Can we prove that there is a maximal complete set of commuting

[pythagoras88]

Hi,

Just wondering, can we prove that there is a maximal complete set of commuting observable?

 

[A. Neumaier]

Hi,

Just wondering, can we prove that there is a maximal complete set of commuting observable?

In the cases of interest, it is easy to write one such set down, based on the way the Hilbert space is defined.

On the other hand, proving that every set of commuting observables on every Hilbert space can be extended to a maximal such set probably requires the axiom of choice.

 

[pythagoras88]

Thanks for the reply.

Do you know of any reference that has the proof for extending every set to a maximal set? Or is this kind of like a trivial fact that does not require much proving?

 

[A. Neumaier]

Do you know of any reference that has the proof for extending every set to a maximal set? Or is this kind of like a trivial fact that does not require much proving?

Given some such set S, if it is not maximal, you can (by definition of maximality) add another operator to get a bigger set. Now apply the Lemma of Zorn (which is equivalent to the axiom of choice).