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I started a bare minimum at quantum probability (redirecting noncommutative probability space etc.)
Some entries have long been secretly referencing such an entry, and I have cross-linked accordingly, for instance from von Neumann algebra and quantum computing.
I had the feeling somewhere we already had a detailed account of probability theory dually in terms of von NNeumann algebras, but if we do I didn’t find it(?)
added the pointer to Segal 65 “Algebraic integration theory”
added also pointer to the recent textbook Landsman 17.
A neat quick introduction is Gleason 09. But at the moment the server with the pdf seems to be down. We should upload a copy to the nLab server. Does anyone have a local copy? (I didn’t save mine, it seems.)
The address is giving me the paper: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf
Okay, that’s strange, I still get no response from this address.
Could you be so kind to upload a copy to the nLab? Or else to mail a copy to my private email address? Thanks!
Is that Ok, now?
Thanks!!
(I took the liberty of uploading it to the main web (here), instead of your personal web. That seems more robust against future changes, such as migrations.)
Can anyone upload to the main web?
Yes. Sorry, I thought this was clear.
Now I was about to point you to the HowTo, only to see that it didn’t have a section on file upload. Now I have added one:
Ok, let’s hope the spammers don’t think to upload their nonsense.
@ Urs #1
I had the feeling somewhere we already had a detailed account of probability theory dually in terms of von NNeumann algebras, but if we do I didn’t find it(?)
You may be thinking of Bayesian interpretation of quantum mechanics (now linked from this article) and its spin-off JBW-algebraic quantum mechanics, both of which have been discussed here before. Or for specifically classical probability and commutative real von Neumann algebras (or equivalently associative JBW-algebras), see measurable locale. There's not a lot of detail there on this subject, but some of Dmitri's posts to MathOverflow cited in the References are relevant.
Thanks, good point to cross-link these entries.
Let me add that I'd love to see a succinct explanation of how to do classical probability theory using operators on von Neumann algebras, whether on the nLab or elsewhere, but I don't actually know one. Nor do I know enough to write one, although I'd like to learn enough.
I think that’s what the references Segal 65 and Whittle 92 are about.
added these pointers:
Jürg Fröhlich, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer 2015 (arXiv:1310.1484, doi:10.1007/978-3-662-46422-9_7)
Jürg Fröhlich, The structure of quantum theory, Chapter 6 in The quest for laws and structure, EMS 2016 (doi, doi:10.4171/164-1/8)
I added Bohr topos as a related concept, but maybe there something more substantial to say about states which are linear and positive only commutative-subalgebra-wise.
By the way, do you still think that suggestion to bring Bohr toposes together with a layered dependent type theory – here – might work?
but maybe there something more substantial to say about states which are linear and positive only commutative-subalgebra-wise.
These are called quasi-states and Gleason’s theorem says that they are equivalent to actual quantum states. From the entry:
In quantum mechanics, a quasi-state on an algebra of observables $A$ is a function $\rho : A \to \mathbb{C}$ that is required to satisfy the axioms of a genuine state (linearity and positivity) only on the poset of commutative subalgebras of $A$.
While therefore the condition on quasi-states is much weaker than that for states, Gleason’s theorem asserts that if $A = B(H)$ for $dim H \gt 2$, then all quasi-states are in fact already genuine quantum states.
Notice that a quasi-state is naturally regarded as an actual state, but internal to the ringed topos over the poset of commutative subalgebras of $A$ – the “Bohr topos”. Therefore Gleason’s theorem is one of the motivations for regarding this ringed topos as the quantum phase space (“Bohrification”.) The other is the Kochen-Specker theorem.
My #18 wasn’t a request for information - I’d read the Bohr topos entry. I was just wondering if it could be useful to a reader of quantum probability to be given a paragraph to encourage them to visit Bohr topos, rather than have a mere link there.
Oh, I see. Sorry. Sure, I’ll add a line.
Great, thanks.
I hadn’t noticed that interesting order-theoretic structure in quantum mechanics page before.
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