Computing path integral with real and Grassmann variables

AI Thread Summary
The discussion focuses on computing path integrals involving real and Grassmann variables, specifically the integral representation of the partition function Z[w]. The initial computation of the integrals for θ and θ̅ leads to a determinant expression, but challenges arise in applying this determinant in further calculations. Alternative approaches discussed include expressing the determinant in terms of partial derivatives and considering the use of the trace logarithm method. There is uncertainty about the feasibility of performing the x integration exactly for Z[w], suggesting that obtaining an explicit value may be difficult. Clarification is sought regarding the specific goals of the computation and the explicit forms of the functions w_i(x).
Geigercounter
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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 $$\det = \exp(\text{Tr} \ln) $$
 
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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?
 
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