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Mike Stay kindly added the standard QM story to path integral.
I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.
added more references to path integral. Also added an Idea-section
much more needs to be added here, eventually.
In general one needs to have integration on paths in phase space, only for special Lagrangeans it reduces to integration only over paths in configuration space. Momenta integration can not be in general skipped over.
I have added to path integral a bunch of on the formalization in quantum mechanics by Wiener integration over stochastic processes.
Then I started a new subsection Realization – As an integral against the Wiener measure with a (very) brief indication of that…
…ending in an observation on what the passage from gauge coupling action functionals to the quantum propagator integral kernel looks like when viewed in higher geometric prequantum theory as an operation on correspondences in the slice (infinity,1)-topos over the moduli of circle bundles and of vector bundles, respectively.
Wiener integral needs to be analytically continued to become a QM path integral, Wiener measure certainly does not exist after the Wick rotation…I do not detect this basic idea in the entry…
Right, I have added a “Eudlidean”-qualifier for the moment.
Added reference
added a remark stating that the expression “holonomy integrated against the Wiener measure” is precisely what appears in the worldline formalism for computing QFT scattering amplitudes.
added pointer to:
Daniel F. Styer, The Errors of Feynman and Hibbs [pdf]
Daniel F. Styer, Richard Feynman, Albert R. Hibbs, Quantum Mechanics and Path Integrals: Emended Edition, Dover (2010) [ISBN:0486477223]
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added a brief remark (here) on operator products corresponding to time-ordered products of their observable values, under the path integral.
For the moment the ambition of the remark is just to record some (original) references for this standard fact. It actually goes back all the way to Feynman 1948, p. 381, and I was looking for modernized accounts. I found Nagaosa 1999, pp. 33 and Ong, which are good but essentially just reiterate Feynman’s original account.
added pointer to the very original articles:
Paul A. M. Dirac, The Lagrangian in Quantum Mechanics, Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933) 64–72 [pdf], reprinted in Brown 2005 [doi:10.1142/9789812567635_0003]
Richard P. Feynman, The Principles of Least Action in Quantum Mechanics, PhD thesis (1942), reprinted in Brown 2005 [doi:10.1142/9789812567635_0001]
(have to admit that, until looking these up now, I didn’t appreciate how close Dirac already got to identifying the path integral, nor how early Feynman, following Dirac, already saw the full picture)
and to this collection of reprints:
No shame in not knowing all the 90yo papers.
On a doubly-related note, I read today that Neumann failed to learn of some stochastic interpretation angle to the QM integrals, in his lifetime, seemingly due to unrelated differences with Wiener. That makes sense to me, given what I e.g. know about the two guys very different war outlooks, but I can’t back it up either. I see there’s a long 1980’s MIT press text that is literally about contrasting the two men. I’m thinking of Wiener atm. as I came across this comfy ETH Zürich affiliated ControlTheory podcast, with some history and some researcher interviews. I know I’m getting from one thing to the next here, but it’s a good listen while in the car and so I thought I’d share.
Thanks. Saw your comment in middle of the night and listened to the first halfs of episodes 6 and 15 now, trying to fall asleep again (but failing at that ;-). This is good stuff.
By the way, looking up 90 year old papers is not just of historical interest.
It happens that insights contained in classical texts fail to be recounted in the textbooks and then may effectively be lost because nobody consults the classics for material that is thought to be in every textbook.
In the present case, what I was really looking for were accounts which make explicit that the non-commutative product of quantum observables reflects the temporal order of making the corresponding observations.
So far, about the only source where I see this stated clearly is Feynman’s 1942 thesis and the resulting 1948 article, as now referenced here. I see Feynman’s original argument recounted also in Nagosa 1999, p. 33 and in a remarkble unpublished (and undated) note by Ong, but not for instance in Kleinert’s book on path integrals. (?)
I actually met Scaramuzza at ICVSS 2019, the “International Computer Vision Summer School” that happens in Sicily every year. He’s a big name but I wonder how you’d get to watch ep.15 first? CV has been 97% AI since many years now, and I’m also drifting more towards the hype now. Which is to say “full circle”, as I came to the podcast through looking at Q-learning.
I’m sure you at one point came across Ron Maimon on the web - he strongly defended the position that one gotta read the original papers too. In the infamous way he defended all of his positions, haha. These angry rants usually came with some nuggets, like here. I also know that he’s the one who wrote the commutation relation section in the path integral Wikipedia page here. He also wrote about it on here. The funny thing is that I wrote this paragraph before I saw that in your note by Ong, the last reference is to this SE thread! Maybe that’s the flow of information here.
In my personal files on the Ito integral, I find these throwaway notes from 2015. Since I don’t see “Euler” mentioned in the paper or website pages, I assume I had more in mind when writing this, but I don’t know what, anymore. (I also don’t know about the extra indices and powers of my \delta’s, but it can’t matter much.) I think the subtle choice of the index foo in of H_{foo} in this definition reflects the Feynman discussion, but in the stochastic context people don’t bother so much as what happens at a single time point for itself, but what these relations mean once you integrate over them. Like for example the covariation of two processes is a whole sum of terms where two indices variants are used. I looked at the Kleinert book at the first paragraph of “18.12 Stochastic Calculus” at least “time order” makes a mention, in this sense.
Thanks for all this.
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Dunno what made me go for episode 15 first, something in the thumbnail blurb. Interesting to hear about cutting-edge theoretical research in a field where to prove your work is to show it fly, which even outsiders can immediately appreciate. (I hope to get to bring my own research to that point, eventually… :-)
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Regarding Ron Maimon: Yes, I have fondly been recording links to his stuff here. Do you know what became of him? I hope he is well.
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Regarding the last reference in Ong’s note: I had missed that, good catch! You are right, it does seem like it is through Ron M.’s online comments that this observation by Feynman survived! Remarkable.
(I am not into editing Wikipedia, but if you or anyone lurking here is: The reference “[10]” to Feynman 1948 here ought to go with a page number (p. 381) and ought to be accompanied by pointer to the earlier Feynman 1942 p. 35, too, from six years earlier. Six years was an eternity, back then.)
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Regarding your 2015 note: Interesting. We should record this or related discussion on some nLab page. The page path integral is badly in need of more attention. Currently there is only my brief Remark 2.1 on this matter. If you have the energy, it would be worth expanding this into a subsection, or even give it its own page!
Last I interacted with Ron was under a youtube video where I talked about constructive theories and he was favorable towards it. Albeit I think he never studied that in isolation.
Btw. I didn’t only see Ong’s reference, but in the longer Feynman text he also notes in a foodnote that Dirac looked at time ordering. Although I think such things are not super explicit in his 1935 book. Since this is all so early already (and I don’t know if we can find out if and where Feynman has his longer discussion from somewhere else) I’d actually be more interested in whether there’s a text where this is analyzed in more depth. As the index choice affects so strongly what calculation rules pop out of a stochastic calculus (how the generalized rule looks, etc.), and since there’s several stochastic integrals and a lot of talk about slicing domains in such books, I expect there’s a dedicated analysis of this somewhere. Whether this then is combined with the Schrödinger-like situation with an i is another question.
Whether QFT or making things fly, I think the math of “path space” comes together to a nice degree. I bought a €100 drone two winters ago and and used it for 45 minutes together with some python scripting. Thought about studying and nicely summarizing all the Lagrangian dynamics of it, but that hasn’t happened yet. There’s many physicists in those fields. But if you work at the page where you show of flight dynamics you maybe need to go through intuitions and empirical findings faster than is comfortable for someone who wants to say he fully got the math of it too. After the 2019 conference I remember also downloading some paper where someone looks at the weight space of neural networks from a QFT POV and used a lot of the language form the latter field. Maybe even talk of renormalization and its relation to relearning or learning neighboring tasks, or something. Can’t find it out, but I might dig it out later, for curiosity sake, and check what the authors did with it since then. Here’s some bonus generative content: slow pop, reggae, slow rock, more pop, electronic dance. For this one, the prompt was “Ein Deutsches Volkslied über Norbert Wiener’s unbekannte und vergessene Beiträge zur Garbentheorie.” I’m not responsible for any cringe lyrics. In the middle of the next year there will be video API’s.
“Last I interacted with Ron…” When was that, though?
Oh, I think right when Covid started in late 2019. I know next to posting about string theory and general philosophy in forums, 10 years ago he was working on DNA sequencing and explained some of his motivation to me, and this was from 10 years before that. He seems private so I don’t want to expound needless details.
Thanks. I am not after anything private, was just wondering about his last sign of life. Good to hear.
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Re #26: This link is also broken.
added pointer to today’s
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