What is a glueball?


Recently I have read a post in Dmitry’s blog by Fabien Buisseret claiming the following conclusion:

“In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states.”

Indeed he considers free gluons interacting each other through a given potential forming bound states. Of course, as all of you may be aware, nobody in the Earth was able to prove that, in the low energy limit, gluons are the right states entering into a quantum Yang-Mills theory. So, this view appears as a well rooted prejudice in the community.

Let me explain what I mean with a classical example. I take the following quartic theory


In the small coupling limit you will get plane waves plus higher order corrections. Assume these plane waves are gluons as we all of us is aware from high-energy QCD. Indeed, these plane waves describe massless excitations. Now I claim that these solutions should hold also when the coupling \lambda becomes increasingly large. But here I have the exact solution

\phi(x)=\mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x,i)

being sn a Jacobi snoidal function and \mu an arbitrary constant. But now

p^2=\mu^2\left(\frac{\lambda}{2}\right)^{1\over 2}

and I am describing massive excitations that are not resembling at all my plane wave solutions given above. The claim is blatantly wrong already at a classical level with this very simple example.

This proves without any doubt that the view of glueballs as bound states of gluons is plainly wrong as nobody knows the behavior of a Yang-Mills theory in the infrared limit and so, nobody knows what are the right glue excitations for the theory here. As you may have realized, if you would know this you will be just  filed for a Millenium Prize. This means that, unless we learn how to treat the theory at low energies, all this kind of approaches are doomed.

Gluon condensate


While I am coping with a revision of a paper of mine asked by a referee, I realized that these solutions of Yang-Mills equations implied by a Smilga’s choice give a proof of existence of a gluon condensate. This in turn means that a lot of phenomenological studies carried out since eighties of the last century are sound as are also their conclusions. E.g. you can check this paper where the authors find a close agreement with my findings about glueball spectrum. The ideas of these authors are founded on the concept of gluon and quark condensates. As they conclusions agree with mine, I have taken some time to think about this. My main conclusion is the following. If you have a gluon condensate, the theory should give \langle F\cdot F\rangle\ne 0 being F_{\mu\nu}^a the usual gluon field. So, let us work out this classically. Let us consider a scalar field mapped on the gluon field in such a way to have

A_\mu^a(t)=\eta_\mu^a \Lambda\left(\frac{2}{3g^2}\right)^\frac{1}{4}{\rm sn}\left[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}t,i\right]

being sn a Jacobi snoidal function, and \eta_\mu^a a constant array of elements obtained by a Smilga’s choice. When you work out the product F\cdot F the main contribution will come from the quartic term producing a term \langle \phi(t)^4 \rangle. Classically, we substitute the average with \frac{1}{T}\int_0^T dt being the period T=4K(i)/[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}]. The integration is quite straightforward and gives

\langle \phi(t)^4 \rangle=\frac{\Gamma(1/4)^2}{18K(i)\sqrt{2\pi}}\frac{\Lambda^4}{4\pi\alpha_s}

I will evaluate this average in order to see if the order of magnitude is the right one with respect to the computations carried out by Kisslinger and Johnson. But the fact that this average is indeed not equal zero is a proof of existence of the gluon condensate directly from Yang-Mills equations.

Phases of a 2D electron gas


A lot of devices today are conceived to work with a 2D electron gas (2DEG). A typical and widespread application is a MOSFET where this gas makes a conducting channel with a neutralizing background of positive ions. A 2DEG is an essential part of any nanoscale device (see my preceding post) and we know that a lot of unexpected effects are seen when the temperature is lowered to few nK°, so very near absolute zero, where a fully quantum behavior should set in but something weird generally happens.

To understand these quite strange behaviors becomes mandatory to have an idea about what happens to a 2DEG changing its temperature. So, there are a lot of studies about. One of these lines of research relies on Montecarlo computations with a fixed number of electrons and taking a proper interaction between them. This people can then obtain a phase diagram of 2DEG and these findings are really interesting. A phase diagram of the 2DEG has a Wigner crystal phase at lower densities while , at higher densities the gas, in its ground state, behaves paramagnetically. This paramagnetic phase is unstable, lowering the density, and the gas enters a ferromagnetic phase! This is quite interesting as ferromagnetic states can produce such excitations as magnons that can make quantum behavior to lose its coherence. I have discussed this here (published on PRB) and here. For supporting these papers I have found a beautiful work of Giovanni Bachelet and his group here (published on PRL) where evidence is found for a ferromagnetic phase. Currently, Giovanni Bachelet has been elected at Italian Parliament for Partito Democratico (Democratic Party). You can find some biographical notes about him (in Italian) here.

The open question about these phases is to know how stable they are. A recent paper on PRL by Drummond and Needs, using the aforementioned Montecarlo methods, try to answer this question (see here).  The main conclusion they arrive is that the ferromagnetic phase does not appear to be stable while they do not find evidence for more exotic phases even if they cannot rule them out. Of course, they confirm all the preceding findings about the very existence of the known phases of 2DEG we mentioned  that since now are all well acquired. Some experimental hint exists for the ferromagnetic phase (see here) but this is not conclusive evidence.

This kind of research is really exciting being at the foundations of our understanding of behavior of matter in exotical physical situations. In the near future we will see how the complete picture will appear.

Edward Witten


Today I was at the Festival of Mathematics 2009 in Rome to listen a talk by Edward Witten. festivaldellamatematica2009Witten is one of the greatest living physicists and his contributions to mathematics were so relevant that he was awarded a Fields medal. This was for me a great chance to see him personally and hear at his way of doing physics for everybody. This is a challenging task for anyone and mostly for the most relevant personalities of our community. I was there with my eleven years old son and two of his friends. Before the start of the talk, John Nash come out near our row of seats and my son and his friends suggested to go to him asking for an autograph. Indeed, he seemed in real difficulty as some people was around him asking for an handshaking or something else. Somebody took him away and this was a significant help.

I showed Witten immediately at my company. He was there speaking and greeting people around. He appeared a tall and a very cordial man.

Marco Cattaneo, deputy director of the Italian edition of Scientific American (Le Scienze), introduced Witten with a very beautiful and well deserved presentation. Witten of course speaks Italian being his wife Chiara Nappi, an Italian physicist. Witten started to talk in Italian saying that he was very happy to be in Italy to meet his wife parents but his Italian was not enough to sustain a talk like the one he was giving.

The talk was addressed to a generic public. It was very well presented and my company found it very interesting. Witten did not use any formulas rather than Einstein’s E=mc^2 and the parabola y=x^2 and this is enough to keep up the attention of the public for all the time.

Witten pointed out that quantum field theory represents the greatest achievement ever for physicists. This theory is so deep and complex that mathematicians still fail to go through it fully and most of these results, widely used by physicists, are presently out of reach for mathematical proofs. He also said clearly, showing it explicitly, that the problem implied in the vertexes of ordinary Feynman diagrams are removed by string theory making all the machinery less singular.

He did a historical excursus starting from Einstein and arriving to string theory. He showed the famous Anderson’s photograph blatantly proving the very existence of antimatter. A great success of the wedding between special relativity and quantum mechanics. This wedding produced such a great triumph as quantum field theory. Witten showed this with the muon magnetic moment, emphasizing the precise agreement between theory and experiment, but saying that the small discrepancy may be or not real new physics being just at 1\sigma.

He emphasized the long path it takes to physicists to achieve our present understanding of quantum field theory and cited several Nobel prize winners that gave key contributions for this goal. He pointed out the relevance of the seventies of the last century that become a cornerstone moment for our current view.

Starting from Gabriele Veneziano‘s insight, Witten arrived to our current view about string theory. He said that this theory has had some frailty aspects that put it, sometime, on the border of a gulch. But, as we know, recoveries happened. He said that strings set the rules and not the other way round as happens with the Standard Model. He gave the example of the Veneziano’s model for strong interactions that was there pretending spin two excitations. This made the model better suited for other aims as indeed happened.

Witten hopes that LHC will unveil supersymmetry. He showed a detector of this great accelerator that we will see at work at the end of this year. Discovery of supersymmetry will be a great achievement for humankind as it will be the first evidence for a world with more than four dimensions. Anyhow, Witten said, string theory put out several elements, quantum gravity and supersymmetry are two of them, that make this theory compelling.

After the talk, some questions by the public were about ten or eleven dimensions in string theory. Witten avoided to be too technical. But the most interesting question was the one by Marco Cattaneo. He asked about critics of string theory and its present inability to do predictions. Witten’s answer was quite unexpected. He said that it is a good fact that a theory has critics. It is some kind of praise for it. But he also said, and his answer was quite similar to the one of Nicola Cabibbo, that there are a lot of things to be understood yet but such a richness physicists run into cannot be just a matter of chance with no significance.

Surely, this has been a very well paid waiting!

Nicola Cabibbo and Arno Penzias


Today I have been at Festival della Matematica 2009 here in Rome to listen at a discussion with Arno Penzias and Nicola Cabibbo.  The moderator was festivaldellamatematica2009Riccardo Chiaberge, a journalist of the italian newspaper Il Sole 24 Ore. Chiaberge wrote a book with an interview to George Coyne and Arno Penzias (in Italian, see here). Cabibbo is currently the President of the Pontificia Accademia delle Scienze and is a believer. This evening he took the role Coyne had in the book in a confrontation between scientists with different beliefs. Asked by Chiaberge if the Pope listens Accademia, Cabibbo said “Sometime.” and cited the case of cerebral death.

Penzias has been put by Chiaberge on the Einstein side, a “non-believer deeply religious”. But what Penzias said has been sometime astonishing revealing a kind of faith elevating humankind well above its nature. Of course, themes like string theory and multiverse were also touched upon in the discussion. Cabibbo said that we should give time to the theory to develop before to conclude anything about, being now too early to draw any conclusion. Stringists will say but surely “mathematics is truly beautiful”. When he said so I thought to Perelman that, copying a functional from string theory, achieved one of the greatest goals of modern mathematics. Penzias is more skeptical about and does not think it should be considered science yet and, on the same line, multiverse is not falsifiable and so to be dismissed. Cabibbo said “till now!” letting us think that the future will deserve some surprise about as always happened in matters like these.

This point was quite entertaining as Cabibbo defended multiverse as the most elegant idea to explain the foundations of quantum mechanics but, as Chiaberge emphasized, this is an escape for atheist to claim the non-existence of God. On the other side, Penzias said that our current understanding of Universe and all this nice fine tuning of its constants appears a serious support for believers.

Galileo, Copernico and Kepler were discussed and the Galileo question, with his trial and abjure, was declared by Cabibbo as a severe error by the Catholic Church that cost too much to Italian science. He said that “Bellarmino, notwithstanding was a fine cultured man, failed to recognize that these were completely new matters” and should have been carefully treated.

About the question of intelligent design and Darwinism, Penzias put forward a nice metaphor. He remembered the last scene of The Wizard of Oz where a dog moves a curtain unveiling the wizard pushing around buttons all the time and this appears to be the god of intelligent design. On the other side, human beings are really cousins of chimpanzees but there is something more, that kind of inexplicable that makes us believe that something like love and free will are real as the rest of our physical world. Cabibbo added that the idea of evolution should enter into the certainties accepted by the Catholic Church as the Copernican system and all that. Evolution is an assessed fact but still some embarrassment is seen from clerics, the same that happens when one talks about the possibility of intelligent life in other solar systems. Giordano Bruno has not been fully digested by Catholic Church yet.

Cabibbo also put forward a nice analogy between phase transitions and the appearance of  intelligence in human beings that completely overwhelmed previous animal species, a threshold effect.

Chiaberge gave a nice summary of the discussion by saying:”Science has limits but no authority should impose limits on it”. All agreed about this but Penzias remembered horrors of some experiments and Cabibbo emphasized that for medicine some kind of limits should be eventually considered or, better, self-imposed.

Tomorrow I will be there to listen Edward Witten. Stay tuned!

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.



Today I have posted a paper on arxiv. It will appear on monday. This paper was required to my by Terry Tao to supplement the proof of the mapping theorem showing that indeed it holds.

If you cannot hold the paper is 09032357v1: preprint. Don’t trust that number as may change.

The argument may be put up with very simple words: If you trust Smilga’s solutions that depend only on time, a Lorentz boost will fit the bill.

I did it for you


It is very easy to show, from Yang-Mills equations, how to obtain a scalar field equation through the Smilga’s choice. Let us write down Yang-Mills equations

\partial^\mu\partial_\mu A^a_\nu-\left(1-\frac{1}{\alpha}\right)\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)+gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0

using the choice A_1^1=A_2^2=A_3^3=\phi. This is really a great simplification. Smilga, in his book, already checked this for us but we give here the full computation. From above eqautions, the only critical term is the following

f^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)

as this term would produce terms deviating from the known form of the scalar theory. For SU(2) we have f^{abc}=\epsilon^{abc} the fully-antisymmetric Levi-Civita tensor. This means that we will have

\epsilon^{a1c}A^{11}(\partial_1A_\nu^c-\partial_\nu A_1^c)+

\epsilon^{a2c}A^{22}(\partial_2A_\nu^c-\partial_\nu A_2^c)+

\epsilon^{a3c}A^{33}(\partial_3A_\nu^c-\partial_\nu A_3^c).

Where we have used largely Smilga’s choice. Now do the following. Take the following components to evolve \nu=1 a=1, \nu=2  a=2 and \nu=3, a=3. It easy to see that the possible harmful term is zero with the Smilgaì’s choice. Now, for the cubic term you should use the useful relation


and you will get back the quartic term.

The gauge fixing term can be easily disposed of through a rescaling of spatial variables while the kinematic term gives the right contribution. You will get three identical equations for the scalar field.

Of course, Smilga in his book already did this and I repeated his computations after the Editor of PLB asked for a revision having the referee already put out this problem. The Editorial work was done very well and two referees read the paper emphasizing errors where they were.

Finally, Tao’s critcism does not apply as I said. This does not mean that what he says is wrong. This means that does not apply to my case.

Update: As the question of the gauge fixing term appears so relevant, let me fix it once and for all. Firstly, I would like to point out that these solutions belong to a class of solutions in the Maximal Abelian Gauge (MAG). But let us forget about this and consider the question of gauge fixing. This term is arbitrarily introduced in the Lagrangian of the field in order to fix the gauge when a quantization procedure is applied. Due to gauge invariance and the fact that becomes an exact differential after partial integration, it useful to have it there for the above aims. The form that it  takes is

\frac{1}{\alpha}(\partial A)^2

and is put directly into the Lagrangian. How does this term become with the Smilga’s choice? One has


and the final effect is a pure rescaling into the space variables of the scalar field. In this way the argument is made consistent. One cannot take the other way around for the very nature of this term and claiming the result is wrong.

This particular class of solutions belongs to the subgroup of SU(N) given by the direct product of U(1). This is a property of MAG and all the matter is really consistent and works.

Finally, I invite people commenting this and other posts to limit herself to polite responses and in the realm of scientific discussion. Of course, doing something wrong happens and happened to anyone working in a scientifc endeavour for the simple reason that she is really doing things. People that only do useless criticisms boiling down to personal offenses are kindly invited to refrain from further interventions.

Update 2: I will get a paper published about this matter. Please, check here.

Is Terry wrong?


I am a great estimator of Terry Tao and a reader of his blog. Tao is a Fields medalist and one of the greatest living mathematicians. Relying on such a giant authority may give someone the feeling of being a kind of dwarf trying to be listened around. Anyhow I will try. Terry come out with an intervention in Wikipedia here claiming:

“It may be relevant to point out that one of the references cited in the disputed section [3] has a significant error in it, despite being published. Namely, in the proof of Theorem 1, the author is assuming that an extremum A for the Yang-Mills action for a special class of connections (namely those in which A^1_1=A^2_2=A^3_3 and all other components vanish) is necessarily an extremum for the Yang-Mills action for all other connections also, but this is not the case (just because YM(A) \geq YM(A'), for instance, for A’ of this special form, does not imply that YM(A) \geq YM(A') for general A’). Since one needs to be an extremiser (or critical point) in the space of all connections in order to be a solution to the Yang-Mills equations, the mapping provided in Theorem 1 has not been shown to actually produce solutions to the Yang-Mills equation (and I suspect that if one actually checks the Yang-Mills equation for this mapping, that one will not in fact get such a solution). Terry (talk) 20:32, 28 February 2009 (UTC)”

This claim of mistake by my side contains a misinterpretation of the mapping theorem. If the theorem would claim that this is true for all connections, as Terry says, it would be istantaneously false. I cannot map a scalar field on all the Y-M connections (think of chaotic solutions). The theorem simply states that there exists a class of solutions of the quartic scalar field that are also solution for the Yang-Mills equations and this can be easily proved by substitution (check Smilga’s book) and Tao is proved istantaneously wrong. So now, what is the point? I have a class of Yang-Mills solutions that Tao is claiming are not. But whoever can check by herself that I am right. So, is Terry wrong?

Physicists and finance


Just to point out an interesting article in the New York Times (see here) about physicists working for financial markets. I have known some years ago a physicist that took this decision rather than keeping on living as a postdoc with very few bucks to maintain his family. I have never seen him again but I think he did not regret his choice.

The article is interesting as points out as a physicist working for certainties can become a scientist on uncertainties. Looking at their salaries you should interchange above adjectives.


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