The Hilbert space is the <fermionic Fock space>
$$
\boxed{\mathcal H\cong\Lambda^\bullet\mathbb C^n}.
$$
In a polarization where $\widehat\eta^a$ acts by exterior multiplication and $\widehat{\bar\eta}_a$ by contraction, a general state is
$$
\Psi(\eta)=\sum_{p=0}^n\frac1{p!}\Psi_{a_1\cdots a_p}\eta^{a_1}\cdots\eta^{a_p}.
$$
The <fermion number operator> $N=\widehat\eta^a\widehat{\bar\eta}_a$ returns the exterior degree, and therefore
$$
\boxed{(-1)^N\Psi(\eta)=\Psi(-\eta)=\sum_p(-1)^p\Psi_{(p)}(\eta)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 2 / b / Solution

{title2=$F$}

The fermion number operator counts occupied fermionic modes. Its parity $(-1)^F$ is positive on even exterior degree and negative on odd exterior degree.


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 Incoming links: Fermion number operator