Yang-Mills theory paper gets published!



Exact solutions of quantum field theories are very rare and, normally, refer to toy models and pathological cases. Quite recently, I put on arxiv a pair of papers presenting exact solutions both of the Higgs sector of the Standard Model and the Yang-Mills theory made just of gluons. The former appeared a few month ago (see here) while the latter has been accepted for publication a few days ago (see here). I have updated the latter just today and the accepted version will appear on arxiv on 2 January next year.

What does it mean to solve exactly a quantum field theory? A quantum field theory is exactly solved when we know all its correlation functions. From them, thanks to LSZ reduction formula, we are able to compute whatever observable in principle being these cross sections or decay times. The shortest way to correlation functions are the Dyson-Schwinger equations. These equations form a set with the former equation depending on the higher order correlators and so, they are generally very difficult to solve. They were largely used in studies of Yang-Mills theory provided some truncation scheme is given or by numerical studies. Their exact solutions are generally not known and expected too difficult to find.

The problem can be faced when some solutions to the classical equations of motion of a theory are known. In this way there is a possibility to treat the Dyson-Schwinger set. Anyhow, before to enter into their treatment, it should be emphasized that in literature the Dyson-Schwinger equations where managed just in one way: Carl BenderUsing their integral form and expressing all the correlation functions by momenta. It was an original view by Carl Bender that opened up the way (see here). The idea is to write the Dyson-Schwinger equations into their differential form in the coordinate space. So, when you have exact solutions of the classical theory, a possibility opens up to treat also the quantum case!

This shows unequivocally that a Yang-Mills theory can display a mass gap and an infinite spectrum of excitations. Of course, if nature would have chosen the particular ground state depicted by such classical solutions we would have made bingo. This is a possibility but the proof is strongly related to what is going on for the Higgs sector of the Standard Model that I solved exactly but without other matter interacting. If the decay rates of the Higgs particle should agree with our computations we will be on the right track also for Yang-Mills theory. Nature tends to repeat working mechanisms.

Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model Eur. Phys. J. Plus (2016) 131: 199 arXiv: 1504.02299v3

Marco Frasca (2015). Quantum Yang-Mills field theory arXiv arXiv: 1509.05292v1

Carl M. Bender, Kimball A. Milton, & Van M. Savage (1999). Solution of Schwinger-Dyson Equations for ${\cal PT}$-Symmetric Quantum Field Theory Phys.Rev.D62:085001,2000 arXiv: hep-th/9907045v1



I tried in different ways to get this paper through the community with standard channels. As far as I can tell, this paper is unpublishable. By this I mean that journals not even send it to referees to start a normal review process or all people try to stop it from making it known. The argument is always the same: A reformulation of quantum mechanics using stochastic processes but using noncommutative geometry this time. I apologize to the community if this unacceptable approach has bothered people around the World but this is the fate of some ideas. Of course, if somebody has the courage and the willing to publish, let me know and I will appreciate the tentative with infinite gratefulness.

Now, back to sane QCD.

Happy new year!

Quantum gravity



Quantum gravity appears today as the Holy Grail of physics. This is so far detached from any possible experimental result but with a lot of attentions from truly remarkable people anyway. In some sense, if a physicist would like to know in her lifetime if her speculations are worth a Nobel prize, better to work elsewhere. Anyhow, we are curious people and we would like to know how does the machinery of space-time work this because to have an engineering of space-time would make do to our civilization a significant leap beyond.

A fine recount of the current theoretical proposals has been rapidly presented by Ethan Siegel in his blog. It is interesting to notice that the two most prominent proposals, string theory and loop quantum gravity, share the same difficulty: They are not able to recover the low-energy limit. For string theory this is a severe drawback as here people ask for a fully unified theory of all the interactions. Loop quantum gravity is more limited in scope and so, one can think to fix theAlain Connes problem in a near future. But of all the proposals Siegel is considering, he is missing the most promising one: Non-commutative geometry. This mathematical idea is due to Alain Connes and earned him a Fields medal. So far, this is the only mathematical framework from which one can rederive the full Standard Model with all its particle content properly coupled to the Einstein’s general relativity. This formulation works with a classical gravitational field and so, one can possibly ask where quantized gravity could come out. Indeed, quite recently, Connes, Chamseddine and Mukhanov (see here and here), were able to show that, in the context of non-commutative geometry, a Riemannian manifold results quantized in unitary volumes of two kind of spheres. The reason why there are two kind of unitary volumes is due to the need to have a charge conjugation operator and this implies that these volumes yield the units (1,i) in the spectrum. This provides the foundations for a future quantum gravity that is fully consistent from the start: The reason is that non-commutative geometry generates renormalizable theories!

The reason for my interest in non-commutative geometry arises exactly from this. Two years ago, I, Alfonso Farina and Matteo Sedehi obtained a publication about the possibility that a complex stochastic process is at the foundations of quantum mechanics (see here and here). We described such a process like the square root of a Brownian motion and so, a Bernoulli process appeared producing the factor 1 or i depending on the sign of the steps of the Brownian motion. This seemed to generate some deep understanding about space-time. Indeed, the work by Connes, Chamseddine and Mukhanov has that understanding and what appeared like a square root process of a Brownian motion today is just the motion of a particle on a non-commutative manifold. Here one has simply a combination of a Clifford algebra, that of Dirac’s matrices, a Wiener process and the Bernoulli process representing the scattering between these randomly distributed quantized volumes. Quantum mechanics is so fundamental that its derivation from a geometrical structure with added some mathematics from stochastic processes makes a case for non-commutative geometry as a serious proposal for quantum gravity.

I hope to give an account of this deep connection in a near future. This appears a rather exciting new avenue to pursue.

Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Quanta of Geometry: Noncommutative Aspects Phys. Rev. Lett. 114 (2015) 9, 091302 arXiv: 1409.2471v4

Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Geometry and the Quantum: Basics JHEP 12 (2014) 098 arXiv: 1411.0977v1

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

Is Higgs alone?



I am back after the announcement by CERN of the restart of LHC. On May this year we will have also the first collisions. This is great news and we hope for the best and the best here is just the breaking of the Standard Model.

The Higgs in the title is not Professor Higgs but rather the particle carrying his name. The question is a recurring one since the first hints of existence made their appearance at the LHC. The point I would like to make is that the equations of the theory are always solved perturbatively, even if exact solutions exist that provide a mass also if the theory is massless or has a mass term with a wrong sign (Higgs model). All you need is a finite self-interaction term in the equation. So, you will have bad times to recover such exact solutions with perturbation techniques and one keeps on living in the ignorance. If you would like to see the technicalities involved just take a cursory look at Dispersive Wiki.

What is the point? The matter is rather simple. The classical theory has exact massive solutions for the potential in the form V(\phi)=a\phi^2+b\phi^4 and this is a general result implying that a scalar self-interacting field gets always a mass (see here and here). Are we entitled to ignore this? Of course no. But today exact solutions have lost their charm and we can get along with them.

For the quantum field theory side what could we say? The theory can be quantized starting with these solutions and I have shown that one gets in this way that these massive particles have higher excited states. These are not bound states (maybe could be correctly interpreted in string theory or in a proper technicolor formulation after bosonization) but rather internal degrees of freedom. It is always the same Higgs particle but with the capability to live in higher excited states. These states are very difficult to observe because higher excited states are also highly depressed and even more hard to see. In the first LHC run they could not be seen for sure. In a sense, it is like Higgs is alone but with the capability to get fatter and present himself in an infinite number of different ways. This is exactly the same for the formulation of the scalar field as originally proposed by Higgs, Englert, Brout, Kibble, Guralnik and Hagen. We just note that this formulation has the advantage to be exactly what one knows from second order phase transitions used by Anderson in his non-relativistic proposal of this same mechanism. The existence of these states appears inescapable whatever is your best choice for the quartic potential of the scalar field.

It is interesting to note that this is also true for the Yang-Mills field theory. The classical equations of this theory display similar solutions that are massive (see here) and whatever is the way you develop your quantum filed theory with such solutions the mass gap is there. The theory entails the existence of massive excitations exactly as the scalar field does. This have been seen in lattice computations (see here). Can we ignore them? Of course no but exact solutions are not our best choice as said above even if we will have hard time to recover them with perturbation theory. Better to wait.

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

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

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v2

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N Physics Reports 526 (2013) 93-163 arXiv: 1210.4997v2

What is going on at NASA?



As a physicist I have been always interested about experiments that can corroborate theoretical findings. Most of these often become important applications for everyday life or change forever the course of the history of mankind. With this in view, I am currently following with great interest the efforts by the NASA group headed by Harold White.Harold "Sonny" White This work has arisen uproar in the web and in the media as it was come to envision the possibility to realize a warp drive, in the way Alcubierre devised it, and the stars were in the reach shortly. As it is well-known, Alcubierre drive implies exotic matter something that does not appear at hand neither in small nor in large quantity. On the other side, it was indirectly observed in the Casimir effect, a beautiful application of quantum field theory to real life. So, it is rather normal to link warp drive with exotic matter. It should be emphasized that nobody on Earth ever managed it in some way and it is not available at your nearest grocery store. The experiment carried out by Harold White and his group is realized with an interference device using lasers on an optical table. The idea is to observe a modification of space-time, a minuscule one, that would modify the paths of the laser beams. This would be comparable to the realization of the Chicago pile by Enrico Fermi that was the starting point for the Manhattan project. I would like to emphasize that such a laboratory small-scale manipulation of space-time would be a huge breakthrough in physics and would open up the way to a new kind of engineering, that of space-time. So, our hopes for a warp drive would be totally fulfilled.

There is an eager desire to obtain any possible information about the progress of White’s work but, of course, there are a couple of hurdles. The first one is that a scientist needs to be certain before to claim a result and we know very well why from some blatant examples in the last years. Extraordinary claims require extraordinary evidence. Last but not least, Harold White is employed at NASA and some restrictions could be required by the organization he is working with. So, it is really interesting a video appeared quite recently where White claims that the effect is there but further work is needed for confirmation. If you have a hour of your spare time, this video is worthwhile to be seen.

This video is interesting per se because Harold White is talking to his colleagues at NASA. But in the question time happens the interesting fact. A White’s colleague asks him “where is the exotic matter?”:

and here something interesting happens. White seems to avoid the question and admits that they talked before in the office. What is more interesting is what the White’s colleague is saying then unveiling some of the machinery behind the experiment. The colleague says that the experiment could be carried out in some strong coupling regime that makes the magic happen without any exotic matter. White denies and disagrees. We know that he is using strong electromagnetic fields in the interference zone. Indeed, the matter of the behaviour of the space-time in a strong perturbation was studied for cosmological aims by Belinski, Kalathnikov and Lifshitz, the BKL trio. David GarfinkleThis scenario was confirmed by numerical studies by David Garfinkle (see here). I was able to derive it by analysing the behaviour of the Einstein equations under a strong perturbation (see here) in analytical way. So, the chance to study such effects in a laboratory would be really striking and would mean an incredible breakthrough for people working in general relativity and related fields. What the exchange between White and his colleague implies is that this could be already at hand and without exotic matter. All the growing concerns about the work at NASA are then not applicable and a different kind of analysis would be needed. Particularly, Alcubierre drive should be devised in a different way. As a physicist, I am eager to learn more about this and to know the real answer, from the horse’s mouth, to the question “where is the exotic matter?”.

Miguel Alcubierre (2000). The warp drive: hyper-fast travel within general relativity Class.Quant.Grav.11:L73-L77,1994 arXiv: gr-qc/0009013v1

David Garfinkle (2003). Numerical simulations of generic singuarities Phys.Rev.Lett. 93 (2004) 161101 arXiv: gr-qc/0312117v4

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys.D15:1373-1386,2006 arXiv: hep-th/0508246v3

Standard Model at the horizon



Hawking radiation is one of the most famous effects where quantum field theory combines successfully with general relativity. Since 1975 when Stephen Hawking uncovered it, this result has obtained a enormousStephen Hawking consideration and has been derived in a lot of different ways. The idea is that, very near the horizon of a black hole, a pair of particles can be produced one of which falls into the hole and the other escapes to infinity and is seen as emitted radiation. The overall effect is to drain energy from the hole, as the pair is formed at its expenses, and its ultimate fate is to evaporate. The distribution of this radiation is practically thermal and a temperature and an entropy can be attached to the black hole. The entropy is proportional to the area of the black hole computed at the horizon, as also postulated by Jacob Bekenstein, and so, it can only increase. Thermodynamics applies to black holes as well. Since then, the quest to understand the microscopic origin of such an entropy has seen a huge literature production with the notable Jacob Bekensteinunderstanding coming from string theory and loop quantum gravity.

In all the derivations of this effect people generally assumes that the particles are free and there are very good reasons to do so. In this way the theory is easier to manage and quantum field theory on curved spaces yields definite results. The wave equation is separable and exactly solvable (see here and here). For a scalar field, if you had a self-interaction term you are immediately in trouble. Notwithstanding this, in  the ’80 UnruhWilliam Unruh and Leahy, considering the simplified case of two dimensions and Schwarzschild geometry, uncovered a peculiar effect: At the horizon of the black the interaction appears to be switched off (see here). This means that the original derivation by Black-hole-model by Kip ThorneHawking for free particles has indeed a general meaning but, the worst conclusion, all particles become non interacting and massless at the horizon when one considers the Standard Model! Cooper will have very bad times crossing Gargantua’s horizon.

Turning back from science fiction to reality, this problem stood forgotten for all this time and nobody studied this fact too much. The reason is that the vacuum in a curved space-time is not trivial, as firstly noted by Hawking, and mostly so when particles interact. Simply, people has increasing difficulties to manage the theory that is already complicated in its simplest form. Algebraic quantum field theory provides a rigorous approach to this (e.g. see here). These authors consider an interacting theory with a \varphi^3 term but do perturbation theory (small self-interaction) probably hiding in this way the Unruh-Leahy effectValter Moretti.

The situation can change radically if one has exact solutions. A \varphi^4 classical theory can be indeed solved exactly and one can make it manageable (see here). A full quantum field theory can be developed in the strong self-interaction limit (see here) and so, Unruh-Leahy effect can be accounted for. I did so and then, I have got the same conclusion for the Kerr black hole (the one of Interstellar) in four dimensions (see here). This can have devastating implications for the Standard Model of particle physics. The reason is that, if Higgs field is switched off at the horizon, all the particles will lose their masses and electroweak symmetry will be recovered. Besides, further analysis will be necessary also for Yang-Mills fields and I suspect that also in this case the same conclusion has to hold. So, the Unruh-Leahy effect seems to be on the same footing and importance of the Hawking radiation. A deep understanding of it would be needed starting from quantum gravity. It is a holy grail, the switch-off of all couplings, kind of.

Further analysis is needed to get a confirmation of it. But now, I am somewhat more scared to cross a horizon.

V. B. Bezerra, H. S. Vieira, & André A. Costa (2013). The Klein-Gordon equation in the spacetime of a charged and rotating black hole Class. Quantum Grav. 31 (2014) 045003 arXiv: 1312.4823v1

H. S. Vieira, V. B. Bezerra, & C. R. Muniz (2014). Exact solutions of the Klein-Gordon equation in the Kerr-Newman background and Hawking radiation Annals of Physics 350 (2014) 14-28 arXiv: 1401.5397v4

Leahy, D., & Unruh, W. (1983). Effects of a λΦ4 interaction on black-hole evaporation in two dimensions Physical Review D, 28 (4), 694-702 DOI: 10.1103/PhysRevD.28.694

Giovanni Collini, Valter Moretti, & Nicola Pinamonti (2013). Tunnelling black-hole radiation with $φ^3$ self-interaction: one-loop computation for Rindler Killing horizons Lett. Math. Phys. 104 (2014) 217-232 arXiv: 1302.5253v4

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

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

Marco Frasca (2014). Hawking radiation and interacting fields arXiv arXiv: 1412.1955v1

Evidence of the square root of Brownian motion



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


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.

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

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