# Path integral over probability functional

1. ### Jezuz

31
Hi. Can anyone tell me how to solve the path integral

$$\int D F \exp \left\{ - \frac{1}{2} \int_{t'}^{t} d \tau \int_{t'}^{\tau} ds F(\tau) A^{-1}(\tau - s) F(s) + i \int_{t'}^{t} d\tau F(\tau) \xi(\tau) \right\}$$

In case my Latex doesn't work the integral is over all possible forces F over the functional

\exp \left\{ - \frac{1}{2} \int_{ t' } ^{ t } d \tau \int_{ t' } ^{ \tau } ds F( \tau ) A^{-1} ( \tau - s ) F( \tau) + i \int_{ t' } ^{t} d \tau F( \tau ) \xi ( \tau ) \right\}

I have tried to solve it by making the discrete Fourier transform of the functions F, A^{-1} and \xi but I run into some trouble when doing that.

/Jezuz

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