A set of exact classical solutions of Yang-Mills equations

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):


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



\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


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.


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.


5 Responses to A set of exact classical solutions of Yang-Mills equations

  1. Rafael says:


    just to point you another possible track:

    Click to access 0105222v1.pdf



  2. mfrasca says:

    Dear Rafael,

    This paper is wonderful. I was not aware of this line of research but people working on it hit the right path.

    I have had some exchange with Oliveira (see

    https://marcofrasca.wordpress.com/2008/09/02/should-we-laugh/ )

    as I did not agree with his findings on lattice. I have no doubt that he is a smart researcher. My view is that some years ago, research in infrared physics was derailed by a wrong understanding about Yang-Mills theory. This can happen when one does research but we are also sure that the emerging of truth can be just a matter of time.


  3. mfrasca says:

    There is a pioneering paper by Ezawa and Iwazaki published in 1982 claiming Abelian dominance is the only relevant part at long distances for QCD:


    They do a further hypothesis, based on wisdom of that time, that solutions to be taken are monopoles. Anyhow this paper is worth reading.


  4. Rafael says:

    Dear Marco,

    superb… a very interesting paper!
    See that they have said: “..the only effect of the non-Abelian component is to smear out short-distance behaviors of the theory…”
    Now, if you give a look at: Phys. B 694, 35 (2004), where authors discuss “universality vs. the size of the non-abelian part of a gauge group”, we see that there seems to be a intricate interplay among abelian and non-abelian parts of gauge-groups to determine the correct order of phase transitions.
    I think it would be also very interesting to investigate how your method deals with finite-temperature spectrum and generalized Lie Groups, as Sp(N) and G2 (since they have same Z2 center as SU(2), but, being really bigger).


  5. mfrasca says:

    Dear Rafael,

    That sentence was a major reason to give this ref. to you. I am working on non-null temperature analysis and I have found some interesting hints in the work about quark-gluon plasma cited here


    I hope to find some time to work this out and post a paper on arxiv.


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