The question of the arrow of time

A recent paper by Lorenzo Maccone on Physical Review Letters (see here) has produced some fuss around. He tries to solve the question of the arrow of time from a quantum standpoint. Lorenzo is currently a visiting researcher at MIT and, together with Vittorio Giovannetti and Seth Lloyd, he produced several important works in the area of quantum mechanics and its foundations. I have had the luck to meet him in a conference at Gargnano on the Garda lake together with Vittorio. So, it is not a surprise to see this paper of him in an attempt to solve one of the outstanding problems of physics.

The question of the arrow of time is open yet. Indeed, one can think that Boltzmann’s H-theorem closed this question definitely but this is false. This theorem has been the starting point for a question yet to be settled. Indeed, Boltzmann presented a first version of his theorem that showed one of the most beautiful laws in physics: the relation between entropy and probability. This proof was criticized by Loschmidt (see here) and this criticism was sound. Indeed, Boltzmann had to modifiy his proof by introducing the so called Stosszahlansatz or molecular chaos hypothesis introducing in this way time asymmetry by hand.  Of course, we know for certain that this theorem is true and so, also the hypothesis of molecular chaos must be true. So, the question of the arrow of time will be solved only when we will know where molecular chaos comes from. This means that we need a mechanism, a quantum one, to explain Boltzmann’s hypothesis. It is important to emphasize that, till today, a proof does not exist of the H-theorem that removes such an assumption.

Quantum mechanics is the answer to this situation and this can be so if we knew how reality forms. An important role in this direction could be given by environmental decoherence and how it relates to the question of the collapse. A collapse grants immediately asymmetry in time and here one has to cope with many-body physics with a very large number of components. In this respect there exists a beautiful theorem by Elliot Lieb and Barry Simon, two of the most prominent living mathematical-physicists, that says:

Thomas-Fermi model is the limit of quantum theory when the number of particles goes to infinity.

For a more precise statement you can look at Review of Modern Physics page 620ff. Thomas-Fermi model is just a semi-classical model and this just means that this fundamental theorem can be simply restated as saying that the limit of a very large number of particles in quantum mechanics is the classical world. In some way, there exists a large number of Hamiltonians in quantum mechanics that are not stable with  respect to such a particle limit losing quantum coherence. For certain we know that there exist other situations where quantum coherence is kept at a large extent in many-body systems. This would mean that exist situations where quantum fluctuations are not damped out with increasing number of particles.  But the very existence of this effect implied in the Lieb and Simon theorem means that quantum mechanics has an internal mechanism producing time-asymmetry. This, together with environmental decoherence (e.g. the box containing a gas is classical and so on), should grant a fully understanding of the situation at hand.

Finally, we can say that Maccone’s attempt, being on this line of thought, is a genuine way to understand from quantum mechanics the origin of time-asymmetry. I hope his ideas will meet with luck.

Update: In Cosmic Variance you will find an interesting post and worthwhile to read discussion involving Sean Carroll, Lorenzo Maccone and others on the questions opened with Lorenzo’s paper.

What is mass?

I should confess that one of the reasons why I have chosen to be a physicist is that physics, like no other sciences, is able to give answers to fundamental open questions that until a few years ago were only discussed by philosophers. Most of these questions are ancient as our species and the possibility that we have means to get truth is too strong to lose our time with other activities. So, I managed to learn such means and today I am here writing on this blog trying to explain you what these truths are. Sometime, I am at the forefront of research and so, what can be believed a truth may lose this quality as we deepen our understanding. Indeed, dynamics of science adds one more element of charm to all this matter.

One of such old questions is: “What are we made of?”. This question has been an open question till the dawn of the 20th century with the fundamental experiments carried out by Ernst Rutherford. Till then we have learned so much about matter that this question changed form becoming: “What is mass?”. This question has become compelling with the birth of the Standard Model due to Sheldon Lee Glashow, Steven Weinberg and Abdus Salam. Indeed, in order to maintain symmetry we must ask all particles to be massless and some mechanism must exist giving mass to them. In the sixties and seventies of last century we moved toward a real understanding of this concept. The idea is to rely on the Higgs mechanism and a scalar particle must exist to grant masses to the other particles in the model. As you may know this particle has not yet been seen and it is the only missing element of an otherwise very successful model. We are confident for several reasons that the Higgs mechanism could turn out the right answer to the question on mass but we are no more so confident that should have the simple aspect given originally in the Standard Model. Indeed, this appears as an open door on a Pandora’s vase of new exciting physics.

But whatever will be the mechanism at work for the masses of leptons and quarks, the answer to the main question is not there. For one reason, both electrons and quarks that form protons and neutrons are really light and do not count too much on the determination of our mass. Most of the mass is in the nuclei and we have to understand where such mass comes from. This arises from bound states of quarks glued together in some way as should yield QCD at low energies. This gives you an idea of why is so important to understand QCD at very low energy. In this way we would be able to answer a fundamental question philosophers discussed for so long time.

So, for our everyday life, it is not so relevant to comprehend the real mechanism that gives mass to elementary particles . What we need is to prove the existence of a mass gap in Yang-Mills theory and so the way bound states form in QCD. As you may know, this is not an easy task and involves a lot of talented people around the World that, with a lot of inventive, is trying to do such computations. So far, only computers succeeded in giving an answer and this is so good that we have the most important observed parameters precise to one percent. The hope is to have a technique to work out such computations analytically, as happens for weak coupled physics. I am deeply involved in such enterprise and I think that what will come out will have a large impact on our knowledge. I can only say: Stay tuned!

Back to work

Today I am back in my office and physics, that I never stopped to think of, is unfriendly urging in my mind. Besides my work as a reviewer for Mathematical Reviews that I try to do at my best, I have not forgotten to look around in the blogosphere. In these days there is a lot of fuss about a recent paper by Fermi Collaboration (see here). You can find some discussion here but it is not the only blog discussing this matter: An inflamed discussion is also here. Is loop quantum gravity dead? I can only spend a few words here by saying that is really too early to draw such conclusions but the results from Fermi Collaboration are really beautiful and open the premises for a bright future in the observation of gamma ray bursts. I would like to remember here the point of view of Steven Weinberg claiming that we finally hit exact truths on Nature (the bitch not the journal as  Tommaso Dorigo uses to say ). These truths are special relativity and quantum mechanics. I share this idea and the more beautiful idea that we are indeed able to reach such exact truths, the same mathematicians are able to achieve. On the same ground, if some experiment should come out claiming that these theories should be modified, I would be glad for one thing: A great opportunity for my generation to put hands on a deeper truth.

I am still working on a perturbation analysis of a non-perturbative Higgs field interacting with a fermion. I am solving Heisenberg equations of motion with a very large coupling for the Higgs. I hope to get some further time to complete the computations that may become really involved.

About QCD there is few to say. I have got my paper accepted for publication as you may know that implied a nice correspondence with Terry Tao. We can say that, after the very good Terry’s intervention, the proof of the mapping theorem is indeed complete. Another paper hit my interest and arose from a Japanese group working on lattice (see here). I have had an email exachange with Hideo Suganuma and surely their results are something to think about in depth. I hope to see other papers in the near future as this area of physics is very active indeed.

Finally, great expectations are for LHC. It will start on November and we hope to see results very soon. Infancy problems should be overcome and it is time to see the face of Higgs (the particle not the professor). Good luck, folks!

A very good justification

There is a plain explanation for my silence so far. The reason is that I am on my holidays yet and there is plenty of beautiful places that I want to share with you here. I am currently in Satriano, a beautiful town near Soverato where I go to the sea (just 7 km away) in the wonderful region of Calabria. Here is Satriano:

As you can see I am sharing the beauty of sea with that of mountains. Indeed,  Soverato is here somewhere hidden by mountains and the Ionian sea is just there in the distance:

A beautiful valley is passed through by the Ancinale river very near Satriano:

The town that you see on the right on the top of a mountain is Gagliato. Finally, I give you a last photo taken at the Mongiana Park inside a fairytale forest:

I have not forgotten Higgs boson but surely these breathtaking views can make one derails!

LHC will start at 3.5 TeV

A communication to press was released by CERN about the start-up of the collider after repairs. You can find it here. LHC will start at half of the promised beam energy on November this year and this only after all the tests on the connections will be ended with success. I have read a recent article on the New York Times here and my personal view is that all will be up and running on the promised time. Meantime, we can only wish the best of lucks at people working on LHC.

Answer to Terry Tao’s criticism will go published

My paper containing the answer to Terry Tao’s criticism will be published in Modern Physics Letters A. You can get a copy of this preprint from arxiv here.

Thank you very much, folks!

The right mathematical question

After my post on the Higgs field (see here) I would like to explain why there is no reason to be afraid. The point is the right mathematical question to be asked.  So, let me state why, from a strict mathematical standpoint, small perturbation theory is not the whole story. Let us consider the differential equation

The exact solution of this equation is the function . So, one can ask : What happens when goes to zero? This is a proper mathematical question and the answer, when it exists, is a Taylor series. This is the most celebrated small perturbation theory and the terms of this Taylor series are computed directly from the given differential equation.

Of course, I can also ask what happens to the function when goes to infinity. This is perfectly legal from a mathematical standpoint. This dual limit, when exists, produces an asymptotic series that has a development parameter . This is a strong perturbation theory when each term is computed directly from the given differential equation.

Indeed, one can build a machinery for this case and prove the very existence of this technique that extends perturbation theory beyond the realm of a research of a small parameter. Rather, one can consider the case of differential equations with a large parameter and solve it producing an analysis of these equations in a range of the parameter space not reachable with small perturbation theory.

Physics is a lucky case for this mathematical question as it is all built on differential equations and having such a technique permits to analyze them in situations never reached before other than with computers.

I would like to emphasize that this is applied mathematics but the solutions one obtains can be interesting for physics. Of course, mathematics cannot be questioned except when is wrong.  But when it is right any discussion  is somewhat grotesque.