There have been a couple of questions on fermionic coherent states, but I didn't find any that covered the following question:

If I define a coherent fermionic state in the 2-level-system spanned by $|0\rangle$ and $|1\rangle$, I will write it as
\begin{align}
|\gamma\rangle=e^{a^\dagger\gamma-\overline{\gamma}a}|0\rangle\,,
\end{align}
where $\gamma$ is a Grassmann variable and $a^\dagger$ and $a$ are fermionic creation and annihilation operators. Such a state has the property that the expectation value of $a^\dagger$ is given by $\gamma$ and the one of $a$ by $\overline{\gamma}$. How can a regular operator have an expectation value that is not a complex number?

To me it seems that we formally extend our Hilbert space to something where vectors cannot just have complex numbers as coefficients, but also polynomials of Grassmann variables. What's the best way to think of this? Can I use such a state to describe a concrete physical state? What happens if the number operator has an expectation value containing Grassmann variables?

This post imported from StackExchange Physics at 2017-09-17 13:03 (UTC), posted by SE-user LFH