Sir Michael Atiyah came out of left field and made sensational claims (proved the Riemann hypothesis, calculated the fine-structure constant) more characteristic of a vixra physicist. And indeed, his Riemann preprint has been uploaded to vixra, though I don't know if it came from him.

I started a forum thread on the physics of his claims. I have not succeeded in properly deciphering his procedure to produce the value of 1/α. The integer part is Eddington meets Bott, but no-one has been able to motivate the next few digits.

But one interesting thing came up in that thread. Atiyah speaks of an iterated process that he calls renormalization, and "Auto-Didact" remarked that it resembles something out of bifurcation theory. And as I have posted here previously, 1/α is approximately equal to 2π times the square of Feigenbaum's constant.

So if - against the odds - there is anything to Atiyah's baroque conceptions, I think it would involve the Feigenbaum connection. But for now, I expect nothing. It was stimulating, it stirred things up, but I think it's a blind alley.

# theory

not endorsed by snarxiv

## Friday, September 28, 2018

## Wednesday, September 5, 2018

### Historical interlude

Yet another subgenre of parodic scholarship has made its first appearance on vixra. An author whose name is Korean for "true history" overturns existing theories of the "Hwan-Suomi hyperwar", a clash of ancient supercivilizations unknown to normies and mundanes, but known to the cognoscenti of 4chan's "History and Humanities" board. Suomi means Finland, but these were hyperborean True Finns, superior to any modern stock; while the Hwan were the ancestral Korean master race, known to us thanks to the

*Handan gogi*, a 20th-century work of "nationalist pseudohistory" (so say the killjoys at RationalWiki). Sometimes, fiction really is stranger than truth.## Wednesday, June 13, 2018

### Koide from S-duality

1) Crackpot idea of the day: "the bottom quark is S-dual to the rho meson".

Gorsky et al conjecture that holographic QCD has a "flavor S-duality" in which vector mesons are dual to baryons. This is to be realized in string theory by a web of 5-branes.

Quark-hadron duality shows a kind of continuity between properties of quarks and properties of hadrons.

And the mass of the rho meson has been estimated at sqrt(6) times the constituent quark mass; while in the simplest version of Rivero's waterfall, the bottom quark mass comes out as 2 . sqrt(6) . sqrt(6) times the constituent quark mass. Also, the Brannen mass scale of the Koide triple containing the bottom quark equals the mass of the proton, the prototypical baryon.

2) Vague bonus idea: Koide relations are an echo of this S-duality.

This just comes from Brannen and Sheppeard's discussion of the discrete Fourier transform, in the context of circulant matrices (they obtain Koide masses as eigenvalues of a circulant). One might start with the realization of geometric Langlands via S-duality, and then look for analogues over finite fields. Sheppeard has occasionally hinted at something like this.

Gorsky et al conjecture that holographic QCD has a "flavor S-duality" in which vector mesons are dual to baryons. This is to be realized in string theory by a web of 5-branes.

Quark-hadron duality shows a kind of continuity between properties of quarks and properties of hadrons.

And the mass of the rho meson has been estimated at sqrt(6) times the constituent quark mass; while in the simplest version of Rivero's waterfall, the bottom quark mass comes out as 2 . sqrt(6) . sqrt(6) times the constituent quark mass. Also, the Brannen mass scale of the Koide triple containing the bottom quark equals the mass of the proton, the prototypical baryon.

2) Vague bonus idea: Koide relations are an echo of this S-duality.

This just comes from Brannen and Sheppeard's discussion of the discrete Fourier transform, in the context of circulant matrices (they obtain Koide masses as eigenvalues of a circulant). One might start with the realization of geometric Langlands via S-duality, and then look for analogues over finite fields. Sheppeard has occasionally hinted at something like this.

## Tuesday, September 5, 2017

### Various news II

I would divide my physics development into two stages, one in which I was studying general frameworks like quantum theory, and another in which I was interested in the details of particle physics. As discussed here, it was Marni Sheppeard who really set me on the second road, when she exhibited her decomposition of the CKM matrix in terms of circulants, and I wondered if this could be obtained from F-theory. But it was Alejandro Rivero who ended up being my biggest stimulus in that way - trying to implement his ideas and constructions led me to learn a lot of orthodox theoretical physics.

Two weeks after Carl Brannen returned to vixra, Sheppeard has come out with her first phenomenology paper in years. My thoughts: I disagree with most of the details, but the spirit of it is something to emulate.

Meanwhile, Rivero's constructions have reached the point where they all but single out specific string vacua for investigation. For me, the most valuable ideas are still some of the earlier versions, but it's impressive that he has come this far.

Two weeks after Carl Brannen returned to vixra, Sheppeard has come out with her first phenomenology paper in years. My thoughts: I disagree with most of the details, but the spirit of it is something to emulate.

Meanwhile, Rivero's constructions have reached the point where they all but single out specific string vacua for investigation. For me, the most valuable ideas are still some of the earlier versions, but it's impressive that he has come this far.

## Wednesday, August 23, 2017

### Various news

1) Carl Brannen once made a great discovery: the circulant representation of the Koide relation, with parameters a phase of 2/9 radians, and a mass scale of about 313 MeV. Koide acknowledged (section 3.1) the phase relation, but deemed it too difficult to explain for now.

Brannen has now returned to the subject. His overall philosophy may be seen at his site, especially in the "Operator Guide". I do not endorse his framework, but it may help the reader understand his latest paper, and whether it does contain any new progress.

2) There was a PF thread on gravitational vacuum polarization in which the author attributed significance to the sum of the squares of all particle masses. That is the LC&P formula for the square of the Higgs vev.

In the "General physics" category at arxiv (its counterpart of vixra), two papers (1 2) on obtaining the fine-structure constant from consideration of "virtual parapositronium in the vacuum". It could be bogus but it reminds me of various speculations here about the criticality of α

Brannen has now returned to the subject. His overall philosophy may be seen at his site, especially in the "Operator Guide". I do not endorse his framework, but it may help the reader understand his latest paper, and whether it does contain any new progress.

2) There was a PF thread on gravitational vacuum polarization in which the author attributed significance to the sum of the squares of all particle masses. That is the LC&P formula for the square of the Higgs vev.

In the "General physics" category at arxiv (its counterpart of vixra), two papers (1 2) on obtaining the fine-structure constant from consideration of "virtual parapositronium in the vacuum". It could be bogus but it reminds me of various speculations here about the criticality of α

_{em}.## Thursday, May 25, 2017

### Feigenbaum meets Feynman II

I am skeptical about the relationship described in the previous post, but it would be beautiful if true. And there has been progress towards making it plausible.

Perhaps the clearest way to classify the available theoretical approaches is how they interpret the doubling cascade. Feigenbaum's constant has the rather abstract meaning, of describing "how quickly" a dynamical system goes from a regime of stasis, to switching between two states, to switching between four states (and so on through powers of two, until chaos is reached), as a control parameter is adjusted. How could that be relevant to the probability that an electron emits a photon?

Angel Garces Doz (who has already appeared many times in this blog) in effect proposes to identify the doubling cascade with the cloud of virtual particles - iterated creation of virtual pairs. He points out that the size of the bulbs budding from the Mandelbrot set also diminishes according to Feigenbaum's constant, and says, let's think of spherical wavefunctions in the virtual cloud in this way. It's a brilliantly vivid intuition.

Meanwhile, I found that Mario Hieb's discovery had already appeared (in a different form) in papers by Vladimir Manasson (2006, 2008). His idea is that there is a prototypical self-organizing system (e.g. think of a soliton), that has 1-state, 2-state, 4-state... forms according to the value of some parameter, as in the doubling cascade. His idea is that the different elementary particles correspond to the different states, and that the levels of the cascade correspond to the fundamental forces. The Feigenbaum ratio, describing how the control parameter changes from one level to the next, maps to the relative strength of the forces!

Finally, I take inspiration from Sergei Gukov, for whom the doubling cascade describes the proliferation of fixed points in a renormalization group flow, as some property of a QFT varies. I am also intrigued by the criticality of the Higgs boson mass in this regard.

I am wondering whether - for example - you could have dissipative "hypermagnetization" (hypercharge magnetization) of the QCD vacuum in the early universe, passing through a doubling cascade of dynamical regimes, and ending in a top quark condensate that breaks electroweak symmetry, leaving only the familiar electromagnetic interaction, with Feigenbaum's constant somehow imprinted on the size of the electromagnetic coupling.

Perhaps the clearest way to classify the available theoretical approaches is how they interpret the doubling cascade. Feigenbaum's constant has the rather abstract meaning, of describing "how quickly" a dynamical system goes from a regime of stasis, to switching between two states, to switching between four states (and so on through powers of two, until chaos is reached), as a control parameter is adjusted. How could that be relevant to the probability that an electron emits a photon?

Angel Garces Doz (who has already appeared many times in this blog) in effect proposes to identify the doubling cascade with the cloud of virtual particles - iterated creation of virtual pairs. He points out that the size of the bulbs budding from the Mandelbrot set also diminishes according to Feigenbaum's constant, and says, let's think of spherical wavefunctions in the virtual cloud in this way. It's a brilliantly vivid intuition.

Meanwhile, I found that Mario Hieb's discovery had already appeared (in a different form) in papers by Vladimir Manasson (2006, 2008). His idea is that there is a prototypical self-organizing system (e.g. think of a soliton), that has 1-state, 2-state, 4-state... forms according to the value of some parameter, as in the doubling cascade. His idea is that the different elementary particles correspond to the different states, and that the levels of the cascade correspond to the fundamental forces. The Feigenbaum ratio, describing how the control parameter changes from one level to the next, maps to the relative strength of the forces!

Finally, I take inspiration from Sergei Gukov, for whom the doubling cascade describes the proliferation of fixed points in a renormalization group flow, as some property of a QFT varies. I am also intrigued by the criticality of the Higgs boson mass in this regard.

I am wondering whether - for example - you could have dissipative "hypermagnetization" (hypercharge magnetization) of the QCD vacuum in the early universe, passing through a doubling cascade of dynamical regimes, and ending in a top quark condensate that breaks electroweak symmetry, leaving only the familiar electromagnetic interaction, with Feigenbaum's constant somehow imprinted on the size of the electromagnetic coupling.

## Friday, April 28, 2017

### Feigenbaum meets Feynman

In quantum field theories, virtual particles cause quantities like the couplings to "run" with energy scale. Amateur physics numerologists find formulas for the low-energy values, but the professionals expect that these quantities will take their simplest form in some high-energy unified theory, so professional physics numerology involves the high-energy values.

The low-energy values - which are the quantities that are measured and listed in the physics databooks - are therefore regarded as being equal to "simple high-energy value + messy correction full of logarithms etc". However, there is the phenomenon of the "infrared fixed point". This occurs when the dynamics of the running (as described by a beta function) converges on the same low-energy value, for a range of starting values at high energy. In the language of dynamical systems theory, this means that the beta function enters an attractor at low energy.

This strikes me as one of the few ways in which amateur physics numerology might be realized within an actual quantum field theory: an attractor might dictate simple relations between the low-energy values. I have no examples of low-energy numerology being realized in this way, but it's a possibility.

It is therefore exceptionally intriguing to see some low-energy numerology which utilizes a famous constant from dynamical systems theory, Feigenbaum's constant. Mario Hieb has noticed that

(2 pi) times the fine-structure constant ~= 1 / (Feigenbaum's constant squared)

to 1 part in 1000.

It's a very attractive formula. It's simple, "2 pi" is a very "physical" factor, and the fine-structure constant is the epitome of what we would like to explain. Still, I wonder how mathematically difficult it is to obtain this within a QFT.

Feigenbaum's constant describes the approach to chaos - the rate at which a point attractor bifurcates, as a control parameter varies. It does show up in the theory of phase transitions, which sounds like QFT, but so far I only see it appearing in an indirect way, as part of a formula for some Lyapunov exponents.

It's unclear how one would go from that, to the constant appearing with such simplicity, in a formula for a coupling. Also, I have not found any work on infrared fixed points in which a weak U(1) coupling is part of the attractor.

But I admit that my survey of the possibilities so far is preliminary and superficial. So, maybe it has a chance of being true.

The low-energy values - which are the quantities that are measured and listed in the physics databooks - are therefore regarded as being equal to "simple high-energy value + messy correction full of logarithms etc". However, there is the phenomenon of the "infrared fixed point". This occurs when the dynamics of the running (as described by a beta function) converges on the same low-energy value, for a range of starting values at high energy. In the language of dynamical systems theory, this means that the beta function enters an attractor at low energy.

This strikes me as one of the few ways in which amateur physics numerology might be realized within an actual quantum field theory: an attractor might dictate simple relations between the low-energy values. I have no examples of low-energy numerology being realized in this way, but it's a possibility.

It is therefore exceptionally intriguing to see some low-energy numerology which utilizes a famous constant from dynamical systems theory, Feigenbaum's constant. Mario Hieb has noticed that

(2 pi) times the fine-structure constant ~= 1 / (Feigenbaum's constant squared)

to 1 part in 1000.

It's a very attractive formula. It's simple, "2 pi" is a very "physical" factor, and the fine-structure constant is the epitome of what we would like to explain. Still, I wonder how mathematically difficult it is to obtain this within a QFT.

Feigenbaum's constant describes the approach to chaos - the rate at which a point attractor bifurcates, as a control parameter varies. It does show up in the theory of phase transitions, which sounds like QFT, but so far I only see it appearing in an indirect way, as part of a formula for some Lyapunov exponents.

It's unclear how one would go from that, to the constant appearing with such simplicity, in a formula for a coupling. Also, I have not found any work on infrared fixed points in which a weak U(1) coupling is part of the attractor.

But I admit that my survey of the possibilities so far is preliminary and superficial. So, maybe it has a chance of being true.

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