This is a question I am wondering about because the answer to it seems to have some interesting - but perhaps already long considered and dismissed because it's been settled - implications for the measurement problem and question of what the "best" interpretation is of the formalism of quantum theory. Yet it may seem oddly elementary to ask.
Of course, I know that observables in quantum theory are Hermitian operators on a Hilbert space, but my question is: given a quantum system, what set of Hermitian operators correspond to its observables? The reason I ask this is that there are arguments, such as those in this video:
that the Hilbert space can be considered in some sense a term that contains redundancy similar to a gauge theory, and that what "really matters" is the observables and associated probabilities around measuring them, which are taken as forming a structure called a C*-algebra.
Now, if we take that view, and the "observables come first", it suggests that the set of observables is something we define in advance, and then the Hilbert space comes later as a sort of "model" thereof (in a sense close to the one used in mathematics, where we talk of a "model of a set of axioms"). However, I also have seen in other places people saying something different: that the set of observables is all Hermitian operators on a Hilbert space. That is, you make a Hilbert space, then boom, you get the set of observables.
And the thing is: these two things are not equivalent. C*-algebras are not in one-to-one correspondence with Hilbert spaces: instead, they may be isomorphic to proper subsets of the sets of all Hermitian operators on a given Hilbert space.
And to top it all off, textbooks often seem to gloss over this point: many often only say that observables are Hermitian operators, then go on to introduce some (e.g. position, momentum, etc.), but "are Hermitian operators" does not imply its converse, any more than "cars are machines" implies "machines are cars".
Is there a consensus on which view is the correct one, and if so, what is it, and why is the other one considered wrong?
You may ask as to what this has to do with the measurement problem. I've asked here earlier regarding Schrodinger's cat and Wigner's friend, and this is why. If it is the case that the set of observables of a system is not necessarily comprehensive of that on some Hilbert space, then there is a very natural interpretation of quantum states as being knowledge states, that goes like this.
Consider a Wigner's friend set up, and think about the experience of both Wigner and his friend. The friend is shut in a Schrodinger cat box with a quantum experiment. He runs the experiment and sees the result. Wigner then opens the box and asks the friend and the friend replies. But if Wigner models the friend using quantum mechanics and the Schrodinger equation, he gets a superposition state. Under the seemingly sensible philosophical assumption that Wigner's friend and Wigner himself both are having experiences throughout this and Wigner's friend experienced a single outcome, then it is rational to say that Wigner's superposition assignment reflects his ignorance of what outcome transpired, leading to the notion the quantum state is proximately a state of knowledge, belief or otherwise of information on an agent's part, as opposed to a property of the system - through the two may be isomorphic in cases.
But this falls apart if Wigner can do a measurement on his friend on a basis incompatible with. Then it seems hard to avoid that the superposition has ontological character, and we have a real conundrum if we accept those philosophical assumptions about the experience.
Yet all that goes away if, in fact, the incompatible measurement is not an observable of his friend, which is possible if the observables of a system are primary to the Hilbert space instead of the other way around. Indeed, pretty much every paradox and no-go result in this vein - not just this one, but results like the Frauchiger-Renner paradox, too - seem to depend on agents being able to do incompatible measurements on their experiences.
I can't help but think this must have been considered before, because it seems just too easy. But I have had a hard time finding it and, moreover, given it is not even commonly referenced in these discussions, that there must be something trivially wrong with it. What is that?