A Has Sir Michael Atiyah Uncovered New Links Between Arithmetic and Physics?

[mitchell porter]

A slightly altered perspective has made Atiyah's claim interesting to me again. I wrote:

@Auto-Didact, you bring up dynamical systems theory. Vladimir Manasson (e.g. eqn 11 here) discovered that 1/α ~ (2π)δ², where δ is Feigenbaum's constant! This is the only way I can imagine Atiyah's calculation actually being based in reality - if it really does connect with bifurcation theory.

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.

 

[Auto-Didact]

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.

Damn, you seem to be picking up on it as well :) which is why I said:

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.

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.

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.

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.

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.

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.

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

Exactly my point.

 

[Auto-Didact]

@Auto-Didact, you bring up dynamical systems theory. Vladimir Manasson (e.g. eqn 11 here) discovered that 1/α ~ (2π)δ2, where δ is Feigenbaum's constant! This is the only way I can imagine Atiyah's calculation actually being based in reality - if it really does connect with bifurcation theory.

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.

 

[Auto-Didact]

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.

 

[arivero]

https://www.uv.es/~azcarrag/pdf/2018 REF Conversation Atiyah English.pdf Here Atiyah mentions that all the four forces and should come from R C O H. In a later obituary azcarraga comments that the argument, using the groups over each plane, reminds of Boya's "Division algebras and Symmetries of Elementary Particles"

 

[ohwilleke]

What is R C O H in this context?

 

[Auto-Didact]

What is R C O H in this context?

Real Complex Octonion Quaternion

 

[lpetrich]

Or in order of object size, R C H O.

These are the first four Cayley-Dickson algebras, starting with the real numbers and with each one of the others being produced from pairs of the elements of the previous one. The next one is he sedenions (size 16): S?

The algebras are rings with unity and with a conjugation operation that is a self-inverse (involution): $(x^*)^* = x$. Each one is constructed from the previous one, if present, with operations

• Addition: $(a,b) + (c,d) = (a+c, b+d)$ (component-by-component)
• Multiplication: $(a,b) (c,d) = (ac - d^*b, da + bc^*)$
• Conjugation: $(a,b)^* = (a^*,-b)$

One can define a norm with $xx^* = x^*x$. Applying to $(a,b)$ gives $(aa^* + bb^*,0)$, and repeating gives $xx^* = (\|x\|, 0, 0, \dots 0) = \|x\| (1, 0, 0, \dots 0)$ where $\|x\| = \sum_i (x_i)^2$. Every element with nonzero norm has a reciprocal: $x^{-1} = x^* / \|x\|$.

The real numbers have a complete set of properties, and higher -ions gradually drop them. Their multiplication is commutative and associative, and interchangeable with norming, and they are self-conjugate.

Complex numbers lose self-conjugacy.

Quaternions lose multiplication commutativity.

Octonions lose multiplication associativity, though they have a form of partial associativity known as alternativity: $(xx)y = x(xy)$, $(xy)x = x(yx)$, and $(yx)x = y(xx)$.

Sedenions lose multiplication alternativity, though they have a limited form of associativity known as power-associativity: $x^m x^n = x^{m+n}$, where m and n can be negative as well as nonnegative integers. They also lose product-norm interchange. In general, $\|x y\| \ne \|x\| \|y\|$. They also acquire nontrivial divisors of zero.

Higher -ions have no change in properties from sedenions.

 

[lpetrich]

This has the consequence that quaternion multiplication can be represented by matrix multiplication, while octonion multiplication cannot.

The algebras' isomorphism groups are: R: identity group, C: Z2, H: SO(3), O and higher: G2. This seems related to the Standard Model with:

• U(1) - complex numbers
• SU(2) - quaternions
• SU(3) - subalgebra of G2, automorphisms of octonions

Is that the connection of this algebra sequence to the Standard Model?

 

[arivero]

What is R C O H in this context?

Hmm, RCHO, to preserve the order, sorry.

In the context of the interview with Atiyah, I am not sure if he wants to play with the proyective planes, which is a very usual bussiness in this field, or just with the straighforward ones. Relevant answer:

Exactly so. The non-commutativity of the quaternions is at the heart of the problem I deal with in my calculation of α. The non-associativity of the octonions is much harder and will be in my next paper. Gravity is much harder than gauge theories of compact Lie groups. The division algebras and the physical forces are a perfect fit. Let me explain. The compact groups that act on ℝ2 , ℂ2 , ℍ2 are SO(2), U(2), U(3). The first gives electromagnetism, the second gives the electroweak theory and the subgroup SU(3) is the gauge group of strong interactions. But 𝕆2 is acted on by octonions which do not give a group because they are non-associative. That is why gravity is harder than gauge theories.

 

[lpetrich]

Meaning algebras R², C², H², and O². R² gives us SO(2) ~ U(1), C² gives us U(2) ~ SU(2) * U(1), and H² gives us SO^*(4) ~ SU(2) * SL(2,R), not SU(3).

[URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez on Octonions[/url] mentions some ways of getting around octonions' non-associativity, IIRC.

 

[Hans de Vries]

Meaning algebras R², C², H², and O². R² gives us SO(2) ~ U(1), C² gives us U(2) ~ SU(2) * U(1), and H² gives us SO^*(4) ~ SU(2) * SL(2,R), not SU(3).

[URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez on Octonions[/URL] mentions some ways of getting around octonions' non-associativity, IIRC.

This is from Hurwitz's original '$\mathbb{RCHO}$' paper:

Note that these arrays can be read in two ways:

1) ... As a multiplication table of the generators.
2) ... As a 2x2, 4x4 or 8x8 matrix where each matrix represents a generator.

In case of 1) for example $x_3$ represents the (entire) generator and the table shows the product of the generator of the first row multiplied with the generator in the first column.

In case of 2) the matrix representing $x_3$ has a +1 for each position which contains $x_3$, a -1 for each position which contains $-x_3$ and a zero for all other positions.

in case of $\mathbb{C}$ and $\mathbb{H}$ each generator is represented by its matrix. The matrices form a group. All matrices are square roots of -I and all matrices anti-commute with each other (except with the I matrix) and they are associative.

Not so with $\mathbb{O}$ however. Both the generators and matrices still anti-commute with each other but:

In case of 1) the generators form a group but are not associative.
In case of 2) the matrices are associative (as matrices are always associative) but they do not form a group anymore.

In the later case the 8x8 matrices can be used to set up the space of a 7-sphere like the 4x4 matrices can be used for a 3-sphere, so:

$$\big(ax_2+bx_3+cx_4+dx_5+ex_6+fx_7+gx_8\big)^2 = -\big(a^2+b^2+c^2+d^2+e^2+f^2+g^2\big)I$$