# Gaussian Functional Integral

1. Nov 26, 2014

### psi*psi

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?
$$\int D \phi e^{-S},$$
where
$$S = \int dx~d \tau [(\frac{\partial \phi}{\partial x})^2+2 i \frac{\partial \theta}{\partial \tau} \frac{\partial \phi}{\partial x} ]$$.
$\theta$ is a function that is not being integrated over.
I know I am supposed to use the formula
$$\int D v(x) \mathrm{exp}[-\frac{1}{2} \int d x d x'~v(x) A(x,x') v(x') + \int d x~j(x) v(x)] \propto \mathrm{exp} [\frac{1}{2} \int d x d x'~j(x) A^{-1}(x,x') j(x')],$$
where $A(x,x')$ is an operator. But I am having trouble in applying this result.