The right mathematical question

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.


7 Responses to The right mathematical question

  1. Kea says:

    Gee, that was a long post on Mottle’s blog, even by his standards. Maybe his string theory friends should tell him that he’s not always right.

    • mfrasca says:

      Hi Kea,

      Really embarrassing, isn’t it? It is almost twenty years that I get published papers on perturbation theory on refereed journals, having widely extended its range of applicability, and this guy did not understand anything. To avoid to be a fool of himself, it would have been enough for him to read some papers of mine. But I think it is to ask too much to such a bright guy.


  2. Ervin Goldfain says:

    Hi Marco,

    Nice posting and debate. I think it would be beneficial if you would write down a summary of these ideas in a format and at a level that is accessible to a larger audience. This way your viewpoint (as well as opposite ones such as Lubos’) would become more transparent to anyone visiting your blog. Keeping readers properly informed on the triumphs and limitations of perturbation theory in Higgs physics or strongly coupled QFT is a well deserved effort.
    I like Tommaso’s blog because of his patience in explaining abstract topics in a crystal clear and highly accessible fashion.



  3. anonymous says:

    just a small comment:

    It’s “expansion”, not “development.” As in expansion parameter, series expansion, and so on.

    Anyways, formally there doesn’t seem to be a problem with replacing lambda with 1/lambda as expansion parameter. It only gets complicated once you realize that typically V is a much more complicated function of \phi than H or \partial_t.
    So while an expansion in \lambda will usually also get rid of nonlinearities in leading order, an expansion in 1/lambda won’t do that in most cases, so you still end up with a much more complicated problem to solve.

    Well, at least there’s not much work that uses a 1/alpha_s expansion for QCD, so there must be some problems with this approach.


    • mfrasca says:

      Here we have a comment from an anonymous coming from Czech Republic. It is the same place where that confused guy writes from.

      Anyhow, coming to technical questions, I should say that referees commented like that more than ten years ago on my papers. Indeed, it holds a duality principle in perturbation theory:

      This paper appeared on Physical Review A on 1998 and it was the starting point for all my work after that date. Then, it is also quite obvious that a strong coupled quantum system is a classical system. This was already proved by Barry Simon and I get a paper published on this. Then, for partial differential equations one uncovers that the strong coupling expansion is just a gradient expansion. This makes equations in a strong coupling regime not so difficult to solve. For the Schroedinger equation this is just the well-known Wigner-Kirkwood expansion that I proved to have eigenvalues given by the same formula as WKB. Again semiclassicality in the strong coupling regime!

      What does it happen in quantum field theory? As you can see above one has all the ingredients to build a quantum field theory in a strong coupling regime. I did this for the scalar theory in 2006 and was published on Physical Review D

      The result is that expected: the theory has the coupling going to zero with energy and the same spectrum of a harmonic oscillator: It is trivial as also happened in dimensions higher than four and in agreement with clues coming from lattice computations.

      Indeed, if you take a look at the exact solutions of these classical scalar field equations you can understand that the appearance of triviality does not mean that there are not non-linear interactions but just that this non-linearities produce the same spectrum of a free theory:

      This is a small excursus of my work in the last ten years that proves that both expectations of people doing research on the lattice are correct and that a strong coupling expansion is always possible for differential equations.



  4. anonymous says:

    No, I’m not from Czech Republic.

    anyways, thanks for your reply, though it’s a bit too technical for me. If we stick to your blog post, what would be a typical V? Something like \psi^2?


    • mfrasca says:

      Sorry, this was your IP: that is University of Zurich.

      Indeed, I wrote that equation with the Schroedinger equation in mind. The technique is quite general and you can use it wherever you need. Just to have an idea, you can look to my old post here with some figures obtained with numerical computations:

      but I think you can work out ton of examples by yourself through Mathematica and a good PC.



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