Chaos and hadronic spectrum

The question of quantum chaos and hadronic spectrum is a relevant matter that I was not able to address in my last post. The reason for this relies on the important fact that quantum chaos in the hadronic spectrum and classical chaotic solutions to build a quantum field theory are just different levels and should not be confused each other. Indeed, I can start with my fully integrable classical solutions to build a quantum field theory and find in the end a fully quantum chaotic spectrum in the bound states of the theory. This is a point that created a lot of confusion and should be clarified properly. But let us state a quite simple example of this. Let us consider QED. We know that the Coulombian potential produces n-body bound states. But already the 3-body problem admits chaotic solutions and the corresponding quantum problem will displays the proper distribution of the energy levels. But QED is a quantum field theory built on perfectly regular solutions of the free equations of motion (Dirac and Maxwell equations). So, we see that quantum chaos, if any, arises naturally for the bound states of the theory due to the properties of the bounding potential.

What can we say about hadronic spectrum? The following papers show example of quantum chaos in the hadronic spectrum: here and here. But what is the potential we obtained from Yang-Mills theory? We gave

being the glueball spectrum. This potential has an infinite number of contributions. Baryons are expected to be chaotic already with a Coulombian approximation being three-body bound states. But a potential like that above could produce classical chaotic dynamics having an infinite number of terms and producing in this way an infinite numbers of resonances in the KAM series. But to obtain this potential we started with perfectly regular solutions in the quantum field theory!

We conclude that our approach produces a consistent potential that agrees fairly well with expectations of quantum chaos in the hadronic spectrum. But this is independent on the way one formulates a quantum field theory. Indeed, bound states could display chaos even with the simplest of the field theories, that is a scalar field theory.

Chaos and quantum field theory

Dmitry is still on is point trying to prove me wrong (see his post here). Of course, I have a theory mathematically sound and he has nothing and so the discussion is somewhat uneven from the start. He is saying that my point

In order to build a meaningful quantum field theory, the initial conditions should be properly chosen.

is wrong. This means that we can start to develop a meaningful QED without electrons or photons and obtain identical results. A fact that is blatantly wrong. QED spectrum is done with plane waves representing the spectrum of the theory. Without this choice you are at odds with experiments. This is a crucial point for quantum field theory and applies wherever you need to compute a cross section or a decay rate. Choosing different solutions at the start (e.g. by changing initial conditions) will produce a different quantum field theory and this can be at odds with experiments. This is a well-known fact and an example is given through the computations generally done e.g. for the Casimir effect.

So, he continues

“I believe, it is actually wrong. Let us again take a hamiltonian classical system with self-interaction. To get the intermittency (i.e., periodic orbits becoming chaotic trajectories and vise versa), it is enough to fix initial conditions and than vary the coupling constant, as I have explained … Since one has running coupling in a QFT without sweating, one will have intermittency as well in the Schwinger-Dyson equations for Keldysh Green functions. So, strictly speaking, one can only get rid of chaos at the RG fixed point which corresponds to CFT anyway (that is — no particles, nor quasiparticles, just unparticles ).”

I would agree with such an argument if intermittency could be a useful solution to build a quantum field theory and would give us a spectrum to start with. E.g. I would like to see how is the glueball spectrum, to be observed into experiments with a lot of people currently eager to detect it, with a chaotic classical solution like this. You are in serious troubles as you do not even have a Fourier expansion. So, no Fourier expansion no spectrum. Indeed, Dmitry has no such result but just a prejudice: Chaotic classical solutions must be important for Yang-Mills theory. This without a proof that, he claims, should rely on me. But I have already shown that the theory is consistent and complete with integrable solutions and this was my aim. He is just claiming the contrary without providing a serious support to his prejudice.

After this rather questionable facts by his side and having him admitted that I am right as

Currently, there is no formulation of a quantum field theory starting with classical chaotic solutions.

he exposes his “would be”s about such a matter.

Let me comment about this discussion and how all this should be interpreted. Whenever you are smart enough to produce a theory and get it published you will get opinions from two different kind of people.  You will find interested people and criticizing people and this is in the matter of things. Criticizing people could be very useful wherever is able to support arguments with serious evidence. But most of times they will just criticize you on the ground of prejudices they have and these prejudices are those you have just demolished with your theory. So, to move an idea from the status of a prejudice to a status of a theory a strong mathematical and experimental support is needed.  A typical historical example has been the question of aether supporting the propagation of electromagnetic waves. A lot of people kept on believing on that till their death even after a strong evidence for relativity was achieved. This behavior belongs to our community, it was never lost, and we have to cope with it anytime we produce something new. It could imply delay into the acceptance of a theory but it is just human behavior and cannot be changed.

Let me repeat again. My approach is well developed and provides a glueball spectrum (to be observed experimentally), propagators and running coupling making the formulation of a Yang-Mills theory in the infrared complete. Propagators and running coupling are in very good agreement with lattice computations that come out in this somewhat unexpected direction. The same can be said with the spectrum but assuming that the lowest glueball is at about 500 GeV and has been already seen as resonance or f0(600). The same interpretation should apply to f0(980). This is in agreement with analysis done with dispersion relations by Narison, Mennessier, Ochs (see here) and Minkowski (see here).

What does one have on the side of classical chaotic solutions and quantum field theory? Substantially nothing as also admitted by Dmitry. No theory, no predictions, nothing. So, it can only be classified as a prejudice and a prejudice generally turns out to be wrong. My aim starts and ends when I have showed that my theory is mathematically sound and consistent and I get predictions that could be confirmed or not.  I do not have the burden to prove that, as one of my hypothesis does not like to someone, I have also to formulate my theory without it.

As a conclusion, I would greatly appreciate a formulation of a quantum field theory starting with chaotic solutions that applies to a realistic model of reality. I do not believe in betting but it would be tempting to put a wager on this.

A wrong argument

Dmitry Podolsky put forward the following argument to claim that chaotic solutions are relevant for Yang-Mills theory at strong coupling (see here):

“First of all, equations of motion of the YM field are non-linear and therefore their solutions admit chaotic behavior. Are all the solutions of these equations of motion chaotic? The answer is of course negative: depending on the coupling strength and initial conditions, one can get whole sets of classical solutions without chaos, which we will call Smilga choices, following Marco. Suppose that we fix coupling and continuously change initial conditions for the YM equations of motion – as a result of this variation, we will first get, say, a chaotic solution,  than a solution without chaos, than again a chaotic soltion, etc.”

This is true if one chooses the wrong initial conditions to build a quantum field theory. Because this is the main point of the question.

Currently, there is no formulation of a quantum field theory starting with classical chaotic solutions.

But let me add this conjecture that I invite anyone to prove false:

A quantum field theory does not exist having as building classical solutions just chaotic solutions.

The reason for this is quite simple. If I choose wrong initial conditions I will not be able to get a leading order spectrum of excitations to build on. For a SU(2) Yang-Mills theory I have e.g. the following starting classical solution:

with and and I am able to get a spectrum of fundamental excitations as the Jacobi function has a Fourier expansion in plane waves with a mass spectrum

that I can call a glueball spectrum. This must be observed in nature. Presently, for SU(3), this is in agreement with lattice computations (see here). But there is also the expectation that f0(600) or is a glueball and another one could be f0(980). If this is proved true, I think that Smilga will be very glad.

Let me state a final point:

In order to build a meaningful quantum field theory, the initial conditions should be properly chosen.

This simple fact seems generally overlooked but try to ask yourselves why one chooses plane waves for QED or other quantum field theories and you will get an easy anwser: these are the excitations seen in experiments.