Classical solutions of Yang-Mills equations

October 9, 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.

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Exact solutions of nonlinear equations

July 15, 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 -<br /> \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!

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Posted by mfrasca

Updated paper

March 18, 2009

After a very interesting analysis about classical solutions of Yang-Mills equations, in this blog and elsewhere in the web, and having recognized that a paper of mine was in great need for corrections (see here) I have finally done it.

I have replaced the paper on arxiv a few moments ago (see here). I do not know if it is immediately available or you have to wait for tomorrow morning. In any case, the only new result added, with respect to material already discussed in this blog, is the first order correction to the propagator of the massless scalar theory. This goes like 1/\lambda making all the argument consistent. This asymptotic series should be modified as the limit \lambda\rightarrow\infty becomes more and more difficult to be applied and this should be in a kind of intermediate region that, presently, I have no technique to manage. This is matter for future work. The perspective is the ability to recover the solution of a scalar field theory for all energy range.

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Posted by mfrasca

Classical scalar theory in D=1+1 and gradient expansion

September 29, 2008


As said before a pde with a large parameter has the spatial variations that are negligible. Let us see this for a very simple case. We consider the following equation

\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}-\lambda\phi^3=0

with the conditions \phi(0,t)=0, \phi(1,t)=0 and \phi(x,0)=x^2-x where the choice of a parabolic profile is arbitrary and can be changed. We also know that, if we can neglect the spatial part, the solution can be written down analytically as (see here and here):

\phi\approx (x^2-x){\rm sn}\left[(x^2-x)\sqrt{\frac{\lambda}{2}}t+x_0,i\right]

being x_0={\rm cn}^{-1}(0,i). Indeed, for \lambda = 5000 we get the following pictures

Numerical Curves - t is chosen as 0=red, 1/8=blue, 1/4=green, 0.3=yellow

Numerical Curves - t is chosen as 0=red, 1/8=blue, 1/4=green, 0.3=yellow

and

Analytical solution - t chosen as above

Analytical solution - t chosen as above

The agreement is excellent confirming the fact that a strong coupling expansion is a gradient expansion. So, a large perturbation entering into a differential equation can be managed much in the same way one does for a small perturbation. In the case of ode look at this post.

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Posted by mfrasca