An answer to another question derives a Hamitonian of the fermionic harmonic oscillator in terms of a pair of position-like and momentum-like operators. These operators are, as expected, defined in terms of the creation and annihilation operators: for the Hamiltonian
$$H = \hbar \omega \left(f^\dagger f - \frac{1}{2}\right)\tag1$$
with $\{f, f^\dagger\}=1$ and $\{f, f\} = 0$, these operators are
\begin{align} \psi_1 &= \sqrt{\frac{\hbar}{2}} \left(f + f^\dagger\right), \\ \psi_2 &= i\sqrt{\frac{\hbar}{2}} \left(f - f^\dagger\right). \tag2 \end{align}
The simplest representation of $f$ is, of course, a 2×2 matrix:
$$f=\begin{pmatrix}0&1\\ 0&0\end{pmatrix},\tag3$$
and the vacuum and one-particle states are, respectively,
\begin{align} |0\rangle&=\begin{pmatrix}1\\ 0\end{pmatrix},\\ |1\rangle&=\begin{pmatrix}0\\ 1\end{pmatrix}. \tag4 \end{align}
This is all fine, but then, if we follow $(2)$, we'll have position-like operator
$$\psi=\sqrt{\frac{\hbar}2}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}, \tag5$$
which has only two degrees of freedom!
But Dirac equation, which is supposed to be quantized by replacing its solutions' Fourier coefficients with ladder operators, works with bispinors whose components are in $\mathbb C$ without restriction to a pair of values. So I suppose that, like it is for projection of angular momentum, where there are only $2\ell+1$ states for each value of $\ell$ but also exists a representation in $\theta$ and $\phi$, there must be some kind of spatial representation for the fermionic harmonic oscillator.
Is there any such representation?
EDIT
I now see that the motivation for this question was not quite correct. First, although Dirac equation works with functions by applying differential operators on them, it's perfectly fine when individual Fourier components are limited in amplitude to e.g. a pair of values: each mode is still a usual differentiable function, so there's no problem in applicability of the Dirac equation to them.
Second, even if a differential representation of the ladder operators is found, it'll be irrelevant to the QFT: the spectrum of $f+f^\dagger$ will still contain only two values (due to its invariance with respect to change of basis), so the position of the fermionic oscillator will still be limited to two locations and nothing in between.
I leave this question here nonetheless, because it still may be useful to have a differential model of such a Hamiltonian.
