## Paper replacement

12/05/2009

I have updated the paper with the answer to Terry Tao on arxiv (see here). No correction was needed, rather I have added a new result giving the next-to-leading order correction for the Yang-Mills field. This result is important as it shows the right approximate solution, in an expansion into the inverse of the coupling constant, for the mapping between the scalar and the Yang-Mills field. As we repeated a lot of times, Smilga’s solutions are all is needed to work out our argument as this relies on a gradient expansion. A gradient expansion at the leading order has a solution depending just on time variable. But, as this has been a reason for discussion, I have also shown to what extent my approach applies to the solution of the quartic scalar field given in the form

$\phi(x) = \mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x,i)$

with $p^2=\mu^2\left(\lambda/2\right)^{1\over 2}$ with $\mu$ an integration constant and $\lambda$ the coupling. But I would like to emphasize that the relevance of these solutions for the Yang-Mills case was just demanded by Tao’s criticism but it is not needed for my argument to work. So, the main result of this paper is that

$A_\mu^a(x)=\eta^a_\mu\phi(x)+O(1/g).$

As it has been noted elsewhere, higher order corrections are zero in the Lorenz gauge. This result is certainly not trivial and worth to be considered in a classical analysis of Yang-Mills equations.

Finally, we note as any concern about gauge invariance is just worthless. Smilga’s solutions are exact solutions of the Yang-Mills equations. Casting doubt on them using gauge invariance arguments should be put on the same ground as casting doubt on Kasner solution of Einstein equations using general covariance reasons. Nothing worth to spend time on but a poor excuse to ignore a good work.

## Exact solutions of Yang-Mills theory: The situation

08/04/2009

Some time passed by since Terry Tao was so kind to take a look to my work. His concern about a main theorem in my paper, the so called mapping theorem, was motivated by the fact that no proof exists that there are common solutions between Yang-Mills equations and the one of the quartic scalar field. This point is quite crucial as, if such solutions do not exist, I cannot do any claim about Yang-Mills theory.

Some people are in confusion yet about this matter and I find occasionally someone, e.g. the Czech guy, claiming that my paper is false also after I have proved that such solutions exist.

Of course, Terry meant to point out a weakness in the proof given in my paper as I gave no evidence whatsoever of the form of these solutions and so the proof is, at least, incomplete. My next preprint proved that such solutions indeed exist and my argument is true already at level of perturbation theory. The conclusion is straightforward: Smilga’s choice select a class of common solutions between Yang-Mills equations and a quartic scalar field. I have not presented them explicitly in my paper and this is the reason why all this arguing was started. Terry’s suggestion was to complete the proof  and this I have done.

Curiously enough, I was able to see such solutions only in the Smilga’s book. I think this was Smilga’s idea and was also my source of inspiration.  I was in need of these solutions to treat classical Yang-Mills equations with a gradient expansion against a lot of unmanageable chaotic solutions. I would like to remember here that this approach is quite common in physics. For interested readers, I invite them to look at this beautiful Wikipedia entry about BKL solution. This is the way this approach is used in general relativity with a widespread example as the Kasner solution. This is an exact solution of Einstein equations that depends solely on time. Exactly as happens to the solutions obtained by a Smilga’s choice from Yang-Mills equations. Indeed, I suspect that Kasner solution may be helpful to quantize Einstein equations in the infrared limit. Currently I have no time to exploit this but I have given a hint about here.

Dmitry Podolsky (see his blog here) hit correctly the point when asked for the fate of chaotic solutions in the infrared quantum field theory. Presently, the fact that they are not relevant has the status of a conjecture: No quantum field theory can be built out of classical chaotic solutions. I do not even know how to face this kind of question as no closed form chaotic solutions exist to start from.

Finally, this gives the current situation about this matter. My paper that started all this is correct and in agreement with current lattice results. People’s mood about lattice computations range from fully convinced to skeptical.  My view is that they represent correctly the infrared physics at hand but I am a supporter of these people working on lattice computations and so, my judgement should not be counted.

## What is a glueball?

31/03/2009

Recently I have read a post in Dmitry’s blog by Fabien Buisseret claiming the following conclusion:

“In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states.”

Indeed he considers free gluons interacting each other through a given potential forming bound states. Of course, as all of you may be aware, nobody in the Earth was able to prove that, in the low energy limit, gluons are the right states entering into a quantum Yang-Mills theory. So, this view appears as a well rooted prejudice in the community.

Let me explain what I mean with a classical example. I take the following quartic theory

$\partial^2\phi+\lambda\phi^3=0.$

In the small coupling limit you will get plane waves plus higher order corrections. Assume these plane waves are gluons as we all of us is aware from high-energy QCD. Indeed, these plane waves describe massless excitations. Now I claim that these solutions should hold also when the coupling $\lambda$ becomes increasingly large. But here I have the exact solution

$\phi(x)=\mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x,i)$

being sn a Jacobi snoidal function and $\mu$ an arbitrary constant. But now

$p^2=\mu^2\left(\frac{\lambda}{2}\right)^{1\over 2}$

and I am describing massive excitations that are not resembling at all my plane wave solutions given above. The claim is blatantly wrong already at a classical level with this very simple example.

This proves without any doubt that the view of glueballs as bound states of gluons is plainly wrong as nobody knows the behavior of a Yang-Mills theory in the infrared limit and so, nobody knows what are the right glue excitations for the theory here. As you may have realized, if you would know this you will be just  filed for a Millenium Prize. This means that, unless we learn how to treat the theory at low energies, all this kind of approaches are doomed.

## Quantum field theory and gradient expansion

21/02/2009

In a preceding post (see here) I showed as a covariant gradient expansion can be accomplished maintaining Lorentz invariance during computation. Now I discuss here how to manage the corresponding generating functional

$Z[j]=\int[d\phi]e^{i\int d^4x\frac{1}{2}[(\partial\phi)^2-m^2\phi^2]+i\int d^4xj\phi}.$

This integral can be computed exactly, the theory being free and the integral is a Gaussian one, to give

$Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}$

where we have introduced the Feynman propagator $\Delta(x-y)$. This is well-knwon matter. But now we rewrite down the above integral introducing another spatial coordinate and write down

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2]+i\int d\tau d^4xj\phi}.$

Feynman propagator solving this integral is given by

$\Delta(p)=\frac{1}{p_\tau^2-p^2-m^2+i\epsilon}$

and a gradient expansion just means a series into $p^2$ of this propagator. From this we learn immeadiately two things:

• When one takes $p=0$ we get the right spectrum of the theory: a pole at $p_\tau^2=m^2.$
• When one takes $p_\tau=0$ and Wick-rotates one of the four spatial coordinates we recover the right Feynman propagator.

All works fine and we have kept Lorentz invariance everywhere hidden into the Euclidean part of a five-dimensional theory. Neglecting the Euclidean part gives us back the spectrum of the theory. This is the leading order of a gradient expansion.

So, the next step is to see what happens with an interaction term. I have already solved this problem here and was published by Physical Review D (see here). In this paper I did not care about Lorentz invariance as I expected it would be recovered in the end of computations as indeed happens. But here we can recover the main result of the paper keeping Lorentz invariance. One has

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}$

and if we want something not trivial we have to keep the interaction term into the leading order of our gradient expansion. So we will break the exponent as

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]-i\int d\tau d^4x\frac{1}{2}[(\partial\phi)^2+m^2\phi^2]+i\int d\tau d^4xj\phi}$

and our leading order functional is now

$Z_0[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}.$

This can be cast into a Gaussian form as, in the infrared limit, the one of our interest, one can use the following small time approximation

$\phi(x,\tau)\approx\int d\tau' d^4y \delta^4(x-y)\Delta(\tau-\tau')j(y,\tau')$

being now

$\partial_\tau^2\Delta(\tau)+\lambda\Delta(\tau)^3=\delta(\tau)$

that can be exactly solved giving back all the results of my paper. When the Gaussian form of the theory is obtained one can easily show that, in the infrared limit, the quartic scalar field theory is trivial as we obtain again a generating functional in the form

$Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}$

being now

$\Delta(p)=\sum_n\frac{A_n}{p^2-m^2_n+i\epsilon}$

after Wick-rotated a spatial variable and having set $p_\tau=0$. The spectrum is proper to a trivial theory being that of an harmonic oscillator.

I think that all this machinery does work very well and is quite robust opening up a lot of possibilities to have a look at the other side of the world.

## Exact solution to a classical spontaneously broken scalar theory

02/08/2008

As promised in my preceding post I said that a classical spontaneously broken scalar theory can be exactly solved. This is true as I will show. Consider the equation

$\ddot\phi -\Delta\phi + \lambda\phi^3-m^2\phi=0.$

You can check by yourself that the exact solution is given by

$\phi(x)=v\cdot{\rm dn}(p\cdot x,i)$

being $v=\sqrt{2m^2/3\lambda}$ the v.e.v. of the field and ${\rm dn}$ an elliptical Jacobi function. As always the following dispersion relation must be true

$p^2=\frac{\lambda v^2}{2}$

giving a consistent classical solution. When one goes to see the spectrum of the theory, the Fourier series of the Jacobi dn function has a zero mass excitation, the Goldstone boson.

Update: A proper full solution is given by

$\phi(x)=v\cdot{\rm dn}(p\cdot x+\varphi,i)$

being $\varphi$ an integration constant.