The Witten’s paradox

17/08/2013

ResearchBlogging.org

Edward Witten is one of the greatest living physicists and also ranks high with mathematicians. He set the agenda for theoretical physics in several areas of research. He is mostly known for championing string theory and being one of few people that revolutionized the field. One of his major contributions to supersymmetry has been a deep understanding of its breaking. In a pair of famous papers (here and here) he put the foundations to our current understanding on the way supersymmetry can break and introduced the well-known Witten index. If a supersymmetric theory breaks supersymmetry then its Witten index is 0. This index is generally very difficult to compute and only perturbative or lattice computations can come to rescue. An important conclusion from Witten’s paper is that the well-known Wess-Zumino model in four dimensions does not break supersymmetry. Witten could rigorously justify this conclusion at small coupling but, at that time, an approach for strong coupling was missing and here Maldacena conjecture cannot help. Anyhow, he concluded that this should be true also for a strongly coupled Wess-Zumino model. Checks to this model in such a regime are rare. After I submitted a paper on arxiv last year (see here) I become aware of an attempt using Dyson-Schwinger equations that confirmed Witten conclusions for small coupling (see here). I have had an interesting mail exchange with one of the authors and this seems a promising approach, given authors’ truncation of Dyson-Schwinger hierarchy. Other approaches consider the Wess-Zumino model in two dimensions on the lattice. So, this appears a rather unexplored area , given the difficulties to cope with a strongly coupled theory, and Witten’s words appear like nails on a coffin to this theory.

I have worked out a lot of techniques to cope with strongly coupled theories and everywhere there is a perturbation going to infinity in a differential equation of any kind and so, I applied these ideas also to this famous model of supersymmetry. The idea is to prove that “supersymmetry has inside itself the seeds of its breaking“. The real issue at stake here is a correct understanding of the way supersymmetry breaks and recover in this way models that now appear to be defeated by data from LHC simply because the idea of symmetry breaking must be applied differently.

Of course, I do not aim to present a claim against the beautiful results given by Witten decades ago but just open up an interesting scientific question. So, considering that the Wess-Zumino model is just a theory of two scalar fields coupled to a Majorana spinor, its equations can be treated classically and so solved both for a strong and a weak coupling limit. I did this in a paper of mine (see here) and this paper has been accepted in these days in the Journal of Nonlinear Mathematical Physics as a letter. The classical solutions contradict the expectations giving a surviving of the supersymmetry at small coupling (as expected from Witten index for the quantum theory) while this does not happen for a strong coupling (formal limit of the coupling going to infinity). This is  a paradox, the Witten paradox, because classical solutions seem to break supersymmetry while the quantum theory does not.  So, we are left with a deep question: How is supersymmetry recovered by quantum corrections?

Marco Frasca (2012). Chiral Wess-Zumino model and breaking of supersymmetry arXiv arXiv: 1211.1039v1

A. Bashir, & J. Lorenzo Diaz-Cruz (1999). A study of Schwinger-Dyson Equations for Yukawa and Wess-Zumino Models J.Phys.G25:1797-1805,1999 arXiv: hep-ph/9906360v1

Marco Frasca (2012). Classical solutions of a massless Wess-Zumino model arXiv arXiv: 1212.1822v2

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Edward Witten

23/03/2009

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!


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