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.

- 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

- 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 CP
_{2}type vacuum extremal, p-adic length scale as dimensional quantity (see this). Needless to say these attempts were premature and a hoc.

- The vision about hierarchy of Planck constants h
_{eff}=n× h and the connection h_{eff}= h_{gr}= GMm/v_{0}, where v_{0}<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.

- 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.

- 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. CP_{2}length scale indeed corresponds to GUT scale.

- 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/[cos^{2}(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.

- 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.

_{F}((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≈ k

^{k}, 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.

## 16 comments:

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

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.

So the relation is the ordering in which they appear and not some other permutation?

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...

wikipedia 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

𝒥^(2,+)x(t)={(p,X)|x(t+z)⩽x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}

𝒥^(2,-)x(t)={(p,X)|x(t+z)⩾u(x)+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?

corrected

𝒥^(2,+)x(t)={(p,X)|x(t+z)⩽x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}

𝒥^(2,-)x(t)={(p,X)|x(t+z)⩾x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}

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;-).

Fejer kernel seems to be average of approximations to delta function at zero. Easier to remember.

a paper on the hot topic of interlacing polynomials is at http://arxiv.org/abs/1306.3969v4

do you know if this applies to only polynomials? does it generalize to infinite dimensional case?

Not a slightest idea;-). My source of frustrations as a physicist is that mathematicians see so totally different things as relevant. They see only the technical challenges related to proving theorems - many of which are "obvious" for physicist and mere technical details. Mathematicians would have ideal skills for challenging the assumptions behind the basic notions- say the notion of state of von Neumann algebra but they are happy to just prove theorems.

Kadison-Singer problem relates to extension of pure states in Abelian algebra of diagonal bounded operators -maximal commuting set of observable for physicist- to the entire algebra. State in the sense used is the one used by C*-algebra people trying to reduce quantum field theory to density matrices.

The profound problem of algebraic quantum field theory -AQFT - is that it only produces the statistical aspects of quantum theory - "thermodynamics". States in this sense indeed are counterparts for density matrices. One can identify pure states as counterparts of quantum states but the description of interference effects etc become difficult and one loses the physical picture so relevant for quantum theory.

Zero energy ontology leading to "complex square root" of thermodynamics is required and this brings in complex hermitian square roots of density matrices and also that of partition function. One obtains generalisation of quantum state and also of AQFT. Basic theorems like Tomita's theorem (I hope I remember correctly) should be generalised. Many things to do but ...

Yeah.. mathematicians are annoying like that. Sorry to go off topic.. I'm trying to understand H=xp .. here is an idea related to coherent states rather than eigenstates. I get what you are saying about the complexification .. it's a nice idea..

https://scholar.google.com/scholar?cluster=16925851478872994728&hl=en&as_sdt=0,44&sciodt=0,44

Concerning H=xp and 1/2 in the non-trivial zeros of zeta. This can be understood also physically in zero energy ontology.

When one constructs tangent space basis in the "world of classical worlds", one has by holography analogs of plane waves at light-cone boundary propagating in radial light-like direction with coordinate r. They have the form r^{q +iy) and preferred extremal condition gives extremely powerful additional constraints quantisation q+iy: the first guess is that non-trivial zeros of zeta are obtained q=1/2.

How to understand q=1/2. The natural scaling invariant integration measure defining inner product for "plane" waves is dr/r =dlog(r). The inner product must be same as for ordinary plane waves and indeed is for plane waves psi= r^(1/2+iy) since in inner product r from Int psi_1*psi_2 dr/r is cancelled by 1/r from integration measure.

One has analogs of ordinary plane waves with delta function normalisation. The identification of 1/2+iy is zero of zeta is natural by generalised conformal invariance.

If one assumes that p-adic primes correspond to zeros of zeta in 1-1 manner in the sense that p^iy(p) is root of unity existing in all number fields (algebraic extension of p-adics) one obtains that the plane wave exists for p at points r= p^n: powers of p one obtains a delta function distribution concentrated on powers of p: logarithmic lattice. This can be seen as space-time correlate for p-adicity. Something very similar is obtained from the Fourier transform of distribution of zeros at critical line (Dyson).

My own "Strategy for Proving Riemann Hypothesis" relies on coherent states instead of eigenstates of Hamiltonian. The above approach in turn absorbs the problematic 1/2 to the integration measure at light cone boundary and conformal invariance is also now central.

Quite generally, I believe that conformal invariance in extended form applying at metrically 2-D light-cone boundary (light-like orbits of partonic 2-surfaces) might be central for understanding why physics requires RH.

For instance, generating elements of extended supersymplectic algebra are labelled by generators having zeros of zeta as conformal weights. The number of generators is infinite. If some generator(s). s=1/2+iy guarantees that the real parts of conformal weights for all states are half integers. By conformal confinement the sum of y:s vanish for physical states. If some weight is not at critical line the situation changes. s= x+iy gives all multiples of x shifted by all half odd integer values. And of course, the realisation as plane waves at boundary fails.

Very nice, Thanks for the comment. For some reason I had the idea this relates to yang-mills mass gap , if , from what I understand that it's the difference between the lowest and next lowest eigenvalue.. in the H=xp paper they say that the RiemannSiegel vartheta function has a symmetry that indicates the position and momentum eigenfunctions are time-reverses of each other and that H=xp "generates dilations"

The answer contained a stupid little euro related to the measure dr/r. I wrote a posting where it is done correctly.

I noticed also that the existence of supersymplectic representations free of pathologies in turn essential for the existence of WCW strongly suggests that zeros are at critical line: assuming that the conformal weights are forced by strong form of holography to be zeros of zeta.

Scaling operators for plane waves at light-cone boundary corresponds to the dilation operator H=xp.

Eigenvalues must however contain strange looking real part 1/2 which would spoil unitarity at real axis. But since the inner product defined by the integral from 0 to infty now ordinary plane waves would be pathological. Presumably so because probability could leak out at origin. Real part 1/2 removes the pathology by reducing the inner product to standard inner product at real line.

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