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

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Discussion Overview

The discussion revolves around the existence of a maximal complete set of commuting observables within the context of Hilbert spaces. Participants explore the theoretical implications and potential proofs related to extending sets of commuting observables.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants propose that it is straightforward to write down a maximal complete set of commuting observables based on the definition of the Hilbert space.
  • Others argue that proving the extension of every set of commuting observables to a maximal set may require the axiom of choice.
  • A participant inquires about references for the proof of extending sets to a maximal set, questioning whether it is a trivial fact.
  • Another participant suggests that if a set is not maximal, one can add another operator to form a larger set, referencing Zorn's Lemma as a method for establishing maximality.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and complexity of proving the extension of commuting observables to a maximal set, indicating that the discussion remains unresolved.

Contextual Notes

The discussion touches on foundational concepts in functional analysis and the implications of the axiom of choice, but does not resolve the specific mathematical steps or assumptions involved in the proofs mentioned.

pythagoras88
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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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