A set of exact classical solutions of Yang-Mills equations

28/02/2009

Following the discussion with Rafael Frigori (see here) and returning to sane questions, I discuss here a class of exact classical solutions that must be considered in the class of Maximal Abelian Gauge.  As usual, we consider the following Yang-Mills action

S=\int d^4x\left[\frac{1}{2}\partial_\mu A^a_\nu\partial^\mu A^{a\nu}+\partial^\mu\bar c^a\partial_\mu c^a\right.

-gf^{abc}\partial_\mu A_\nu^aA^{b\mu}A^{c\nu}+\frac{g^2}{4}f^{abc}f^{ars}A^b_\mu A^c_\nu A^{r\mu}A^{s\nu}

\left.+gf^{abc}\partial_\mu\bar c^a A^{b\mu}c^c\right]

being c,\ \bar c the ghost field, g the coupling constant and, for the moment we omit the gauge fixing term. Let us fix the gauge group being SU(2). We choose the following (Smilga’s choice, see the book):

A_1^1=A_2^2=A_3^3=\phi

being \phi a scalar field. The other components are taken to be zero. It easy to see that the action becomes

S=-6\int d^4x\left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+\partial\bar c\partial c\right]+6\int d^4x\frac{2g^2}{4}\phi^4.

This is a very nice result as if we have a solution of the scalar field theory we get immediately a classical solution of Yang-Mills equations while the ghost field decouples and behaves as that of a free particle. But such solutions do exist. We can solve exactly the  equation

\partial_t^2\phi-\partial_x^2\phi+2g^2\phi^3=0

by

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

being sn Jacobi snoidal function, \mu,\ \theta two arbitrary constants, if holds

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

We see that the field acquired a mass notwithstanding it was massless and the same happens to the Yang-Mills field. These are known as non-linear waves. These solutions do not represent a new theoretical view. A new theoretical view is given when they are used to build a quantum field theory. This is the core of the question.

What happens when we keep the gauge fixing term as

\frac{1}{\xi}(\partial\cdot A)^2?

If you substitute Smilga’s choice in this term you will find a correction to the kinematic term implying a rescaling of space variables. This is harmless for the obtained solutions resulting in the end into a multiplicative factor for the action of the scalar field.

The set of Smilga’s choice is very large and increases with the choice of the gauge group. But such solutions always exist.

If you want more information about their use in QCD see the following:

Infrared gluon and ghost propagators Phys. Lett. B

Yang-Mills propagators and QCD to appear in Nucl. Phys. B: Proc. Suppl.

Update: Together with Terry Tao, we agreed that these solutions hold in a perturbative sense, i.e.

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

being \eta_\mu^a a constant and g the coupling taken to be very large. These become exact solutions when just time dependence is retained. So, the theorem contained in the above papers is correct for this latter case and approximate in the general case. As the case of interest is that of a large coupling, these results permit to say that all the conclusions drawn in the above papers are correct. This completed proof will appear shortly in Modern Physics Letters A (see http://arxiv.org/abs/0903.2357 ).

Thanks a lot to Terry for the very helpful criticism.


Wikipedia and abuses

28/02/2009

Till now, I have avoided to feed this flooding about abuses on Wikipedia. But I think that a few words are needed in order to clarify my position and to let my own point of view widely known.

The problem started when Peter Woit took a look at the Yang-Mills entry of Wikipedia. He has found a section apparently self-promoting my work. What was about this section? The title said “Integrable solutions of classical Yang-Mills equations and QFT “. I think that there is a lot to say about this matter as classical solutions of Y-M exist and is well acquired matter. But in this section a class of solutions were put, cited in the Smilga’s book, that I have generalized and introduced in my papers.  Should they be there? I think yes as they belong to the class of solutions stated in the title. They appear to be too recent for inclusion but this is plain mathematics. Mathematics is a two-way switch: It is either right or wrong and so, if these solutions are right, they should be there as a bookkeeping for the readers.

The worst question is anyhow self-promotion. On this ground Peter Woit did worst: He is self-promoting his book (see here , thank you Lubos). Promoting a book means to earn money for the author while promoting a scientific idea may have the right side that, being the idea good, a good service has been done to the community.

The worst aspect of the story has been the intervention of Woit through his blog. This is a perfect war machine that when activated may leave a lot of casualties. People should be smart at their defense as otherwise the risk is to be counted in that number. A flood of people moved toward Wikipedia with any means trying to remove the questioned section and attacking me and whoever has written it. A lot of comments in Woit’s blog was posted attacking me. I was forced to introduce moderation for comments in my blog. Of course this appears like a kind of lynching without any understanding of scientific merit. Curators of Wikipedia decided that majority was right and Woit have had his win: The section was finally removed.

What next? This situation is quite interesting by my side. The reason is that the physical matter is Yang-Mills theory that is one of the biggest open problems both in phyics and mathematics. There is a lot of very good people working on that in this moment and my view is that a complete understanding is at hand. Ask yourself this question: What would be Woit’s position if I am right? This is not like string theory that we do not know when a confirmation will be at hand. Here we have computers, accelerators and a lot of smart people crunching this problem. In a very short time an eventual Woit’s error will be exposed. And by irony, Wikipedia’s entry will be updated with my ideas. Much better than now.

The question to be asked is: Should Wikipedia support new material? Since the editors of scientific entries in Wikipedia are scientists themselves one cannot ask them impartiality. Science is a dynamic endeavor and Wikipedia a dynamic source of information. They should be merged to meet each other in the right way.


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