Does Riemann Zeta Code for Generic Coupling Constant Evolution?

The understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.

1. The first idea dates back to the discovery of WCW Kähler geometry defined by Kähler function defined by Kähler action (this happened around 1990) (see this). The only free parameter of the theory is Kähler coupling strength α_K analogous to temperature parameter α_K postulated to be is analogous to critical temperature. Whether only single value or entire spectrum of of values α_K is possible, remained an open question.

   About decade ago I realized that Kähler action is complex receiving a real contribution from space-time regions of Euclidian signature of metric and imaginary contribution from the Minkoswkian regions. Euclidian region would give Kähler function and Minkowskian regions analog of QFT action of path integral approach defining also Morse function. Zero energy ontology (ZEO) (see this) led to the interpretation of quantum TGD as complex square root of thermodynamics so that the vacuum functional as exponent of Kähler action could be identified as a complex square root of the ordinary partition function. Kähler function would correspond to the real contribution Kähler action from Euclidian space-time regions. This led to ask whether also Kähler coupling strength might be complex: in analogy with the complexification of gauge coupling strength in theories allowing magnetic monopoles. Complex α_K could allow to explain CP breaking. I proposed that instanton term also reducing to Chern-Simons term could be behind CP breaking

2. p-Adic mass calculations for 2 decades ago (see this) inspired the idea that length scale evolution is discretized so that the real version of p-adic coupling constant would have discrete set of values labelled by p-adic primes. The simple working hypothesis was that Kähler coupling strength is renormalization group (RG) invariant and only the weak and color coupling strengths depend on the p-adic length scale. The alternative ad hoc hypothesis considered was that gravitational constant is RG invariant. I made several number theoretically motivated ad hoc guesses about coupling constant evolution, in particular a guess for the formula for gravitational coupling in terms of Kähler coupling strength, action for CP₂ type vacuum extremal, p-adic length scale as dimensional quantity. Needless to say these attempts were premature and a hoc.
3. The vision about hierarchy of Planck constants h_eff=n× h and the connection h_eff= h_gr= GMm/v₀, where v₀<c=1 has dimensions of velocity (see this>) forced to consider very seriously the hypothesis that Kähler coupling strength has a spectrum of values in one-one correspondence with p-adic length scales. A separate coupling constant evolution associated with h_eff induced by α_K∝ 1/hbar_eff ∝ 1/n looks natural and was motivated by the idea that Nature is theoretician friendly: when the situation becomes non-perturbative, Mother Nature comes in rescue and an h_eff increasing phase transition makes the situation perturbative again.

   Quite recently the number theoretic interpretation of coupling constant evolution (see this> or this in terms of a hierarchy of algebraic extensions of rational numbers inducing those of p-adic number fields encouraged to think that 1/α_K has spectrum labelled by primes and values of h_eff. Two coupling constant evolutions suggest themselves: they could be assigned to length scales and angles which are in p-adic sectors necessarily discretized and describable using only algebraic extensions involve roots of unity replacing angles with discrete phases.

4. Few years ago the relationship of TGD and GRT was finally understood (see this>) . GRT space-time is obtained as an approximation as the sheets of the many-sheeted space-time of TGD are replaced with single region of space-time. The gravitational and gauge potential of sheets add together so that linear superposition corresponds to set theoretic union geometrically. This forced to consider the possibility that gauge coupling evolution takes place only at the level of the QFT approximation and α_K has only single value. This is nice but if true, one does not have much to say about the evolution of gauge coupling strengths.
5. The analogy of Riemann zeta function with the partition function of complex square root of thermodynamics suggests that the zeros of zeta have interpretation as inverses of complex temperatures s=1/β. Also 1/α_K is analogous to temperature. This led to a radical idea to be discussed in detail in the sequel.

   Could the spectrum of 1/α_K reduce to that for the zeros of Riemann zeta or - more plausibly - to the spectrum of poles of fermionic zeta ζ_F(ks)= ζ(ks)/ζ(2ks) giving for k=1/2 poles as zeros of zeta and as point s=2? ζ_F is motivated by the fact that fermions are the only fundamental particles in TGD and by the fact that poles of the partition function are naturally associated with quantum criticality whereas the vanishing of ζ and varying sign allow no natural physical interpretation.

   The poles of ζ_F(s/2) define the spectrum of 1/α_K and correspond to zeros of ζ(s) and to the pole of ζ(s/2) at s=2. The trivial poles for s=2n, n=1,2,.. correspond naturally to the values of 1/α_K for different values of h_eff=n× h with n even integer. Complex poles would correspond to ordinary QFT coupling constant evolution. The zeros of zeta in increasing order would correspond to p-adic primes in increasing order and UV limit to smallest value of poles at critical line. One can distinguish the pole s=2 as extreme UV limit at which QFT approximation fails totally. CP₂ length scale indeed corresponds to GUT scale.

6. One can test this hypothesis. 1/α_K corresponds to the electroweak U(1) coupling strength so that the identification 1/α_K= 1/α_U(1) makes sense. One also knows a lot about the evolutions of 1/α_U(1) and of electromagnetic coupling strength 1/α_em= 1/[cos²(θ_W)α_U(1). What does this predict?

   It turns out that at p-adic length scale k=131 (p≈ 2^k by p-adic length scale hypothesis, which now can be understood number theoretically (see this ) fine structure constant is predicted with .7 per cent accuracy if Weinberg angle is assumed to have its value at atomic scale! It is difficult to believe that this could be a mere accident because also the prediction evolution of α_U(1) is correct qualitatively. Note however that for k=127 labelling electron one can reproduce fine structure constant with Weinberg angle deviating about 10 per cent from the measured value of Weinberg angle. Both models will be considered.

7. What about the evolution of weak, color and gravitational coupling strengths? Quantum criticality suggests that the evolution of these couplings strengths is universal and independent of the details of the dynamics. Since one must be able to compare various evolutions and combine them together, the only possibility seems to be that the spectra of gauge coupling strengths are given by the poles of ζ_F(w) but with argument w=w(s) obtained by a global conformal transformation of upper half plane - that is Möbius transformation (see this) with real coefficients (element of GL(2,R)) so that one as ζ_F((as+b)/(cs+d)). Rather general arguments force it to be and element of GL(2,Q), GL(2,Z) or maybe even SL(2,Z) (ad-bc=1) satisfying additional constraints. Since TGD predicts several scaled variants of weak and color interactions, these copies could be perhaps parameterized by some elements of SL(2,Z) and by a scaling factor K.

   Could one understand the general qualitative features of color and weak coupling contant evolutions from the properties of corresponding Möbius transformation? At the critical line there can be no poles or zeros but could asymptotic freedom be assigned with a pole of cs+d and color confinement with the zero of as+b at real axes? Pole makes sense only if Kähler action for the preferred extremal vanishes. Vanishing can occur and does so for massless extremals characterizing conformally invariant phase. For zero of as+b vacuum function would be equal to one unless Kähler action is allowed to be infinite: does this make sense?. One can however hope that the values of parameters allow to distinguish between weak and color interactions. It is certainly possible to get an idea about the values of the parameters of the transformation and one ends up with a general model predicting the entire electroweak coupling constant evolution successfully.

See the new chapter Does Riemann Zeta Code for Generic Coupling Constant Evolution? or the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?.