Evaluating Functional integrals

[Klaus_Hoffmann]

HOw can you compute a Gaussian functional integral?

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

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

$$\frac{c}{(Det(a\partial +b)}$$ a,b,c constant

so $$Det(a\partial +b)}= exp^{-\zeta '(0)}$$

$$\zeta (s) = \sum_{1 \le n}\lambda_{n} ^{-s}$$

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

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

so you get finite results ??

 

[f-h]

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.

 

[f-h]

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.

 

[Klaus_Hoffmann]

thank you everybody.. f-h hit the correct answer (thankx) however still we have the problem that how could you evaluate the functional Determinant.

$$Det( a\partial ^{2} +b)$$ of the operator:

$$( a\partial ^{2} +b)(\phi +J(x))$$