^{4}projections of magnetic flux tubes.

**Dicrete coupling constant evolution: from quantum criticality or from the convergence of perturbation series?**

- h
_{eff}/h_0 =n identifiable as dimension for extension of rationals has integer spectrum. This allows the generalization of the formula for Newton's constant as G_{eff}= R^{2}/&hbar_{eff}with Planck length l^{P}identifies much longer CP_{2}size so that TGD involves only single fundamental length in accordance with the assumption held for about 35 years before the emergence of twistor lift of TGD. Therefore Newton's constants is varies and is different at different levels of dark matter hierarchy identifiable in terms of hierarchy of extensions of rationals

- As a special case one has hbar
_{eff}= h_{gr}= GMm/v_{0}. Is this case case the gravitational coupling become G_{eff}Mm= v_{0}and does not depend on masses or G at all. In quantum scattering amplitudes a dimensionless parameter (1/4π)v_{0}/c would appear in the role of gravitational fine structure constant and would be obtained from hbar_{eff}= h_{gr}= GMm/v_{0}consistent with Equivalence Principle. The miracle would be that G_{eff}would disappear totally from the perturbative expansion in terms of GMm as one finds by looking what α_{gr}= GMm/ℏ_{gr}is! This picture would work when GMm is larger than perturbative expansion fails to converge. For Mm above Planck mass squared this is expected to be the case. What happens below this limit is yet unclear (n is integer).

Could v

_{0}be fundamental coupling constant running only mildly? This does not seem to be the case: Nottale's original work proposing hbar_{gr}proposes that v_{0}for outer planets is by factor 1/5 smaller than for the inner planets (see this and this).

- This picture works also for other interactions (see this). Quite generally, Nature would be theoretician friendly and induce a phase transition increasing hbar when the coupling strength exceeds the value below which perturbation series converges so that perturbation series converges. In adelic physics this would mean increase of the algebraic complexity since h
_{eff}/h=n is the dimension of extension of rationals inducing the extensions of various p-adic number fields and defining the particular level in the adelic hierarchy (see this). The parameters characterizing space-time surfaces as preferred extremals of the action principle would be numbers in this extension of rationals so that the phase transition would have a well-defined mathematical meaning. In TGD the extensions of rationals would label different quantum critical phases in which coupling constants would not run so that coupling constant evolution would be discrete as function of the extension.

- When convergence is lost, a phase transition increasing algebraic complexity takes place and increases n. Extensions of rationals have also other characteristics than the dimension n.

For instance, each extension is characterized by ramified primes and the the proposal is that favoured p-adic primes assignable to cognition and also to elementary particles and physics in general correspond to so called ramified primes analogous to multiple zeros of polynomials. Therefore number theoretic evolution would also give rise to p-adic evolution as analog of ordinary coupling constant evolution with length scale.

- At quantum criticality coupling constant evolution is trivial. In QFT context this would mean that loops vanish separately or at least they sum up to zero for the critical values of coupling constants. This argument however seems to make theargument about the convergence of coupling constant expansion obsolete unless one allows only the quantum critical values of coupling constants guaranteeing that quantum TGD is quantum critical. There are strong reasons to believe that the TGD analog of twistor diagrammatics involves only tree diagrams and there are strong number theoretic argument for this: infinite sum of diagrams does not in general give a number in given extension of rationals. Quantum criticality would be forced by number theory.

- The consistency would be achieved if ordinary continuous coupling constant evolution is obtained as a completion of the discrete coupling constant evolution to real number based continuous evolution. Similar completions should make sense in p-adic sectors. These perturbation series should converge and this condition would force the phase transitions for the critical values of coupling constant strength the sum over the loop corrections would vanish and the outcome would be in the extension of rationals and make sense in extension of fany number field induced by the extension of rationals. Quantum criticality would boil down to number theoretical universality. The completions to continuous evolution are not unique and this would correspond to a finite measurement resolution achieved only at the limit of algebraic numbers.

One can ask whether one should regard this hierarchy as a hierarchy of approximations for space-time surfaces in M

^{8}represented as zero loci for real or imaginary part (in quaternionic sense) of octonion analytic functions obtained by replacing them with polynomials of finite degree. The picture based on the notion of WCW would correspond to this limit and the hierarchy of rational extensions to what cognitive representations can provide.

**Evolution of cosmological constant**

The goal is to understand the evolution of cosmological constant number theoretically and correlate it with the intuitive idea that cosmic expansion corresponds to the thickening of the M^{4} projections of cosmic strings in discrete phase transitions changing the value of the cosmological contant and other coupling parameters.

First some background is needed.

- The action for the twistor lift is 6-D analog of Kähler action for the 6-D surfaces in 12-D product of twistor spaces for M
^{4}and CP_{2}. The twistor space for M^{4}is the geometric variant of twistor space and simply the product M^{4}× S^{2}. For the allowed extremals of this action 6-D surfaces dimensionally reduces to twistor bundle over X^{4}having S^{2}as fiber. The action for the space-time surface is sum of Kähler action and 4-volume term. The coefficient of four-volume term has interpretation terms of cosmological constant and I have considered explicitly its p-adic evolution as function of p-adic length scale.

- The vision is that cosmological constant Λ behaves in average sense as 1/a
^{2}as function of the light-cone proper time assignable to causal diamond (CD) and serving as analog cosmological time coordinate. One can say that Λ is function of the scale of the space-time sheet and a as cosmological time defines this scale. This solves the problem due to large values of Λ at very early times. The size of Λ is reduced in p-adic length scale evolution occurring via phase transitions reducing Λ.

- p-Adic length scales are given by L
_{p}= kR p^{1/2}, where k is numerical constant and R is CP_{2}size - say radius of geodesic sphere. An attractive interpretation (see this) is that the real Planck length actually corresponds to R although it is by a factor of order 10^{-3.5}shorter. The point is that one identifies G_{eff}= R^{2}/hbar_{eff}with hbar_{eff}∼ 10^{7}for G_{N}. G_{eff}would thus depend on h_{eff}/h=n, which is essentially the dimension of extension of rationals, whose hierarchy gives rise to coupling constant evolution. Also evolution of Λ which is indeed coupling constant like quantity.

- p-Adic length scale evolution predicts discrete spectrum for Λ ∝ L
_{p}^{-2}∝ 1/p and p-adic length scale hypothesis stating that p-adic primes p≈ 2^{k}, k some (not arbitrary) integer, are preferred would reduce the evolution to phase transitions in which Λ changes by a power of 2. This would replace the

continuous cosmic expansion with a sequence of this kind of phase transitions. This would solve the paradox due to the fact that stellar objects participate to cosmic expansion but do not seem to expand themselves. The objects would expand in discrete jerks and so called Expanding Earth hypothesis would have TGD variant: in Cambrian explosion the radius of Earth increased by factor of 2 (see this and this).

^{2}× Y

^{2}⊂ M

^{4}× CP

_{2}, where X

^{2}is minimal surface - string world sheet- and Y

^{2}is complex sub-manifold of CP

_{2}- homologically non-trivial or trivial geodesic sphere in the simplest situation. In homologically non-trivial case there is monopoke flux of Kähler magnetic field along the string. M

^{4}projection of cosmic string is unstable against perturbations and gradually thickens during cosmic evolution.

- The Kähler magnetic energy of the flux tube is proprtional to B
^{2}SL where B is Kähler magnetic field, whose flux is quantized and does no change. By flux quantization B itself is roughly proportional to 1/S, S the area of M^{4}projection of the flux tube, which gradually thickens. Kähler energy is proportional to L/S and thus decreases as S increases unless L increases. In any case B weakens and this suggests that Kähler magnetic energy transforms to ordinary particles or their dark counterparts and part of particles remains inside flux tube as dark particles with h_{eff}/h_{0}=n characterizing the dimension of extension of rationals.

- What happens to the volume energy E
_{vol}? One E_{vol}∝ Λ LS and increases as S increases. This cannot make sense but the p-adic evolution of Λ as Λ ∝ 1/p saves the situation. Primes p possible or given extension of rationals would correspond to ramified primes for the extension. Cosmic expansion would take place as phases transition changing extension of rationals and the larger extension should posses larger ramified primes.

- The total energy of the flux tube would be of the form E= (a/S+bS)L corresponding to Kähler contribution and volume contribution. Physical intuition tells that also volume energy decreases during the sequence of the phase transitions thickening the string but also increasing the length of the string. The problem is that if bS is essentially constant, the volume energy of string like objects increases like L
_{p}if so long string like objects in cosmic scales are allowed. Situation changes if the string like objects are much shorter.

To understand whether this is possible, one must consider an alternative but equivalent view about cosmological constant (see this). The density ρ

_{vol}of the volume energy has dimensions 1/L^{4}and can be parametrized also by p-adic length scale L_{p1}: one would have ρ_{vol}∝ 1/L_{p1}^{4}. The p-adic prime p of the previous parametrization corresponds to cosmic length scale and one would have p∝ p_{1}^{2}, which for the size scale assignable to the age of the Universe observable to us corresponds to neutrino Compton length roughly. Galactic strings would however correspond to much longer strings and TGD indeed predicts a Russian doll fractal hierarchy of cosmologies.

- The condition that the value of volume part of the action for 4-volume L
_{p1}^{4}remains constant under p-adic evolution gives Λ ∝ 1/L_{p}^{4}. Parameterize volume energy as E_{vol}=bSL. Assume that string length L scales as L_{p}and require that the volume energy of flux tube scales as 1/L_{p}(Uncertainty Principle). Parameter b (that is Λ) would scale as 1/L_{p1}^{2}S. Consistency requires that S scales as L_{p1}^{2}. As a consequence, both volume and Kähler energy would decrease like 1/L_{p1}. Both Kähler and volume energy would transform to ordinary particles and their dark variants part of which would remain inside flux tube. The transformations would occur as phase transitions changing p_{1}and generate burst of radiation. The result looks strange at first but is due to the fact that L_{p1}is much shorter than L_{p}: for L_{p}the result is not possible.

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