## Quantum gravity

27/12/2015

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 the 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

## Back to work

02/02/2014

I would like to have a lot more time to write on my blog. Indeed, time is something I have no often and also the connection is not so good as I would like in the places I spend most of it. So, I take this moment to give an update of what I have seen around in these days.

LHC has found no evidence of dark matter so far (see here). Dark matter appears even more difficult to see and theory is not able to help the search. This is also one of our major venues to go beyond the Standard Model. On the other side, ASACUSA experiment at CERN produced the first beam of antihydpogen atoms (see here, this article is free to read). We expect no relevant news about the very nature of Higgs until, on 2015, LHC will restart. It must be said that the data collected so far are saying to us that this particle is behaving very nearly as that postulated by Weinberg on 1967.

In these days there has been some fuss about the realization in laboratory of a Dirac magnetic monopole (see here).  Notwithstanding this is a really beautiful experiment, nobody has seen a magnetic monopole so far. It is a simulation performed with another physical system: A BEC. This is a successful technology that will permit us an even better understanding of physical systems that are difficult to observe. Studies are ongoing to realize a simulation of  Hawking radiation in such a system.  Even if this is the state of affairs, I have read in social networks and in the news that a magnetic monopole was seen in laboratory. Of course, this is not true.

The question of black holes is always at the top of the list of the main problems in physics. Mostly when a master of physics comes out with a new point of view. So, a lot of  fuss arose from this article in Nature involving a new idea from Stephen Hawking that the author published in a paper on arxiv (see here). Beyond the resounding title, Hawking is just proposing a way to avoid the concept of firewalls that was at the center of a hot debate in the last months. Again we recognize that a journalist is not making a good job but is generating a lot of noise around and noise can hide a signal very well.

Finally, we hope in a better year in science communication. The start was somewhat disappointing.

Kuroda N, Ulmer S, Murtagh DJ, Van Gorp S, Nagata Y, Diermaier M, Federmann S, Leali M, Malbrunot C, Mascagna V, Massiczek O, Michishio K, Mizutani T, Mohri A, Nagahama H, Ohtsuka M, Radics B, Sakurai S, Sauerzopf C, Suzuki K, Tajima M, Torii HA, Venturelli L, Wu Nschek B, Zmeskal J, Zurlo N, Higaki H, Kanai Y, Lodi Rizzini E, Nagashima Y, Matsuda Y, Widmann E, & Yamazaki Y (2014). A source of antihydrogen for in-flight hyperfine spectroscopy. Nature communications, 5 PMID: 24448273

M. W. Ray,, E. Ruokokoski,, S. Kandel,, M. Möttönen,, & D. S. Hall (2014). Observation of Dirac monopoles in a synthetic magnetic field Nature, 505, 657-660 DOI: 10.1038/nature12954

Zeeya Merali (2014). Stephen Hawking: ‘There are no black holes’ Nature DOI: 10.1038/nature.2014.14583

S. W. Hawking (2014). Information Preservation and Weather Forecasting for Black Holes arXiv arXiv: 1401.5761v1

## A first paper on square root of a Brownian motion and quantum mechanics gets published!

20/11/2012

Following my series of posts on the link between the square root of a stochastic process and quantum mechanics (see here, here, here, here, here), that I proved to exist both theoretically and experimentally, I am pleased to let you know that the first paper of my collaboration with Alfonso Farina and Matteo Sedehi was finally accepted in Signal, Image and Video Processing. This paper contains the proof of what I named the “Farina-Frasca-Sedehi proposition” in my paper that claims that for a well localized free particle there exists a map between the wave function and the square root of binomial coefficients. This finally links the Pascal-Tartaglia triangle, given through binomial coefficients, to quantum mechanics and closes a question originally open by Farina and collaborators on the same journal (see here). My theorem about the square root of a stochastic process also appears in this article but without a proof.

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v2

Farina, A., Giompapa, S., Graziano, A., Liburdi, A., Ravanelli, M., & Zirilli, F. (2011). Tartaglia-Pascal’s triangle: a historical perspective with applications Signal, Image and Video Processing DOI: 10.1007/s11760-011-0228-6

## Quantum mechanics and the square root of Brownian motion

25/01/2012

There is a very good reason why I was silent in the past days. The reason is that I was involved in one of the most difficult article to write down since I do research (and are more than twenty years now!).  This paper arose during a very successful collaboration with two colleagues of mine: Alfonso Farina and Matteo Sedehi. Alfonso is a recognized worldwide authority in radar technology and last year has got a paper published here about the ubiquitous Tartaglia-Pascal triangle and its applications in several areas of mathematics and engineering. What was making Alfonso unsatisfied was the way the question of Tartaglia-Pascal triangle fits quantum mechanics. It appeared like this is somewhat an unsettled matter. Tartaglia-Pascal triangle gives, in the proper limit, the solution of the heat equation typical of Brownian motion, the most fundamental of all stochastic processes. But when one comes to the Schroedinger equation, notwithstanding the formal resemblance between these two equations, the presence of the imaginary term changes things dramatically. So, a wave packet of a free particle is seen to spread like the square of time rather than linearly. Then, Alfonso asked to me to try to clarify the situation and see what is the role of Tartaglia-Pascal triangle in quantum mechanics. This question is old almost as quantum mechanics itself. Several people tried to explain the probabilistic nature of quantum mechanics through some kind of Brownian motion of space and the most famous of these attempts is due to Edward Nelson. Nelson was able to show that there exists a stochastic process producing hydrodynamic equations from which the Schroedinger equation can be derived. This idea turns out to be a description of quantum mechanics similar to the way David Bohm devised it. So, this approach was exposed to criticisms that can be summed up in a paper by Peter Hänggi, Hermann Grabert and Peter Talkner (see here) denying any possible representation of quantum mechanics as a classical stochastic process.

So, it is clear that the situation appears rather difficult to clarify with such notable works. With Alfonso and Matteo, we have had several discussions and the conclusion was striking: Tartaglia-Pascal triangle appears in quantum mechanics rather with its square root! It appeared like quantum mechanics is not itself a classical stochastic process but the square root of it. This could explain why several excellent people could have escaped the link.

At this point, it became quite difficult to clarify the question of what a square root of a stochastic process as Brownian motion should be. There is nothing in literature and so I tried to ask to trained mathematicians to see if something in advanced research was known (see here). MathOverflow is a forum of discussion for advanced research managed by the community of mathematicians. It met a very great success and this is testified by the fact that practically all the most famous mathematicians give regular contributions to it. Posting my question resulted in a couple of favorable comments that informed me that this question was not known to have an answer. So, I spent a lot of time trying to clarify this idea using a lot of very good books that are available about stochastic processes. So, last few days I was able to get a finite answer: The square of Brownian motion is computable in a standard way with Itō integral reducing to a Brownian motion multiplied by a Bernoulli process. The striking fact is that the Bernoulli process is that of tossing a coin! The imaginary factor emerges naturally out of this mathematical procedure and now the diffusion equation is the Schroedinger equation. The identification of the Bernoulli process came out thanks to the help of Oleksandr Pavlyk after I asking this question at MathStackexchange. This forum is also for well-trained mathematicians but the kind of questions one can put there can also be at a student level. Oleksandr’s answer was instrumental for a complete understanding of what I was doing.

Finally, I decided to verify with the community of mathematicians if all this was nonsense or not and I posted again on MathStackexchange a derivation of the square root of a stochastic process (see here).  But, with my great surprise, I discovered that some concepts I used for the Itō calculus were not understandable at all. I gave them for granted but these were not defined in literature! So, after some discussions, I added important clarifications there and in my paper making clear what I was doing from a mathematical standpoint. Now, you can find all this in my article. Itō calculus must be extended to include all the ideas I was exploiting.

The link between quantum mechanics and stochastic processes is a fundamental one. The reason is that, if one get such a link, an understanding of the fundamental behavior of space-time is obtained. This appears a fluctuating entity but in an unexpected way. This entails a new reformulation of quantum mechanics with the language of stochastic processes. Given this link, any future theory of quantum gravity should recover it.

I take this chance to give publicly a great thank to all these people that helped me to reach this important understanding and that I have cited here. Also mathematicians that appeared anonymously were extremely useful to improve my work. Thank you very much, folks!

Update: After an interesting discussion here with Didier Piau and George Lowther, we reached the conclusion that the definitions I give in my paper to extend the definition of the Ito integral are not mathematically consistent. Rather, when one performs the corresponding Riemann sums one gets diverging results for the interesting values of the exponent $0<\alpha<1$ and the absolute value. Presently, I cannot see any way to get a sensible definition for this and so this paper should be considered mathematically not consistent. Of course, the idea of quantum mechanics as the square root of a stochastic process is there to stay and to be eventually verified, possibly with different approaches and better mathematics.

Further update:  I have posted a revised version of the paper with a proper definition of this generalized class of Ito integrals (see here).

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v1

Farina, A., Giompapa, S., Graziano, A., Liburdi, A., Ravanelli, M., & Zirilli, F. (2011). Tartaglia-Pascal’s triangle: a historical perspective with applications Signal, Image and Video Processing DOI: 10.1007/s11760-011-0228-6

Grabert, H., Hänggi, P., & Talkner, P. (1979). Is quantum mechanics equivalent to a classical stochastic process? Physical Review A, 19 (6), 2440-2445 DOI: 10.1103/PhysRevA.19.2440

## A box seat at OPERA

26/09/2011

While at Bari Conference (see here), the news was spreading that OPERA Collaboration, a long baseline experiment using muon neutrino beams launched by CERN by CNGS Project, detected a possible Lorentz violating effect. Initially, it started as a rumor in the comment area at Jester’s blog (see here). Then, Tommaso Dorigo provided a full account on his blog well before the Collaboration come out with its results (see here and here) so, he was kindly advised to remove his ill-timed report by his management at CERN. This post, as you can see, is now back and gives, as usual for Tommaso, a very good description of facts. Similarly, you can find good posts about at vixra and Jester’s blog. Meantime, OPERA Collaboration published its paper on arxiv (see here) and, on Friday, gave a seminar at CERN that was broadcast through all the web. Today, we know that they performed very well at the measurement and, after a struggle with the data lasting about three years, they forcefully published the result waiting for all the community to scrutinize it. Indeed, at first it is very difficult to find some drawback in this work and, being all well-trained physicists, it appears quite difficult to expect this. So, the next and more important step is just to have this result replicated or not by independent labs.

It is interesting to note that this experiment was originally conceived to observe neutrino oscillations. What they should observe are tau neutrinos arising from the muon neutrinos coming from CERN. But this has proved really difficult and, after about 16000 events, they were able to get a possible serendipitous discovery. The spokesperson of the Collaboration is Antonio Ereditato and you can find an interview here.

My first impressions about this result were a couple of important points that surely should have been emphasized: It could be the first evidence of a Lorentz-violating effect and string theory could be put in some difficulty if this result should be confirmed. Theories introducing Lorentz-violating terms have been known for years. These are generally connected to possible formulations of quantum gravity and someone claimed them incorrect just because string mainstream needed a perfect Lorentz symmetry. Besides, in the sixties of the last century, tachyons were introduced by Gerald Feinberg (see here). These are particle with an immaginary mass and so, they could never be seen at rest. But their quantum field theory has an instability in the ground state that would change their nature from superluminal to subluminal breaking symmetry. You can realize this immediately if you have in mind a Higgs field. Neutrinos are Fermions but this does not change too much such a conclusion as for these particles a formulation of spin-statistics theorem could be a mess. But even if we accept their existence, a paper today on arxiv by Giovanni Amelino-Camelia, Giulia Gubitosi, Niccoló Loret, Flavio Mercati, Giacomo Rosati and Paolo Lipari (see here) rules them out as a possible explanation for the OPERA effect. This paper and the other by Giacomo Cacciapaglia, Aldo Deandrea, Luca Panizzi (see here) show that a proper analysis should be accomplished using a modified dispersion relation between energy and momenta. This is perfectly in line with the recently proposal for a modified special relativity that has Amelino-Camelia as one of the proponents. Anyhow, as emphasized by these authors, an in-depth scrutiny of the OPERA experiment is in need as the fits seem to point toward a somewhat exotic dispersion relation even if a kind of fit can be found. On the other side, Cacciapaglia&al. seem to find a fit with non-integer exponent putting OPERA result somewhat out of the theoretical proposals of these last years.

From a string theory standpoint, it appears a rather strange situation even if it is possible to propose modified formulations accounting for the Lorentz-violation and the reason relies on the fact that, essentially, one starts from a fully-fledged quantum field theory preserving all the cherished symmetries. We just point out that what appears today in view is a world with no strings and supersymmetry not even in sight but this is a rapidly changing scenario having LHC at full steam.

Finally, this appears the first significant move toward new physics and a great one indeed arising from an important collaboration. With LHC at full power and other labs now tuned, the future appears quite exciting.

The OPERA Collaboraton: T. Adam, N. Agafonova, A. Aleksandrov, O. Altinok, P. Alvarez Sanchez, S. Aoki, A. Ariga, T. Ariga, D. Autiero, A. Badertscher, A. Ben Dhahbi, A. Bertolin, C. Bozza, T. Brugiére, F. Brunet, G. Brunetti, S. Buontempo, F. Cavanna, A. Cazes, L. Chaussard, M. Chernyavskiy, V. Chiarella, A. Chukanov, G. Colosimo, M. Crespi, N. D’Ambrosios, Y. Déclais, P. del Amo Sanchez, G. De Lellis, M. De Serio, F. Di Capua, F. Cavanna, A. Di Crescenzo, D. Di Ferdinando, N. Di Marco, S. Dmitrievsky, M. Dracos, D. Duchesneau, S. Dusini, J. Ebert, I. Eftimiopolous, O. Egorov, A. Ereditato, L. S. Esposito, J. Favier, T. Ferber, R. A. Fini, T. Fukuda, A. Garfagnini, G. Giacomelli, C. Girerd, M. Giorgini, M. Giovannozzi, J. Goldberga, C. Göllnitz, L. Goncharova, Y. Gornushkin, G. Grella, F. Griantia, E. Gschewentner, C. Guerin, A. M. Guler, C. Gustavino, K. Hamada, T. Hara, M. Hierholzer, A. Hollnagel, M. Ieva, H. Ishida, K. Ishiguro, K. Jakovcic, C. Jollet, M. Jones, F. Juget, M. Kamiscioglu, J. Kawada, S. H. Kim, M. Kimura, N. Kitagawa, B. Klicek, J. Knuesel, K. Kodama, M. Komatsu, U. Kose, I. Kreslo, C. Lazzaro, J. Lenkeit, A. Ljubicic, A. Longhin, A. Malgin, G. Mandrioli, J. Marteau, T. Matsuo, N. Mauri, A. Mazzoni, E. Medinaceli, F. Meisel, A. Meregaglia, P. Migliozzi, S. Mikado, D. Missiaen, K. Morishima, U. Moser, M. T. Muciaccia, N. Naganawa, T. Naka, M. Nakamura, T. Nakano, Y. Nakatsuka, D. Naumov, V. Nikitina, S. Ogawa, N. Okateva, A. Olchevsky, O. Palamara, A. Paoloni, B. D. Park, I. G. Park, A. Pastore, L. Patrizii, E. Pennacchio, H. Pessard, C. Pistillo, N. Polukhina, M. Pozzato, K. Pretzl, F. Pupilli, R. Rescigno, T. Roganova, H. Rokujo, G. Rosa, I. Rostovtseva, A. Rubbia, A. Russo, O. Sato, Y. Sato, A. Schembri, J. Schuler, L. Scotto Lavina, J. Serrano, A. Sheshukov, H. Shibuya, G. Shoziyoev, S. Simone, M. Sioli, C. Sirignano, G. Sirri, J. S. Song, M. Spinetti, N. Starkov, M. Stellacci, M. Stipcevic, T. Strauss, P. Strolin, S. Takahashi, M. Tenti, F. Terranova, I. Tezuka, V. Tioukov, P. Tolun, T. Tran, S. Tufanli, P. Vilain, M. Vladimirov, L. Votano, J. -L. Vuilleumier, G. Wilquet, B. Wonsak, J. Wurtz, C. S. Yoon, J. Yoshida, Y. Zaitsev, S. Zemskova, & A. Zghiche (2011). Measurement of the neutrino velocity with the OPERA detector in the CNGS
beam arXiv arXiv: 1109.4897v1

Feinberg, G. (1967). Possibility of Faster-Than-Light Particles Physical Review, 159 (5), 1089-1105 DOI: 10.1103/PhysRev.159.1089

Giovanni Amelino-Camelia, Giulia Gubitosi, Niccoló Loret, Flavio Mercati, Giacomo Rosati, & Paolo Lipari (2011). OPERA-reassessing data on the energy dependence of the speed of neutrinos arXiv arXiv: 1109.5172v1

Giacomo Cacciapaglia, Aldo Deandrea, & Luca Panizzi (2011). Superluminal neutrinos in long baseline experiments and SN1987a arXiv arXiv: 1109.4980v1

## Ashtekar and the BKL conjecture

18/02/2011

Abhay Ashtekar is a well-known Indian physicist working at Pennsylvania State University. He has produced a fundamental paper in general relativity that has been the cornerstone of all the field of research of loop quantum gravity. Beyond the possible value that loop quantum gravity may have, we will see in the future, this result of Ashtekar will stand as a fundamental contribution to general relativity. Today on arxiv he, Adam Henderson and David Sloan posted a beautiful paper where the Ashtekar’s approach is used to reformulate the Belinski-Khalatnikov-Lifshitz (BKL) conjecture.

Let me explain why this conjecture is important in general relativity. The question to be answered is the behavior of gravitational fields near singularities. About this, there exist some fundamental theorems due to Roger Penrose and Stephen Hawking. These theorems just prove that singularities are an unavoidable consequence of the Einstein equations but are not able to state the exact form of the solutions near such singularities. Vladimir Belinski, Isaak Markovich Khalatnikov and Evgeny Lifshitz put forward a conjecture that gave them the possibility to get the exact analytical behavior of the solutions of the Einstein equations near a singularity: When a gravitational field is strong enough, as near a singularity, the spatial derivatives in the Einstein equations can be safely neglected and only derivatives with respect to time should be retained. With this hypothesis, these authors were able to reduce the Einstein equations to a set of ordinary differential equations, that are generally more treatable, and to draw important conclusions about the gravitational field in these situations. As you may note, they postulated a gradient expansion in a regime of a strong perturbation!

Initially, this conjecture met with skepticism. People simply have no reason to believe to it and, apparently, there was no reason why spatial variations in a solution of a non-linear equation with a strong non-linearity should have to be neglected. I had the luck to meet Vladimir Belinski at the University of Rome “La Sapienza”. I was there to follow some courses after my Laurea and Vladimir was teaching a general relativity course that I took. The course showed the BKL approach and gravitational solitons (another great contribution of Vladimir to general relativity). Vladimir is also known to have written some parts of the second volume of the books of Landau and Lifshitz on theoretical physics. After the lesson on the BKL approach I talked to him about the fact that I was able to get their results as their approach was just the leading order of a strong coupling expansion. It was on 1992 and I had just obtained the gradient expansion for the Schroedinger equation, also known in literature as the Wigner-Kirkwood expansion, through my approach to strong coupling expansion. The publication of my proof happened just on 2006 (see here), 14 years after our colloquium.

Back to Ashtekar, Henderson and Sloan’s paper, this contribution is relevant for a couple of reasons that go beyond application to quantum gravity. Firstly, they give a short but insightful excursus on the current situation about this conjecture and how computer simulations are showing that it is right (a gradient expansion is a strong coupling expansion!). Secondly, they provide a sound formulation using Ashtekar variables of the Einstein equations that is better suited for its study. In my proof too I use a Hamiltonian formulation but through ADM formalism. These authors have in mind quantum gravity instead and so ADM formalism could not be the best for this aim. In any case, such a different approach could also reveal useful for numerical simulations.

Finally, all this matter is a strong support to my view started with my paper on 1992 on Physical Review A. Since then, I have produced a lot of work with a multitude of applications in almost all areas of physics. I hope that the current trend of confirmations of the goodness of my ideas about perturbation theory will keep on. As a researcher, it is a privilege to be part of this adventure of humankind.

Ashtekar, A. (1986). New Variables for Classical and Quantum Gravity Physical Review Letters, 57 (18), 2244-2247 DOI: 10.1103/PhysRevLett.57.2244

Abhay Ashtekar, Adam Henderson, & David Sloan (2011). A Hamiltonian Formulation of the BKL Conjecture arxiv arXiv: 1102.3474v1

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

Frasca, M. (1992). Strong-field approximation for the Schrödinger equation Physical Review A, 45 (1), 43-46 DOI: 10.1103/PhysRevA.45.43

## Ted Jacobson’s deep understanding

05/03/2009

A few weeks ago I published a post about Ted Jacobson and his deep understanding of general relativity (see here).  Jacobson proved in 1995 that Einstein equations can be derived from thermodynamic arguments as an equation of state. To get the proof, Jacobson used Raychaudhuri equation and the proportionality relation between area and entropy holding for all local acceleration horizons. This result implies that exist some fundamental quantum degrees of freedom from which Einstein equations are obtained by properly managing the corresponding partition function. To estabilish such a connection is presently not at all a trivial matter and there are a lot of people around the World trying to achieve this goal even if we lack any experimental result that could lead the way.

Today in arxiv appeared a nice paper by Ram Brustein and Merav Hadad that generalize Jacobson’s result to a wider class of gravitational theories having Einstein equations as a particular case (see here). This result appears relevant in view of the fact that a theory exploiting quantum gravity could have as a low-energy limit some kind of modified Einstein equations, containing at least coupling with matter. Anyhow, we see how vacuum of quantum field theory seems to become even more important in our understanding of behavior of space-time.

## Liouville theory in the infrared limit

05/02/2009

Today I want to report a quite interesting result that I have discussed in the comments of a preceding post of mine (see here):  2d general relativity has no confinement as a quantum field theory. 2d general relativity can be written down as

$R=-\Lambda$

being $\Lambda$ a cosmological constant. This equation is the same as the Liouville equation

$\partial_t^2\phi-\partial_x^2\phi+\Lambda e^{b\phi}=0$

and all the problem is to find the scalar function $\phi$. As you know this equation can be solved exactly. About quantum field theory for 2d gravity there is really a large body of literature due the importance of this equation. I just point out to you this paper but there is much more about.

So, if you want to study this equation in the infrared limit, you have just to take the cosmological constant going to infinity. Then, to solve this problem we have to use strong perturbation theory (or a gradient expansion) giving at the leading order the equation for the propagator

$\partial_t^2G+\Lambda e^{bG}=\delta(t)$

and this equation can be solved exactly:

$G(t)=\theta(t)\frac{2}{b}\ln\left[\sqrt{\frac{2b\epsilon}{\Lambda}}\frac{1}{\cosh(b\sqrt{\epsilon}t)}\right]+\theta(-t)\frac{2}{b}\ln\left[\sqrt{\frac{2b\epsilon}{\Lambda}}\frac{1}{\cosh(b\sqrt{\epsilon}t+\phi)}\right]$

being $\epsilon$ and $\phi$ two arbitrary constants that may depend on the spatial coordinate. This Green function solves for the propagator after we have rescaled time by$\sqrt{\epsilon}\tanh\phi$ and the $\Lambda$ constant as $\Lambda/\epsilon\tanh^2\phi$. What can we learn from it? We see that this is not a periodic function and so it cannot be expressed through a Fourier series. This implies that the quantum spectrum is not discrete and so the theory has no bound states in the infrared limit of an increasingly large cosmological constant. This is a substantial difference with respect to a quartic scalar field theory that has a discrete spectrum in the same limit producing confinement.

As shocking as this result may seem, it can be straightforwardly extended to general relativity. We know that the solution, in the gradient expansion of the Einstein equations, is the Kasner solution that is not periodic at all. The situation is made more complicate by BKL scenario. In this case we have a sequence of oscillatory epochs making an overall chaotic scenario. So, we cannot find a class of periodic solutions to build an infrared quantum field theory that in this way seems to have no bound state again in a regime of strong nonlinearities (strong gravitational fields). I should say that a more detailed analysis would be helpful here opening the possibility to have an infrared formulation of QFT for Einstein equations.

## Cramer-Rao bound and Ricci flow

04/02/2009

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, $R=\Lambda$ being $\Lambda$ 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

$\Delta_2\phi+\Lambda e^{\phi}=0$

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.

## Ricci flow as a stochastic process

30/01/2009

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.