## Yang-Mills and string theory

09/12/2010

As I pointed out in a recent post, the question of the mass gap for Yang-Mills theory should be considered settled. This implies an understanding of the way mass arises in our world. It is seen that mass is a derived concept and not a fundamental one. I have given an explanation of this here. In a Yang-Mills theory, massive excitations appear due to the presence of a finite nonlinearity. The same effect is seen for a massless quartic scalar field and, indeed, these fields map each other at a classical level. It is interesting to note that a perturbation series with a coupling going to zero can miss this conclusion and we need a dual perturbation with the coupling going to infinity to uncover it. The question we would like to ask here is: What does all this mean for string theory?

As we know, string theory has been claimed not to have any single proposal for an experimental verification. But, of course, without entering into a neverending discussion, there are some important points that could give strong support to the view string theory entails. Indeed, so far there are two essential points that research on string theory produced and that should be confirmed as soon as possible: AdS/CFT correspondence and supersymmetry. Both these theoretical results are strongly supported by the research pursued by our community. For the first point, understanding of QCD spectrum, with or without quarks, through the use of AdS/CFT correspondence is a very active field of research with satisfactory results. I have discussed here this matter several times and I have pointed out the very good work of Stan Brodsky and Guy de Teramond as an example for this kind of research (e.g. see this). Soft-wall model discussed by these authors seems in a very good agreement with the current scenario that is arisen in our understanding of Yang-Mills theory that I emphasized several times in this blog.

About supersymmetry I should say that I am at the forefront since I have presented this paper. The mass gap obtained in Yang-Mills theory arising from nonlinearities has an interesting effect when considered for the quartic scalar field interecting with a gauge field and spinor fields. Taking a coupling for the self-interaction of the scalar field being very large, all the conditions for supersymmetry are fulfilled and all the interacting fields get identical masses and coupling. This implies that, if the mechanism that produces mass in QCD and Standard Model is the same, the Higgs field must be supersymmetric. I call this field Higgs, notwithstanding it has lost some important characteristics of a Higgs field, because is again a scalar field inducing masses to the other fields interacting with it. So, if current experiments should confirm this scenario this would be a big hit for physics ending with a complete understanding of the way mass arises in our world both for the macroscopic and the microscopic world.

So, we can conclude that our research area is producing some relevant conclusions that could address research in more fundamental areas as quantum gravity in a well-defined direction. I think we will get some great news in the near future. As for the present, I am happy to have given an important contribution to this research line.

## Hawking’s successor

20/10/2009

Michael Green is the new Lucasian Professor (see here). Best congratulations to a really worthy appointment and wishes for future success to Professor Green. Green is a string theorist famous for being one of the authors of the first string revolution. He was already full professor at Cambridge University and represents a perfect choice as Hawking’s successor.

## 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!

## 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.

Update: I have three beautiful links about this matter that is very well explained leaving readers with a puzzle:

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!

## Ted Jacobson and quantum gravity

15/01/2009

There are some days when concepts are there running round and round in my head. I have taken a look at the Poincare’ conjecture and was really impressed by the idea of the Ricci’s flow. People with some background in mathematics should read this paper that contains a 493 pages long discussion of the Perelman proof and gives all technical details about that and the mathematics behind Ricci’s flow. If you have a manifold endowed with a metric g then Ricci’s flow satisfies the equation

$\frac{\partial g_{ik}}{\partial t}=-2R_{ik}$

being $R_{ik}$ the Ricci tensor and t is taken to be time for convention. People knowing differential geometry should be accustomed with the fact that a flat manifold is not given by taking the Ricci tensor to be zero, rather is the Riemann tensor that should be null. But Einstein equations in vacuum are given by $R_{ik}=0$ whose most known exact solution is Schwarschild solution. So, what has the Ricci’s flow so shocking to interest physicists?

Consider a two dimensional manifold that has only conformal metrics. In this case the Ricci’s flow takes a very simple form

$\frac{\partial g}{\partial t}=\triangle g$

where $\triangle$ is the Laplace-Beltrami operator. This is a Fokker-Planck equation or, if you prefer, the heat equation. Fokker-Planck equations enter into statistical physics to describe a system approaching equilibrium and are widely discussed in the study of Brownian motion. So, Einstein equations seem to be strongly related to some kind of statistical equilibrium given by the solution of a Fokker-Planck like equation taking $\frac{\partial g}{\partial t}=0$ and, in some way, a deep relation seems to exist between thermodynamics and Einstein equations .

Indeed Einstein equations are an equation of state! This striking result has been obtained by Ted Jacobson. I point out to you a couple of papers by him where this result is given here and here. This result has the smell of a deep truth as also happens for the Bekenstein-Hawking entropy of a black hole. The next question should be what is the partition function producing such an equation of state?  Here enters the question of quantum gravity in all its glory.

So, an equilibrium solution of an heat equation produces Einstein equations as seen from the Ricci’s flow. Does it exist in physics a fundamental model producing a Ricci’s flow? The answer is a resounding yes and this is the non-linear sigma model. This result was firstly obtained by Daniel Friedan in a classical paper that was the result of his PhD work. You can get a copy of the PhD thesis at his homepage. Ricci’s flow appears as a renormalization group equation in the quantum theory of the non-linear sigma model with energy in place of time and the link with thermodynamics and equations of state does not seem so evident. This result lies at the foundations of string theory.

Indeed, one can distinguish between a critical string and a non-critical string. The former corresponds to a non-linear sigma model in 26 dimensions granting a consistent quantum field theory. The latter is under study yet but il va sans dire that the greatest success went to the critical string. So, we can see that if we want to understand the heat operator describing Ricci’s flow in physics we have to buy string theory at present.

Is this an unescapable conclusion? We have not yet an answer to this question. Ricci’s flow seems to be really fundamental to understand quantum gravity as it represents a typical equation of  a system moving toward equilibrium in quest for the identification of microstates. Fundamental results from Bekenstein, Hawking and Jacobson prove without doubt that things stay this way, that is, there is a more fundamental theory underlying general relativity that should have a similar link as mechanical statistics has with thermodynamics. So, what are quanta of space-time?

## Rumors on SUSY

25/07/2008

In these days there have been some rumors in the blogosphere about string theorists and SUSY (see Motl, Woit and Dorigo) due to a recent preprint appeared on arxiv. Indeed SUSY is a relevant ingredient of string theory and the latter was the vehicle for the uncovering of this concept that obtained such a fortune in the community. I have listened a talk of Sergio Ferrara at Accademia dei Lincei in Rome a few months ago. Ferrara is one of the discoverers of supergravity and he gave a nice talk on the argument of supersymmetry. He was confident that supersymmetric particles will be seen at LHC. I would like to say that this was also the expectation for LEP and Tevatron but nothing has been seen so far. So the paper above seems like an attempt by a string theorist to be pessimistic and save the day.

Supersymmetry has some problems that still are in need for a satisfactory answer. One is philosophical as one can say that the number of particles simply doubles and so why should we expect such an anti-economical behavior by Nature? Ferrara argued against this question by saying that also with antimatter Nature doubled the number of particles so this would not be the first time that, in order to keep a symmetry, one needs such a doubling. One can say anyhow that for antimatter one has a discrete symmetry on a single field while for supersymmetry is the number of fields that doubles. The other point is about breaking of supersymmetry. There is no satisfactory model so far and such symmetry is not seen at low energies as we know. But this could be just a matter of time before someone finds a way out.

My view is that even if there is no supersymmetry at large, one can save supergravity. Indeed, all one needs is to observe a gravitino, that is a spin 3/2 particle, and we will have a theory of quantum gravity while, at large, no supersymmetry can exist. But this would not be enough for string theory. As a theoretical physicist I would like to see the discover of a gravitino and the failure of supersymmetry at large as this would imply a lot of interesting work to do and an incredible new scenario to understand.