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

$V(r)=-\alpha_s\sum_{n=0}^\infty A_n \frac{e^{-m_n r}}{r}$

being $m_n$ 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.

### 2 Responses to Chaos and hadronic spectrum

1. Dmitry says:

Hi Marco

If I follow your logic, glueballs are the bound states of gluons (and hardons are bound states of quarks and gluons), so I should see chaos already at the level of their spectrum.

Cheers,
Dmitry.

2. mfrasca says:

Hi Dmitry,

nice to hear from you again. For glueballs the question is somewhat different. I try to explain my point of view. You are in a situation with a strong coupling and the question is: What are the fundamental excitations in this limit? We know that in the high energy limit, where asymptotic freedom rules, one can consider at the start the spectrum of free particles and so one has gluons and quarks. The main aim one has in the infrared is to find the right starting point. This is quite similar to the situation in condensed matter and the great intuition behind that is due to Landau. Indeed, Landau’s idea was so successful that condensed matter experiments still struggle to get a proof of something different from a Fermi liquid. You should ask yourself why, in condensed matter, something that is so strongly coupled in the end appears harmless when you choose the right excitations.

The question to be answered here is a foundational one as, if we are able to answer to this, one should be able to build a QFT in the infrared for Yang-Mills theory and derive a potential. Indeed, in the infrared limit, these excitations will be exchanged rather than gluons as happens instead in the UV limit.

So, your question implies somehow a confusion between the carries and the particles that exchange them.

Ciao,

Marco