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A Atiyah's arithmetic physics

  1. Sep 27, 2018 #21
    I'm pleasantly surprised at how well the neutrino mass and mixing parameters are now known.

    I also neglected to note the error of the quarks' CP-violating phase. it is about 0.05 radians.

    The data for the CKM matrix (from review at pdg.lbl.gov): |Vud| = 0.94720 +- 0.00021, |Vus| = 0.2243 +- 0.0005, |Vcd| = 0.218 +- 0.004, |Vcs| = 0.997 +- 0.017, |Vcb| = (42.2 +- 0.8)*10^(-3), |Vub| = (3.94 +- 0.36)*10^(-3), |Vtd| = (8.1 +- 0.5)*10^(-3), |Vts| = (39.4 +- 2.3)*10^(-3), |Vtb| = 1.019 +- 0.025

    The matrix's errors range from 0.00021 to 0.025 (absolute) and 0.00021 to 0.09 (relative) -- very good.

    So there is a lot for BSM theories to try to predict.
  2. Sep 28, 2018 #22
    This approach should not just give the electromagnetic coupling constant, but all possible coupling constants including Newton's constant.

    It is also claimed that the precision of the numbers can be given to arbitrarily high precision. It seems that no one so far has actually been able to reproduce an explicit numerical calculation of the function, let alone evaluate it for ##\pi##.
  3. Sep 29, 2018 #23
    To compare to GUT predictions, one has to extrapolate Standard-Model parameters to above the energy scale where electroweak symmetry breaking happens. That energy scale is roughly the Higgs particle's vacuum expectation value, 246 GeV, and for definiteness, we may use that value.

    The most precise input available for the Standard Model's numerical values is the fine-structure constant, but it is measured at essentially zero momentum transfer. To get up to the EWSB energy scale requires renormalization, and to lowest order, that is by calculating one-loop corrections to the photon propagator -- a photon turns into two charged particles, which then turn back into a photon again. This can be done precisely for charged leptons, but quarks are another story. It is difficult to do the calculations at color-confinement energy scales, because quarks' interactions turn superstrong there. I recall from somewhere that one has to do the expedient of using measurements of the rate of e+e- -> hadrons as inputs. But once one gets far enough above that energy scale, quarks can be treated as almost free particles. For color-confinement-scale calculations with a precision of 0.1 - 0.01, this means that renormalizing the FSC up to EWSB energy scales will only have a precision of 10^(-3) - 10^(-4) (1000 to 100 ppm). Renormalizing the low-energy weak-interaction rate may have similar precision. This is fairly close to how well we know the W and Z masses, and those masses don't need renormalization through the color-confinement energy scale. So we have five quantities that are mainly determined by three Standard-Model ones, the two electroweak gauge-coupling constants and the Higgs vev. This gives us consistency checks for the Standard Model, or alternately, a way of measuring BSM effects.

    Electron masses are better, since hadronic effects set in at the two-loop level, making them 10^(-5) - 10^(-6) (10 to 1 ppm). Muon and tau masses are also good in this way, though the tau's mass error is larger than that. Quark masses are more difficult, though the top quark's mass is known to within about 0.2%.

    So in summary, several parameters of the unbroken Standard Model are known to 1% or better.
  4. Sep 29, 2018 #24
    The method of calculation used by Atiyah is claimed to supersede the entire Feynman diagram loop correction scheme, it is instead based on a much more general mathematical version of renormalization than used in QED and other QFTs.

    The form of renormalization Atiyah opts for is an algebraic renormalization scheme involving infinitely iterated complex exponentials, giving a much higher convergence speed in the calculation.

    He cooked up this particular numerical scheme based on an analogy of how Euler significantly improved upon the convergence speed towards obtaining the digits of ##\pi## using ##n^2## opposed to Archimedes' classical equation ##\pi(n)=n{\frac {\sin 180°} n}## with the convergence speed scaling with ##n##.

    In other words, Atiyah's scheme isn't merely another way of doing renormalization, it is a completely new branch of physics, predicting among other things all coupling constants at all possible energy scales in physics. This is of course given anyone can actually reproduce his numerics.
  5. Sep 29, 2018 #25
    Even if they did, it would all only be numerology. There is nothing in the paper like an equation of motion, let alone one employing Atiyah's new constant. No alternative method of calculating anything physical is provided.

    His idea seems to be: the forces correspond to an algebraic hierarchy (levels I through III), the coupling constants are mathematical constants that appear at the different levels, and exactly how this comes together as physics will be figured out later.

    The most positive thing I can say, is that this is a lesson in imagination and in thinking big. The idea that the various couplings will arise as "the noncommutative counterpart of pi" or "the nonassociative counterpart of pi", in the context of a novel algebraic ordering of the fundamental forces, is visionary and systematic. One should hope for and aim for ideas so striking and clear. Nonetheless, this particular idea also seems to be completely wrong.
  6. Sep 29, 2018 #26
    Its good to see that you are skeptical, I am as well. Having said that I am also a bit more optimistic, or - more correctly stated - more excited; there is a larger theme surrounding the ideas which most people seem to not have picked up on yet.

    Quite honestly it's been quite a long while, that I was actually this excited about some mathematical method. The nice thing about Atiyah's equations is that they both tie in quite nicely with some already existing proposals in theoretical physics as well as imply some new things about old theories.

    I myself am also trying to reproduce all of his numerics; it may be a wild goose chase but what is there to lose? If it works, this will be the first real progress in theoretical physics in 40 years, and if it doesn't we'll have learned some new potentially useful rapid convergence computational techniques.

    I would love to be much more specific about what it is I'm exactly on about, but I don't wanna jump the gun. Needless to say I'd prefer to have the competition for potential new discoveries in theoretical physics based on this remain at a bare minimum as well :wink:
  7. Sep 30, 2018 #27
    Something like Series acceleration - Wikipedia? Or Nima Arkani-Hamed's amplituhedron?

    If he had some construction that is mathematically equivalent to evaluating several Feynman diagrams together, that would be very valuable. Even if it was for some simplified theory, like a pure gauge theory. But getting the value of the fine structure constant requires the full complexity of the Standard Model and whatever GUT produces it.
  8. Sep 30, 2018 #28
    Yes, the method is exactly a form of nonlinear series acceleration and at the same time something new like the amplituhedron.

    Like the amplituhedron, it is fully constructed in terms of algebraic geometry and complex manifolds, but unlike the amplituhedron, the specific mapping also seems to serve as a bridge directly connecting number theory to analysis through among other things the Riemann zeta function.
    The connection to (SM) physics comes in through von Neumann algebras, more specifically the (hyperfinite) factors therein.
    Last edited: Sep 30, 2018
  9. Oct 4, 2018 #29
    The use of von Neumann hyperfinite factors/Bott periodicity and conformal/complex structures sounds like Tony Smith's idea for linking Armand Wyler's math for the fine structure constant to diffusion equations in an 8-dim Kaluza Klein spacetime.
  10. Oct 8, 2018 #30


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    That boat sailed long before you were born.
  11. Oct 25, 2018 #31
    A slightly altered perspective has made Atiyah's claim interesting to me again. I wrote:
    In other words, there is some existing fine-structure-constant numerology in which the mathematical constant employed is something from chaos theory.

    Now here's the thing. Feigenbaum's constant shows up in dynamical systems in a variety of contexts. What if Atiyah has simply discovered another example, this time in the context of von Neumann algebras? This suggests a different way of looking at what he wrote. One may remain agnostic or skeptical about the claimed connection with the fine-structure constant. The immediate focus should instead be on whether he could have found a new occurrence of Feigenbaum's constant.

    Here we should face again the fact that no participant in the Internet discussions around Atiyah's claims has understood his two papers in anything like a comprehensive way. People just focus on some little part that they think they understand. For example, it's only now that I really noticed the actual formula for "ж"! ... equation 8.11, a double limit of a sum of "Bernoulli numbers of higher order". And when I check the reddit attempt to reproduce Atiyah's calculation, 8.11 is all but ignored.

    So are there any formulas for Feigenbaum's constant? I haven't found anything like a series that converges on it. Instead, I find purely numerical (and thus quasi-empirical) ways to obtain it, by simulating the behavior of specific dynamical systems; also a collection of weird approximations, and I can't tell if any of them derive from the deep properties of the constant, or if they are just approximations. I also have not found any discussion prior to this, relating Feigenbaum's constant to the Bernoulli numbers. But the latter are combinatorial and do show up in some "branching tree" contexts reminiscent of period doubling.

    Anyway, this gives new meaning to some of Atiyah's propositions. For example, ж is supposed to play the role of π in a kind of quaternionic Euler equation. He also implies (section 7 of "The Fine-Structure Constant" preprint) that the pieces of the sum that converges on ж, come from homotopy groups. Well, the quaternionic Hopf fibration, which e.g. "gives an element in the 7th homotopy group of the 4-sphere", can in fact be used to analyze some kinds of Hopf bifurcation, where a fixed point of a dynamical system is replaced by a periodic orbit.

    These fragmentary connections are just straws in the wind. Perhaps they don't ultimately cohere. But at this point, there's still something to investigate here.
  12. Oct 25, 2018 #32
    Damn, you seem to be picking up on it as well :) which is why I said:
    By existing proposals I was also specifically referring to among other things the dynamical systems/chaos theory aspect. This may be somewhat confusing because to me it is de facto physics, while in most circles and academic classifications it is usually classified as applied math, leading most physicists to just ignore it or view it as some kind of numerology instead of an extension of theoretical dynamics.
    Exactly what I have been saying here. I had been trying to reproduce everything in section 8 up to and including 8.11 using Mathematica, but the computer just gives up on me before I am even able to reach 7 iterations.
    From what I recall from rereading the literature a few years ago, Feigenbaum's constant is actually a numerical factor characterising the Mandelbrot set.

    Apart from Feigenbaum 1979, in which Feigenbaum defined his ##\alpha##, ##\delta## and the other exponents using renormalization from statistical physics, the correspondence with the Mandelbrot set is the only other kind of pure derivation of the factor that I could do/find.
    Exactly my point.
  13. Oct 26, 2018 #33
    I just started reading Manasson's paper. Just made it to the end of page 2 and my mind has already been blown several times, but in particular that SU(2) symmetry can be viewed as an instance of period doubling! This simple realization never occurred to me before.

    I'm going to continue reading but I got so excited I had to comment here first: viewing QT as a dissipative open system makes far more sense to me then anything I have ever heard from any string/loop/GUT proponent! From the moment I began learning dynamical systems theory, I have always suspected that there was some linearizable nonlinearity underlying QT, which of course is traditionally assumed to be completely linear due to superposition and unitarity.

    It just so happens that I am reading a recent work by Schuh on a nonlinear reformulation of QT. Schuh is trying to find a nonlinear differential equation which can be exactly linearized so it both can respect the superposition principle as well as have sensitive dependence on initial conditions. Fortunately there already even exists an NDE with all the above properties, namely the complex Riccati equation and this equation also plays a role in QM!

    Both Schuh as well as Manasson seem to also reach the same conclusions: taking such nonlinearities seriously also just so happens to be able to mathematically offer a model for the arising of discontinuities within the equations and so unintentionally giving an actual prediction/explanation of state vector reduction, i.e. offering an actual resolution of the measurement problem, incidentally also confirming both Penrose and 't Hooft's intuitions on these matters. As far as I'm concerned that part is just icing on the cake.
  14. Nov 15, 2018 #34
    Minor development: Penrose (close colleague/friend of Atiyah, studied together under Todd) recently gave a talk where someone asked his opinion on this preprint by Atiyah on the fine structure constant.

    Penrose however said he hasn't read any of these preprints and therefore couldn't comment; he just said something along the lines of Atiyah just wanting to get the ideas out there.

    The next time I see Penrose, I'll try to see if I can get him to give me his intuition/opinion on the properties of the most important equation of section 8 or why Mathematica can't seem to handle it, without saying where it is from.
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