Download 2.85 Mb.

archived as http://www.stealthskater.com/Documents/TGDBlog_2015.doc (also …TGDBlog_2015.pdf) => ^{doc} ^{pdf} ^{URL}^{doc} ^{URL}^{}^{pdf} more from Matti Pitkänen is on the /Pitkanen.htm page at ^{doc} ^{pdf} ^{URL} note: because important websites are frequently "here today but gone tomorrow", the following was archived from http://matpitka.blogspot.com on November 16, 2015. This is NOT an attempt to divert readers from the aforementioned website. Indeed, the reader should only read this backup copy if the updated original cannot be found at the original author's site. [note: listed in descending date order (latest is listed first and oldest is listed last) ] 11/18/2015  http://matpitka.blogspot.com/2015/11/doesriemannzetacodeforgeneric.html#comments Does Riemann Zeta Code for Generic Coupling Constant Evolution? Understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During the years I have made several attempts to understand coupling evolution.
About decade ago, I realized that Kähler action is complex receiving a real contribution from spacetime 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 spacetime 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 ChernSimons term could be behind CP breaking.
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 padic 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 padic sectors necessarily discretized and describable using only algebraic extensions involve roots of unity replacing angles with discrete phases.
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 padic 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. CP_{2} length scale indeed corresponds to GUT scale.
It turns out that at padic length scale k=131 (p≈ 2^{k} by padic 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.
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. To sum up, the big idea is the identification of the spectra of coupling constant strengths as poles of ζ_{F}((as+b/)(cs+d)) identified as a complex square root of partition function with motivation coming from ZEO, quantum criticality, and superconformal symmetry; the discretization of the RG flow made possible by the padic length scale hypothesis p≈ k^{k}, k prime; and the assignment of complex zeros of ζ with padic primes in increasing order. These assumptions reduce the coupling constant evolution to four real rational or integer valued parameters (a,b,c,d). One can say that one of the greatest challenges of TGD has been overcome. For details see the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?. For a summary of earlier postings see Links to the latest progress in TGD. posted by Matti Pitkanen @ 10:31 PM 9 Comments: At 2:50 PM, Anonymous said... The part I have doubts about is the validity of assigning a prime to each zero. I really doubt there is a one to one correspondence.. unless I missed something Stephen At 7:19 PM, Matpitka6@gmail.com said... I do not see as a question of whether to believe or not. Number theoretical universality (one of the basic principles of QuantumTGD) states that for given prime p p^iy exists for some set C(p) of zeros y. The strong form (supported now by the stunning success of the identification of zeros as inverses of U(1) coupling constant strength) states that correspondence is 11: C(p) contains only one zero. Another support for the hypothesis is that it works and predicts U(1) coupling at electron scale with accuracy of .7 per cent without any further assumptions and that it leads to a parametrisation of generic coupling constant evolution in terms of rational or integer parameter real Mobius transformation. This is incredibly powerful prediction: number theoretical universality would provide highly detailed overall view about physics in all length scales. No one has dared even to dream of anything like this. Dyson speculated that zeros and primes and their powers form quasicrystals. Ordinary crystal is such and zeros and primes would be analogous to lattice and reciprocal lattice and therefore in 11 correspondence naturally. At 11:35 PM, Anonymous said... So the relation is the ordering in which they appear and not some other permutation? At 1:55 PM, Stephen said... https://en.wikipedia.org/wiki/Fej%C3%A9r_kernel Unitary Correlations and the Fiejer kernel https://statistics.stanford.edu/sites/default/files/200101.pdf You might be on to something here from what I can tell with my mathematical understanding... Wwikipedia has something about "almostHermitian operators" I think this might be found in the last section where I briefly mention the possibility http://vixra.org/pdf/1510.0475v6.pdf on the last page in section 2.3 ��^(2,+)x(t)={(p,X)x(t+z)⩽x(t)+p⋅z+(X:z⊗z)/2+o(z^2) as z→0} ��^(2,)x(t)={(p,X)x(t+z)⩾x(t)+p⋅z+(X:z⊗z)/2+o(z^2) as z→0} what I think is so cool is that the error term just so happens to be smallo z^2 hapybe the 'approximation error' is also a (randomly.. at what level?) complex wavefunction? At 6:58 PM, Matti Pitkanen said... Ordering by size is essential for obtaining realistic coupling constant evolution. About almostHermitian operators. The problem of standard approach is that zeros s= 1/2+iy are not eigenvalues of a unitary operator. In Zero Energy Ontology wave function is replaced with a couple square root of density matrix and vacuum functional with a complex square root of partition function. This interpretational problem disappears. It is a pity that a too restricted view about quantum theory leads to misguided attempts to understand RH in terms of physical analogies. But this is not my problem;). At 7:06 PM, Matpitka6@gmail.com said... Fejer kernel seems to be average of approximations to delta function at zero. Easier to remember. 11/04/2015  http://matpitka.blogspot.com/2015/11/aboutfermidiracandboseeinstein.html#comments 
search 