vanhees71 said:
What do you mean by the "operators can't be prepared".
I was referring to the common practice of calling self-adjoint operators observables.
vanhees71 said:
Of course not, because there are no operators in the lab, but there are observables, and I can measure them in the lab with real-world devices with more or less accuracy.
Some observables can be measured in the lab, and they
sometimes correspond to operators, more often only to POVMs.
However,
most observables corresponding to operators according to Born's rule
cannot be observed in the lab!
vanhees71 said:
I'm criticizing your approach as a foundational description of what quantum mechanics is. For a physicist it must be founded in phenomenology,
Why? It must only
reproduce phenomenology, not be founded in it.
When the atomic hypothesis was proposed (or rather revitalized), atoms were conceptual tools, not observable items. Theey simplified and organized the understanding of chemistry, hence their introduction was good science - though not founded in phenomenology beyond the requirement of reproducing the known phenomenology.
Similarly, energy is basic in the foundations of physics but has no direct phenomenological description. You need already theory founded on concepts prior to phenomenology to be able to tell how to measure energy differences.
Of course these prior concepts are motivated by phenomenology, but they are not founded in it. Instead they determine how phenomenology is interpreted.
vanhees71 said:
i.e., you never say what your expectation values are if not defined in the standard way.
They are numbers associated to operators. This is enough to working with them and to obtain all quantum phenomenology.
Tradition instead never says what the probabilities figuring in Born's rule (for arbitrary self-adjoint operators) are in terms of phenomenology since for most operators these probabilities cannot be measured. Thus there is the same gap that you demand to be absent, only at another place.
vanhees71 said:
If you discard the standard definition, which is usually understood and founded in an operation way to phenomena, you have to give an alternative operational definition, which you however don't do. I'm pretty sure that the edifice is sound and solid mathematically, but it doesn't make sense to me as an introduction of quantum theory as a physical theory.
As
@gentzen mentioned in post #72, Callen's criterion provides the necessary and sufficient connection to phenomenology. Once Callen's criterion is satisfied you can do all of physics - which proves that nothing more is needed.
If you require more
you need to justify why this more should be essential for physics to be predictive and explanative.