_{K}= g

_{K}

^{2}/2h

_{eff}gives the evolution of α

_{K}as a function of dimension n of EQ: α

_{K}= g

_{K}

^{2}/2nh

_{0}. If g

_{K}

^{2}corresponds to electroweak U(1) coupling, it is expected to evolve also with respect to PLS so that the evolutions would factorize.

Note that the original proposal that g_{K}^{2} is renormalization group invariant was later replaced with a piecewise constancy: α_{K} has indeed interpretation as piecewise constant critical temperature

- In the TGD framework, coupling constant as a continuous function of the continuous length scale is replaced with a function of PLS so that coupling constant is a piecewise constant function of the continuous length scale.
PLSs correspond to p-adic primes p, and a hitherto unanswered question is whether the extension determines p and whether p-adic primes possible for a given extension could correspond to ramified primes of the extension appearing as factors of the moduli square for the differences of the roots defining the space-time surface.

In the M

^{8}picture the moduli squared for differences r_{i}-r_{j}of the roots of the real polynomial with rational coefficients associated with the space-time surfaces correspond to energy squared and mass squared. This is the case of p-adic prime corresponds to the size scale of the CD.The scaling of the roots by constant factor however leaves the number theoretic properties of the extension unaffected, which suggests that PLS evolution and dark evolution factorize in the sense that PLS reduces to the evolution of a power of a scaling factor multiplying all roots.

- If the exponent Δ K/log(p) appearing in p
^{Δ K/log(p))}=exp(Δ K) is an integer, exp(Δ K) reduces to an integer power of p and exists p-adically. If Δ K corresponds to a deviation from the Kähler function of WCW for a particular path in the tree inside CD, p is fixed and exp(Δ K) is integer. This would provide the long-sought-for identification of the preferred p-adic prime. Note that p must be same for all paths of the tree. p need not be a ramified prime so that the trouble-some correlation between n and ramified prime defining padic prime p is not required. - This picture makes it possible to understand also PLS evolution if Δ K is identified as a deviation from the Kähler function. p
^{Δ K/log(p))}=exp(Δ K) implies that Δ K is proportional to log(p). Since Δ K as 6-D Kähler action is proportional to 1/α_{K}, log(p)-proportionality of Δ K could be interpreted as a logarithmic renormalization factor of α_{K}∝ 1/log(p). - The universal CCE for α
_{K}inside CDs would induce other CCEs, perhaps according to the scenario based on M"obius transformations.

**Dark and p-adic length scale evolutions of Kähler coupling strength**

The original hypothesis for dark CCE was that h_{eff}=nh is satisfied. Here n would be the dimension of EQ defined by the polynomial defining the space-time surface X^{4}subset M^{8}_{c} mapped to H by M^{8}-H correspondence. n would also define the order of the Galois group and in general larger than the degree of the irreducible polynomial.

** Remark**: The number of roots of the extension is in general smaller and equal to n for cyclic extensions only. Therefore the number of sheets of the complexified space-time surface in M^{8}_{c} as the number of roots identifiable as the degree d of the irreducible polynomial would in general be smaller than n. n would be equal to the number of roots only for cyclic extensions (unfortunately, some former articles contain the obviously wrong statement d=n).

Later the findings of Randell Mills, suggesting that h is not a minimal value of h_{eff}, forced to consider the formula h_{eff}=nh_{0}, h_{0}=h/6, as the simplest formula consistent with the findings of Mills. h_{0} could however be a multiple of even smaller value of h_{eff}, call if h_{0} and the formula h_{0}=h/6 could be replaced by an approximate formula.

The value of h_{eff}=nh_{0} can be understood by noticing that Galois symmetry permutes "fundamental regions" of the space-time surface so that action is n times the action for this kind of region. Effectively this means the replacement of α_{K} with α_{K}/n and implies the convergence of the perturbation theory. This was actually one of the basic physical motivations for the hierarchy of Planck constants. In the previous section, it was argued that h_{0} is given by the square of the ratio l_{P}/R of Planck length and CP_{2} length scale identified as dark scale and equals to n_{0}=(7!)^{2}.

The basic challenge is to understand p-adic length scale evolutions of the basic gauge couplings. The coupling strengths should have a roughly logarithmic dependence on the p-adic length scale p≈ 2^{k/2} and this provides a strong number theoretic constraint in the adelic physics framework.

Since Kähler coupling strength α_{K} induces the other CCEs it is enough to consider the evolution of α_{K}.

**p-Adic CCE of α from its value at atomic length scale?**

If one combines the observation that fine structure constant is rather near to the inverse of the prime p=137 with PLS, one ends up with a number theoretic idea leading to a formula for α_{K} as a function of p-adic length scale.

- The fine structure constant in atomic length scale L(k=137) is given α (k)=e
^{2}/2h ≈ 1/137. This finding has created a lot of speculative numerology. - The PLS L(k)= 2
^{k/2}R(CP_{2}) assignable to atomic length scale p≈ 2^{k}corresponds to k=137 and in this scale α is rather near to 1/137. The notion of fine structure constant emerged in atomic physics. Is this just an accident, cosmic joke, or does this tell something very deep about CCE?Could the formula

α(k)= e

^{2}(k)/2h= 1/khold true?

- α is length scale dependent and the formula in the electron length scale is only approximate. In the weak boson scale one has α≈ 1/127 rather than α= 1/89.
- There are also other interactions and one can assign to them coupling constant strengths. Why electromagnetic interactions in electron Compton scale or atomic length scales would be so special?

**p-Adic CCE from the condition α _{K}(k=137)= 1/137**

In the TGD framework, Kähler coupling strength α_{K} serves as the fundamental coupling strength. All other coupling strengths are expressible in terms of α_{K}, and I have proposed that M"obius transformations relate other coupling strengths to α_{K}. If α_{K} is identified as electroweak U(1) coupling strength, its value in atomic scale L(k=137) cannot be far from 1/137.

The factorization of dark and p-adic CCEs means that the effective Planck constant h_{eff}(n,h,p) satisfies

h_{eff}(n,h,p)=h_{eff}(n,h) = nh .

and is independent of the p-adic length scale. Here n would be the dimension of the extension of rationals involved. h_{eff}(1,h,p) corresponding to trivial extension would correspond to the p-adic CCE as the TGD counterpart of the ordinary evolution.

The value of h need not be the minimal one as already the findings of Randel Mills suggest so that one would have h=n_{0}h_{0}.

h_{eff}= nn_{0}h ,

α_{K,0}= g_{K,max}^{2}/2h_{0} =n_{0} .

This would mean that the ordinary coupling constant would be associated with the non-trivial extension of rationals.

Consider now this picture in more detail.

- Since dark and p-adic length scale evolutions factorize, one has
α

_{K}(n)= g_{K}^{2}(k)/2h_{eff},h

_{eff}= nh_{0}.U(1) coupling indeed evolves with the p-adic length scale, and if one assumes that g

_{K}^{2}(k,n_{0}) (h=n_{0}h_{0}) is inversely proportional to the logarithm of p-adic length scale, one obtainsg

_{K}^{2}(k,n_{0}) =g_{K}^{2}(max)/k ,α

_{K}= g_{K}^{2}(max)/2kh_{eff}. - Since k=137 is prime (here number theoretical physics shows its power!), the condition α
_{K}(k=137,h_{0})=1/137 givesg

_{K}^{2}(max)/2h_{0}}= α_{K}(max) =(7!)^{2}.The number theoretical miracle would fix the value of α

_{K}(max) to the ratio of Planck mass and CP_{2}mass n_{0}= M^{2}_{P}/M^{2}(CP_{2})= (7!)^{2}if one takes the argument of the previous section seriously.The convergence of perturbation theory could be possible also for h

_{eff}=h_{0}if the p-adic length scale L(k) is long enough to make α_{K}= n_{0}/k small enough. - The outcome is a very simple formula for α
_{K}α

_{K}(n,k) = n_{0}/kn ,which is a testable prediction if one assumes that it corresponds to electroweak U(1) coupling strength at QFT limit of TGD. This formula would give a practically vanishing value of α

_{K}for very large values of n associated with h_{gr}. Here one must have n>n_{0}.For h

_{eff}=nn_{0}h characterizing extensions of extension with h_{eff}=h one can writeα

_{K}(nn_{0},k) = 1/kn . - The almost vanishing of α
_{K}for the very large values of n associated with ℏ_{gr}would practically eliminate the gauge interactions of the dark matter at gravitational flux tubes but leave gravitational interactions, whose coupling strength would be beta_{0}/4pi. The dark matter at gravitational flux tubes would be highly analogous to ordinary dark matter.

For a summary of earlier postings see Latest progress in TGD.

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