## The XV Workshop on Statistical Mechanics and nonperturbative Field Theory

25/09/2011

This week I was in Bari as the physics department of that university organized a major event: SM&FT 2011. This is a biennial conference having the aims to discuss recent achievements in fields as statistical mechanics and quantum field theory that have a lot of commonalities. The organizers are well-known physicists and so it was a pleasure for me to see my contribution accepted. Leonardo Cosmai wrote to me confirming my partecipation. Leonardo, together with Paolo Cea, Alessandro Papa and Massimo d’Elia produced a lot of significant works in quantum field theory and a recent paper by Cosmai and Cea arose some fuzz also in the blogosphere (see here). Their forecast for the Higgs boson agrees quite well with my view about this matter. They were also part of the organizing committee. Of course, I was in Bari with my friend Marco Ruggieri that lived there for more than twelve years gaining a PhD in that university.

The scientific content was really interesting an I have had the chance to learn something more about lattice field theory. You can find all the talks here. About this, it should be said that people work with small lattices yet. While this has been a natural way to manage the QCD on the lattice due to missing computational resources, things are rapidly changing due to CUDA as I discussed a lot in my blog and was presented in some talks at this conference. Small groups will be able, with very few bucks of their budgets, to reach a significant ability to analyze increasingly lattice volumes. Besides, also large scale projects in this direction, mostly due to INFN and extending the APE project originated by Nicola Cabibbo and Giorgio Parisi, were presented (see talks by Francesco di Renzo e Piero Vicini). A typical situation in this kind of lattice analysis, improved using CUDA,  was also pointed out by Massimo D’Elia in his talk.  Thanks to this new technology they are increasing significantly the volumes. You can compare the content of his talk with that of his collaborator Francesco Negro, discussing a really interesting problem on the lattice (and a promise for the future with CUDA), with smaller volumes due to reduced computational resources. The interest for the activity of this group and Francesco’s work is strongly linked to a paper that I and Marco Ruggieri wrote together about the QCD vacuum in presence of a magnetic field (see here). The work by Francesco, even if for small volumes, provides interesting conclusions. It should be said that the Nambu-Jona-Lasinio model is there well alive and kicking.

Petruzzelli Theater

From a strictly theoretical side, I would like to point out the talks by Giuseppe Mussardo, with which I have had a nice mail exchange and is author of some beautiful books (e.g. see here), and the ones by Adriano Di Giacomo and Valentin Zakharov that seem to have some relevant contact points with my work. There was also a talk by Edward Shuryak, one of the proponents of the instantons liquid for the QCD vacuum that is strongly supported by lattice simulations and theoretical works like mine.

At the end of the social dinner, we have had some interesting discussions with Di Giacomo and Cosmai. There was some excitation about the announced seminar about neutrinos by CERN and INFN. In a pub after the dinner, I have had some interesting discussions about a proposal by Michele Pepe and others (see his talk) that holds the promises to improve significantly lattice computations removing artifacts. It was also the chance to hear the point of view of Owe Philipsen (see his talk) about the current situation on lattice simulations on QCD at finite temperature. As I have discussed in some posts in this blog, this kind of simulations are plagued by the infamous sign problem and most of the work turns back to try to get rid of it. My friend Marco expressed the somewhat pessimistic view that a critical endpoint will never be seen on lattice computations. Indeed, he is the proponent of the use of a chiral chemical potential that does not display this stumbling block on the lattice (see his talk). This approach holds the promises to reach the goal as he showed in a recent paper. His proposal is under scrutiny by the lattice community. The QCD critical endpoint is a Holy Grail for all of us working in this area as QCD displays a quite rich phase diagram and we have also a lot of experimental data in heavy ion collisions to understand. You should take a look both at the talks of Marco and Alessandro Papa.

I would like to have cited all the talks and I apologize for omissions. If my readers have some time to spend usefully just read it all, as the conference was well organized and with very interesting contents in a really nice atmosphere somehow excited by neutrino news in the last two days.

P. Cea, & L. Cosmai (2011). The trivial Higgs boson: first evidences from LHC arXiv arXiv: 1106.4178v1

Marco Frasca, & Marco Ruggieri (2011). Magnetic Susceptibility of the Quark Condensate and Polarization from Chiral Models Phys.Rev.D83:094024,2011 arXiv: 1103.1194v1

Marco Ruggieri (2011). The Critical End Point of Quantum Chromodynamics Detected by Chirally
Imbalanced Quark Matter Phys.Rev.D84:014011,2011 arXiv: 1103.6186v2

## Solving Dyson-Schwinger equations

15/09/2009

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.

## Who fears a non-perturbative Higgs field?

28/07/2009

One of my preferred readings in the blogosphere is Tommaso Dorigo’s blog. I think this is a widely known blog for people interested about physics and got some citation also at New York Times. Quite recently he published a very interesting post (see here) about the fate of our loved Standard Model taking the move from a very nice paper by J.Ellis, J.R.Espinosa, G.F.Giudice, A.Hoecker, and A.Riotto (see here). These authors are well known and really smart at their work and, indeed, I have noticed this paper as it appeared in arxiv. My readers know that I work on a small part (QCD) of the whole picture arisen in sixties and seventies and I have never taken a look from outside. So, while I appreciated this paper I thought it was not the case to comment on it  in my blog. But reading Tommaso’s post some thoughts come to my mind and these are really pertinent.

People put out two kind of constraints on the Higgs part of the standard model to have an idea of what to expect. I give you here the Higgs potential for your needs

$V_H=\frac{1}{4}\lambda(\phi^\dagger\phi-v^2)^2$

and one immediately realizes that it introduces two free parameters. The critical one is $\lambda$ and let me explain why. When one does quantum field theory, the only real tool that she has to do any meaningful computation is small perturbation theory. The word “small” is never said but it should be said in any circumstance as this technique only works if you have a small parameter in your theory (a coupling) to use as a development parameter. Otherwise we are lost and all starts to become foggy and not so well-defined. Today, nobody knows how to manage a theory with a strong coupling. Parameter $\lambda$ is exactly such a coupling and we are able to manage a Higgs field when this parameter is small. But when you do small perturbation theory in quantum field theory you realize immediately that infinities come out and you are not able to obtain meaningful results going beyond the first order. For the most interesting theories around we are lucky:  Schwinger, Tomonaga, Feynman and Dyson invented renormalization and this works to remove infinities at each order of perturbation theory in the Standard Model and also for the Higgs, if the coupling is small. We are so accustomed to such a situation that we think that this is all one needs to know to understand quantum field theory: Perturbation theory and renormalization. We think that small perturbation theory is the perturbation theory and nothing else. So, we hope also the Higgs field should fulfill such requirements. Indeed, we are already in trouble in QCD for these same reasons but I have discussed at lengthy such a situation before here and I do not want to repeat myself.

There is no reason whatsoever to believe that we know all one has to know to manage a quantum field theory. Higgs could as well be not that light and strong coupled and there is no reason to think that Nature chose the small coupling case to favor us. Of course, if things will not stay this way I will be happy as a light Higgs is favored by supersymmetry and I like supersymmetry. But I would like also to emphasize that we already have all we need to manage analytically a strong coupled Higgs field. This matter I have discussed widely here and in my published papers.

So, while we all agree that a light Higgs is favored my view is that we should not have any fear of a non-perturbative Higgs field.

## Exact solutions on arxiv

24/07/2009

As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator

$-\Box\Delta+3\lambda\phi_c^2(x)\Delta=\delta^D(x)$

being $\phi_c$ the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation

$\ddot\phi(t)+3\lambda\phi_c^2(t,0)\phi(t)=\delta(t)$

that I know when the phase in $\phi_c(t,0)$ is quantized as $(4n+1)K(i)$, being $n$ an integer and $K(i)$ an elliptic integral. This gives back a consistent result in the strong coupling limit, $\lambda\rightarrow\infty$, with my preceding paper on Physical Review D (see here).

The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.

## Edward Witten

23/03/2009

Today I was at the Festival of Mathematics 2009 in Rome to listen a talk by Edward Witten. Witten 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!

## Updated paper

18/03/2009

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.

## 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.

## 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.

07/02/2009

Due to the relevance of the argument, after a nice discussion with a contribution of Carl Brannen, I decided to pursue this matter further. Indeed, the only way to have a covariant formulation of a gradient expansion is adding a time variable and taking the true time variable Wick rotated. In this way, for d=1+1 wave equation you will use d=2+1 wave equation and so on. In d=3+1 you will use d=4+1 wave equation. Let me explain with some equations what I mean. I consider again d=1+1 case as

$\partial^2_{tt}u-\partial^2_{xx}u=0$

but, instead to apply a gradient expansion to it, I apply this to the equation

$\partial^2_{tt}u-\Delta_2u=0$

being $\Delta_2 = \partial_{xx}+\partial_{yy}$. As usual, I rescale time variable as $t\rightarrow\sqrt{\lambda}t$ and I take a solution series

$u=u_0+\frac{1}{\lambda}u_1+\frac{1}{\lambda^2}u_2+\ldots.$

Now I will get the set of equations

$\partial^2_{tt}u_0=0$

$\partial^2_{tt}u_1=\Delta_2u_0$

$\partial^2_{tt}u_2=\Delta_2u_1$

and so on. Let us note that, in this case, we can introduce two new spatial variables as $z=x+iy$ and $\bar z=x-iy$. These are conjugate variables as you know. So, already at the leading order I have solved my equation. Indeed, I note that

$\Delta_2=\partial_z\partial_{\bar z}$

and so the Laplacian has the solution $f(z)+g(\bar z)$ being f and g arbitrary functions. In this case the gradient expansion gives immediately the exact result making its application trivial as should be. Indeed, I take $t=0$ in the perturbation series and put $iy=t$ and I get

$u=f(x+t)+g(x-t)$

that is the exact solution. Nice, it works! This means that a quantum field theory using gradient expansion exists and it is a strong coupling expansion. This result is surely less trivial than the one obtained above.

## Gradient expansions and quantum field theory

01/02/2009

It is more than two years that I am working on quantum field theory in the strong coupling limit and I am generally very satisfied with the acceptance by the community about my views. Of course, these are new ideas and may take some time to be accepted. So, I keep on working on them trying to clarify them at best so that people can have a clear understanding of their strengths and weaknesses. One of the ways we researchers have to know how our colleagues consider our views is peer-review. This system is indeed crucial to any serious scientific endeavor and, indeed, I am proud of my achievements only when my peers agree about their value. But peer-review is also useful to my work to know what are the main objections to it. It can happen that sometime these objections are deeply wrong and may be worthwhile to discuss them at length also to have an idea on how such a prejudice arose.

We should know that when a mathematical theory enters into the description of nature, whatever mathematical method one uses to exploit it is always correct. So, natural laws in physics are described by differential equations and  whatever method you know to solve them is good provided is also mathematically legal. You should consider mathematics for physicists as a severe judge that grants no appeal. You are right or wrong depending on the correctness of your computation. But in physics there is something more and these are assumptions we start with. You can do the beautiful mathematics in the world but if you started with a wrong concept about how nature works your computations are simply rubbish.

One of the criticisms I have received on trying to get my papers published is that one cannot do a gradient expansion because this breaks Lorentz/Poincare’ invariance. This is completely wrong from a mathematical standpoint. As an exercise  you can consider the wave equation in two dimensions as

$\partial^2_{tt}u-\partial^2_{xx}u=0$

and consider the case where the spatial part is not so important. This can be easily obtained by rescaling time as $t\rightarrow\sqrt{\lambda}t$ and taking the limit $\lambda\rightarrow\infty$. One gets the solution series

$u=u_0+\frac{1}{\lambda}u_1+\frac{1}{\lambda^2}u_2+\ldots$

solving the equations

$\partial^2_{tt}u_0=0$

$\partial^2_{tt}u_1=\partial^2_{xx}u_0$

$\partial^2_{tt}u_2=\partial^2_{xx}u_1$

and so on. All this is perfectly legal from a mathematical standpoint and I get a true solution of the wave equation. But, as you can see, I have broken Lorentz invariance, a symmetry of this equation. So, mathematics says yes while physics seems to say no. The answer is quite simple and is known since a long time: The computation is right but Lorentz invariance is no more manifest. This is due to the fact that I have separated time and space. But if I am able to resum all the terms of the expansion series I will get the right answer

$u=f_1(x-t)+f_2(x+t)$

that is Lorentz invariant. So, both physics and mathematics give the same answer and is a resounding yes, it works and it works so well that we are left with a kind of strong coupling expansion.

So, what should do a smart referee with such a doubt, admitting that a smart referee does not know such mundane facts of physics and mathematics? It should realize that here one is facing a really interesting problem of physics: Could we formulate a gradient expansion in such a way to have Lorentz invariance manifest? I have not an answer yet to this question but I grant to you that is a matter I would like to publish a paper about  somewhere. This is an interesting mathematical problem as well. We know that people met a similar problem at the start of the deep understanding of QED due to Feynman, Schwinger, Tomonaga and Dyson. I think that an answer to this question would have the same scientific value.