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.(adsbygoogle = window.adsbygoogle || []).push({});

Manasson observed that the fine-structure constant is approximately 1/(2π δ^{2}), 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 constantin the deep infraredwould 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) acoupling constantat 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 thatthere existsa 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.

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# A Fine structure and fixed points

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