I am new to path integral and struggling with the computation involving Gaussian functional integrals. Could anyone show me the steps of computing the following integral?(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int D \phi e^{-S},[/tex]

where

[tex] S = \int dx~d \tau [(\frac{\partial \phi}{\partial x})^2+2 i \frac{\partial \theta}{\partial \tau} \frac{\partial \phi}{\partial x} ][/tex].

[itex] \theta [/itex] is a function that is not being integrated over.

I know I am supposed to use the formula

[tex] \int D v(x) \mathrm{exp}[-\frac{1}{2} \int d x d x'~v(x) A(x,x') v(x') + \int d x~j(x) v(x)] \propto \mathrm{exp} [\frac{1}{2} \int d x d x'~j(x) A^{-1}(x,x') j(x')],[/tex]

where [itex] A(x,x') [/itex] is an operator. But I am having trouble in applying this result.

Thanks in advance.

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# Gaussian Functional Integral

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