Interpretation of the fermionic path integral

Question

The bosonic path integral computes transition amplitudes. E.g. for a scalar field $\phi$, the amplitude between state $|\phi_1\rangle$ on Cauchy surface $\Sigma_1$ and $|\phi_2\rangle$ on $\Sigma_2$ is given by \begin{equation} \langle \phi_2|U_{\Sigma_1\to\Sigma_2}|\phi_1\rangle=\int_{\phi|_{\Sigma_1}=\phi_1}^{\phi|_{\Sigma_2}=\phi_2}D\phi e^{iS[\phi]}.\tag{1} \end{equation} (I'm writing $U_{\Sigma_1\to\Sigma_2}$ for the unitary evolution between the Cauchy surfaces, and $S$ for the action).

I'd like to know whether the fermionic path integral admits a similar interpretation. More precisely, if $\psi_1, \psi_2$ are Grassman-valued fields on $\Sigma_1,\Sigma_2$ (resp.), let us define: \begin{equation} Z[\psi_1,\psi_2]\equiv \int_{\psi|_{\Sigma_1}=\psi_1}^{\psi|_{\Sigma_2}=\psi_2}D\psi D\bar{\psi}e^{iS[\psi,\bar{\psi}]},\tag{2} \end{equation} where the path integral is understood to be fermionic.

What is the meaning of $Z[\psi_1,\psi_2]$? It's not clear how it can be a transition amplitude, since $\psi_1$ and $\psi_2$ don't seem to label states in the Hilbert space in any obvious way. (Compare this with the scalar case, where $\phi_1$ and $\phi_2$ label corresponding field eigenstates). But perhaps there is some nice way to associate $\psi_1$ and $\psi_2$ with states in Hilbert space, in such a way that $Z[\psi_1,\psi_2]$ gives the amplitude between the associated states.

If there is no interpretation of $Z[\psi_1,\psi_2]$ as a transition amplitude, then my question becomes: what is the reason for introducing such path integrals at all?

Answer 1

OP provides no references, so the context of OP's eq. (2) is not completely clear, but let us make the following general warning about supernumbers:

1. An observable/measurable quantity can only consist of ordinary numbers (belonging to $\mathbb{C}$). It does not make sense to measure a soul-valued output in an actual physical experiment, cf. e.g. this related Phys.SE post.

2. The soul-part of a supernumber [and in particular a Grassmann-odd variable like $\psi_1$ and $\psi_2$ in OP's eq. (2)] is an indeterminate/a placeholder/has no value. So e.g. (the absolute square of) OP's eq. (2) has no meaning as a (relative) probability as it stands.

3. The only way to achieve a measurable quantity from a Grassmann-odd variable is to integrate it out, cf. e.g. this related Phys.SE post.

4. In other words, OP's fermionic construction (2) should eventually include the Berezin integrations $\int\!\mathrm{d}\psi_1\int\!\mathrm{d}\psi_2$ in order to produce a physically measurable quantity, like an overlap or a probability.