Computing path integral with real and Grassmann variables

[Geigercounter]

Homework Statement

I want to compute the following path integral
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \prod_{i=1}^{n}d\overline{\theta}_id\theta \: \exp{\left(-\overline{\theta}_i \partial_j w_i(x)\theta_j -\frac{1}{2}w_i(x)w_i(x)\right)}.$$ Here $w_i(x)$ are functions of the $n$ real variables $x_i$ and $\theta_i$ and $\overline{\theta}_i$ are $n$ independent Grassmann variables.

Relevant Equations

See below.

The first step seems easy: computation of the $\theta$ and $\overline{\theta}$ integrals give
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \det(\partial_j w_i(x)) \exp{\left(-\frac{1}{2}w_i(x)w_i(x)\right)}.$$

From here, I tried using that $$\det(\partial_j w_i (x)) = \det\left(\partial_j w_i \left(\frac{d}{db}\right)\right) \exp\left(b_i x_i\right)\bigg\vert_{b=0}.$$ But I don't seem to be able to apply this step.

Other ideas I had included writing out the determinant as $$det(\partial_j w_i(x)) = \frac{1}{n!}\varepsilon_{i_1...i_n}\varepsilon_{j_1...j_n} \partial_{j_1} w_{i_1}(x) ... \partial_{j_n} w_{i_n}(x)$$ to then use some kind of partial integration.
Another, similar, idea was to use the fact that $$\det = \exp(\text{Tr} \ln) $$

 

[julian]

Performing the $x$ integration exactly for $Z[w]$ could very well not be possible. You may very well not be able to obtain the exact value even if you were just considering this integral:

\begin{align*}
\frac{1}{(2 \pi)^{n/2}} \int d^nx \exp \left( - \frac{1}{2} w_i (x) w_i (x) \right)
\end{align*}

Could you clarify what it is you are aiming to achieve exactly? Do we know what the functions $w_i (x)$ are explicitly? Are you wanting to put the integral into some nice form rather than explicitly evaluating it? Could you show us the source of the question?