# Fine structure and fixed points

## Main Question or Discussion Point

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.

ohwilleke, Klystron and Auto-Didact

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king vitamin
Gold Member
"the largest eigenvalue of the derivative of the renormalisation operator at its unique fixed point"
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?

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).
Is the thread or post you mention still up? I looked in the list of citations in those papers and didn't see anything.

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.
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.
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.
Thanks, I'm gonna look at these as soon as I find some time. Not thanks for destroying the rest of my free time.
But maybe something as arcane as Atiyah's formula is really needed.
It would be nice if things did turn out to eventually vindicate Atiyah

Andrea Panza
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?
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.

Auto-Didact
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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ohwilleke
Another perspective is provided by the possible "triviality" of quantum electrodynamics. This is the idea that in a rigorously defined QED, electric charge would be completely screened by virtual particles, leaving the effective charge as zero. The idea is not proven, but it would mean that QED with a nonzero electric charge can only exist in the context of a bigger theory.

In the present context, the question is whether we could motivate a QED in which, at the scale of the lightest charged fermion, the running electric charge is about √2/δ, as some kind of deformation of trivial pure QED, in which the charge is just zero. I have come across the observation that "the Dirac sea model was originally proposed by Dirac in Hartree-Fock approximation". This is interesting because limit cycles can arise in Hartree-Fock. One could then look for analogous behaviors in a more advanced formalism, like Dyson-Schwinger equations.

I also note the existence of "an analytic estimate of the Feigenbaum ratio" (pdf), due to May and Oster, which is simply that δ ~ 2(1+√2). This isn't just numerology, they explain why δ should be near this value. So this is also a "conceptual approximation" which may help in thinking about the problem.

Auto-Didact
Another perspective is provided by the possible "triviality" of quantum electrodynamics. This is the idea that in a rigorously defined QED, electric charge would be completely screened by virtual particles, leaving the effective charge as zero. The idea is not proven, but it would mean that QED with a nonzero electric charge can only exist in the context of a bigger theory.
Heard of this before, but never checked up on it. Any good references for this?
This is interesting because limit cycles can arise in Hartree-Fock.
To play devil's advocate: did you (or anyone else) try Arnold's last suggestion?
I also note the existence of "an analytic estimate of the Feigenbaum ratio" (pdf), due to May and Oster, which is simply that δ ~ 2(1+√2).
In the final paragraph they mention an improved estimate can be obtained going to next order in the iterative scheme outlined in (May and Oster, 1976). Any idea if this has been achieved yet?

Maybe the main challenge in getting Feigenbaum's constant to show up as part of a coupling constant, is that coupling constants are controlled by continuum dynamics (renormalization group flow), whereas Feigenbaum's constant is a signature of discrete dynamics (iterated mappings). I now think the way to do it, is to look for the point attractor of the continuum dynamics (the infrared fixed point of the RG flow) to exhibit the right discrete symmetry (a kind of self-similarity).

What is Feigenbaum's constant? It is the amount you must rescale the control parameter of certain iterated systems, in order to move one level in the period doubling cascade. And what is the electromagnetic coupling constant? It is the probability amplitude associated with an electron-photon vertex.

In particular, if you start with a multi-particle state that contains n photons, to which you already associate a particular amplitude, then a corresponding state with (n-1) photons and an electron-positron pair (so pair production has occurred), should have an associated amplitude smaller by one factor of the coupling constant. The amplitude is analogous to the control parameter that gets rescaled.

As for the period doubling, I notice that for n instances of pair creation - which according to the analogy, should move you n steps along the period doubling cascade - you have 2n extra fermions in your multi-particle state. The new state is in a specific part of Fock space. So I'm thinking there's some kind of operator, a permutation or rotation operator, which induces a 2^n step cycle, in the "2n fermions" part of Fock space.

The idea is that the value of the coupling constant imply a corresponding ratio between the amplitudes of states at different levels of Fock space. And when the coupling constant is a specific function of Feigenbaum's constant, then there is a discrete symmetry, acting on multi-particle states, with respect to which Fock space is self-similar (in the way that a doubling cascade is self-similar). And we are to think of the RG flow through theory space, as being a flow through something like possible norms on Fock space, that converges on that critical value of the coupling constant.

(P.S. with respect to the questions in the previous comment: I haven't looked into those technical issues yet, first I wanted to get a sensible overall framework, which maybe I now have. As for the possible triviality of pure QED, I don't have a preferred technical reference, but the informal discussion in Kerson Huang's "A Critical History of Renormalization" looks like a good start.)

eloheim, ftr and Auto-Didact
Maybe the main challenge in getting Feigenbaum's constant to show up as part of a coupling constant, is that coupling constants are controlled by continuum dynamics (renormalization group flow), whereas Feigenbaum's constant is a signature of discrete dynamics (iterated mappings).
Actually, period doubling bifurcations can and do occur in continuous systems; the only criteria such continuous systems need to satisfy are 1) be at least three dimensional (due to the existence and uniqueness theorem), 2) have a nonlinearity in at least one of the differential equations, and 3) have a parameter in at least one of the differential equations.

Given that the above criteria hold, one can then numerically integrate one of the equations in time and then use the Lorenz map technique to construct a discrete recurrence map of the local maxima of the numerical integration.

This is where the miracle occurs: if the resulting Lorenz map is unimodal for a given parameter, then the continuous system will display period doubling. This mapping doesn't even have to be approximatable by a proper function i.e. uniqueness isn't required!

Incidentally, this unimodal Lorenz map miracle only applies for any strange attractor with fractal dimension close to 2 and Lorenz map dimension close to 1.
I now think the way to do it, is to look for the point attractor of the continuum dynamics (the infrared fixed point of the RG flow) to exhibit the right discrete symmetry (a kind of self-similarity).
I'll wait and see what you come up with.

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By way of clarifying my previous comment...

What would be simplest, is if we could derive the observed value of α just by looking at beta functions. A beta function is just a differential equation showing how a quantity in QFT depends on energy scale (the beta function is deduced from the fundamental equations of the QFT). A set of beta functions (for several interrelated quantities) are then just a system of coupled differential equations. In the GUT paradigm, the values of couplings at the unification energy are regarded as more fundamental, and the beta functions are then used to extrapolate their values at observable energies. But if the coupled beta functions define the right kind of dynamical system, there can be an attractor at low energies, leading to a prediction of observable values, even when the high-energy behavior is unknown.

For example, pages 15-16 here describe an attractor (infrared fixed point) for α, when QED is coupled to gravity in the framework of asymptotic safety. Another, more orthodox example for QED may be found in the "ε expansion", where one considers QED in a space-time with a non-integer number of dimensions, e.g. section IIB here. Essentially, in space-time of dimension 4-ε, α*, the fixed point of α, is proportional to ε/Nf, where Nf is the number of fermion flavors. So if you could find some quantum-gravitational reason why space-time, at the scale of the electron, deviated from exact four-dimensionality by just the right amount (by my calculation, it would need to be about 3.9996-dimensional), you could obtain the observed value of α.

But let us recall what Feigenbaum's δ is, in dynamical systems theory. It generally does not show up just as a value of one of the controlling or evolving parameters. Instead, it measures the rate at which the dynamical regime becomes more complicated, as one of the control parameters is adjusted. I guess one can say that the value of that parameter, at the onset of chaos, equals the value of that parameter, at the onset of period doubling, plus the sum of a geometric series in δ... that is a way in which δ can enter directly, albeit a little abstractly, into a special value of one of the system parameters.

And what is α? α is e2/4π, where e is the photon-electron coupling that appears in the QED lagrangian. Interestingly, a focus on e rather than α produces an expression that is even simpler as a function of δ. α ~ 1/(2π δ2), which means that e ~ √2/δ. For an example of this value in print, see chapter 58 ("Spinor Electrodynamics") of Mark Srednicki's popular web-textbook. (Srednicki's value for e is -0.302822; one may calculate that √2/δ ~ 0.30288, a little off, but still close enough to maintain my interest.)

e ~ √2/δ is about as simple a function of δ as we are going to get. So what is the physical meaning of e? In terms of perturbative QFT, it contributes to the amplitude for photon emission. See the end of Srednicki's chapter 58, where he gives rules for constructing Feynman diagrams: "for each [electron-photon] vertex, [a factor of] ieγμ".

If we take Manasson's observation seriously, this means that the amplitude for emission of n photons (or the amplitude for n instances of positron-electron pair creation) is proportional to 1/δn. Well, in dynamical systems theory that number means something too. If you are in the regime of period doubling, and the first doubling involved adding C to the control parameter, then adding C/δn should produce n further doublings.

On the physical side of this correspondence, shrinking by a factor of 1/δ is associated with the probability amplitude for a particular event, so it seems that the "control parameter" must also be some kind of probability amplitude. For example, suppose you have a state with an electron, and then you evolve it under a QED interaction Hamiltonian, and then you ask, what is the amplitude for the out state to have an electron and one photon, an electron and two photons, an electron and three photons... Then you should be looking at amplitudes which are proportional to e, e2, e3... i.e. proportional to 1/δ, 1/δ2, 1/δ3... The more interaction vertices a perturbative process contains, the smaller the amplitude for that process.

Here you may ask two things - what about the √2, and what about the periods? That is: e is not equal to 1/δ, it is (approximately) equal to √2/δ. So the amplitudes are actually proportional to √2/δ, 2/δ2, 2√2/δ3... And: under an "approach to chaos" interpretation, there should be some kind of periodic dynamical behavior associated with each of these out states, e.g. a 2-step cycle for the "electron + photon" state, a 4-step cycle for the "electron + 2 photons" state, an 8-step cycle for the "electron + 3 photons" state... Where and what are these "cycles"?

I don't have a clear answer. But I am thinking that the cycles could be, not evolution in time, but rather, dynamical trajectories under iterated application of some operator acting on Hilbert space. The operator might permute states in the different sectors of Fock space, e.g. there might be four states in the "electron + 2 photons" sector (perhaps they would differ by polarizations, or they might be some special set of orthogonal vectors) that get swapped around under the action of the operator. I also find it intriguing that we have these factors of √2 unaccounted for - one for each extra photon, in the case of "electron + multiple photons" - it's the kind of factor which could show up in constructing a special set of vectors, that have to be normalized appropriately.

In other words, the idea is there's an operator which acts differently on different sectors of QED Fock space, e.g. permutations of order 2n, and when e is tuned to the vicinity of √2/δ, a symmetry emerges that implements or imitates period doubling and the approach to chaos. (There is a mild precedent for QED to contain a hidden symmetry of infinitely many levels, in that it contains a hidden Kac-Moody symmetry, but that is apparently for all values of α.) This symmetry would be a property of the infrared fixed point of "QED + extra ingredient", but to understand why, you would have to go deeper than the usual beta functions, e.g. into the measure on the space of field configurations, how it changes under RG flow, and how that affects the relative magnitude of perturbative amplitudes.

As a postscript, I will note that I also found an obscure string theory paper claiming to obtain the log of the Feigenbaum constant in RG equations.

ftr and Auto-Didact
Then you should be looking at amplitudes which are proportional to e, e2, e3... i.e. proportional to 1/δ, 1/δ2, 1/δ3... The more interaction vertices a perturbative process contains, the smaller the amplitude for that process.
It seems clear to me that this route is promising; in order to carry this answer out to its full completion however it seems that you are quickly getting stuck, mainly due to trying to stick to perturbative reasoning for no good mathematical reason. This directly implies that there is a physically illegitimate reduction of a second order equation to a first order equation taking place here.
So the amplitudes are actually proportional to √2/δ, 2/δ2, 2√2/δ3... And: under an "approach to chaos" interpretation, there should be some kind of periodic dynamical behavior associated with each of these out states, e.g. a 2-step cycle for the "electron + photon" state, a 4-step cycle for the "electron + 2 photons" state, an 8-step cycle for the "electron + 3 photons" state... Where and what are these "cycles"?
This question seems to be key.
The obvious answer would be: limit cycles in the phase space of such a system.

In other words, the idea is there's an operator which acts differently on different sectors of QED Fock space, e.g. permutations of order 2n, and when e is tuned to the vicinity of √2/δ, a symmetry emerges that implements or imitates period doubling and the approach to chaos.
I need to summarize a few things for myself, because things are too disjoint for me to see anything directly resembling a solution. Feel free to correct me if I am mischaracterizing your argument, but I think you are saying the following:

Assuming one can use QED to evolve an initial state ##| \psi_i \rangle## with an electron into a final state ##| \psi_f \rangle## with an electron and ##n## photons, with a probability amplitude for that final state proportional to ##e^n \approx (\frac {\sqrt {2}} {\delta})^n##, then:
- there is some discrete or continous dynamical system for such states which has ##\alpha## or ##e## in one of the parameters
- this parameter is the control parameter, and is the probability amplitude i.e. is proportional to ##e^n## of the final state ##| \psi_f \rangle##
- this implies the existence of a period doubling ##2^n##-cycle for a final state ##| \psi_f \rangle## with an electron and ##n## photons

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This was an attempt to make the possible role of δ within QED, imitate the known role of δ within dynamical systems theory.

Let's suppose that a is the control parameter in a system which exhibits a period-doubling approach to chaos, with a1, a2, ... aN, ... being the least value of a at which there is a limit cycle with 21, 22, ... 2N, ... fixed points. Then we can say (here I am imitating eqn 14.26 of Hassani's "Mathematical Methods")

aN ~ (a2-a1) (1/δ + 1/δ2 + ... + 1/δ(N-1)) + a2

Meanwhile, if you approximate the QED coupling constant as e ~ √2/δ, then the perturbation expansion of the S-matrix looks like

S = 1 - i √2/δ ∫.. + (√2/δ)2 / 2! ∫∫.. + i (√2/δ)3 / 3! ∫∫∫.. - (√2/δ)4 / 4! ∫∫∫∫.. - ...

where the space-time integrals are over 1, 2, 3, 4... interactions.

So the real idea is that when the coupling constant runs to this value in the infrared, it is in some sense gravitating to the edge of chaos. For example, there ought to be limit cycles somewhere, and pursuing the analogy, the limit cycle with 2N fixed points should somehow be associated to states created by N applications of the QED interaction term, e.g. N emissions of a photon from a single electron.

Some other challenges for this analogy: In the S-matrix expansion one has, not just factors of 1/δ, but factors of √2/δ. Also, you have factors of i. And, those integrals are highly nontrivial too, since they involve time-ordered products of interactions at arbitrarily large separations. Well, maybe the √2 can be absorbed by a field redefinition, and the factors of i can go away if you use imaginary time, and the time-ordered products can simplify if you focus just on multiple interactions that are infinitesimally close.

I find the analogy with turbulence appealing. A QFT vacuum state is highly entangled, maybe there's a kind of "entanglement turbulence" that sets in at this coupling.

Whether or not some version of this works, it's worth emphasizing how it differs from Manasson's approach. In Manasson, the values of the control parameter a1, a2, a3... are couplings of different forces (though ultimately just different regimes of a novel kind of unified force), and the fixed points of the associated limit cycles are different elementary particle states. Here, those values are the absolute values of probability amplitudes, and correspond to processes in which increasingly many interactions occur.

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Auto-Didact
A lattice study alleges quantum chaos in QED. But the criterion of quantum chaos employed is that the distribution of eigenvalues of the Dirac operator, resembles that of a random matrix. I believe the authors say in one of their papers, that the Dirac operator has no classical counterpart. (John Baez has recently publicized the notion of a "classical spin-1/2 particle", and there are papers that talk about "classical Dirac operator", but I don't know if any of this extends to classical counterparts of fermion fields. Just last year a paper from China announced that relativistic quantum chaos is a theoretically distinct topic, including "chiral scars", a fermionic version of the quantum scars that correspond to classical periodic orbits; perhaps that is relevant.)

I have also so far failed to find a clear appearance of Feigenbaum's constant in a quantum-chaotic context. The best bet would seem to be the dissipative Henon map. The Henon map is two-dimensional, and it turns out there are two values of Feigenbaum's constant, one for when the map is conservative, the other for when it is has a sufficiently large dissipative component. Apparently the dissipative Henon map approximates the logistic map, and so the Feigenbaum constant for this case, is the one that we are concerned with in this thread. The dissipative Henon map has been quantized, as a kicked harmonic oscillator; but I just can't find the Feigenbaum constant in any of the resulting literature. In fact there seem to be a number of basic lines of research in quantum chaos, first developed in the 1980s, that have not been pursued since; though perhaps I am just looking in the wrong places.

Something that is ubiquitous in the quantum chaos literature, is the "Gutzwiller trace formula" and its relatives. For this reason, the most promising leads I have found, when it comes to bridging quantum chaos and Feigenbaum's constant, are references 5 and 6 in these course notes from Cvitanovic. Of them he says, "Traces weighted by ±1 sign of the derivative of the fixed point have been used to study the period doubling repeller, leading to high precision estimates of the Feigenbaum constant δ". The papers in reference 5 are available from his own website, 1 2. The second paper (eqn 15) contains a "cycle expansion" for δ, the cycle terms being associated with special periodic orbits that allow to efficiently approximate the properties of the chaotic phase.

I presume that something like these concepts will need to be elaborated, in order to really understand the presence of "eigenvalue chaos" in quantum fields. There's still the problem that finding δ in the coupling constant seems too crude; δ should show up at some more abstract level. But this might just mean that we need to motivate "QED with effective charge √2/δ", as being the limit of some other theory in which δ does enter in that more abstract way.

A stringy candidate for this would be the Sugimoto-Takahashi implementation of QED via D-branes, studied using Dmitry Polyakov's extension of NSR formalism to include D-branes, simply because that is the stringy formalism that I mentioned earlier, in which log δ allegedly showed up in a coefficient of the beta function for the dilaton (which determines the string coupling). One could then look e.g. for how the Butcher group, which controls the Hopf-algebraic approach to renormalization, shows up in this stringy context. But I mention this only as an illustratively concrete example of how to proceed.