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 $\Psi_0[\chi]\propto\operatorname{det}^{-\frac{1}{4}}\left(\Omega\right)\; e^{\frac{1}{2}\chi\cdot\Omega\cdot \chi}$?



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