Ashtekar and the BKL conjecture


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

Paper replacement


I have updated the paper with the answer to Terry Tao on arxiv (see here). No correction was needed, rather I have added a new result giving the next-to-leading order correction for the Yang-Mills field. This result is important as it shows the right approximate solution, in an expansion into the inverse of the coupling constant, for the mapping between the scalar and the Yang-Mills field. As we repeated a lot of times, Smilga’s solutions are all is needed to work out our argument as this relies on a gradient expansion. A gradient expansion at the leading order has a solution depending just on time variable. But, as this has been a reason for discussion, I have also shown to what extent my approach applies to the solution of the quartic scalar field given in the form

\phi(x) = \mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x,i)

with p^2=\mu^2\left(\lambda/2\right)^{1\over 2} with \mu an integration constant and \lambda the coupling. But I would like to emphasize that the relevance of these solutions for the Yang-Mills case was just demanded by Tao’s criticism but it is not needed for my argument to work. So, the main result of this paper is that


As it has been noted elsewhere, higher order corrections are zero in the Lorenz gauge. This result is certainly not trivial and worth to be considered in a classical analysis of Yang-Mills equations.

Finally, we note as any concern about gauge invariance is just worthless. Smilga’s solutions are exact solutions of the Yang-Mills equations. Casting doubt on them using gauge invariance arguments should be put on the same ground as casting doubt on Kasner solution of Einstein equations using general covariance reasons. Nothing worth to spend time on but a poor excuse to ignore a good work.

Quantum field theory and gradient expansion


In a preceding post (see here) I showed as a covariant gradient expansion can be accomplished maintaining Lorentz invariance during computation. Now I discuss here how to manage the corresponding generating functional

Z[j]=\int[d\phi]e^{i\int d^4x\frac{1}{2}[(\partial\phi)^2-m^2\phi^2]+i\int d^4xj\phi}.

This integral can be computed exactly, the theory being free and the integral is a Gaussian one, to give

Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}

where we have introduced the Feynman propagator \Delta(x-y). This is well-knwon matter. But now we rewrite down the above integral introducing another spatial coordinate and write down

Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2]+i\int d\tau d^4xj\phi}.

Feynman propagator solving this integral is given by


and a gradient expansion just means a series into p^2 of this propagator. From this we learn immeadiately two things:

  • When one takes p=0 we get the right spectrum of the theory: a pole at p_\tau^2=m^2.
  • When one takes p_\tau=0 and Wick-rotates one of the four spatial coordinates we recover the right Feynman propagator.

All works fine and we have kept Lorentz invariance everywhere hidden into the Euclidean part of a five-dimensional theory. Neglecting the Euclidean part gives us back the spectrum of the theory. This is the leading order of a gradient expansion.

So, the next step is to see what happens with an interaction term. I have already solved this problem here and was published by Physical Review D (see here). In this paper I did not care about Lorentz invariance as I expected it would be recovered in the end of computations as indeed happens. But here we can recover the main result of the paper keeping Lorentz invariance. One has

Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}

and if we want something not trivial we have to keep the interaction term into the leading order of our gradient expansion. So we will break the exponent as

Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]-i\int d\tau d^4x\frac{1}{2}[(\partial\phi)^2+m^2\phi^2]+i\int d\tau d^4xj\phi}

and our leading order functional is now

Z_0[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}.

This can be cast into a Gaussian form as, in the infrared limit, the one of our interest, one can use the following small time approximation

\phi(x,\tau)\approx\int d\tau' d^4y \delta^4(x-y)\Delta(\tau-\tau')j(y,\tau')

being now


that can be exactly solved giving back all the results of my paper. When the Gaussian form of the theory is obtained one can easily show that, in the infrared limit, the quartic scalar field theory is trivial as we obtain again a generating functional in the form

Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}

being now


after Wick-rotated a spatial variable and having set p_\tau=0. The spectrum is proper to a trivial theory being that of an harmonic oscillator.

I think that all this machinery does work very well and is quite robust opening up a lot of possibilities to have a look at the other side of the world.

Covariant gradient expansion


Due to the relevance of the argument, after a nice discussion with a contribution of Carl Brannen, I decided to pursue this matter further. Indeed, the only way to have a covariant formulation of a gradient expansion is adding a time variable and taking the true time variable Wick rotated. In this way, for d=1+1 wave equation you will use d=2+1 wave equation and so on. In d=3+1 you will use d=4+1 wave equation. Let me explain with some equations what I mean. I consider again d=1+1 case as


but, instead to apply a gradient expansion to it, I apply this to the equation


being \Delta_2 = \partial_{xx}+\partial_{yy}. As usual, I rescale time variable as t\rightarrow\sqrt{\lambda}t and I take a solution series


Now I will get the set of equations




and so on. Let us note that, in this case, we can introduce two new spatial variables as z=x+iy and \bar z=x-iy. These are conjugate variables as you know. So, already at the leading order I have solved my equation. Indeed, I note that

\Delta_2=\partial_z\partial_{\bar z}

and so the Laplacian has the solution f(z)+g(\bar z) being f and g arbitrary functions. In this case the gradient expansion gives immediately the exact result making its application trivial as should be. Indeed, I take t=0 in the perturbation series and put iy=t and I get


that is the exact solution. Nice, it works! This means that a quantum field theory using gradient expansion exists and it is a strong coupling expansion. This result is surely less trivial than the one obtained above.

Gradient expansions and quantum field theory


It is more than two years that I am working on quantum field theory in the strong coupling limit and I am generally very satisfied with the acceptance by the community about my views. Of course, these are new ideas and may take some time to be accepted. So, I keep on working on them trying to clarify them at best so that people can have a clear understanding of their strengths and weaknesses. One of the ways we researchers have to know how our colleagues consider our views is peer-review. This system is indeed crucial to any serious scientific endeavor and, indeed, I am proud of my achievements only when my peers agree about their value. But peer-review is also useful to my work to know what are the main objections to it. It can happen that sometime these objections are deeply wrong and may be worthwhile to discuss them at length also to have an idea on how such a prejudice arose.

We should know that when a mathematical theory enters into the description of nature, whatever mathematical method one uses to exploit it is always correct. So, natural laws in physics are described by differential equations and  whatever method you know to solve them is good provided is also mathematically legal. You should consider mathematics for physicists as a severe judge that grants no appeal. You are right or wrong depending on the correctness of your computation. But in physics there is something more and these are assumptions we start with. You can do the beautiful mathematics in the world but if you started with a wrong concept about how nature works your computations are simply rubbish.

One of the criticisms I have received on trying to get my papers published is that one cannot do a gradient expansion because this breaks Lorentz/Poincare’ invariance. This is completely wrong from a mathematical standpoint. As an exercise  you can consider the wave equation in two dimensions as


and consider the case where the spatial part is not so important. This can be easily obtained by rescaling time as t\rightarrow\sqrt{\lambda}t and taking the limit \lambda\rightarrow\infty. One gets the solution series


solving the equations




and so on. All this is perfectly legal from a mathematical standpoint and I get a true solution of the wave equation. But, as you can see, I have broken Lorentz invariance, a symmetry of this equation. So, mathematics says yes while physics seems to say no. The answer is quite simple and is known since a long time: The computation is right but Lorentz invariance is no more manifest. This is due to the fact that I have separated time and space. But if I am able to resum all the terms of the expansion series I will get the right answer


that is Lorentz invariant. So, both physics and mathematics give the same answer and is a resounding yes, it works and it works so well that we are left with a kind of strong coupling expansion.

So, what should do a smart referee with such a doubt, admitting that a smart referee does not know such mundane facts of physics and mathematics? It should realize that here one is facing a really interesting problem of physics: Could we formulate a gradient expansion in such a way to have Lorentz invariance manifest? I have not an answer yet to this question but I grant to you that is a matter I would like to publish a paper about  somewhere. This is an interesting mathematical problem as well. We know that people met a similar problem at the start of the deep understanding of QED due to Feynman, Schwinger, Tomonaga and Dyson. I think that an answer to this question would have the same scientific value.

Gradient expansions, strong perturbations and classicality


It is a common view that when in an equation appears a very large term we cannot use any perturbation approach at all. This is a quite common prejudice and forced physicists, for a lot of years, to invent exotic approaches with very few luck to unveil physics behind equations. The reason for this relies on a simple trick generally overlooked by mathematicians and physicists and here is my luck. This idea can be easily exposed for the Schroedinger equation. So, let us consider the case

(H_0+\lambda V)|\psi\rangle=i\hbar\frac{\partial|\psi\rangle}{\partial t}

with \lambda\rightarrow\infty. This is a very unlucky case both for a physicist and a mathematician as the only sure approach that come to our rescue is a computer program with all the difficulties this implies. Of course, it would be very nice if we could find a solution in the form of an asymptotic series like


but we know quite well that if we insert such a solution into the Schroedinger equation we get meaningless results. But there is a very smart trick that can get us out of this dark and can produce the required result. I have exposed this since 1992 on Physical Review A (see here) and this paper was not taken too seriously by the community so that I had time enough to be able to apply this idea to all fields of physics. The paper producing the turning point has been published on Physical Review A (thank you very much, Bernd Crasemann!). You can find it here and here. The point is that when you have a strong perturbation, an expansion is not enough. You also need a rescaling in time like \tau=\lambda t. If you do this and insert the above expansion into the original Schroedinger equation, this time you will get meaningful results: A dual Dyson series that, being now the perturbation independent of time, becomes a well-known gradient expansion: Wigner-Kirkwood series. But this series is a semiclassical one and you get the striking result that a strongly perturbed quantum system is a semiclassical system! So, if you want to change a quantum system into a classical one just perturb it strongly. This is something that happens when one does a measurement in quantum mechanics using just electromagnetic fields that are the only means we know to accomplish such a task.

This result about strong perturbations and semiclassicality has been published on a long time honored journal: Proceedings of the Royal Society A (see here and here). I am pleased of this also because of my estimation for Michael Berry, the Editor. I have met him at a Garda lake’s Conference some years ago and I have listened a beautiful talk by him about the appearance of a classical world out of the quantum conundrum. I remember he asked me how to connect to internet from the Conference site but there there was just a not so cheap machine from Telecom Italia and then my help was quite limited.

So, I just removed a prejudice and was lucky enough to give sound examples in all branches of physics. Sometime, looking in some dusty corners of physics and mathematics can be quite rewarding!

Classical scalar theory in D=1+1 and gradient expansion


As said before a pde with a large parameter has the spatial variations that are negligible. Let us see this for a very simple case. We consider the following equation

\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}-\lambda\phi^3=0

with the conditions \phi(0,t)=0, \phi(1,t)=0 and \phi(x,0)=x^2-x where the choice of a parabolic profile is arbitrary and can be changed. We also know that, if we can neglect the spatial part, the solution can be written down analytically as (see here and here):

\phi\approx (x^2-x){\rm sn}\left[(x^2-x)\sqrt{\frac{\lambda}{2}}t+x_0,i\right]

being x_0={\rm cn}^{-1}(0,i). Indeed, for \lambda = 5000 we get the following pictures

Numerical Curves - t is chosen as 0=red, 1/8=blue, 1/4=green, 0.3=yellow

Numerical Curves - t is chosen as 0=red, 1/8=blue, 1/4=green, 0.3=yellow


Analytical solution - t chosen as above

Analytical solution - t chosen as above

The agreement is excellent confirming the fact that a strong coupling expansion is a gradient expansion. So, a large perturbation entering into a differential equation can be managed much in the same way one does for a small perturbation. In the case of ode look at this post.

Physics laws and strong coupling


It is a well acquired fact that all the laws of physics are expressed through differential equations and our ability as physicists is to unveil their solutions in a way or another. Indeed, almost all these equations are really difficult to solve in a straightforward way and are very far from the exercises at undergraduate courses. During the centuries people invented several techniques to manage such equations and the most generally known is surely perturbation theory. Perturbation theory applies when a small parameter enters into the equations and a series solution is so allowed. I remember I have seen this method the first time at the third year of my “laurea” course and was Giovanni Jona-Lasinio that showed it to me and other fellow students.

Presently, we see as small perturbation theory has become so pervasive that conclusions derived just at a perturbation level are sometime believed always true. An example of this is the Landau pole or, generally, what implies the renormalization program. It is not generally stated but it is quite common the prejudice that when a large parameter enters into a differential equation we are stuck and nothing can be done than using our physical intuition or numerical computation. This is true despite the fact that the inverse of such a large parameter is indeed a small parameter and most known functions have both a small parameter and a large parameter series as well.

As I said elsewhere this is just a prejudice and I have proved it wrong in a series of papers on Physical Review A (see here, here and here). I have given an overview in a recent paper. With such a great innovation to solve differential equations at hand is really tempting to try to apply it to all fields of physics. Indeed, I have worked for a lot of years in quantum optics testing the approach in a lot of successful ways and I have also found applications to condensed matter physics appeared on Physical Review B and Physica E.

The point is quite clear. How to apply all this to partial differential equations? What is the effect of a large perturbation on such equations? Indeed, I have had this understanding under my nose since the start but I have not been so able to catch it immediately. The reason is that the result is really counterintuitive. When a physical system is strongly perturbed all the terms that imply spatial variation can be neglected. So, a strong perturbation series is a gradient expansion and the converse is true as well. I have proved it numerically in a quite easy way using two or three lines of Maple. These results can be found in my very recent paper on quantum field theory (see here and here). Other results can be found by yourself with similar simple means and are very easy to verify.

As strange as may seem this conclusion, it has obtained a striking confirmation through numerical computations in general relativity. Indeed, I have applied this method also to general relativity (see here and here). Indeed this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz or BKL conjecture on the behavior of space-time approaching a singularity. Indeed, BKL conjecture has been analyzed numerically by David Garfinkle with a very beatiful paper published on Physical Review Letters (see here and here). It is seen in a striking way how all the gradient contributions from Einstein equations become increasingly irrelevant as the singularity is approached. This is a clear proof of BKL conjecture and our approach of strong perturbations at work. Since then Prof. Garfinkle has done a lot of other very good work on general relativity (see here).

We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here).

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