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I have been touching and editing a bit more the circle of entries on the foundations of quantum mechanics which all revolve around the phenomenon that the space of states in quantum mechanics is all determined (just) by the Jordan algebra structure on the algebra of observables, and notably by the poset of commutative subalgebras of the algebra of observables:
The last of these entries is new, but essentially just split-off from “poset of commutative subalgebras” for the moment. The other entries in the list I have mildly edited, mainly cross-linking them with each other. At Kochen-Specker theorem I did a bit more editing, but mainly just trying to prettify the formatting and the layout of the paragraphs and cross-links..
I wanted to do more, but I am running of out time now.
Anyway, I think together these theorems paint a picture that is noteworthy and hasn’t been highlighted much. The proponents of looking at QM through the ringed topos over os poset of commutative subalgebras highlight Kochen-Specker, but I find Gleason’s theorem is actually a stronger argument for this approach, while Kochen-Specker is then more of a nice spin-off. Also Alfsen-Shultz combined with Harding-Döring-Hamhalter is essentially a re-formulation of Gleason that amplifies more the poset structure on the poset of commutative subalgebras.
Here Gleason and, via Jordan, Alfsen-Shultz of course go back to the very roots of QM in the 1950s, whereas Döring et al is recent. This is maybe noteworthy.
More later. Have to run now.
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