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.

# theory

not endorsed by snarxiv

## Thursday, May 25, 2017

## 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.

## Monday, April 10, 2017

### vixra watch: April fools

arxiv sees a few joke papers on April 1st every year. vixra was created as a repository for papers blocked from arxiv. We may now have the first parody paper posted to vixra because it wouldn't survive on arxiv. It's a proposal for an "Un-collider", that not only mocks numerous aspects of contemporary physics culture, but also today's political and geopolitical situation. The authors are "Snowden" and "Ellsberg" (the latter promoted it at "Not Even Wrong"), and it has all the professionalism of a proper arxiv parody paper.

There was another parody uploaded at the same time, on "gauge theology", but it's merely clever, and doesn't have the sting of the Un-collider.

Finally (for now), yet another paper has appeared, promising "a Hodge-theoretic analysis of reinforcement learning". I thought that one might be real - making such a connection is not beyond the reach of vixra authors, or of arxiv authors, or even of reality. But the paper merely reproduces the abstract, which says "we begin with a diagram" illustrating the connections. That there is no diagram, is perhaps a way of saying that there is no connection. Then talk of inducing entropy in an economy makes it sound fake, and the final straw is that it's classified as "Relativity and Cosmology". So, another joke paper; perhaps someone testing the vixra submission procedure.

There was another parody uploaded at the same time, on "gauge theology", but it's merely clever, and doesn't have the sting of the Un-collider.

Finally (for now), yet another paper has appeared, promising "a Hodge-theoretic analysis of reinforcement learning". I thought that one might be real - making such a connection is not beyond the reach of vixra authors, or of arxiv authors, or even of reality. But the paper merely reproduces the abstract, which says "we begin with a diagram" illustrating the connections. That there is no diagram, is perhaps a way of saying that there is no connection. Then talk of inducing entropy in an economy makes it sound fake, and the final straw is that it's classified as "Relativity and Cosmology". So, another joke paper; perhaps someone testing the vixra submission procedure.

## Tuesday, March 7, 2017

### Two problems

There was unexpected progress, posted at Physics Stack Exchange, on two problems that were low on my list.

First, numerology of the charge radius. See my 2017 update: I ran across a model of the nucleon in which the radius is 4 natural units, divided by the mass. That doesn't explain why the radius comes out a little different for muonic hydrogen compared to electronic hydrogen; but it can explain why dandb's ratio is approximately 4 in both cases.

Second, mystery of the Z0 decay width - that it lies on the same curve as a number of mesons. It's one of @arivero's minor observations, and not one that I spent any time on. I was just going through the motions of investigating it, when to my surprise, something turned up.

First, numerology of the charge radius. See my 2017 update: I ran across a model of the nucleon in which the radius is 4 natural units, divided by the mass. That doesn't explain why the radius comes out a little different for muonic hydrogen compared to electronic hydrogen; but it can explain why dandb's ratio is approximately 4 in both cases.

Second, mystery of the Z0 decay width - that it lies on the same curve as a number of mesons. It's one of @arivero's minor observations, and not one that I spent any time on. I was just going through the motions of investigating it, when to my surprise, something turned up.

## Friday, February 10, 2017

### tHWZ - latest formulation

During a discussion at PF, I found the following interesting way to think of these quantities:

m

m

H

m

The last one may look a little odd, but it allows us to approximate sin

The impetus was a comment by @arivero in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies

(m

Now in many GUTs, at the GUT scale, we have that

m

So it's as if (m

We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)

Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.

Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that m

m

_{H}~ √2 m_{Z}m

_{t}~ 2 m_{Z}H

_{vev}~ 2 √2 m_{Z}m

_{W}~ √7 / 3 m_{Z}The last one may look a little odd, but it allows us to approximate sin

^{2}of the Weinberg angle as 2/9.The impetus was a comment by @arivero in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies

(m

_{W}^{2}- m_{H}^{2}) / (m_{Z}^{2}- m_{t}^{2}) = 3/8Now in many GUTs, at the GUT scale, we have that

m

_{W}^{2}/ m_{Z}^{2}= 3/8So it's as if (m

_{W}^{2}- m_{H}^{2}) / (m_{Z}^{2}- m_{t}^{2}) is almost invariant under renormalization group flow, with m_{H}= m_{t}= 0 at the GUT scale.We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)

Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.

Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that m

_{Z}is very close to the standard model's μ parameter.## Wednesday, August 24, 2016

### Multifractal worldsheet

The opinion is spreading that the real discovery of the LHC was that the Higgs boson mass is special. The most impressive prediction was Shaposhnikov and Wetterich 2006, which got the right value from the assumption that gravity is asymptotically safe.

This creates cognitive dissonance for anyone who appreciates the string-theoretic model of quantum gravity. Asymptotic safety isn't even consistent with the holographic principle, is it?

Well, asymptotic safety is one of several heterodox approaches to quantum gravity in which the dimension of spacetime seems to change from 4 to 2 at the smallest scales. Sabine Hossenfelder lists a few others and says, "It is difficult to visualize what is happening with the dimensionality of space if it goes down continuously, rather than in discrete steps".

Fractals can have non-integer dimensionality. But they are typically embedded in a larger space. Meanwhile, in string theory, one has a 2d worldsheet embedded in a "target space" that usually has more than two dimensions. So what if the world sheet embeds in the target space as a multifractal surface that is 4d on large scales but 2d on small scales?

This creates cognitive dissonance for anyone who appreciates the string-theoretic model of quantum gravity. Asymptotic safety isn't even consistent with the holographic principle, is it?

Well, asymptotic safety is one of several heterodox approaches to quantum gravity in which the dimension of spacetime seems to change from 4 to 2 at the smallest scales. Sabine Hossenfelder lists a few others and says, "It is difficult to visualize what is happening with the dimensionality of space if it goes down continuously, rather than in discrete steps".

Fractals can have non-integer dimensionality. But they are typically embedded in a larger space. Meanwhile, in string theory, one has a 2d worldsheet embedded in a "target space" that usually has more than two dimensions. So what if the world sheet embeds in the target space as a multifractal surface that is 4d on large scales but 2d on small scales?

## Saturday, July 9, 2016

### A formula for α

On page 4 of "Naturally Speaking" by G.F. Giudice, after a short list of numerological formulas for the fine-structure constant α, one finds a formula for α according to physical orthodoxy, i.e. grand unification. I reproduce it here for the edification of passing numerologists:

α = { α

"Here, the fine-structure constant α, the strong coupling constant α

α = { α

_{s}sin^{2}θ_{W}(b_{1}−b_{3})+3/5 cos^{2}θ_{W}(b_{3}−b_{2}) } / (b_{1}−b_{2}) + higher-order terms."Here, the fine-structure constant α, the strong coupling constant α

_{s}and the weak mixing angle θ_{W}are evaluated at the same renormalization scale and b_{1,2,3}are the gauge β-function coefficients. Higher-order terms cannot be neglected to achieve a prediction that matches the experimental accuracy."
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