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

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The discussion centers on the existence of a maximal complete set of commuting observables in quantum mechanics, specifically within the framework of Hilbert spaces. It is established that while one can easily construct such sets, proving that every set of commuting observables can be extended to a maximal set necessitates the use of the axiom of choice. The Lemma of Zorn is referenced as a critical tool in this proof, indicating its foundational role in extending sets within mathematical structures.

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  • Understanding of Hilbert spaces in quantum mechanics
  • Familiarity with the concept of commuting observables
  • Knowledge of the axiom of choice in set theory
  • Acquaintance with Zorn's Lemma and its applications
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Hi,

Just wondering, can we prove that there is a maximal complete set of commuting observable?
 
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pythagoras88 said:
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.
 


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?
 


pythagoras88 said:
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).
 

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