## A wrong argument

03/11/2008

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:

$A_\mu^a=\eta^a_\mu\left(\frac{\Lambda}{\sqrt{g}}\right){\rm sn}(p\cdot x,i)$

with $\eta^a_\mu=((0,1,0,0),(0,0,1,0),(0,0,0,1))$ and $p^2=g\Lambda^2$ 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

$m_n=(2n+1)\frac{\pi}{2K(i)}\sqrt{g}\Lambda$

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 $\sigma$ 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.