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Evaluating Functional integrals

  1. Aug 11, 2007 #1
    HOw can you compute a Gaussian functional integral?

    i mean integral of the type [tex] e^{-iS_{0}[\phi]+i(J,\phi)} [/tex]

    if J=0 then i believe that we can describe the Functional integral as

    [tex] \frac{c}{(Det(a\partial +b)} [/tex] a,b,c constant

    so [tex] Det(a\partial +b)}= exp^{-\zeta '(0)} [/tex]

    [tex] \zeta (s) = \sum_{1 \le n}\lambda_{n} ^{-s} [/tex]

    my problem comes when J(x) is different from 0 so we have a functional determinant which is also a function (functional ?? ) of J(x) , then how could yo evaluate it?, also a Fourier transform of a gaussian is again another Gaussian.

    Another question..how can you once you have obtained the functional integral Z[J] expressions of the form

    [tex] \frac{ \delta ^{n}Z[J]}{ \delta J(x1) \delta J(x2)........} [/tex]

    so you get finite results ??
    Last edited: Aug 11, 2007
  2. jcsd
  3. Aug 11, 2007 #2


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    You can't except perturbatively. The perturbation expansion is explained in any good book on QFT, try Zee's QFT in a Nutshell for example.
  4. Aug 11, 2007 #3


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    Sorry I see I misread, the trick is to complete the square. You add J^2 - J^2 write the whole thing as (phi + J)^2 and pull the J^2 term out of the integral.
  5. Aug 11, 2007 #4
    thank you everybody.. f-h hit the correct answer (thankx) however still we have the problem that how could you evaluate the functional Determinant.

    [tex] Det( a\partial ^{2} +b) [/tex] of the operator:

    [tex] ( a\partial ^{2} +b)(\phi +J(x))[/tex]
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