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Circumstances prompted me to write a kind of pamphlete pointing out some aspects that seem worth taking notice of have not found much appreciation yet:
This surveys how basic theorems about the standard foundation of quantum mechanics imply an accurate geometric incarnation of the “phase space in quantum mechanics” by an order-theoretic structure that combines with an algebraic structure to a ringed topos, the “Bohr topos”. While the notion of Bohr topos has been motivated by the Kochen-Specker theorem, the point here is to highlight that taking into account further theorems about the standard foundations of quantum mechanics, the notion effectively follows automatically and provides an accurate and useful description of the geometry of “quantum phase space” also in quantum field theory.
added second section Relation to the traditional non-commutative geometry
That’s a great page. I just heard Samson Abramsky yesterday on sheaf theory as measuring non-locality and contextuality. He sees a common structure between QM, databases and natural language. I should follow that up.
This observation indeed puts doubt on the long and widely held believe that the quantum phase space is an object in noncommutative geometry, a believe that in fact motivated much of the development of noncommutative geometry in the first place.
Are you suggesting Connes’ programme might not be as important as we were led to believe?
If ordinary quantum mechanics is the holographic dual of the A-model on certain D-branes/2-dimensional Poisson sigma-model, do we see the holographic relation between Bohr toposes for the former and whatever the Bohr thing is for the latter (local nets of Bohr toposes?)?
Are you suggesting Connes’ programme might not be as important as we were led to believe?
Good question. This is an important subtlety to sort out.
A spectral triple is effectively an axiomatization of a quantum particle, as a 1d QFT. One of the ingredients is a Hilbert space, so that’s already all the states and observables. Next a spectral triple in addition has a possibly non-commutative algebra densely embedded into the algebra of bounded operator on that Hilbert space.
Kontsevich and Soibelman highlighted under the headline “graph field theory”, that this associative (but possibly non-commutative) algebra in a spectral triple is to be identified as the data encoding the interaction, namely the data associated to a 1-dimensional “cobordism with singularities” which has two edges coming in and one coming out, see on the nLab here.
So in total quantum mechanics certainly has a non-commutative algebra in it. I tried to make this clear in the subsection order theoretic structure in QM – Relation to the traditional non-commutative geometry.
But the point that I think needs to be made is that it is not right to say that quantization deforms phase space to a non-commutative space. If at all, then quantization turns phase space instead into a non-associative “Jordan space”. The non-commutativity is instead part of the dynamics/interaction.
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