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A Fine structure and fixed points

  1. Nov 1, 2018 #1
    I apologize for creating a new thread which has significant overlap with two other ongoing threads ("Quantization isn't fundamental", "Atiyah's arithmetic physics"). But both those threads discuss theories or paradigms of extreme breadth, whereas here I want to focus on a very specific bundle of issues.

    Manasson observed that the fine-structure constant is approximately 1/(2π δ2), where δ is the Feigenbaum universality constant. Technically this constant is

    "the largest eigenvalue of the derivative of the renormalisation operator at its unique fixed point"

    where renormalization does not specifically mean renormalization as in QFT, but a more general concept in which one is moving among a family of parametrized dynamical systems.

    Now, coupling constants run with energy, in a way that depends on the logarithm of the energy, and the famous value of the fine structure constant is its value only at the lowest energies - in the infrared, as the low energy realm is called. So to believe Manasson, one needs a reason why the fine structure constant in the deep infrared would be a function of the Feigenbaum constant.

    The running of QFT parameters, their renormalization group flow, is a kind of dynamics, and it can be analyzed in terms of concepts from dynamical systems theory, such as attractors and repellors. In particular, renormalized quantum field theories have fixed points - ultraviolet fixed points at high energies, and infrared fixed points at low energies.

    This slightly clarifies what we want: we want to obtain Manasson's formula at an infrared fixed point. But that still is not an answer. Feigenbaum's constant is a phenomenon of fixed points, but how does it become (part of) a coupling constant at a fixed point?

    Atiyah recently gave us a formula which supposedly derives from a kind of iteration that he calls renormalization, and which supposedly produces the fine structure constant, but no-one else has been able to reproduce the calculation he claims.

    However, there is another such formula, which does work quite well, and which obtains the fine-structure constant as a fixed point. It is due to Hans de Vries, and may have first appeared in this very forum, though by now it has at least two citations (Poelz 2012, eqn 39; Chiatti 2017, eqns 12-15).

    This ansatz looks quite promising as a way to justify Manasson's formula. For example, it has all those factors of 2π built in.

    In a recent paper, Cecotti and Vafa pointed out that the extreme infrared of our universe, consisting only of those particles that are strictly massless, should contain only photons and gravitons. And a handful of papers (1 2 3, as cited here, "Is there an infrared fixed point?") have reported the existence of an infrared fixed point in a particular form of quantum gravity.

    So the question now is, can the de Vries ansatz be obtained as the infrared fixed point of electromagnetism coupled to some form of gravity? I cannot yet answer that question, but it is the most concrete way I know to probe the plausibility of Manasson's formula.

    Now, at this point I have ceased to mention the Feigenbaum constant. Interestingly, there does not seem to be a simple formula for it. The proof of its universality just shows that there exists a single number which describes a broad class of dynamical systems, but doesn't tell you what the number is.

    The previous line of thought suggests that, if the de Vries ansatz works, then he not only found a formula for the fine-structure constant, but also a formula for the Feigenbaum constant! One should somehow be able to associate the de Vries fixed point, even with something as simple as the logistic map, the chaotic system where Feigenbaum first observed his constant.

    It may be that some or all of this leads nowhere. The de Vries ansatz, especially in the context of a photon-graviton infrared fixed point, just seems a nice concrete way to investigate the possibilities. But maybe something as arcane as Atiyah's formula is really needed. Or maybe none of this works at any level. Hopefully the truth will become clearer with some discussion.
  2. jcsd
  3. Nov 1, 2018 #2

    king vitamin

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    Can you unpack this definition for people who understand renormalization in physics but are not familiar with its more general definition that you refer to after this quote?
  4. Nov 2, 2018 #3
    Is the thread or post you mention still up? I looked in the list of citations in those papers and didn't see anything.
  5. Nov 4, 2018 #4
    I really wish I had more time to work on this, but work and my own research comes first. I tried to informally tag this off to a few math and comp sci undergrad/grad students without success. Its really interesting to me, because the reason they gave for not wanting to work on it was because the mathematicians in faculty are saying Atiyah has basically lost his marbles directly stating everything he said w.r.t. mathematics in the proof is just nonsense, while the physics side of the pre-prints (renormalization in QFT) is way too specialized to actually grasp properly.
    Thanks, I'm gonna look at these as soon as I find some time. Not thanks for destroying the rest of my free time.
    It would be nice if things did turn out to eventually vindicate Atiyah :smile:
  6. Nov 5, 2018 #5


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  7. Nov 9, 2018 #6
    I don't have a good way to say it, but Sfondrini's review contains an exposition. In section 3, he introduces a "renormalization operator" for a simple class of iterated dynamical systems. In the final paragraph of that section, he says that the δ eigenvector corresponds (in the jargon of Wilsonian renormalization) to a relevant perturbation. But I'm still not sure what the "physical interpretation" of δ itself would be.
  8. Nov 15, 2018 at 5:41 PM #7
    It's worth pointing out that if this is real, it needs to involve the whole electroweak sector. The following diagram from wikipedia is helpful...
    The point is that the electromagnetic coupling is actually a function of the hypercharge coupling and the weak isospin coupling. Since those quantities run with energy, one should imagine that right-angled triangle changing in proportion as one descends from higher energies. And then the question would be why, at the end, it arrives at proportions such that e ~ √(4πα) ~ √2/δ.

    It's the Higgs field, a charged condensate with hypercharge and weak isospin quantum numbers, which removes 3/4 of the U(2) electroweak gauge bosons from long-range physics, leaving only the photon. And it is widely observed that, in various ways, the Higgs field is "critical": the electroweak vacuum is at the edge of stability, the Higgs mass appears to be finetuned. There have been suggestions that the Higgs is "quantum critical" and even that it exhibits self-organized criticality.

    I find that second option - self-organized quantum criticality - attractive in this context. That second paper describes a Higgs-like system in which something called holographic BKT scaling occurs. Two critical points merge in a bifurcation, a coupling constant becomes complex-valued, and you get "discrete scale invariance". The discrete part is appealing because the only dynamical systems I know so far, where there are quantities described by 1/δ, are those discrete-time iterated maps.
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