We know that quantum mechanics requires self-adjoint operators, not only symmetric. Can we say that this follows ONLY from the two following axioms of quantum mechanics, namely that
- each observable $a$ corresponds to a linear operator $A$
- an expectation value of a measurement of $a$ must be real
I thought these two imply only the need for a Hermitian (i.e. symmetric) operators (because a linear Hermitian operator has real eigenvalues) and that the need for self-adjointness was somehow connected to an additional requirement such as unitarity of the time-evolution operator. What is the missing piece?
(I know how the two terms are defined, e.g. here.)