DICE 2014

21/09/2014

I have spent this week in Castiglioncello participating to the Conference DICE 2014. This Conference is organized with a cadence of two years with the main efforts due to Thomas Elze.

Castello Pasquini at Castiglioncello  (DICE 2014)

I have been a participant to the 2006 edition where I gave a talk about decoherence and thermodynamic limit (see here and here). This is one of the main conferences where foundational questions can be discussed with the intervention of some of the major physicists. This year there have been 5 keynote lectures from famous researchers. The opening lecture was held by Tom Kibble, one of the founding fathers of the Higgs mechanism. I met him at the registration desk and I have had the luck of a handshake and a few words with him. It was a recollection of the epic of the Standard Model. The second notable lecturer was Mario Rasetti. Rasetti is working on the question of big data that is, the huge number of information that is currently exchanged on the web having the property to be difficult to be managed and not only for a matter of quantity. What Rasetti and his group showed is that topological field theory yields striking results when applied to such a case. An application to NMRI for the brain exemplified this in a blatant manner.

The third day there were the lectures by Avshalom Elitzur and Alain Connes, the Fields medallist. Elitzur is widely known for the concept of weak measurement that is a key idea of quantum optics. Connes presented his recent introduction of the quanta of geometry that should make happy loop quantum gravity researchers. You can find the main concepts here. Connes explained how the question of the mass of the Higgs got fixed and said that, since his proposal for the geometry of the Standard Model, he was able to overcome all the setbacks that appeared on the way. This was just another one. From my side, his approach appears really interesting as the Brownian motion I introduced in quantum mechanics could be understood through the quanta of volumes that Connes and collaborators uncovered.

Gerard ‘t Hooft talked on Thursday. The question he exposed was about cellular automaton and quantum mechanics (see here). It is several years that ‘t Hoof t is looking for a classical substrate to quantum mechanics and this was also the point of other speakers at the Conference. Indeed, he has had some clashes with people working on quantum computation as ‘t Hooft, following his views, is somewhat sceptical about it. I intervened on this question based on the theorem of Lieb and Simon, generally overlooked in such discussions, defending ‘t Hoof ideas and so, generating some fuss (see here and the discussion I have had with Peter Shor and Aram Harrow). Indeed, we finally stipulated that some configurations can evade Lieb and Simon theorem granting a quantum behaviour at macroscopic level.

This is my talk at DICE 2014 and was given the same day as that of  ‘t Hooft (he was there listening). I was able to prove the existence of fractional powers of Brownian motion and presented new results with the derivation of the Dirac equation from a stochastic process.

The Conference was excellent and I really enjoyed it. I have to thank the organizers for the beautiful atmosphere and the really pleasant stay with a full immersion in wonderful science. All the speakers yielded stimulating and enjoyable talks. For my side, I will keep on working on foundational questions and look forward for the next edition.

Marco Frasca (2006). Thermodynamic Limit and Decoherence: Rigorous Results Journal of Physics: Conference Series 67 (2007) 012026 arXiv: quant-ph/0611024v1

Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Quanta of Geometry arXiv arXiv: 1409.2471v3

Gerard ‘t Hooft (2014). The Cellular Automaton Interpretation of Quantum Mechanics. A View on the Quantum Nature of our Universe, Compulsory or Impossible? arXiv arXiv: 1405.1548v2

The question of the mass gap

10/09/2014

Some years ago I proposed a set of solutions to the classical Yang-Mills equations displaying a massive behavior. For a massless theory this is somewhat unexpected. After a criticism by Terry Tao I had to admit that, for a generic gauge, such solutions are just asymptotic ones assuming the coupling runs to infinity (see here and here). Although my arguments on Yang-Mills theory were not changed by this, I have found such a conclusion somewhat unsatisfactory. The reason is that if you have classical solutions to Yang-Mills equations that display a mass gap, their quantization cannot change such a conclusion. Rather, one should eventually expect a superimposed quantum spectrum. But working with asymptotic classical solutions can make things somewhat involved. This forced me to choose the gauge to be always Lorenz because in such a case the solutions were exact. Besides, it is a great success for a physicist to find exact solutions to fundamental equations of physics as these yield an immediate idea of what is going on in a theory. Even in such case we would get a conclusive representation of the way the mass gap can form.

Finally, after some years of struggle, I was able to get such a set of exact solutions to the classical Yang-Mills theory displaying a mass gap (see here). Such solutions confirm both the Tao’s argument that an all equal component solution for Yang-Mills equations cannot hold in any gauge and also my original argument that an all equal component solution holds, in a general case, only asymptotically with the coupling running to infinity. But classically, there exist solutions displaying a mass gap that arises from the nonlinearity of the equations of motion. The mass gap goes to zero as the coupling does. Translating this in the quantum realm is straightforward as I showed for the Lorenz (Landau) gauge. I hope all this will help to better elucidate all the physics around strong interactions. My efforts since 2005 went in that direction and are still going on.

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v1

Higgs what?

06/09/2014

In these days it has been announced the new version of Review of Particle Physics by the Particle Data Group (PDG). This is the bread and butter of any particle physicist and contains all the relevant data about this area of research. It is quite common for us to search the on-line version or using the booklet to know a mass or a decay rate. After the first run of LHC data gathering about Higgs particle, this edition contains a bunch of fundamental informations about it and I post a part of them.

It is Standard Model Higgs! No, not so fast. Take a look at the WW final state. It is somewhat low but yes, it is perfectly consistent with the Standard Model. Also, error bars are somewhat large to conclude something definitive. So, let us take a look nearer at these strengths.

We discover that the strengths measured by CMS are really low and takes down this value. Indeed, this is consistent with my proposal here. I get 0.68 for both channels WW and ZZ. On the other side, ATLAS moves all upward consistently and there is this strange behaviour compensating each other. So, let us also take a look at the ZZ strength. PDG yields

again CMS agrees with my conclusions and ATLAS moves all upward to compensate. But both these results, due to the large error bars, agree rather well with the Standard Model. So, I looked for the publication by CMS  that were produced till today if one or both these analyses were improved. The result was that CMS improved the measure of the strength in the WW channel to leptons (see here). What they measure is

$\frac{\sigma}{\sigma_{SM}}=0.72^{+0.20}_{-0.18}.$

The error is significantly smaller and the result striking. It is bending in the “wrong” turn loosing higgsness. It would be interesting to understand why CMS appear to get results downward for these strengths and ATLAS more upward compensating each other toward the Standard Model. On the other side, I should admire the more aggressive approach by CMS with their results more and more similar to my expectations. I am just curious to see with the restart of LHC what will happen to these data that CMS sharpened to such a point.

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

CMS Collaboration (2013). Measurement of Higgs boson production and properties in the WW decay
channel with leptonic final states JHEP 01 (2014) 096 arXiv: 1312.1129v2

Do quarks grant confinement?

21/07/2014

In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

• Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
• Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
• The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
• The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with $N_f$ the number of quarks (flavours):

From this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal
window of QCD arXiv arXiv: 1405.7340v1

f0(500) and f0(980) are not tetraquarks

27/06/2014

Last week I have been in Giovinazzo, a really beautiful town near Bari in Italy. I participated at the QCD@Work conference. This conference series is now at the 7th edition and, for me, it was my second attendance. The most striking news I heard was put forward in the first day and represents a striking result indeed. The talk was given by Maurizio Martinelli on behalf of LHCb Collaboration. You can find the result on page 19 and on an arxiv paper . The question of the nature of f0(500) is a vexata quaestio since the first possible observation of this resonance. It entered in the Particle Data Group catalog as f0(600) but was eliminated in the following years. Today its existence is no more questioned and this particle is widely accepted. Also its properties as the mass and the width are known with reasonable precision starting from a fundamental work by Irinel Caprini, Gilberto Colangelo and Heinrich Leutwyler (see here). The longstanding question around this particle and its parent f0(980) was about their nature. It is generally difficult to fix the structure of a resonance in QCD and there is no exception here.

The problem arose from famous papers by Jaffe on 1977 (this one and this one) that using a quark-bag model introduced a low-energy nonet of states made of four quarks each. These papers set the stage for what has been the current understanding of the f0(500) and f0(980) resonances. The nonet is completely filled with all the QCD resonances below 1 GeV and so, it seems to fit the bill excellently.

Someone challenged this kind of paradigm and claimed that f0(500) could not be a tetraquark state (e.g. see here and here but also papers by Wolfgang Ochs and Peter Minkowski disagree with the tetraquark model for these resonances). The answer come out straightforwardly from LHCb collaboration: Both f0(500) and f0(980) are not tetraquark and the original view by Jaffe is no more supported. Indeed, people that know the Nambu-Jona-Lasinio model should know quite well where the f0(500) (or $\sigma$ ) comes from and I would also suggest that this model can also accommodate higher states like f0(980).

I should say that this is a further striking result coming from LHCb Collaboration. Hopefully, this should give important hints to a better understanding of low-energy QCD.

$\overline{B}^0\rightarrow J/ψπ^+π^-$ decays arXiv arXiv: 1404.5673v2
Irinel Caprini, Gilberto Colangelo, & Heinrich Leutwyler (2005). Mass and width of the lowest resonance in QCD Phys.Rev.Lett.96:132001,2006 arXiv: hep-ph/0512364v2
Jaffe, R. (1977). Multiquark hadrons. I. Phenomenology of Q^{2}Q[over ¯]^{2} mesons Physical Review D, 15 (1), 267-280 DOI: 10.1103/PhysRevD.15.267
Jaffe, R. (1977). Multiquark hadrons. II. Methods Physical Review D, 15 (1), 281-289 DOI: 10.1103/PhysRevD.15.281
G. Mennessier, S. Narison, & X. -G. Wang (2010). The sigma and f_0(980) from K_e4+pi-pi, gamma-gamma scatterings, J/psi,
phi to gamma sigma_B and D_s to l nu sigma_B Nucl.Phys.Proc.Suppl.207-208:177-180,2010 arXiv: 1009.3590v1

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

Looking for the Lucasian Professor

09/04/2014

Michael Green, the current Lucasian Professor, is now 67 and Cambridge University is looking for his substitute (see here). It is interesting that Wikipedia is now reporting rumours from researchers at DAMPT (see here)

As of 3rd April 2014, the University of Cambridge is recruiting the 19th Lucasian Professor. Rumour amongst DAMPT researchers is that the process is a required formality, but that unless there’s a last minute surprise the Lucasian Professorship will be held for the first time by a woman. It may also be the first time that the chair is not held by somebody born in the British Isles.

Wikipedia is generally not doing gossip so I report this here because I think this will not last too long. Anyhow, the procedure will be brief. The candidates should send their CV for 15 April and the chosen one should take up the appointment on 1 September this year.  I would like to remember that this is a really prestigious chair held by Newton, Dirac and Hawking to name just a few. Surely, Cambridge University will choose for the best once again.

Evidence of the square root of Brownian motion

06/03/2014

A mathematical proof of existence of a stochastic process involving fractional exponents seemed out of question after some mathematicians claimed this cannot exist. This observation is strongly linked to the current definition and may undergo revision if nature does not agree with it. Stochastic processes are very easy to simulate on a computer. Very few lines of code can decide if something works or not. I and Alfonso Farina, together with Matteo Sedehi,  have introduced the idea that the square root of a Wiener process yields the Schroedinger equation (see here or download a preprint here). This implies that one has to attach a meaning to the equation

$dX=(dW)^\frac{1}{2}.$

In a paper appeared today on arxiv (see here) we finally have provided this proof: We were right. The idea is to solve such an equation by numerical methods. These methods are themselves a proof of existence. We used the Euler-Maruyama method, the simplest one and we compared the results as shown in the following figure

a) Original Brownian motion. b) Same but squaring the formula for the square root. c) Formula of the square root taken as a stochastic equation. d) Same from the stochastic equation in this post.

There is now way to distinguish each other and the original Brownian motion is completely recovered by taking the square of the square root process computed in three different ways. Each one of these completely supports the conclusions we have drawn in our published paper. You can find the code to recover this figure in our arxiv paper. It is obtained by a Monte Carlo simulation with 10000 independent paths. You can play with it changing the parameters as you like.

This paper has an important consequence: Our current mathematical understanding of stochastic processes should be properly extended to account for our results. As a by-product, we have shown how, using Pauli matrices, this idea can be generalized to include spin introducing a new class of stochastic processes in a Clifford algebra.

In conclusion, we would like to remember that, it does not matter what your mathematical definition could be, a stochastic process is always a well-defined entity on a numerical ground. Tests can be easily performed as we proved here.

Farina, A., Frasca, M., & Sedehi, M. (2013). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y

Marco Frasca, & Alfonso Farina (2014). Numerical proof of existence of fractional Wiener processes arXiv arXiv: 1403.1075v1