Two dimensional Ricci flow is really easy to manage. In this case the equation takes a very simple form and a wealth of results can be extracted. As you know from my preceding posts, I have been able to prove in a rigorous way that in this case the Ricci flow arises from Brownian motion (see here). So, the equation for Einstein manifolds in this case takes the very simple form, being a constant, that is also the equation for a Ricci soliton. This equation is rather well-knwon to physicists as is the equation of 2d Einstein gravity. This equation is nothing else than Liouville equation

that admits an exact solution notwithstanding being non-linear. There is an unexpected application of all this machinery of Riemann geometry to the case of statistics. Statistics has a wide body of application fields as radar tracking, digital communications and so on. Then, any new result about can be translated into a wealthy number of applications.

The problem one meets in this case is that of parameter estimation of a given probability distribution. For a sample of measured data the question is to determine the best probability distribution with respect to the spread of the data themselves with a proper choice of the parameters. A known result in this area is the so called Cramer-Rao bound. This inequality gives limit for the optimality of the chosen estimators of the data entering into the distribution. The result I have found is that, for a probability distribution with two parameters, an infinite class of optimal estimators exists that are all efficient. These estimators are given by the solution of Liouville equation! The result can be extended to the n-dimensional case granted the existence of isothermal coordinates that are the conformal ones.

This result arises from the deep link between differential geometry and statistics that was put forward by Calayampudi Radhakrishna Rao. My personal interest in this matter was arisen working in radar tracking but one can think on a large number of other areas. I should say, as a final consideration, that the work of Hamilton and Perelman can have a deep impact in a large body of our knowledge. We are just at the beginning.

On the Ricci flow and emergent quantum mechanics
Authors: J.M. Isidro, J.L.G. Santander, P. Fernandez de Cordoba
(Submitted on 1 Feb 2009)

Abstract: The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential function that is proportional to the Ricci scalar curvature of configuration space. This allows one to obtain Schroedinger quantum mechanics from Perelman’s action functional: the quantum-mechanical wavefunction is the exponential of $i$ times the conformal factor of the metric on configuration space. We explore links with the recently discussed emergent quantum mechanics.

Thank you for pointing me out this paper. I know the works of these authors and I have exchanged some interesting emails with Jose’ Isidro. I think these authors should be considered pioneers on the way to give an understanding of Ricci flow for physics. Of course, they work in 2d. Presently this dimensionality is the most treatable one from a physics standpoint and we have several serious clues that the reality underlying our 4-dimensional manifold is 2-d. Indeed, both string theory and loop quantum gravity take or obtain this same conclusion.

On the other side, there is the most general case of Perelman functional in 2-d that is looking for a serious application. Physicists know this as non-linear sigma model in a curved manifold and you can find a few pages treatment in Polchinski’s books about strings. In this case there is no limitation about dimensions (otherwise being forced to d=26) to get a sensible quantum field theory.

So, there are several reasons to take all these mathematical developments quite seriously and worthwhile to be exploited in the wide framework of quantum gravity.

Fractal Structure of Loop Quantum Gravity
Authors: Leonardo Modesto
(Submitted on 11 Dec 2008)

Abstract: In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decrees from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at hight energy. Our result is consistent with other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity.

Fractal properties of quantum spacetime
Authors: Dario Benedetti
(Submitted on 10 Nov 2008)

Abstract: We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of $\k$-Minkowski, the latter being relevant in the context of quantum gravity.

[…] bound and Ricci flow II The paper I presented about this matter (see here) has been accepted by EuRad 2009 Conference. This will result in a publication in IEEE Proceedings. […]

Given that QM is a theory of statistical realization, this might be interesting to you:

http://arxiv.org/abs/0902.0143

On the Ricci flow and emergent quantum mechanics

Authors: J.M. Isidro, J.L.G. Santander, P. Fernandez de Cordoba

(Submitted on 1 Feb 2009)

Abstract: The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential function that is proportional to the Ricci scalar curvature of configuration space. This allows one to obtain Schroedinger quantum mechanics from Perelman’s action functional: the quantum-mechanical wavefunction is the exponential of $i$ times the conformal factor of the metric on configuration space. We explore links with the recently discussed emergent quantum mechanics.

Hi Daniel,

Thank you for pointing me out this paper. I know the works of these authors and I have exchanged some interesting emails with Jose’ Isidro. I think these authors should be considered pioneers on the way to give an understanding of Ricci flow for physics. Of course, they work in 2d. Presently this dimensionality is the most treatable one from a physics standpoint and we have several serious clues that the reality underlying our 4-dimensional manifold is 2-d. Indeed, both string theory and loop quantum gravity take or obtain this same conclusion.

On the other side, there is the most general case of Perelman functional in 2-d that is looking for a serious application. Physicists know this as non-linear sigma model in a curved manifold and you can find a few pages treatment in Polchinski’s books about strings. In this case there is no limitation about dimensions (otherwise being forced to d=26) to get a sensible quantum field theory.

So, there are several reasons to take all these mathematical developments quite seriously and worthwhile to be exploited in the wide framework of quantum gravity.

Marco

Hi Marco,

Yes, there are clues that 2d undelies 4d. Check these out:

http://arxiv.org/abs/0812.2214

Fractal Structure of Loop Quantum Gravity

Authors: Leonardo Modesto

(Submitted on 11 Dec 2008)

Abstract: In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decrees from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at hight energy. Our result is consistent with other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity.

Comments: 5 pages, 5 figures

http://arxiv.org/abs/0811.1396

Fractal properties of quantum spacetime

Authors: Dario Benedetti

(Submitted on 10 Nov 2008)

Abstract: We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of $\k$-Minkowski, the latter being relevant in the context of quantum gravity.

Also, there are hints, in these articles, that LQG is closely related to gravitational assymptotic freedom and Causal Dynamical Triangulations.

[…] bound and Ricci flow II The paper I presented about this matter (see here) has been accepted by EuRad 2009 Conference. This will result in a publication in IEEE Proceedings. […]