# What's the momentum-space vacuum wave-functional of a fermion?

In the Schrödinger picture, the field eigenstates of a real scalar field $$\hat\phi(\mathbf x)$$ with $$\mathbf x \in\mathbb R^3$$ are the states $$\hat\phi(\mathbf x)|\phi\rangle=\phi(\mathbf x)|\phi\rangle$$, with field configurations $$\phi$$ in the space $$\mathbb R^3\to\mathbb R\;$$. They define the wave-functional $$\Psi[\phi]=\langle\phi|\Psi\rangle$$, and express the free field vacuum wave-functional $$\Psi_0[\phi]=\langle\phi|\Psi_0\rangle$$ (derived from the momentum-space eigenstates $$\tilde\phi(\mathbf k)$$) as (Jackiw 89)

$$\begin{array}{cl} \Psi_0[\phi] &= C\prod_k e^{-\omega(\mathbf k)\frac{\tilde\phi(\mathbf k)^2}{2}\epsilon^3} \\ &\to C e^{-\frac{1}{2}\int\frac{d^3k}{(2\pi)^3} \omega_{\mathbf k}|\tilde\phi(\mathbf k)|^2} \\ &= \operatorname{det}^{\frac{1}{4}}\left(\frac{K}{\pi}\right)\; e^{-\frac{1}{2}\int d\mathbf{x} \int d\mathbf{y}\, \phi(\mathbf{x}) K(\mathbf{x},\mathbf{y}) \phi(\mathbf{y}) } = \operatorname{det}^{\frac{1}{4}}\left(\frac{K}{\pi}\right)\; e^{-\frac{1}{2}\phi\cdot K\cdot\phi}.\\ \end{array}$$

Jackiw (and Symanzik 81, and Hatfield 92) also discuss the wave-functional $$\Psi[\chi]=\langle\chi|\Psi\rangle$$ of a fermionic field $$\hat\chi(\mathbf x)$$ with field eigenstates $$\hat\chi(\mathbf x)|\chi\rangle = \chi(\mathbf x)|\chi\rangle$$, and derives its vacuum state $$\Psi_0[\chi]\propto\operatorname{det}^{-\frac{1}{4}}\left(\Omega\right)\; e^{\frac{1}{2}\chi\cdot\Omega\cdot \chi}$$. I'm still not sure what space $$\chi:\mathbb R^3\to\mathrm (\,\_\,)\,$$ the field configuration $$\chi(\mathbf x)$$ is in, but

Question: What is the momentum-space representation $$\Psi_0[\tilde{\chi}]$$ of the vacuum state fermionic wave-functional $$\Psi_0[\chi]\propto\operatorname{det}^{-\frac{1}{4}}\left(\Omega\right)\; e^{\frac{1}{2}\chi\cdot\Omega\cdot \chi}$$?