I have a somewhat naive question about Majorana fermions. Typically, two Majorana fermion modes $\gamma_{i,1}$ and $\gamma_{i,2}$ are defined by writing a single ordinary fermion $c_i$ in terms of "real and imaginary parts": $$ c_i = \frac{\gamma_{i,1} + i \gamma_{i,2}}{\sqrt{2}}, \quad c^{\dagger}_i = \frac{\gamma_{i,1} - i \gamma_{i,2}}{\sqrt{2}} $$ We therefore obtain $2N$ Majorana fermions from $N$ fermionic modes. These Majorana fermions satisfy the commutation relations $\{ \gamma_i, \gamma_j \} = \delta_{ij}$, but since they are Hermitian the naive occupation number one might write down is trivial: $\gamma_i^{\dagger} \gamma_i = \gamma_i^2 = 1/2$. Instead, typically pairs Majoranas into pairs and measures the operator $i \gamma_i \gamma_j$, which has eigenvalues $\pm 1/2$. Fair enough.
Now, here's the question: is there anything in particular which stops me from considering the eigenstates and eigenvalues of the single Majorana fermion $\gamma_i$? It's a Hermitian operator, so it can certainly be diagonalized. It's easy to see by inspection that $\gamma_i$ has eigenvalues $\pm 1/\sqrt{2}$. Additionally, if I think about my fermions in the sense of Jordan-Wigner, it appears that $\gamma_{i,1}$ plays the role of $\sigma^x_i$ in the spin language while $\gamma_{i,2}$ plays the role of $\sigma^y_i$, albeit with Jordan-Wigner strings attached.
One potential reason I can think of is that we clearly cannot assign each of $2N$ Majoranas different eigenvalues, since our Hilbert space is only $2^N$ dimensional. But this would not be much of a complaint in the corresponding spin language (nobody would be upset to hear that $\sigma^x_i$ and $\sigma^y_i$ are not independent); it's really just a reminder that the $2N$ Majoranas are not all independent. Is this the only reason not to consider the eigenvalues of single Majoranas alone, or have I missed something else? For example, is there a physical reason why $\gamma_i$ might not be a well-defined observable?
