Ricci flow as a stochastic process


Yesterday I have posted a paper on arxiv (see here). In this work I prove a theorem about Ricci flow. The question I give an answer is the following. When you have a heat equation you have always a stochastic process from which such an equation can be derived. In two dimensions the Ricci flow takes the straightforward form of a heat equation. So, could it be derived from a stochastic process? The answer is affirmative and can be obtained through a generalization of path integrals (Wiener integrals) on a Riemannian manifold given here. One can write for the metric something like

g=\int [dq]\exp[-{\cal L}(q)]g_0

so, what is \cal L? The really interesting answer is that this is Perelman \cal L-length functional. A similar expression was derived by Bryce DeWitt in the context of Feynman’s path integrals in a non-Euclidean manifold in 1957 (see here) but in this case we are granted of the existence of the integral.

This result shows a really interesting conclusion that underlying Ricci flow there is a stochastic process (Wiener process), at least in two dimensions. So, we propose a more general conjecture: Ricci flow is generated by a Wiener process independently on the dimensionality of the manifold.

I’ll keep on working on this as this result provide a clear path to quantum gravity. Mostly, I would like to understand how Ricci flow and the non-linear sigma model are connected. Also here, I guess, Perelman will play a leading role.

Ricci solitons in two dimensions


Today I have read recent changes to DispersiveWiki. This is a beautiful site about differential equations that is maintained at University of Toronto by Jim Colliander and has notable contributors as the Fields medallist Terence Tao. Terry introduced a new page about Liouville’s equation as he got involved with it in a way you can read here. Physicists working on quantum gravity has been aware of this equation since eighties as it is the equation of two-dimensional quantum gravity and comes out quite naturally in string theory. A beautiful paper about quantum field theory of Liouville equation is due to Roman Jackiw and one of his collaborators Eric D’Hoker (see here). But what people could have overlooked is that Liouville’s equation is the equation of the Ricci soliton in two dimensions. The reason is that in this case a set of isothermal coordinates can always be found and the metric is always conformal, that is


being g_0 the Euclidean metric. The Ricci tensor takes here a quite simple form


being \epsilon_{ik}=diag(1,1) . Then the Ricci flow is

\frac{\partial\phi}{\partial t}=e^{-2\phi}(\partial^2_x+\partial^2_y)\phi

and finally for the Ricci soliton one has

(\partial^2_x+\partial^2_y)\phi = H e^{2\phi}

being H a constant. After a simple rescaling we are left with the Euclidean Liouville’s equation

(\partial^2_x+\partial^2_y)u = \Lambda e^{u}.

Turning back to the Jackiw and D’Hoker paper, we can see that a 2D gravity theory emerges naturally as the equilibrium (Ricci soliton) solution of a Fokker-Planck (Ricci flow) equation. This scenario seems a beautiful starting point to build an understanding of quantum gravity. I am still thinking about a lot and I will put all this on a paper one day.

Ricci solitons


These days I am looking at all this area of mathematical research born with Richard Hamilton and put at maturity with the works of Grisha Perelman. As all of you surely know the conclusion was that the Thurston conjecture, implying Poincare’ conjecture, is a theorem. These results present the shocking aspect of a deep truth waiting for an understanding by physicists and, I think that this comes out unexpectedly, statisticians (do you know Fischer information matrix and Cramer-Rao bound?).

One of the most shocking concept mathematicians introduced working with Ricci flow is a Ricci soliton. I will use some mathematics to explain this. A Ricci flow is given by

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

a Ricci soliton is a metric solving the equation

R_{ik}-L_X g_{ik} = \Lambda g_{ik}

where I have used an awkward notation for the Lie derivative along a field X but if this field is a scalar than one has a gradient soliton. I think that all of you will recognize these equations that for a Lorentzian metric are just Einstein equations in vacuum with a cosmological constant! Now, I have found a beautiful paper about all this question on arxiv (see here). This paper gives the first meaningful application to physics of this striking concept. Ricci solitons are resembling a kind of behavior of the metric under the flow that can be expanding, collapsing or static depending on the cosmological constant.

As time goes by we learn something deeper about Einstein equations. Their very nature seems rooted in quite recent concepts coming from differential geometry and it is my personal view that whatever quantum gravity theory we will formulate, these are the questions we have to cope with.

0.7 anomaly and the Fermi liquid


Nanophysics is one of the research acitvities  full of promises for the improvement of our lives through the realization of new devices. This application of solid state physics becomes relevant when quantum mechanics comes into play in conduction phenomena. The aspect people may not be aware is that these researches produced several unexpected results. One of these is the so called 0.7 anomaly. This effect appears in the QPC or quantum point contacts. This can be seen as a waveguide for the wavefunction of the electrons. As such, the main effect is that conductance is quantized in integer multiples of an universal constant 2e^2/\hbar. qpc
Measurements on these devices are realized at very low temperature so to have quantum effects at work. The result of such measurements come out somewhat unexpected. Indeed, the quantization of conductance appeared as due but a further step occurred at 0.7\times 2e^2/\hbar and was called the 0.7 anomaly.

Theoretical physicists proposed two alternatives to explain this effect. The first one claimed that the Fermi liquid of conduction electrons was spin polarized while the second claimed that the Kondo effect was at work. Kondo effect appears in presence of magnetic impurities modifying the resistance curve of the material. In any case, both proposals have effects on the electron conductance and are able to explain the observed anomaly. The only way to achieve an understanding is then through further experimental work.

I have found a recent paper by Leonid Rokhinson at Purdue University, and Loren Pfeiffer and Kenw West both at Bell Lab producing a consistent result that proves that the conducting electrons are spin polarized (see here). I cannot expect a different result also in view of my paper about another problem in nanophysics and this is the appearence of a finite coherence time in nanowires, a rather shocking result for the community as the standard result should be an infinite coherence time (see here). Indeed, I have accomoned both effects as due to the same reason and this is the polariztion of the Fermi liquid (see here). This matter is still open and under hot debating in the nanophysics community. What I see here are the premises of a relevant new insight into condensed matter physics.

Ted Jacobson and quantum gravity


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?



People really interested about this matter should read the preprint by Elias Kiritsis (see here). This paper gives a full account about this matter and is a recollection of conferences’ contributions yielded by the author.

My point of view about this question, as the readers of the blog may know, is that a general technique to strong coupling problems should be preferred to more aimed approaches. This by no means diminishes the value of these works. Another point I have discussed about the spectrum of AdS/QCD is what happens if one takes the lower state at about 1.19, does one recover the ground state seen in lattice QCD for the glueball spectrum as the next state?

The value of this approach relies on a serious possibility to verify, with a low energy theory, a higher level concept connecting gravity and gauge theories. Both sides have something to be earned.

Narison, Ochs, Mennessier and the width of the sigma


In order to understand what is going on in the lower part of the meson spectrum of QCD that is currently seen in experiments one would like to have an explicit formula for the width of the sigma. The reason is that we would like to have an idea of its broadness. Being this the infrared limit the only known way to get this would be lattice computations but in this case there is no help. Lattice computations see no sigma resonance anywhere. Narison, Ochs and Mennessier were able to obtain an understanding of this quantity by QCD spectral sum rules here and here. They get the following phenomenological equation

\Gamma_\sigma=\frac{|g_{\sigma\pi^+\pi^-}|^2}{16\pi m_\sigma}\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}

being the coupling |g_{\sigma\pi^+\pi^-}|\approx (4\sim 5)\ GeV explaining in this way why this resonance is so broad. Their main conclusion, after computing the width of the reaction \sigma\rightarrow\gamma\gamma, is that this resonance is a glueball.

In our latest paper (see here) we computed the width of the sigma directly from QCD. We obtained the following equation

\Gamma_\sigma = \frac{6}{\pi}\frac{G^2_{NJL}}{4\pi\alpha_s} m_\sigma f_\pi^4\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}

being G_{NJL}\approx 3.76\frac{4\pi\alpha_s}{\sigma}, \sigma the string tension that we take about 410 MeV, and f_\pi\approx 93\ MeV the pion decay constant. The mass was given by

m_\sigma\approx 1.198140235\sqrt{\sigma}.

This permits us to give the coupling in the Narison, Ochs and Mennessier formula as

|g_{\sigma\pi^+\pi^-}|\approx 156.47\sqrt{\frac{\alpha_s}{\sigma}}f^2_\pi

giving in the end

|g_{\sigma\pi^+\pi^-}|\approx 3.3\sqrt{\alpha_s}\ GeV

in very nice agreement with their estimation. We can conclude that their understanding of \sigma is quite precise. An interesting conclusion to be drawn here is about how good turn out to be these techniques based on spectral sum rules. The authors call these methods with a single acronym QSSR. They represent surely a valid approach for the understanding of the lower part of QCD spectrum. Indeed, QCD calculations prove that this resonance is a glueball.


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