Classical solutions of Yang-Mills equations

09/10/2009

So far, I have posted several posts in this blog about the question of classical solutions to Yang-Mills equations. This has produced some fuzz, mostly arisen from my published papers, as to such solutions may not be correct. Thanks to a wise intervention of Terry Tao, I was able to give a complete understanding of my solutions and a theorem was fully proved in a recent paper of mine to appear in Modern Physics Letters A (see here), agreed with Terry in a private communication. So, I think it is time to give a description of this result here as it appears really interesting showing how, already at a classical level, this theory can display massive solutions and a mass gap is already seen in this case. Then, it takes a really small step to get the corresponding proof in quantum field theory.

To understand how these solutions are obtained, let us consider the following equation for a scalar field

\Box\phi+\lambda\phi^3=0.

This is a massless self-interacting field. We can select a class of solutions by looking at the case of a rest reference frame. So, we put any dependence on spatial variables to zero and solve the equation

\partial_{tt}\phi+\lambda\phi^3=0

whose solutions are known and given by

\phi(t,0)=\mu\left(\frac{2}{\lambda}\right)^{\frac{1}{4}}{\rm sn}\left[\left(\frac{\lambda}{2}\right)^{\frac{1}{4}}\mu t+\theta,i\right]

being \mu and \theta two integration constants and sn a Jacobi elliptical function. Then, boosting this solution will produce an exact solution of the equation we started from given by

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

provided the following dispersion relation holds

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

and we see that, although we started with a massless field, self-interaction provided us massive solutions!

Now, the next question one should ask is if such a mechanism may be at work for classical Yang-Mills equations. These can be written down as

\partial^\mu\partial_\mu A^a_\nu-\left(1-\frac{1}{\alpha}\right)\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)

+gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0

being \alpha chosen depending on the gauge choice, g the coupling and f^{abc} the structure constants of the gauge group taken to be SU(N). The theorem I proved in my paper above states that the solution given for the scalar field theory is an exact solution of Yang-Mills equations, provided it will not depend on spatial coordinates, for a given choice of Yang-Mills components (Smilga’s choice) and \lambda=Ng^2, otherwise the following identity holds

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

Here \eta_\mu^a is a set of constants arising with the Smilga’s choice. This theorem has the following implications: Firstly, when the coupling become increasingly large, a massless scalar field theory and Yang-Mills theory can be mapped each other. Secondly, already at the classical level, for a coupling large enough, a Yang-Mills theory gets massive solutions. We can see here that a mass gap arises already at a classical level for these theories. Finally, we emphasize that the above mapping appears to hold only in a strong coupling regime while, on the other side, these theories manifest different behaviors. Indeed, we know that Yang-Mills theory has asymptotic freedom while the scalar theory has not. The mapping theorem just mirrors this situation.

We note that these solutions are wave-like ones and describe free massive excitations. This means that these classical theories have to be considered trivial in some sense as these solutions seem to behave in the same way as the plane waves of a free theory.

One can build a quantum field theory on these classical solutions obtaining a theory manifesting a mass gap in some limit. This is has been done in several papers of mine and I will not repeat these arguments here.


Exact solutions on arxiv

24/07/2009

As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator

-\Box\Delta+3\lambda\phi_c^2(x)\Delta=\delta^D(x)

being \phi_c the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation

\ddot\phi(t)+3\lambda\phi_c^2(t,0)\phi(t)=\delta(t)

that I know when the phase in \phi_c(t,0) is quantized as (4n+1)K(i), being n an integer and K(i) an elliptic integral. This gives back a consistent result in the strong coupling limit, \lambda\rightarrow\infty, with my preceding paper on Physical Review D (see here).

The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.


Exact solutions of nonlinear equations

15/07/2009

Recently, I have posted on the site of Terry Tao and Jim Colliander, Dispersive Wiki. I am a regular contributor to this beautiful effort to collect all available knowledge about differential equations and dispersive phenomena. Of course, I can give contributions as a solver of differential equations in the vein of a pure physicist. But mathematicians are able to give rigorous theorems on the behavior of the solutions without really solving them. I invite you to take some time to look at this site and, if you are an expert, to register and contribute to it.

My recent contribution is about exact solutions of nonlinear equations. This is a really interesting field and most of the relevant results come from soliton theory.  Terry posted on his blog about Liouville equation (see here). This equation is exactly solvable and is widely known to people working in string theory. But also one of the most known equations in physics literature can be solved exactly. My preprint shows this. Indeed I have to update it as, working on KAM theorem, I have obtained the exact solution to the following equation (check here on Dispersive Wiki):

\Box\phi +\mu_0^2\phi+\lambda\phi^3 = 0

that can be written as

\phi(x) = \pm\sqrt{\frac{2\mu^4}{\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}{-\mu_0^2 -  \sqrt{\mu_0^4 + 2\lambda\mu^4}}}\right)

being now the dispersion relation

p^2=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4}}.

As always \mu is an arbitrary parameter with the dimension of a mass. You can see here an example of mass renormalization due to interaction. Indeed, from the dispersion relation we can recognize the following renormalized mass

m^2(\mu_0,\mu,\lambda)=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4}}

that depends on the coupling. This class of solutions clearly show how the nonlinearities produce contributions to mass. Either by modifying it or by generating it. So, it is not difficult to imagine that Nature may have adopted them to display mass wherever there is not.

As a by-product, I am now able to give a consistent quantum field theory in the infrared for the scalar field (always thank to my work on KAM theorem), obtaining the needed corrections to the propagator and the spectrum. I hope to find some time in the next days to add all this new material to my preprint. Meanwhile, enjoy Dispersive Wiki!


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.


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