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Thread: Gaussian integrals in QFT

  1. #1 Gaussian integrals in QFT 
    . DrRocket's Avatar
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    This is a question for salsaonline.

    In Zee's book Quantum Field Theory in a Nutshell he wastes no time in talking about Gaussian integrals (pp 13-14).

    It is pretty trivial to show that .

    To integrate a general quadratic polynomial by completing the square is not much more difficult, and then one gets.

    .

    This all makes perfect sense, since the integrand goes to zero rather rapidly.

    But then he manages to conclude by simply replacing by that

    .

    Now, this replacement is clearly not legitimate. I can get this same expression using a contour integral if, letting we evaluate the "improper integral"

    .

    However the integrand is clearly not absolutely integrable (the modulus being 1), and so, given any complex number z whatsoever, there is a measure-preserving transformation of the line such that




    This brings up the question as to why one should accept that



    from a physical point of view since the value of the integral depends not only on the values of the integrand but more importantly on the manner in which the contributions to "area" are added up.

    Is this perhaps justified by some later step in which he is only going to evaluate such integrals after invoking some sort of regularization in the name of an "ultraviolet cutoff" "? If so, has this been checked to make sure that approximations on top of approximations are not hiding something ?

    This is starting to remind me of a problem that a physics friend was doing in college. The problem was to show that some function grew without bound near a given point. I looked at the function and immediately produced an upper bound. Undeterred the physicist did a couple of standard physics approximations and concluded that, yep, it went to infinity there. I could only conclude that .


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  3. #2  
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    Zee is trying to motivate the path integral formalism. Path integrals typically take the form



    where Q is some quadratic form (or a quadratic form plus a linear source term like J).

    For a more convincing discussion by Witten of the Gaussian integrals Zee is talking about, see "Quantum Fields and Strings: A Course for Mathematicians", volume 1, page 671. (solution to problem FP11)

    Keep in mind that what we're doing is looking for a convincing *definition* of these Gaussian integrals that we can carry over to the infinite-dimensional situation.

    Ultimately the path integral formalism is just that--a formalism. There is no rigorous definition for path integrals in general. In some sense, the path integral is the place where math ends and physics begins.


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  4. #3  
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    Quote Originally Posted by salsaonline
    Zee is trying to motivate the path integral formalism. Path integrals typically take the form



    where Q is some quadratic form (or a quadratic form plus a linear source term like J).

    For a more convincing discussion by Witten of the Gaussian integrals Zee is talking about, see "Quantum Fields and Strings: A Course for Mathematicians", volume 1, page 671. (solution to problem FP11)

    Keep in mind that what we're doing is looking for a convincing *definition* of these Gaussian integrals that we can carry over to the infinite-dimensional situation.

    Ultimately the path integral formalism is just that--a formalism. There is no rigorous definition for path integrals in general. In some sense, the path integral is the place where math ends and physics begins.
    Thanks. That makes sense.

    I don't have any great problem with the use of formalisms by the physicists, given that nobody knows how to do Feynman path integrals rigorously and given that they seem to produce correct physical answers, at least in most cases. But it would be nice if they clearly announced when they are just waving their arms. Such notices would alert the reader than one is entering uncharted territory, as opposed to giving the impression that the author has lost his mind. I think they need to take a cue from the old map makers and clearly note on the chart that "Here there be dragons".

    Witten clearly knows better, but I get the distinct impression that a large number of the theoretical physicists don't really know, or care, when they are doing things that are rigorously jutifiable and when they are on shaky ground. It is not as though QFTs are always accurate. They seem to overpredict the zero point energy of the vacuum, which they can interpret as a cosmological constant or dark energy, by a factor of . That is considered a significant error in some circles.
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by salsaonline
    Zee is trying to motivate the path integral formalism. Path integrals typically take the form



    where Q is some quadratic form (or a quadratic form plus a linear source term like J).

    For a more convincing discussion by Witten of the Gaussian integrals Zee is talking about, see "Quantum Fields and Strings: A Course for Mathematicians", volume 1, page 671. (solution to problem FP11)

    Keep in mind that what we're doing is looking for a convincing *definition* of these Gaussian integrals that we can carry over to the infinite-dimensional situation.

    Ultimately the path integral formalism is just that--a formalism. There is no rigorous definition for path integrals in general. In some sense, the path integral is the place where math ends and physics begins.
    Thanks. That makes sense.

    I don't have any great problem with the use of formalisms by the physicists, given that nobody knows how to do Feynman path integrals rigorously and given that they seem to produce correct physical answers, at least in most cases. But it would be nice if they clearly announced when they are just waving their arms. Such notices would alert the reader than one is entering uncharted territory, as opposed to giving the impression that the author has lost his mind. I think they need to take a cue from the old map makers and clearly note on the chart that "Here there be dragons".

    Witten clearly knows better, but I get the distinct impression that a large number of the theoretical physicists don't really know, or care, when they are doing things that are rigorously jutifiable and when they are on shaky ground. It is not as though QFTs are always accurate. They seem to overpredict the zero point energy of the vacuum, which they can interpret as a cosmological constant or dark energy, by a factor of . That is considered a significant error in some circles.
    Regarding this cosmological constant prediction: Is that a problem with QFT per se, or just with string theory? For example, I don't know of any predictive flaws in QED, but let me know if there are any.

    Regarding your comment about theoretical physicists: I think, as you probably suspect, many of them simply don't know the math well enough to know when they're being rigorous and when they're not. Zee himself unwittingly reveals his lack of background in math when he says things like "dx is not really d applied to a function". Locally, it certainly is--it's d applied to the coordinate variable x. Somewhere else in the book he writes something like



    whatever that means. Given the context, it sounded like he wanted to talk about



    A big difference. I may be misrepresenting the exact details here, but my point is that I don't think Zee has a lot of formal mathematical training.
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  6. #5  
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    Quote Originally Posted by salsaonline

    Regarding this cosmological constant prediction: Is that a problem with QFT per se, or just with string theory? For example, I don't know of any predictive flaws in QED, but let me know if there are any.
    As I understand it the problem is in the value predicted by QED for the zero point energy.

    Here is an article that discusses it a bit, and also makes reference to some work that might resolve the problem.

    http://www.calphysics.org/zpe.html
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    interesting, thanks.
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