# Completeness relation and scalar product for Grassmann coherent states

I was wondering, given two fermionic canonical operators: $$\{\hat{\chi}_{\alpha a}(x),\hat{\bar{\chi}}_{\beta b}(y)\} = \delta_{\alpha\beta}\delta_{ab}\delta^{(3)}(x-y)$$ they should have as eigenstates the Grassmann coherent states which give rise to the usual completness relation that maps a Fock space into a Grassmann algebra: $$\mathbb{I}_{Fock} = \int \mathcal{D}[\chi]\mathcal{D}[\bar{\chi}]\ e^{-\bar{\chi}\cdot \chi}\ \rvert \bar{\chi},\chi \rangle\langle \bar{\chi},\chi \rvert$$ I was wondering, is it possible to define in the same Hilbert space a scalar product like: $$\langle \Phi\rvert \Psi \rangle = \int \mathcal{D}[\chi]\mathcal{D}[\bar{\chi}]\ e^{-\bar{\chi}A\cdot \chi}\ \Phi^*(\bar{\chi},\chi) \Psi(\bar{\chi},\chi)$$ where $A$ is a positive definite matrix? Or should I need to redefine the $\bar{\chi},\chi$ operators and their coherent states?