## 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.

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.

## Quote of the day

27/02/2009

Today let me quote Einstein:

Great spirits have often encountered violent opposition from weak minds

My dedication is for people working on string theory.

## Osaka and Berlin merge their data!

26/02/2009

Today in arxiv appeared a relevant paper by Osaka and Berlin groups (see here). This is a really important paper as these two groups merged their data for the lattice computation of the gluon and ghost propagators for SU(3) in the Coulomb gauge. As usual I give here a picture summing up their results about gluon propagator

As you can see one has again the propagator reaching a finite, non-null value in the infrared limit. About the ghost propagator they obtain again a result very near to the case of a free particle. In this case the agreement was perfect for SU(2). For SU(3) there is a tiny disagreement.

I would like to emphasize a couple of points that should be discussed with these results at hand. There is a paper, published on Physical Review Letters, that was claiming that the gluon propagator in the Coulomb gauge should take the Gribov form going to zero at lower momenta. You can find this paper here and here. I think that authors should reconsider their computations as the disagreement with lattice is really serious. All the research lines aimed at a proof of confinement scenarios heavily relying on Gribov ideas seem to have reached a failure point. There could be a lot of reasons for this but it seems to me that, as lattice computations improve, we are left with the only option that the starting points of all these studies are to be reconsidered.

A second point to be made is the completely missing link between people working on the computation of propagators and those working on the spectrum of QCD. I think this is the moment to try to connet these two relevant areas as times are mature to try a consistency check between them. After the failure in view of some functional methods do we have to believe yet that Kaellen-Lehman formula does not apply in the infrared limit?

## Scholarpedia

24/02/2009

I would like to point out to my readers Scholarpedia. This represents a significant effort of the scientific community to grant a wiki-like resource with the benefit of peer-review. This means that articles are written on invitation and reviewed by referees chosen by the Editorial Board. This resource is important as correctness of information is granted by the review process and by the choice of the authors that are generally main contributors to the considered fields. It is interesting to point out that, currently, there are articles written by 15 Nobelists and 4 Fields medalists. The most relevant aspect to be emphasized is that the information is freely accessible to everybody exactly in the spirit of Wikipedia.

## 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.

## Most extreme gamma-ray blast yet

20/02/2009

As my blog’s readers know, I follow as far as I can space missions that can have a deep impact on our knowledge of universe. Most of them are from NASA. One of these missions is Fermi-GLAST that has produced a beautiful result quite recently.  It has seen the greatest gamma-ray burst ever (see here). The paper with the results is appeared on Science (see here). The burst was seen in Carina constellation.  These explosions are the most energetic processes in the universe and were uncovered by chance with military satellites named Vela used to find nuclear explosions in the atmosphere in the sixties of the last century. Understanding gamma-ray bursts implies a deeper understanding of stellar explosions.

## Too hilarious to not cite

19/02/2009

I have found this very nice piece of humor about Higgs particle at Cosmic Variance (see here). This text is a combination of quotes from famous movies. E.g., the last paragraph is taken from Ocean’s eleven and runs like this:

Terry: All right. Now I have complied with your every request, would you agree?
Rusty: I would.
Terry: Good, ’cause now I have one of my own. Run and hide, asshole. Run and hide. If you should be picked up next week buying a hundred-thousand dollar sports car in Newport Beach, I am going to be supremely disappointed. Because I want my people to find you, and when they do, rest assured we are not going to hand you over to the police. So my advice to you again is this: run and hide. That is all that I ask.

The first paragraph is taken from The Matrix and runs like this:

Neo: I know you’re out there. I can feel you now. I know that you’re afraid… you’re afraid of us. You’re afraid of change. I don’t know the future. I didn’t come here to tell you how this is going to end. I came here to tell you how it’s going to begin. I’m going to hang up this phone, and then I’m going to show these people what you don’t want them to see. I’m going to show them a world without you. A world without rules and controls, without borders or boundaries. A world where anything is possible. Where we go from there is a choice I leave to you.

Nobody at Cosmic Variance was able to hit the citation for the second paragraph. It is very well accustomed to LHC situation so not that easy to identify. If you know it please, share your knowledge.

Movie quotes are from Internet Movie DB.

## Lubos and divergent series

12/02/2009

I have read this nice post and I have found it really interesting. The reason is the kind of approach of Lubos Motl, being physicist-like, on such somewhat old mathematical matter. The question of divergent series and their summation is as old as at least Euler and there is a wonderful book written by a great British mathematician, G. H. Hardy, that treats this problem here. Hardy is well-known for several discoveries in mathematics and one of this is Ramanujan. He had a long time collaboration with John Littlewood.

Hardy’s book is really shocking for people that do not know divergent series. In mathematics several well coded resummation techniques exist for these series. With a proper choice of one of these techniques a meaning can be attached to them. A typical example can be

$1-1+1-1+\ldots=\frac{1}{2}$

and this is true exactly in the same way is true that the sum of all  integers is -1/12. Of course, this means that discoveries by string theorists are surely others and most important than this one that is just good and known mathematics.

I agree with Lubos that these techniques are not routinely taught to physics students and come out as a surprise also to most mathematics students. I am convinced that Hardy’s book can be used for a very good series of lectures, for a short time, to make people acquainted with this deep matter that can have unexpected uses.

I think that mathematicians have something to teach us that is really profound: Do not throw anything out of the window. It could turn back in an unexpected way.

http://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/

http://cornellmath.wordpress.com/2007/07/30/sum-divergent-series-ii/

http://cornellmath.wordpress.com/2007/08/02/sum-divergent-series-iii/

Enjoy!

## Quantum field theory and prejudices

10/02/2009

It happened somehow, discussing in this and others blogs, that I have declared that today there are several prejudices about quantum field theory. I never justified this claim and now I will try to do this making the arguments as clearest as I can.

The point is quite easy to be realized when we recognize that the only way we know to manage a quantum field theory is small perturbation theory. So, all we know about this matter is obtained through this ancient mathematical technique. You should try to think about our oldest thinkers looking at the Earth and claiming no other continent exists rather than Europe. Whatever variation you get is about Europe. After a lot of time, a brave man took the sea and uncovered America. But this is another story.

There is people in our community that is ready to claim that a strong coupling expansion is not possible at all. I have read this on a beautiful textbook, Peskin and Scroeder. This is my textbook of choice but is plainly wrong about this matter. It is like our ancient thinkers claiming that nothing else is possible other than Europe because this is the best of all the possible worlds. Claims like this come from our renormalization group understanding of quantum field theory. As you may know, such understandings are realized through small perturbation theory and we are taken back to the beginning. How is the other side of the World?

Well, I am not Columbus but I can say that here we have an entire continent to explore. As happened to Columbus, there are a lot of thinkers against such an idea but maybe it is worth a try taking into account the possible pay-off.

What should one say about renormalization? Renormalization appears in any attempt to use small perturbation theory in quantum field theory. It naturally arises from the product of distributions at the same point. This is not quite a sensible mathematical situation. The question to ask is then: Is this true with any kind of perturbation technique? One could answer: we haven’t any other and so the requirement of renormalizability becomes a key element to have a valid quantum field theory. The reason for this is the same again: The only technique we believe to exist to do computations in quantum field theory is small perturbation technique and this must work to have a sensible theory. Meantime, we have also learned to work with non-renormalizable theories and called them effective theories. All this is well-known matter.

Of course, the wrong point about such a question is the claim that there are no other techniques than small perturbation theory. There is always another face to a medal as the readers of my blog know. But this implies a great cultural jump to be accepted. The same that happened to Columbus.