Could the basic parameters of TGD be fixed by a number theoretical miracle?

If the v₀ deduced to have value v₀=2^-11 appearing in the expression for gravitational Planck constant hbar_gr=GMm/v₀ is identified as the rotation velocity of distant stars in galactic plane, it is possible to express it in terms of Kähler coupling strength and string tension as v₀^-2= 2×α_KK,

α_K(p)= a/log(pK) , K= R²/G .

The value of K is fixed to a high degree by the requirement that electron mass scale comes out correctly in p-adic mass calculations. The uncertainties related to second order contributions in p-adic mass calculations however leave the precise value open. Number theoretic arguments suggest that K is expressible as a product of primes p ≤ 23: K= 2×3×5×...×23 .

If one assumes that α_K is of order fine structure constant in electron length scale, the value of the parameter a cannot be far from unity. A more precise condition would result by identifying α_K with weak U(1) coupling strength α_K = α_U(1)=α_em/cos²(θ_W)≈ 1/105.3531 ,

sin²(θ_W)≈ .23120(15),

α_em= 0.00729735253327 .

Here the values refer to electron length scale. If the formula v₀= 2^-11 is exact, it poses both quantitative and number theoretic conditions on Kähler coupling strength. One must of course remember, that exact expression for v₀ corresponds to only one particular solution and even smallest deformation of solution can change the number theoretical anatomy completely. In any case one can make following questions.

1. Could one understand why v₀≈ 2^-11 must hold true.
2. What number theoretical implications the exact formula v₀= 2^-11 has in case that it is consistent with the above listed assumptions?

1. Are the ratios π/log(q) rational?

The basic condition stating that gravitational coupling constant is renormalization group invariant dictates the dependence of the Kähler coupling strength of p-adic prime exponent of Kähler action for CP₂ type extremal is rational if K is integer as assumed: this is essential for the algebraic continuation of the rational physics to p-adic number fields. This gives a general formula α_K= a π/log(pK). Since K is integer, this means that v₀² is of form

v₀²= qlog(pK)/π, q rational.

if a is rational.

1. Since v₀² should be rational for rational value of a, the minimal conclusion would be that the number log(pK)/π should be rational for some preferred prime p=p₀ in this case. If this miracle occurs, the p-adic coupling constant evolution of Kähler coupling strength, the only coupling constant in TGD, would be completely fixed. Same would also hold true for the ratio of CP₂ to length characterized by K^1/2.

2. A more general conjecture would be that log(q)/π is rational for q rational: this conjecture turns out to be wrong as discussed in the previous posting. The rationality of π/log(q) for single q is however possible in principle and would imply that exp(π) is an algebraic number. This would indeed look extremely nice since the algebraic character of exp(π) would conform with the algebraic character of the phases exp(iπ/n). Unfortunately this is not the case. Hence one loses the extremely attractive possibility to fix the basic parameters of theory completely from number theory.

The condition for v₀=2^-m, m=11, allows to deduce the value of a as

a= (log(pK)/π) × (2^2m/K).

The condition that α_K is of order fine structure constant for p=M₁₂₇= 2¹²⁷-1 defining the p-adic length scale of electron indeed implies that m=11 is the only possible value since the value of a is scaled by a factor 4 in m→ m+1.

The value of α_K in the length scale L_p0 in which condition of the first equation holds true is given by

1/α_K= 2²¹/K≈ 106.379 .

2. What is the value of the preferred prime p₀?

The condition for v₀ can hold only for a single p-adic length scale L_p0. This correspondence would presumably mean that gravitational interaction is mediated along the space-time sheets characterized by p₀, or even that gravitons are characterized by p₀.

1. If same p₀ characterizes all ordinary gauge bosons with their dark variants included, one would have p₀=M₈₉=2⁸⁹-1.

2. One can however argue that dark gravitons and dark bosons in general can correspond to different Mersenne prime than ordinary gauge bosons. Since Mersenne primes larger than M₁₂₇ define super-astrophysical length scales, M₁₂₇ is the unique candidate. M₁₂₇ indeed defines a dark length scale in TGD inspired quantum model of living matter. This predicts 1/α_U(1)(M₁₂₇)= 106.379 to be compared with the experimental estimate 1/α_U(1)(M₁₂₇)= 105.3531 deduced above. The deviation is smaller than one percent, which indeed puts bells ringing!

This agreement seems to provide dramatic support for the general picture but one must be very cautious.

1. The identification of Kähler coupling strength as U(1) coupling strength poses strong conditions on the p-adic length scale evolution of Weinberg angle using the knowledge about the evolution of the electromagnetic coupling constant. The condition

   cos²(θ_W)(89)= [log(M₁₂₇K)/log(M₈₉K)] × [α_em(M₁₂₇)/α_em(M₈₉)]× cos²(θ_W)(127) .

   Using the experimental value 1/α_em(M₈₉)≈ 128 as predicted by standard model one obtains sin²(θ_W)(89)=.0479. There is a bad conflict with experimental facts unless the experimentally determined value of Weinberg angle corresponds to M₁₂₇ space-time sheet.

   I will leave leave the implications of this conflict to the future posting.

The reader interested in details is recommended to look previous postings and the new chapter Can TGD Predict the Spectrum of Planck Constants? of the book "TGD: an Overview" and the chapter TGD and Astrophysics of the book "Physics in Many-Sheeted Space-Time".