Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
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 $f$ such that $\psi$ is eigenket of an observable $A$ iff $d_\psi f=0$? I am surprised why such natural fact cannot be seen in standard textbooks!
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”$d_\psi$”?
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 $d_\psi f$ I mean just the derivative of $f$ at point $\psi$ 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.
In fact, the function is
$f(\psi)=\langle\psi,A\psi\rangle/\langle\psi,\psi\rangle.$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.
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.
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.
My second question: We know that any complex Hilbert space $H$ is a real symplectic manifold via the imaginary part of its inner product
$-2\hbar Im\langle,\rangle.$Now, let $A$ be an observable with discrete eigenvectors $|\psi_n\rangle$ with eigenvalue $a_n$ which form an orthonormal basis $|\psi_n\rangle$ of $H$. Now consider a bra $u_n:=\langle\psi_n|$ as a complex function over $H$. Then we have
$i\hbar\{u_n,\langle A\rangle\}=a_n u_n$Is this result well-known? Please introduce me a reference? Thanks
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:
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?
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.
Does anybody know nontrivial examples of functions $f(x,p)$ over classical phase spaces such that the following equation
$i\hbar\{u,f\}=au$has compactly-supported differentiable solution $u$ for some unknown number a? Namely, $f$ is known and $u$ and a are unknowns.
By nontriviality of $f$, I mean $f$ is not $f=f(x)$ or $f=f(p)$ or $f=Harmonic oscillator$.
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.
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.
1 to 18 of 18