With the development of the vision about number theoretically universal view about functional integration in WCW , a concrete vision about the exponent of Kähler action in Euclidian and Minkowskian space-time regions. The basic requirement is that exponent of Kähler action belongs to an algebraic extension of rationals and therefore to that of p-adic numbers and does not depend on the ordinary p-adic numbers at all - this at least for sufficiently large primes p. Functional integral would reduce in Euclidian regions to a sum over maxima since the troublesome Gaussian determinants that could spoil number theoretic universality are cancelled by the metric determinant for WCW.
The adelically exceptional properties of Neper number e, Kähler metric of WCW, and strong form of holography posing extremely strong constraints on preferred extremals, could make this possible. In Minkowskian regions the exponent of imaginary Kähler action would be root of unity. In Euclidian space-time regions expressible as power of some root of e which is is unique in sense that ep is ordinary p-adic number so that e is p-adically an algebraic number - p:th root of ep.
These conditions give conditions on Kähler coupling strength αK= gK2/4π (hbar=1)) identifiable as an analog of critical temperature. Quantum criticality of TGD would thus make possible number theoretical universality (or vice versa).
What about Minkowskian regions? It is difficult to say anything definite. For cosmic string like objects the action is non-vanishing but proportional to the area A of the string like object and the conditions would give quantization of the area. The area of geodesic sphere of CP2 is proportional to π. If the value of gK is same for Minkowskian and Euclidian regions, gK2∝ π2 implies SK ∝ A/R2π so that A/R2∝ π2 is required.
- In Euclidian regions the natural starting point is CP2 vacuum extremal for which the maximum value of Kähler action is
SK= π2/2gK2= π/8αK .
The condition reads SK=q =m/n if one allows roots of e in the extension. If one requires minimal extension of involving only e and its powers one would have SK=n. One obtains
1/αK= 8q/π ,
where the rational q=m/n can also reduce to integer. One cannot exclude the possibiity that q depends on the algebraic extension of rationals defining the adele in question.
For CP2 type extremals the value of p-adic prime should be larger than pmin=53. One can consider a situation in which large number of CP2 type vacuum extremals contribute and in this case the condition would be more stringent. The condition that the action for CP2 extremal is smaller than 2 gives
1/αK≤ 16/π ≈ 5.09 .
It seems there is lower bound for the p-adic prime assignable to a given space-time surface inside CD suggesting that p-adic prime is larger than 53× N, where N is particle number.
This bound has not practical significance. In condensed matter particle number is proportional to (L/a)3 - the volume divided by atomic volume. On basis p-adic mass calculations p-adic prime can be estimated to be of order (L/R)2. Here a is atomic size of about 10 Angstroms and R CP2 "radius. Using R≈ 104 LPlanck this gives as upper bound for the size L of condensed matter blob a completely super-astronomical distance L≤ a3/R2 ∼ 1025 ly to be compared with the distance of about 1010 ly travelled by light during the lifetime of the Universe. For a blackhole of radius rS= 2GM with p∼ (2GM/R)2 and consisting of particles with mass above M≈ hbar/R one would obtain the rough estimate M>(27/2)× 10-12mPlanck ∼ 13.5× 103 TeV trivially satisfied.
- The physically motivated expectation from earlier arguments - not necessarily consistent with the recent ones - is that the value αK is quite near to fine structure constant at electron length scale: αK≈ αem≈ 137.035999074(44).
The latter condition gives n=54=2× 33 and 1/αK≈ 137.51. The deviation from the fine structure constant is Δ α/α= 3× 10-3 -- .3 per cent. For n=53 one obtains 1/αK= 134.96 with error of 1.5 per cent. For n=55 one obtains 1/αK= 150.06 with error of 2.2 per cent. Is the relatively good prediction could be a mere accident or there is something deeper involved?
This approach leads to different algebraic structure of αK than the earlier arguments.
- αK is rational multiple of π so that gK2 is proportional to π2. At the level of quantum TGD the theory is completely integrable by the definition of WCW integration(!) and there are no radiative corrections in WCW integration. Hence αK does not appear in vertices and therefore does not produce any problems in p-adic sectors.
- This approach is consistent with the proposed formula relating gravitational constant and p-adic length scale. G/Lp2 for p=M127 would be rational power of e now and number theoretically universally. A good guess is that G does not depend on p. As found this could be achieved also if the volume of CP2 type extremal depends on p so that the formula holds for all primes. αK could also depend on algebraic extension of rationals to guarantee the independence of G on p. Note that preferred p-adic primes correspond to ramified primes of the extension so that extensions are labelled by collections of ramified primes, and the ramimified prime corresponding to gravitonic space-time sheets should appear in the formula for G/Lp2.
- Also the speculative scenario for coupling constant evolution could remain as such. Could the p-adic coupling constant evolution for the gauge coupling strengths be due to the breaking of number theoretical universality bringing in dependence on p? This would require mapping of p-adic coupling strength to their real counterparts and the variant of canonical identification used is not unique.
- A more attractive possibility is that coupling constants are algebraically universal (no dependence on number field). Even the value of αK, although number theoretically universal, could depend on the algebraic extension of rationals defining the adele. In this case coupling constant evolution would reflect the evolution assignable to the increasing complexity of algebraic extension of rationals. The dependence of coupling constants on p-adic prime would be induced by the fact that so called ramified primes are physically favored and characterize the algebraic extension of rationals used.
- One must also remember that the running coupling constants are associated with QFT limit of TGD obtained by lumping the sheets of many-sheeted space-time to single region of Minkowski space. Coupling constant evolution would emerge at this limit. Whether this evolution reflects number theoretical evolution as function of algebraic extension of rationals, is an interesting question.
See the chapter Coupling Constant Evolution in Quantum TGD and the chapter Unified Number Theoretic Vision of "Physics as Generalized Number Theory" or the article Could one realize number theoretical universality for functional integral?.