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nLab > Latest Changes: foundational theorems of quantum mechanics

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 3rd 2013
    • (edited Oct 3rd 2013)

    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.

    • CommentRowNumber2.
    • CommentAuthorAkrami
    • CommentTimeSep 2nd 2022
    • (edited Sep 2nd 2022)

    Having a math PhD in the area of quantum groups, NCG and TQFT, I started to think on the mathematical foundation of quantum mechanics 16 years ago after my PhD and postdoc. So I have faced by some questions which I did not find answers on the standard texts of quantum physics. So I start to ask them here and I will be very glad if somebody guide me. I think my questions are in the area of “foundational theorems of quantum mechanics”

    First question: Are the stationary states (eigenkets) of a quantum observable (Hermitian operator with discrete spectrum) as the stationary (critical) points of a suitable function? I mean is there a function ff such that ψ\psi is eigenket of an observable AA iff d ψf=0d_\psi f=0? I am surprised why such natural fact cannot be seen in standard textbooks!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2022

    Maybe you could add more details on the question you raise. Since finite-dimensional linear subspaces of a Hilbert space are closed, they are the zero-sets of (smooth) functions, by general facts – I would think. But maybe you mean something else. Which differential operator do you mean when you write”d ψd_\psi”?

    • CommentRowNumber4.
    • CommentAuthorAkrami
    • CommentTimeSep 2nd 2022

    Thanks for comment. But unfortunately, I do not see any relation between my question and your comment that finite-dimensional subspaces are closed. Could you please explain more your comment. By the differential operator d ψfd_\psi f I mean just the derivative of ff at point ψ\psi as a smooth function over whole Hilbert space or the space of quantum states, i.e. normalized kets or better saying rays in the Hilbert space.

    • CommentRowNumber5.
    • CommentAuthorAkrami
    • CommentTimeSep 2nd 2022
    • (edited Sep 2nd 2022)

    In fact, the function is

    f(ψ)=⟨ψ,Aψ⟩/⟨ψ,ψ⟩.f(\psi)=\langle\psi,A\psi\rangle/\langle\psi,\psi\rangle.

    Have you ever seen this result somewhere? Please introduce a reference for it, since I am going to bring it in my paper and I have to cite the references on it.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2022

    Now I understand which question you are after. Off the top of my head I don’t know a reference that would say this explicitly – but I suspect people will find that this is obvious. Maybe somebody else here can say more.

    • CommentRowNumber7.
    • CommentAuthorAkrami
    • CommentTimeSep 2nd 2022

    Thanks. So, you mean I don’t need to cite a reference on this fact in my paper. Although it is an obvious fact, but I think it may have rich consequences in understanding quantum mechanics.

    • CommentRowNumber8.
    • CommentAuthorAkrami
    • CommentTimeSep 2nd 2022
    • (edited Sep 2nd 2022)

    My second question: We know that any complex Hilbert space HH is a real symplectic manifold via the imaginary part of its inner product

    −2ℏIm⟨,⟩.-2\hbar Im\langle,\rangle.

    Now, let AA be an observable with discrete eigenvectors |ψ n⟩|\psi_n\rangle with eigenvalue a na_n which form an orthonormal basis |ψ n⟩|\psi_n\rangle of HH. Now consider a bra u n:=⟨ψ n|u_n:=\langle\psi_n| as a complex function over HH. Then we have

    iℏ{u n,⟨A⟩}=a nu ni\hbar\{u_n,\langle A\rangle\}=a_n u_n

    Is this result well-known? Please introduce me a reference? Thanks

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2022
    • (edited Sep 2nd 2022)

    I don’t know, off the top of my head, a reference making this exact statement – but this reminds me of the discussion in:

    advertised in:

    • CommentRowNumber10.
    • CommentAuthorAkrami
    • CommentTimeSep 3rd 2022

    Thanks for introducing very useful reference. I am very surprised that the results of the paper of Ashtekar and Schilling are closely related to my research but unfortunately I was unaware while they have gotten their results at least 25 years ago. Also I am surprised why such important theory was ignored. Does somebody know whether this geometric formulation of quantum mechanics has been expanded or not?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2022

    Glad to hear that this helped.

    I am not sure if this was “ignored”: the article has 319 citations.You can see them listed by going to https://arxiv.org/abs/gr-qc/9706069, scrolling down to “References & Citations” and then clicking on one of the three services offered.

    For instance, clicking on “GoogleScholar” first takes you to https://scholar.google.com/scholar_lookup?arxiv_id=gr-qc/9706069 which shows again the abstract, now with a link “cited by 319”. Clicking on that shows a list of all citing articles.

    In this case, many of them have “loop” in their title, these you can safely ignore. A few potentially relevant ones remain, such as maybe Geometrization of quantum mechanics (2007) or Remarks on geometric quantum mechanics (2004), and some more (I haven’t checked them out, but I would if I were to dig into this).

    Generally, there are many different perspectives on quantum physics, and to some extent every researcher has to make their own spiritual journey through them.

    • CommentRowNumber12.
    • CommentAuthorAkrami
    • CommentTimeSep 3rd 2022
    Thanks very much Urs. Very useful comments. As I said before, I'm a mathematician and I love to construct new concepts in math and prove math theorems originated from quantum theory. Until now I have been concentrated on quantum mechanics but I think after 16 years it is now time to enter the ocean of QFT to search for foundational theorems of QFT, where you are active in establishing mathematical foundation of nonperturbative QFT.
    • CommentRowNumber13.
    • CommentAuthorAkrami
    • CommentTimeSep 3rd 2022
    • (edited Sep 3rd 2022)

    Does anybody know nontrivial examples of functions f(x,p)f(x,p) over classical phase spaces such that the following equation

    iℏ{u,f}=aui\hbar\{u,f\}=au

    has compactly-supported differentiable solution uu for some unknown number a? Namely, ff is known and uu and a are unknowns.

    By nontriviality of ff, I mean ff is not f=f(x)f=f(x) or f=f(p)f=f(p) or f=Harmonicoscillatorf=Harmonic oscillator.

    • CommentRowNumber14.
    • CommentAuthorAkrami
    • CommentTimeSep 4th 2022
    @Ursm you said "I am not sure if this was “ignored”: the article has 319 citations..." But I think these references are not seriously citing the geometric method of Ashtekar and Schilling. Also up to my search, even Ashtekar and Schilling have not continued this project seriously, have they?
    Schilling has very few publications. Does anybody know what he is doing these days?

    @Urs, are you insisting on your opinion mentioned on the Secret Blogging Seminar in 2009, on this project, yet?
    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2022

    You write:

    these references are not seriously citing the geometric method of Ashtekar and Schilling.

    I am not sure about this. Just googling for “geometric quantum mechanics” gives plenty of hits which seem to focus on just this picture.

    Curiously, at Surrey they have a whole research center devoted to the idea (here)!

    I am suspecting that this is indicative of what’s going on generally: Probably, people keep getting enchanted by this geometric picture, but then fail to discover substantial new results.

    Of course, if you see substantial results that the community has missed, then that’s your chance to go all in. Let us know what you come up with, I’ll be interested in taking a look. But that’s probably all I have to say about the topic, for the time being.

    • CommentRowNumber16.
    • CommentAuthorAkrami
    • CommentTimeSep 5th 2022
    Thanks again for useful comments. Ok, I'll try to share here my ideas on this subject to get the comments of the members of nforum and in particular yours.
    • CommentRowNumber17.
    • CommentAuthorAkrami
    • CommentTimeSep 5th 2022
    p.s.
    If you know any other research groups on geometric quantum mechanics, please introduce me. Of course, I'll send similar request to Surrey's group and follow their work, I was not aware of their group.
    • CommentRowNumber18.
    • CommentAuthorAkrami
    • CommentTimeSep 6th 2022

    My correspondence with Prof. Brody:

    Dear Prof. Brody I am very interested in geometric quantum mechanics. I have visited the page of your group at Surray and your old paper 2001 on this subject. I am writhing to ask whether you have published new paper on this subject and also introduce me other groups working on this area. Is this subject active as it was twenty years ago. Has it achieved its promised goals? Could it attracted the quantum theory community?
    Many thanks Best wishes Seyed Ebrahim Akrami Math Faculty of Semnan University

    Dear Seyed,

    I list below some of the papers along the programme set out in our 2001 paper, which you might find interesting.

    Regards, Dorje

    Brody, D.C. & Hughston, L.P. (2021) “Quantum measurement of space-time events” Journal of Physics A54, 235304.

    Brody, D.C., Gibbons, G.W. & Meier, D.M. (2015) “Time-optimal navigation through quantum wind” New Journal of Physics 17, 033048.

    Brody, D.C. (2013) “Geometry of the complex extension of Wigner’s theorem” Journal of Physics A46, 395301.

    Brody, D.C. (2011) “Information geometry of density matrices and state estimation” Journal of Physics A44, 252002.

    Brody, D.C. & Graefe, E.M. (2010) “Coherent states and rational surfaces” Journal of Physics A43, 255205

    Brody, D.C. & Hughston, L.P. (2005) “Theory of quantum space-time” Proceedings of the Royal Society London A462, 2679-2699.

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