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- How to interpret the large variety of observables QT allows us to construct?

So generally in most literature observables are represented via self-adjont (or equivalently real-valued) linear operators in QT. But that definition leaves it open for a wide variety of operators that can be view as observables. I was always a bit uncertain how to understand this freedom, so i want to scrutinize a bit its implication.

For that let's assume we have some observable ##O## and we pick some orthonormal basis of the Hilbert space ##|\psi_i\rangle##, then we can construct a new operator ##O_{\psi} :=\sum_i \langle \psi_i |O| \psi_i \rangle | \psi_i \rangle \langle \psi_i|## via a sum of projection operators multiplied by real valued constants. But on the other hand these constants ##\langle \psi_i |O| \psi_i \rangle## are the expectation values of ##O## for a state ##|\psi_i \rangle## and so in general we can check that ##\langle \phi |O| \phi \rangle = \langle \phi |O_{\psi}| \phi \rangle## for any state ##|\phi\rangle##. Hence, if we are just interested in getting the expectation value of ##O## we have actually a large family of operators at our disposal to do that. While this family has in common to always yield the same expectations, the uncertainty ##\Delta O_{\psi}## differs as per construction the choice of basis makes it certain/sharp for states of that basis. This also means that while ##O_{\psi}## will match in expectation, it will have a different spectrum i.e. represent the same variable via different quantum numbers. However this makes each a real valued operator and therefore self-adjoint, hence they are technically valid observables.

Here is the thing, that I don't know anything in QT that would forbid calling ##O_{\psi}## an observable, but then what is its interpretation? If we were just associate observables by their expectation to a variable they represent, then this leaves us with some ambiguity for that variable. On the other hand there is the correspondence principle that specifically picks out one operator from its family for each variable. But while it does make an unique correspondence, does it actually disqualify other possibilities? The correspondence principle was historically intended to bridge the gap between the new and old theory and i haven't heard it making any assumptions about the uniqueness of this mapping - nor anyone considering that it might not be.

When it comes to conservation laws written in terms of expectation values, we can exchange any observable by one from its same-expectation-family of operators and it changes nothing. So for this purpose all observables of the same expectation can be viewed as equivalent.

Going back to the spectrum of ##O_{\psi}##, isn't quantization in QT actually a part of the measurement process specifically? If a quantum system is undisturbed, then it supports superposition of states like Schrödinger's cat and the quantization towards a "dead" or "alive" cat quantum number only happens when the system is measured. So wouldn't it make sense to consider that observable operators don't just represent what variable they measure but also how they acquire it/how its measurement interacts with the system and therefore governs how it quantizes? However that would also mean that we might figure out a way to measure the cats state via a different method/observable and potentially have a scenario where we find that it has a cat quantum number of 0.5*dead + 0.5*alive with 100% certainty after repeated measurements (yet consequently such a measurement won't be able to measure the cat in a alive state with certainty).

EDIT: typo in formula

For that let's assume we have some observable ##O## and we pick some orthonormal basis of the Hilbert space ##|\psi_i\rangle##, then we can construct a new operator ##O_{\psi} :=\sum_i \langle \psi_i |O| \psi_i \rangle | \psi_i \rangle \langle \psi_i|## via a sum of projection operators multiplied by real valued constants. But on the other hand these constants ##\langle \psi_i |O| \psi_i \rangle## are the expectation values of ##O## for a state ##|\psi_i \rangle## and so in general we can check that ##\langle \phi |O| \phi \rangle = \langle \phi |O_{\psi}| \phi \rangle## for any state ##|\phi\rangle##. Hence, if we are just interested in getting the expectation value of ##O## we have actually a large family of operators at our disposal to do that. While this family has in common to always yield the same expectations, the uncertainty ##\Delta O_{\psi}## differs as per construction the choice of basis makes it certain/sharp for states of that basis. This also means that while ##O_{\psi}## will match in expectation, it will have a different spectrum i.e. represent the same variable via different quantum numbers. However this makes each a real valued operator and therefore self-adjoint, hence they are technically valid observables.

Here is the thing, that I don't know anything in QT that would forbid calling ##O_{\psi}## an observable, but then what is its interpretation? If we were just associate observables by their expectation to a variable they represent, then this leaves us with some ambiguity for that variable. On the other hand there is the correspondence principle that specifically picks out one operator from its family for each variable. But while it does make an unique correspondence, does it actually disqualify other possibilities? The correspondence principle was historically intended to bridge the gap between the new and old theory and i haven't heard it making any assumptions about the uniqueness of this mapping - nor anyone considering that it might not be.

When it comes to conservation laws written in terms of expectation values, we can exchange any observable by one from its same-expectation-family of operators and it changes nothing. So for this purpose all observables of the same expectation can be viewed as equivalent.

Going back to the spectrum of ##O_{\psi}##, isn't quantization in QT actually a part of the measurement process specifically? If a quantum system is undisturbed, then it supports superposition of states like Schrödinger's cat and the quantization towards a "dead" or "alive" cat quantum number only happens when the system is measured. So wouldn't it make sense to consider that observable operators don't just represent what variable they measure but also how they acquire it/how its measurement interacts with the system and therefore governs how it quantizes? However that would also mean that we might figure out a way to measure the cats state via a different method/observable and potentially have a scenario where we find that it has a cat quantum number of 0.5*dead + 0.5*alive with 100% certainty after repeated measurements (yet consequently such a measurement won't be able to measure the cat in a alive state with certainty).

EDIT: typo in formula

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