Wonderful QCD!


On Science this week appeared a milestone paper showing two great achievements by lattice QCD:

  • QCD gives a correct description of low energy phenomenology of strong interactions.
  • Most of the ordinary mass (99%) is due to the motion of quarks inside hadrons.

The precision reached has no precedent. The authors are able to get a control of involved  errors in such a way to reach an agreement of about 1% into the computation of nucleon masses. Frank Wilczek gives here a very readable account of these accomplishments and is worthwhile reading. These results open a new era into this kind of method to extract results to be compared with experiments for QCD and give an important confirmation to our understanding of strong interactions. But I would like to point out Wilczek’s concern: Until we will not have a theoretical way to obtain results from QCD in the low energy limit, we will miss a great piece of understanding of physics. This is a point that I discussed largely with my posts in this blog but it is worthwhile repeating here coming from such an authoritative voice.

An interesting point about these lattice computations can be made by observing that again no \sigma resonance is seen. I would like to remember that in these computations entered just u, d and s quarks as the authors’ aims were computations of bound states of such quarks. Some authoritative theoretical physicists are claiming that this resonance should be a tetraquark, that is a combination of u and d quarks and their antiparticles. What we can say about from our point of view? As I have written here some time ago, lattice computations of the gluon propagator in a pure Yang-Mills theory prove that this can be fitted with a Yukawa form


being m\approx 500 MeV. This is given in Euclidean form. This kind of propagators says to us that the potential should be Yukawa-like, that is


if this is true no tetraquark state can exist for lighter quarks. The reason is that a Yukawa-like potential heavily damps any van der Waals kind of residual potential. But, due to asymptotic freedom, this is no more true for heavier quarks c and b  as in this case the potential is Coulomb-like and, indeed, such kind of states could have been seen at Tevatron.

We expect that the glueball spectrum should display itself in the observed hadronic spectrum. This means that a major effort in lattice QCD computations should be aimed in this direction now that such a deep understanding of known hadronic states has been reached.

Gradient expansions, strong perturbations and classicality


It is a common view that when in an equation appears a very large term we cannot use any perturbation approach at all. This is a quite common prejudice and forced physicists, for a lot of years, to invent exotic approaches with very few luck to unveil physics behind equations. The reason for this relies on a simple trick generally overlooked by mathematicians and physicists and here is my luck. This idea can be easily exposed for the Schroedinger equation. So, let us consider the case

(H_0+\lambda V)|\psi\rangle=i\hbar\frac{\partial|\psi\rangle}{\partial t}

with \lambda\rightarrow\infty. This is a very unlucky case both for a physicist and a mathematician as the only sure approach that come to our rescue is a computer program with all the difficulties this implies. Of course, it would be very nice if we could find a solution in the form of an asymptotic series like


but we know quite well that if we insert such a solution into the Schroedinger equation we get meaningless results. But there is a very smart trick that can get us out of this dark and can produce the required result. I have exposed this since 1992 on Physical Review A (see here) and this paper was not taken too seriously by the community so that I had time enough to be able to apply this idea to all fields of physics. The paper producing the turning point has been published on Physical Review A (thank you very much, Bernd Crasemann!). You can find it here and here. The point is that when you have a strong perturbation, an expansion is not enough. You also need a rescaling in time like \tau=\lambda t. If you do this and insert the above expansion into the original Schroedinger equation, this time you will get meaningful results: A dual Dyson series that, being now the perturbation independent of time, becomes a well-known gradient expansion: Wigner-Kirkwood series. But this series is a semiclassical one and you get the striking result that a strongly perturbed quantum system is a semiclassical system! So, if you want to change a quantum system into a classical one just perturb it strongly. This is something that happens when one does a measurement in quantum mechanics using just electromagnetic fields that are the only means we know to accomplish such a task.

This result about strong perturbations and semiclassicality has been published on a long time honored journal: Proceedings of the Royal Society A (see here and here). I am pleased of this also because of my estimation for Michael Berry, the Editor. I have met him at a Garda lake’s Conference some years ago and I have listened a beautiful talk by him about the appearance of a classical world out of the quantum conundrum. I remember he asked me how to connect to internet from the Conference site but there there was just a not so cheap machine from Telecom Italia and then my help was quite limited.

So, I just removed a prejudice and was lucky enough to give sound examples in all branches of physics. Sometime, looking in some dusty corners of physics and mathematics can be quite rewarding!

A quite effective QCD theory


As far as my path toward understanding of QCD is concerned, I have found a quite interesting effective theory to work with that is somewhat similar to Yukawa theory. Hideki Yukawa turns out to be more in depth in his hindsight than expected.

yukawa Indeed, I have already showed as the potential in infrared Yang-Mills theory is an infinite sum of weighted Yukawa potentials with the range, at each order, decided through a mass formula for glueballs that can be written down as


being \sigma the string tension, an experimental parameter generally taken to be (440 MeV)^2, and K(i) is an elliptic integral, just a number.

The most intriguing aspect of all this treatment is that an effective infrared QCD can be obtained through a scalar field. I am about to finish a paper with a calculation of the width of the \sigma resonance, a critical parameter for our understanding of low energy QCD. Here I put the generating functional if someone is interested in doing a similar calculation (time is rescaled as t\rightarrow\sqrt{N}gt)

Z[\eta,\bar\eta,j] \approx\exp\left\{i\int d^4x\sum_q \frac{\delta}{i\delta\bar\eta_q(x)}\frac{\lambda^a}{2\sqrt{N}}\gamma_i\eta_i^a\frac{\delta}{i\delta j_\phi(x)}\frac{i\delta}{\delta\eta_q(x)}\right\} \times
\exp\left\{-\frac{i}{Ng^2}\int d^4xd^4y\sum_q\bar\eta_q(x)S_q(x-y)\eta_q(y)\right\}\times
\exp\left\{\frac{i}{2}(N^2-1)\int d^4xd^4y j_\phi(x)\Delta(x-y)j_\phi(y)\right\}.

As always, S_q(x-y) is the free Dirac propagator for the given quark q=u,d,s,\ldots and \Delta(x-y) is the gluon propagator that I have discussed in depth in my preceding posts. People seriously interested about this matter should read my works (here and here).

For a physical understanding of this you have to wait my next posting on arxiv. Anyhow, anybody can spend some time to manage this theory to exploit its working and its fallacies. My hope is that, anytime I post such information on my blog, I help the community to have an anticipation of possible new ways to see an old problem with a lot of prejudices well grounded.

Wikipedia and physics


Wikipedia is doing a relevant service to our community. I have contributed to some voices both for the english and the italian version and I have used it a lot of time to make myself acquainted with some unknown parts of physics. I think that this is worthwhile a support:

As you will know since now I am a strong supporter of any kind of revolution and this is a big one.

So, long life to Wikipedia!

First photo of an exoplanet


I think that is worthwhile to put here this first photo of an exoplanet. This is the result of a joint effort of three research groups, one of them using Hubble of course. Here is the photo


but this is more explicative (via Corriere della Sera)


A fine article about, as always on Physics World, can be found here.

Quantum mechanics and gravity


Reading the daily by arxiv today I cannot overlook a quite interesting paper that will appear soon on Physical Review Letters. This paper (see here), written by Saurya Das and Elias Vagenas, presents some relevant conclusions about the effects of gravity in quite common quantum mechanical systems. The authors rely their conclusions on an acquired result, due mostly to string theory, that a fundamental length must exist and this fundamental length modifies in a well defined way the indeterminacy principle. So, one can quantify this effect on whatever quantum mechanical system through a correcting Hamiltonian term and evaluating the effect of gravity on this system. In this way one can obtain an estimation on how relevant is the effect and how far can be an experimental measurement of this. The conclusions the authors reached are quite interesting. Of course, all of the cases imply a too small effect to be in the reach of a laboratory observation but, the most not trivial conclusion is that could exist an intermediate fundamental length that could be observed e.g. at LHC. This intermediate length should be placed between the electroweak and the Planck scale.

It is the first time that I see such estimations on quite simple quantum mechanical models and I would expect more extended analysis on a similar line. Surely, it would be striking to see in laboratory such a tiny effect correcting the Lamb shift. But, working in quantum optics, I learned that progress experimentalists are able to put out can be very impressive in a very short time. So, I would not be surprised if in some years Physical Review Letters should publish some experimental letter about this matter being the first evidence of a quantum gravity effect in a laboratory.

Screening masses in SU(3) Yang-Mills theory


Thanks to a useful comment by Rafael Frigori (see here) I become aware of a series of beautiful papers by an Italian group at Universita’ della Calabria. I was mostly struck by a recent paper written by R. Fiore, R. Falcone, M. Gravina and A. Papa (see here) that appeared in Nuclear Physics B (see here). This paper belongs to a long series of works about the behavior of Yang-Mills theory at non-null temperature and its critical behavior. Indeed, using high-temperature expansion and Polyakov loops one arrives at the main conclusion that the ratio between the lowest and the higher state of the theory must be 3/2. This ratio depends on the universality class the theory belongs to and so, on the kind of effective theory one has in the proper temperature limit (below or above T_c). It should be said that, in order to get a proper verification of the above prediction, people use lattice computations. Fiore et al. use a lattice of 16^3 \times 4 points and, as all this kind of computations are done on lattices having such a dimension, one can cast some doubt about the fact that the true ground state of the theory is really hit. Indeed, this happens in all this kind of computations done to get a glueball spectrum that seem at odd with those giving the gluon propagator producing a lower screening mass at about 500 MeV (see my post here). A state at about 500 MeV is seen at accelerator facilities as \sigma resonance or f0(600) but is not predicted by any lattice computation. One of the reasons to reduce lattice volume is that one can reach higher values of \beta granting the reaching of a non-perturbative regime, the one interesting for us.

What can we say about this ratio with our theory? We have put on arxiv a paper that answer this question (see here). These results were also presented at QCD 08 in Montpellier (see here). We assume that the \sigma cannot be seen at such small volumes but its excited state \sigma^* can be obtained. This implies that one can exchange the \sigma^* with the lowest state and 0^+ as the higher one. Then this ratio gives exactly 3/2 as expected. We can conclude on the basis of this analysis that this ratio is the same independently on the temperature but, the one to be properly measured is given in the paper of Craig McNeile (see here) that gives close agreement between lattice and theoretical predictions.

So, we would like to see lattice computations of Yang-Mills spectra at lower lattice spacing and increased volumes granting in this way the proper value of the ground state. This is overwhelming important in view of the fact that no real understanding exists of the existence of the \sigma resonance with lattice computations. This will implies, as discussed above, a deeper understanding of the spectrum of the theory also at higher temperatures.

Chaos and hadronic spectrum


The question of quantum chaos and hadronic spectrum is a relevant matter that I was not able to address in my last post. The reason for this relies on the important fact that quantum chaos in the hadronic spectrum and classical chaotic solutions to build a quantum field theory are just different levels and should not be confused each other. Indeed, I can start with my fully integrable classical solutions to build a quantum field theory and find in the end a fully quantum chaotic spectrum in the bound states of the theory. This is a point that created a lot of confusion and should be clarified properly. But let us state a quite simple example of this. Let us consider QED. We know that the Coulombian potential produces n-body bound states. But already the 3-body problem admits chaotic solutions and the corresponding quantum problem will displays the proper distribution of the energy levels. But QED is a quantum field theory built on perfectly regular solutions of the free equations of motion (Dirac and Maxwell equations). So, we see that quantum chaos, if any, arises naturally for the bound states of the theory due to the properties of the bounding potential.

What can we say about hadronic spectrum? The following papers show example of quantum chaos in the hadronic spectrum: here and here. But what is the potential we obtained from Yang-Mills theory? We gave

V(r)=-\alpha_s\sum_{n=0}^\infty A_n \frac{e^{-m_n r}}{r}

being m_n the glueball spectrum. This potential has an infinite number of contributions. Baryons are expected to be chaotic already with a Coulombian approximation being three-body bound states. But a potential like that above could produce classical chaotic dynamics having an infinite number of terms and producing in this way an infinite numbers of resonances in the KAM series. But to obtain this potential we started with perfectly regular solutions in the quantum field theory!

We conclude that our approach produces a consistent potential that agrees fairly well with expectations of quantum chaos in the hadronic spectrum. But this is independent on the way one formulates a quantum field theory. Indeed, bound states could display chaos even with the simplest of the field theories, that is a scalar field theory.

Chaos and quantum field theory


Dmitry is still on is point trying to prove me wrong (see his post here). Of course, I have a theory mathematically sound and he has nothing and so the discussion is somewhat uneven from the start. He is saying that my point

In order to build a meaningful quantum field theory, the initial conditions should be properly chosen.

is wrong. This means that we can start to develop a meaningful QED without electrons or photons and obtain identical results. A fact that is blatantly wrong. QED spectrum is done with plane waves representing the spectrum of the theory. Without this choice you are at odds with experiments. This is a crucial point for quantum field theory and applies wherever you need to compute a cross section or a decay rate. Choosing different solutions at the start (e.g. by changing initial conditions) will produce a different quantum field theory and this can be at odds with experiments. This is a well-known fact and an example is given through the computations generally done e.g. for the Casimir effect.

So, he continues

“I believe, it is actually wrong. Let us again take a hamiltonian classical system with self-interaction. To get the intermittency (i.e., periodic orbits becoming chaotic trajectories and vise versa), it is enough to fix initial conditions and than vary the coupling constant, as I have explained … Since one has running coupling in a QFT without sweating, one will have intermittency as well in the Schwinger-Dyson equations for Keldysh Green functions. So, strictly speaking, one can only get rid of chaos at the RG fixed point which corresponds to CFT anyway (that is — no particles, nor quasiparticles, just unparticles ;-)).”

I would agree with such an argument if intermittency could be a useful solution to build a quantum field theory and would give us a spectrum to start with. E.g. I would like to see how is the glueball spectrum, to be observed into experiments with a lot of people currently eager to detect it, with a chaotic classical solution like this. You are in serious troubles as you do not even have a Fourier expansion. So, no Fourier expansion no spectrum. Indeed, Dmitry has no such result but just a prejudice: Chaotic classical solutions must be important for Yang-Mills theory. This without a proof that, he claims, should rely on me. But I have already shown that the theory is consistent and complete with integrable solutions and this was my aim. He is just claiming the contrary without providing a serious support to his prejudice.

After this rather questionable facts by his side and having him admitted that I am right as

Currently, there is no formulation of a quantum field theory starting with classical chaotic solutions.

he exposes his “would be”s about such a matter.

Let me comment about this discussion and how all this should be interpreted. Whenever you are smart enough to produce a theory and get it published you will get opinions from two different kind of people.  You will find interested people and criticizing people and this is in the matter of things. Criticizing people could be very useful wherever is able to support arguments with serious evidence. But most of times they will just criticize you on the ground of prejudices they have and these prejudices are those you have just demolished with your theory. So, to move an idea from the status of a prejudice to a status of a theory a strong mathematical and experimental support is needed.  A typical historical example has been the question of aether supporting the propagation of electromagnetic waves. A lot of people kept on believing on that till their death even after a strong evidence for relativity was achieved. This behavior belongs to our community, it was never lost, and we have to cope with it anytime we produce something new. It could imply delay into the acceptance of a theory but it is just human behavior and cannot be changed.

Let me repeat again. My approach is well developed and provides a glueball spectrum (to be observed experimentally), propagators and running coupling making the formulation of a Yang-Mills theory in the infrared complete. Propagators and running coupling are in very good agreement with lattice computations that come out in this somewhat unexpected direction. The same can be said with the spectrum but assuming that the lowest glueball is at about 500 GeV and has been already seen as \sigma resonance or f0(600). The same interpretation should apply to f0(980). This is in agreement with analysis done with dispersion relations by Narison, Mennessier, Ochs (see here) and Minkowski (see here).

What does one have on the side of classical chaotic solutions and quantum field theory? Substantially nothing as also admitted by Dmitry. No theory, no predictions, nothing. So, it can only be classified as a prejudice and a prejudice generally turns out to be wrong. My aim starts and ends when I have showed that my theory is mathematically sound and consistent and I get predictions that could be confirmed or not.  I do not have the burden to prove that, as one of my hypothesis does not like to someone, I have also to formulate my theory without it.

As a conclusion, I would greatly appreciate a formulation of a quantum field theory starting with chaotic solutions that applies to a realistic model of reality. I do not believe in betting but it would be tempting to put a wager on this.

A wrong argument


Dmitry Podolsky put forward the following argument to claim that chaotic solutions are relevant for Yang-Mills theory at strong coupling (see here):

“First of all, equations of motion of the YM field are non-linear and therefore their solutions admit chaotic behavior. Are all the solutions of these equations of motion chaotic? The answer is of course negative: depending on the coupling strength and initial conditions, one can get whole sets of classical solutions without chaos, which we will call Smilga choices, following Marco. Suppose that we fix coupling and continuously change initial conditions for the YM equations of motion – as a result of this variation, we will first get, say, a chaotic solution,  than a solution without chaos, than again a chaotic soltion, etc.”

This is true if one chooses the wrong initial conditions to build a quantum field theory. Because this is the main point of the question.

Currently, there is no formulation of a quantum field theory starting with classical chaotic solutions.

But let me add this conjecture that I invite anyone to prove false:

A quantum field theory does not exist having as building classical solutions just chaotic solutions.

The reason for this is quite simple. If I choose wrong initial conditions I will not be able to get a leading order spectrum of excitations to build on. For a SU(2) Yang-Mills theory I have e.g. the following starting classical solution:

A_\mu^a=\eta^a_\mu\left(\frac{\Lambda}{\sqrt{g}}\right){\rm sn}(p\cdot x,i)

with \eta^a_\mu=((0,1,0,0),(0,0,1,0),(0,0,0,1)) and p^2=g\Lambda^2 and I am able to get a spectrum of fundamental excitations as the Jacobi function has a Fourier expansion in plane waves with a mass spectrum


that I can call a glueball spectrum. This must be observed in nature. Presently, for SU(3), this is in agreement with lattice computations (see here). But there is also the expectation that f0(600) or \sigma is a glueball and another one could be f0(980). If this is proved true, I think that Smilga will be very glad.

Let me state a final point:

In order to build a meaningful quantum field theory, the initial conditions should be properly chosen.

This simple fact seems generally overlooked but try to ask yourselves why one chooses plane waves for QED or other quantum field theories and you will get an easy anwser: these are the excitations seen in experiments.


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