Dumbness does not pay


Today I have read on New York Times (see here) that the lawsuit to halt LHC has been dismissed by a federal judge in Honolulu. A wasting of time and resource against missing of common sense and serching for limelight. Very bad. A story to be forgotten as fastly as possible.

Terry Tao and gauge theories


I have found a beatiful post by Terry Tao, a Fields medallist, about gauge theories. See here for a worthwhile reading. This post is truly elucidating and so well written that I thought it was worthing a larger audience.

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


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


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.

A shocking reality


I am currently a member of American Mathematical Society and when I have the chance I support the activity of this venerable and fundamental association by working as a reviewer for Mathematical Reviews. So, I have the opportunity to read Notices that is a beatiful journal published by AMS and send freely to all members. It can be found here. In the August issue I have read an Opinion by Melvyn B. Nathanson of CUNY that shocked me and mostly my fundations on mathematical truth. The article is here. I think that the core of this article resides on the idea that bosses of the community make truth, mostly when proof implies thousands pages of published material very difficult to be checked carefully. We all know that, for the Wiles’s proof of Fermat theorem, the community was lucky enough to catch such a bug in a long proof. This was conveniently corrected and all agreed that such a demonstration indeed was given. But when, for any reason, we have to rely on authorities to get the truth we are in serious troubles. This means that whatever they said should be taken as granted but in this case does seem that no protestation is indeed possible. As scientists we cannot accept “ipse dixit” position and if mathematics is in such a situation something must be done to correct it.

The next question to be answered is: What is the situation in physics? Physics, differently from mathematics, is an experimental science. This means that experiments should grant our ability to tell where the truth is. Today things do not seem to be that easy for a lot of reasons but I think other sources can give judgments better than mine.

Physics laws and strong coupling


It is a well acquired fact that all the laws of physics are expressed through differential equations and our ability as physicists is to unveil their solutions in a way or another. Indeed, almost all these equations are really difficult to solve in a straightforward way and are very far from the exercises at undergraduate courses. During the centuries people invented several techniques to manage such equations and the most generally known is surely perturbation theory. Perturbation theory applies when a small parameter enters into the equations and a series solution is so allowed. I remember I have seen this method the first time at the third year of my “laurea” course and was Giovanni Jona-Lasinio that showed it to me and other fellow students.

Presently, we see as small perturbation theory has become so pervasive that conclusions derived just at a perturbation level are sometime believed always true. An example of this is the Landau pole or, generally, what implies the renormalization program. It is not generally stated but it is quite common the prejudice that when a large parameter enters into a differential equation we are stuck and nothing can be done than using our physical intuition or numerical computation. This is true despite the fact that the inverse of such a large parameter is indeed a small parameter and most known functions have both a small parameter and a large parameter series as well.

As I said elsewhere this is just a prejudice and I have proved it wrong in a series of papers on Physical Review A (see here, here and here). I have given an overview in a recent paper. With such a great innovation to solve differential equations at hand is really tempting to try to apply it to all fields of physics. Indeed, I have worked for a lot of years in quantum optics testing the approach in a lot of successful ways and I have also found applications to condensed matter physics appeared on Physical Review B and Physica E.

The point is quite clear. How to apply all this to partial differential equations? What is the effect of a large perturbation on such equations? Indeed, I have had this understanding under my nose since the start but I have not been so able to catch it immediately. The reason is that the result is really counterintuitive. When a physical system is strongly perturbed all the terms that imply spatial variation can be neglected. So, a strong perturbation series is a gradient expansion and the converse is true as well. I have proved it numerically in a quite easy way using two or three lines of Maple. These results can be found in my very recent paper on quantum field theory (see here and here). Other results can be found by yourself with similar simple means and are very easy to verify.

As strange as may seem this conclusion, it has obtained a striking confirmation through numerical computations in general relativity. Indeed, I have applied this method also to general relativity (see here and here). Indeed this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz or BKL conjecture on the behavior of space-time approaching a singularity. Indeed, BKL conjecture has been analyzed numerically by David Garfinkle with a very beatiful paper published on Physical Review Letters (see here and here). It is seen in a striking way how all the gradient contributions from Einstein equations become increasingly irrelevant as the singularity is approached. This is a clear proof of BKL conjecture and our approach of strong perturbations at work. Since then Prof. Garfinkle has done a lot of other very good work on general relativity (see here).

We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here).

Emerging scenario


Reading arxiv dailys today I have found three different papers on the gluon and ghost propagators for Yang-Mills (see here, here and here). These papers prove that this line of research is very strongly alive and that there exist a lot of points to be settled down before to carry on. In this post I would like to point out several evidences that should not be forgotten when one talks about this matter. First of all there are the results of Yang-Mills theory in D=1+1. We know that, for this dimensionality, Yang-Mills theory has no dynamics. Anyhow, several people tried to solve it on the lattice or modified it to try to relate these solutions of the ones of Dyson-Schwinger equations with a given truncation. The bad news is that they find agreement with such solutions of Dyson-Schwinger equations. Why is this bad news? Because this gives, beyond any doubt, a proof that such a truncation of Dyson-Schwinger equations is fault as it removes any dynamics from Yang-Mills theory in higher dimensionality and appears to agree with numerical results just when such a dynamics does not exist. This is already a severe indicator that lattice computations done in higher dimensionality are right. What do they say us about ghost and gluon propagators?

  • Gluon propagator reaches a non-null finite value at zero momenta.
  • Ghost propagator is that of a free particle.
  • Running coupling goes to zero at lower momenta.

This means that the confinement scenarios that are normally considered are faulty and do not work at all. These results demand for a better understanding of the physical situation at hand. It we are not ourselves convinced that they are right, we will keep on fumbling in the dark losing precious resources and time. Evidences are really heavy already at this stage and should be combined with spectra computations carried out so far. Also in this case a lot of work still must be carried out. You can read the beatiful paper of Craig McNeile about (contribution to QCD 08). It is a mistery to me why these ways are seen as different into the understanding of Yang-Mills theory.

A formula I was looking for


As usual I put in this blog some useful formulas to work out computations in quantum field theory. My aim in these days is to compute the width of the \sigma resonance. This is a major aim in QCD as the nature of this particle is hotly debated. Some authors think that it is a tetraquark or molecular state while others as Narison, Ochs, Minkowski and Mennessier point out the gluonic nature of this resonance. We have expressed our view in some posts (see here and here) and our results heavily show that this resonance is a glueball in agreement with the spectrum we have found for a pure Yang-Mills theory.

Our next step is to understand the role of this resonance in QCD. Indeed, we have shown in our recent paper (see here) that, once the gluon propagator is known, it is possible to derive a Nambu-Jona-Lasinio model from QCD with all parameters properly fixed. We have obtained the following:

S_{NJL} \approx \int d^4x\left[\sum_q \bar q(x)(i\gamma\cdot\partial-m_q)q(x)\right.

-\frac{1}{2}g^2\sum_{q,q'}\bar q(x)\frac{\lambda^a}{2}\gamma^\mu q(x)\times

\left.\int d^4yG(x-y)\bar q'(y)\frac{\lambda^a}{2}\gamma_\mu q'(y)\right]

being G(x-y) the gluon propagator with all indexes in color and space-time already saturated. This in turn means that we can use the following formula (see my paper here and here):

e^{\frac{i}{2}\int d^4xd^4yj(x)G(x-y)j(y)}\approx {\cal N}\int [d\sigma]e^{-i\int d^4x\left[\sigma\left(\frac{1}{2}\partial^2\sigma+\frac{Ng^2}{4}\sigma^3\right)-j\sigma\right]}

being again G(x-y) the gluon propagator for SU(N) and {\cal N} a normalization factor. This formula does hold only for infrared limit, that is when the theory is strongly coupled. We plan to extract physical results from this formula and define in this way the role of \sigma resonance.


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