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Mathematical foundation of quantum field theory

  1. Dec 27, 2003 #1

    I have question regarding the mathematica foundations of QFT. As I understand, the "regular" QM (Schrödinger, Heisenberg...) been developped so that the math underlying it checks out. Is this the case for QFT, or is the theory still "iffy" at points? I know it works well experimentally, but are the theories in itself consistent and well-known?

    I would really like to know if current research in theoretical physics focusses mainly on quantum gravity, or if there are still alot of people completing QFT.

  2. jcsd
  3. Dec 28, 2003 #2
    still iffy.

    it is not known whether the theories are consistent. anyone who shows that they are (or even just makes significant progress in this area) is in for a million bucks from clay math.

    certainly there are people working on the mathematical foundations of QFT. i just think those people are mathematicians, not physicists.
  4. Dec 28, 2003 #3
    The "iffy" points are primarily hand-waving at infinities, or renormalization, as I understand.
  5. Dec 28, 2003 #4
    Strange that you would say that only mathematicians are working on it. It seems to me that this would be an interesting topic for theoretical, or at least mathematical physicists.

    Besides the renormalizations, are there other major inconsistencies?
  6. Dec 28, 2003 #5


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    Not at all; this is a popular misconception. Regularization and renormalization are not the problem, perturbative expansion is. And also the handling of interacting fields (Haag's theorem).

    There are ways to get around Haag's theorem but the results as to the definitions of particles and fields are pretty iffy themselves: you can have fairly well-defined particles in the distant past or in the distant future, but not, or not exactly, in the interaction itself.

    The perturbative expansion problem is that the series may not converge. There is some (shaky) evidence that it doesn't; this goes by the name "Landau Pole".
  7. Dec 28, 2003 #6


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    Because of the modern view of QFTs as approximations at lower energies of an as yet unknown or unproven "correct" theory which probably isn't a QFT, together with what we've learned from renormalization group ideas about the relation between a theory's behaviour at different energy scales (namely, that the behaviour of a system at lower energies doesn't depend on it's behaviour at higher energies. Unfortunately this also means that inferences can't be safely drawn about the behaviour of a system at high energies from it's behaviour at lower energies), questions about the ultimate status of QFT as a basis for physical theories don't seem as relevant as they did as late as 30 years ago, and whatever residual concern remains about such issues certainly isn't driving mainstream research in high energy theory.
    Last edited: Dec 29, 2003
  8. Dec 28, 2003 #7
  9. Dec 28, 2003 #8

    Tom Mattson

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