Solving Dyson-Schwinger equations


Sunday I posted a paper of mine on arxiv (see here). I was interested on managing a simple interacting theory with the technique of Dyson-Schwinger equations. These are a set of exact equations that permit to compute all the n-point functions of a given theory. The critical point is that a lower order equation depends on higher order n-point functions making the solution of all set quite difficult. The most common approach is to try a truncation at some order relying on some physical insight. Of course, to have a control on such a truncation could be a difficult task and the results of a given computation should be carefully checked. The beauty of these equations relies on their non-perturbative nature to be contrasted with the severe difficulty in solving them.

In my paper I consider a massless \phi^4 theory and I solve exactly all the set of Dyson-Schwinger equations. I am able to do this as I know a set of exact solutions of the classical equation of the theory and I am able to solve an apparently difficult equation for the two point function. At the end of the day,  one gets the exact propagator, the spectrum and the beta function. It is seen that this theory has only trivial fixed points. I was able to get these results on another paper of mine. So, it is surely comforting to get identical results with different approaches.

Finally,  I can apply  the mapping theorem with Yang-Mills theories, recently proved thanks also to Terry Tao intervention, to draw conclusions on them in the limit of a very large coupling. In the paper you can find a formulation of this theorem as agreed with Terry, a direct consequence of my latest accepted paper on this matter (see here).

I think this paper adds an important contribution to our understanding of Dyson-Schwinger equations presenting an exact non-trivial solution of them.

Paper replacement


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


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


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.

Updated paper


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.

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 ).

Thanks a lot to Terry for the very helpful criticism.

Exact solution to a classical spontaneously broken scalar theory


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.


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