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# What Hermitian operators can be observables?

We can construct a Hermitian operator $$O$$ in the following general way:

1. find a complete set of projectors $$P_\lambda$$ which commute,
2. 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?

Update:

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?