely determined. Later I concluded that Kähler coupling strength could depend on the p-adic length scale. The reason was that the prediction for the gravitational coupling strength was otherwise non-sensible. This motivated the assumption that gravitational coupling is RG invariant in the p-adic sense.

The expression of the basic parameter v0=2-11 appearing in the formula of hbargr=GMm/v0 in terms of basic parameters of TGD leads to the unexpected conclusion that αK in electron length scale can be identified as electro-weak U(1) coupling strength αU(1). This identification, or actually something slightly complex (see below), is what group theory suggests but I had given it up since the resulting evolution for gravitational coupling predicted G to be proportional to Lp2 and thus completely un-physical. However, if gravitational interactions are mediated by space-time sheets characterized by Mersenne prime, the situation changes completely since M127 is the largest non-super-astrophysical p-adic length scale.

The second key observation is that all classical gauge fields and gravitational field are expressible using only CP2 coordinates and classical color action and U(1) action both reduce to Kähler action. Furthermore, electroweak group U(2) can be regarded as a subgroup of color SU(3) in a well-defined sense and color holonomy is abelian. Hence one expects a simple formula relating various coupling constants. Let us take αK as a p-adic renormalization group invariant in strong sense that it does not depend on the p-adic length scale at all.

The relationship for the couplings must involve αU(1), αs and αK. The formula 1/αU(1)+1/αs = 1/αK states that the sum of U(1) and color actions equals to Kähler action and is consistent with the decrease of the color coupling and the increase of the U(1) coupling with energy and implies a common asymptotic value 2αK for both. The hypothesis is consistent with the known facts about color and electroweak evolution and predicts correctly the confinement length scale as p-adic length scale assignable to gluons. The hypothesis reduces the evolution of αs to the calculable evolution of electro-weak couplings: the importance of this result is difficult to over-estimate.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? of "TGD: an Overview".

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