The right mathematical question

01/08/2009

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

\partial_t\psi=(H+\lambda V)\psi.

The exact solution of this equation is the function \psi(t;\lambda). So, one can ask : What happens when \lambda 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 \psi(t;\lambda) when \lambda 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 1/\lambda. 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.



Mathematica and KAM tori reforming

28/06/2009

Some days ago I received an email from Wolfram Research asking to me to produce a demonstration for their demonstrations project based on my last proved theorem about KAM tori reforming (see here). WolframDemosBeing aware of the power of Mathematica I have found the invitation quite stimulating.  The idea behind these demonstrations is to use Mathematica’s command Manipulate that permits to have interactive presentations. A typical application is exactly in the area of differential equations where you can have some varying parameters. But the possibilities are huge for this method and, indeed, you can find almost 5000 demonstrations at that site. Indeed, in this way you are able to explore the behavior of mathematical models interactively and this appears as a really helpful tool. If you mean to send a demo of yours, be advised that it will undergo peer-review. So, such a publication has exactly the same value of other academic titles.

In a few days I prepared the demo and I have sent it to Wolfram. It was accepted for publication last friday. You can find it here. You can check by yourself the truthfulness of my theorem. The advantage to work in classical mechanics is that you can have immediately an idea of what is going on by numerics. Manipulate of Mathematica is a powerful tool in your hands to accomplish such an aim.

Finally, you can download the source code and modify it by yourself changing ranges, equations and so on. I tried the original Duffing oscillator without dissipation and, granted the validity of KAM theorem, one can verify an identical behavior with tori reforming for a very large perturbation. A shocking evidence without experiments, isn’t it?

Update: My demonstration has been updated (see here). The code has been improved, and so the presentation, due to a PhD student, Simon Tyler, that did this work. Thank you very much, Simon!


Quantum field theory and gradient expansion

21/02/2009

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

\Delta(p)=\frac{1}{p_\tau^2-p^2-m^2+i\epsilon}

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

\partial_\tau^2\Delta(\tau)+\lambda\Delta(\tau)^3=\delta(\tau)

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

\Delta(p)=\sum_n\frac{A_n}{p^2-m^2_n+i\epsilon}

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.


Gradient expansions and quantum field theory

01/02/2009

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

\partial^2_{tt}u-\partial^2_{xx}u=0

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

u=u_0+\frac{1}{\lambda}u_1+\frac{1}{\lambda^2}u_2+\ldots

solving the equations

\partial^2_{tt}u_0=0

\partial^2_{tt}u_1=\partial^2_{xx}u_0

\partial^2_{tt}u_2=\partial^2_{xx}u_1

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

u=f_1(x-t)+f_2(x+t)

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

24/11/2008

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

|\psi\rangle=|\psi_0\rangle+\frac{1}{\lambda}|\psi_1\rangle+\frac{1}{\lambda^2}|\psi_2\rangle+\ldots

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!


Physics laws and strong coupling

28/09/2008

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).


A stunning summer

03/09/2008

It is customary for me, during my holidays, to read some technical books that I bear with me at Satriano, a small town very near Soverato. This summer I have read the book of Stephan Narison on QCD as QCD is currently my work and this is a really beautiful book. But I have written papers on almost all fields of physics and in the summer of 2005 I was struggling with general relativity. That year I took with me a wonderful book by James Hartle: “Gravity: An Introduction to Einstein’s General Relativity” . This book has been a source of inspiration for my work. Indeed, in a few days I was trying to apply my well developed approach for strong perturbations to Schwarzschild solution. This was a blind alley as I was manipulating an exact solution that has no much to say for perturbation techniques. As I looked at the problem more in depth, I was able to make this application to general relativity in a paper published on International Journal of Modern Physics D (see here for a preprint). I have to thank the editor Jorge Pullin for appreciating my work. I met him in Piombino at DICE 2006 Conference where he gave a very nice talk.

This paper has been a fundamental starting point for all my succesive work for two main reasons: Firstly, I have understood that to solve strongly perturbed partial differential equations one has to recur to a gradient expansion and secondly, this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz (BKL) conjecture that permits to understand why near a singularity the space-time is homogeneous. Vladimir Belinski has been a professor of mine at University “La Sapienza” of Rome in 1992. He taught me general relativity with the more exotic solutions: BKL and gravisolitons. He is also well-known for having written down parts of Landau-Lifshitz book about this matter. I remember a nice talk with him in those days about BKL and for this reason I acknowledged him in my paper. Indeed, after speaking with him I should have been able to catch immediately the answer but times were not so mature and I have had to do a lot of work and thinking before to arrive at this successful point.

After understanding how to do strong perturbation theory to partial differential equations, my  next step was to find a way to generalize it to quantum field theory. In a few days I was able to do it and this became a paper for Physical Review D (see here for a preprint). In this paper I improve on the Bender’s approach (see here) obtaining the propagator and the spectrum for a massless scalar theory showing that in the infrared this theory acquires a mass gap. A real successful improvement!

After that stunning summer, I have been able to extend all this to Yang-Mills and QCD but we are talking about today…


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