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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 back-up 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/does-riemann-zeta-code-for-generic.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. 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 αK analogous to temperature parameter αK postulated to be is analogous to critical temperature. Whether only single value or entire spectrum of of values αK 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 αK could allow to explain CP breaking. I proposed that instanton term also reducing to Chern-Simons term could be behind CP breaking. 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 (see this). Needless to say, these attempts were premature and ad hoc. The vision about hierarchy of Planck constants heff=n× h and the connection heff= hgr= GMm/v0, where v0this>) 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 heff induced by αK∝ 1/hbareff ∝ 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 heff 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/αK has spectrum labelled by primes and values of heff. 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. 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 αK has only single value. This is nice. But if true, one does not have much to say about the evolution of gauge coupling strengths. 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/αK is analogous to temperature. This led to a radical idea to be discussed in detail in the sequel. Could the spectrum of 1/αK reduce to that for the zeros of Riemann zeta or - more plausibly - to the spectrum of poles of fermionic zeta ζF(ks)= ζ(ks)/ζ(2ks) giving for k=1/2 poles as zeros of zeta and as point s=2? ζF 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 ζF(s/2) define the spectrum of 1/αK 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/αK for different values of heff=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. One can test this hypothesis. 1/αK corresponds to the electroweak U(1) coupling strength so that the identification 1/αK= 1/αU(1) makes sense. One also knows a lot about the evolutions of 1/αU(1) and of electromagnetic coupling strength 1/αem= 1/[cos2(θW)αU(1). What does this predict? It turns out that at p-adic length scale k=131 (p≈ 2k 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 αU(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. 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 ζF(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 ζF((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. To sum up, the big idea is the identification of the spectra of coupling constant strengths as poles of ζF((as+b/)(cs+d)) identified as a complex square root of partition function with motivation coming from ZEO, quantum criticality, and super-conformal symmetry; the discretization of the RG flow made possible by the p-adic length scale hypothesis p≈ kk, k prime; and the assignment of complex zeros of ζ with p-adic 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 Quantum-TGD) 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 1-1: 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 1-1 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/2001-01.pdf You might be on to something here from what I can tell with my mathematical understanding... Wwikipedia has something about "almost-Hermitian 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 small-o |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 almost-Hermitian 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/about-fermi-dirac-and-bose-einstein.html#comments

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