Is there a way to motivate, retrospectively, that observables must be representable by
- linear operators
- on a Hilbert space?
Specifically, there seems to be a hint to something in the accepted answer to this question. There, the author writes that
"[...] and since these commutators satisfy the Jacobi identity, they can be represented by linear operators on a Hilbert space."
- Is this true? If an observable can be written as a commutator (like the three coordinates of angular momentum), does it automatically follow that it corresponds to some linear operator? If yes, how / is this a theorem with a name?
Thanks for further hints, and for all hints so far. Quantum mechanics is still really strange to me, additional motivation for one of the postulates would be edit: is nice.
If you want me to clarify further, please say so.
This question: "How did the operators come about" seems related, but the answer doesn't help since it starts from the postulates. My question is about possible motivations one could give for one of the postulates.
Revision history part 1: The question was quite general before, it also asked "where did the operators come from" which aready has a very good answer here.
For completeness / Revision history part 2: Another approach might be via Wigner's theorem, that any symmetry operator on a Hilbert space is either linear unitary or antilinear antiunitary (and then one could maybe go from symmetry operators to their generators, and find out that they correspond to observables). Originally the question was also whether this line of argumentation works as well. But probably, this is covered in the historical sources, given here and in the comments.