In the Schrödinger picture of quantum field theory, 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 $\mathbb R^3\to\mathbb R\;$.
This field eigenbasis defines the wave-functional $\Psi[\phi]=\langle\phi|\Psi\rangle$, and expresses the vacuum state, free field wave-functional $\Psi_0[\phi]=\langle\phi|\Psi_0\rangle$ 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 discusses the wave-functional $\Psi[\chi]=\langle\chi|\Psi\rangle$ of a fermionic field $\hat\chi(\mathbf x)$ with field eigenstates $\hat\chi(\mathbf x)|\chi\rangle = \chi(\mathbf x)|\chi\rangle$, but in less detail. (Though he does derive a form of the fermionic vacuum state wave-functional, $\Psi_0[\chi]\propto\operatorname{det}^{-\frac{1}{4}}\left(\Omega\right)\; e^{\frac{1}{2}\chi\cdot\Omega\cdot \chi}$.)
Question: In what space are the field configurations $\chi(\mathbf x)$?
In other words, what fills in $\chi:\mathbb R^3\to\mathrm (\,\_\,)\,$? (For both spin-½ and spin-3/2, and for the Majorana and Dirac cases?)
Assumptions:
Ignore UV/IR issues and assume Minkowski space is the well-defined limit of the cyclic lattice $\mathbf x\equiv a\mathbf n, \mathbf n\in \mathbb{Z}^4_N$ as $(a,N)\to(0,\infty)$.
Assume the Grassmann algebra over vector space V with basis $\{\mathbf{e}_i\}$ is denoted $\Lambda\,\mathrm{V}$, with generating elements $\{\theta_{\mathbf{e}_i}\}$.
for example, the dual numbers are $\Lambda\,\mathbb R$, with generator $\theta_1$ and general element $z=c_0+c_1\theta_1$
the infinite-dimensional Grassmann algebra over the complex free vector space of $\mathbb R^3$ is $\Lambda\,\mathbb{C}^{\mathbb R^3}$, with generators $\theta_{\mathbf x}|\mathbf x\in\mathbb R^3$ and a general element $z=\sum_{k=0}^\infty \sum_{\{\mathbf{x}_1,\cdots,\mathbf{x}_k\}\,\subset\, \mathbb R^3} \frac{1}{k!}c_{\mathbf{x}_1\cdots\mathbf{x}_k} \theta_{\mathbf{x}_1}\cdots\theta_{\mathbf{x}_k}$. Example functions $f:\mathbb R^3\to\Lambda\,\mathbb{C}^{\mathbb R^3}$ are $f(\mathbf{x})=e^{i\mathbf{k}\cdot\mathbf{x}}\theta_{\mathbf{x}}$ and $f(\mathbf{x})=e^{i\mathbf{k}\cdot\mathbf{x}}\theta_{\mathbf{k}}$ for some $\mathbf{k}\in\mathbb{R}^3$.
two generators per point in $\mathbb R^3$ would be $\Lambda\,\mathbb{C}^{2\mathbb R^3}$, with generating elements $\theta_{\mathbf x,a}|\mathbf x\in\mathbb R^3,a\in\{0,1\}$. An example function $f:\mathbb R^3\to\Lambda\,\mathbb{C}^{2\mathbb R^3}$ is $f(\mathbf{x})=e^{i\mathbf{k}\cdot\mathbf{x}}\theta_{\mathbf{x},0}+e^{-i\mathbf{k}\cdot\mathbf{x}}\theta_{\mathbf{x},1}$.
one generator per function $\psi:\mathbb R^3\to\mathbb C$ would be $\Lambda\,\mathbb{C}^{\mathbb{C}^{\mathbb{R}^3}}$
