We can construct a Hermitian operator $O$ in the following general way:
- find a complete set of projectors $P_\lambda$ which commute,
- assign to each projector a unique real number $\lambda\in\mathbb R$.
By this, each projector defines an eigenspace of the operator $O$, and the corresponding eigenvalues are the real numbers $\lambda$. In the particular case in which the eigenvalues are non-degenerate, the operator $O$ has the form $$O=\sum_\lambda\lambda|\lambda\rangle\langle\lambda|$$
Question: what restrictions which prevent $O$ from being an observable are known?
For example, we can't admit as observables the Hermitian operators having as eigenstates superpositions forbidden by the superselection rules.
a) Where can I find an exhaustive list of the superselection rules?
b) Are there other rules?
c) Is the particular case when the Hilbert space is the tensor product of two Hilbert spaces (representing two quantum systems), special from this viewpoint?