I am trying to calculate
$$Z = \int\limits_{\phi(\beta) = \phi(0) =0} D \phi\ e^{-\frac{1}{2} \int_0^{\beta} d\tau \dot{\phi}^2}$$
without transforming it to the Matsubara frequency space, I can show that $Z = \sqrt{\frac{1}{2\pi \beta}}$. However, I have a problem in obtaining the same result in the Matsubara frequency space:
\begin{equation}
\phi (\tau) = \frac{1}{\sqrt{\beta}} \left( \sum_{n} \phi_n \ e^{i\omega_n\tau} \right),
\end{equation}
with $\sum_n \phi_n =0, \omega_n = \frac{2\pi n}{\beta}$. And
\begin{equation}
Z = \int \prod_n D\phi_n\ \delta\left(\sum_n \phi_n\right)\ e^{-\frac{1}{2} \sum_n \phi_n \phi_{-n} \omega_n^2 }
\end{equation}
which, I think, vanishes.

I guess the problem lies in the measure. Any comments?

This post imported from StackExchange Physics at 2014-07-11 17:23 (UCT), posted by SE-user L. Su