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Timeline for Any real contribution of functional analysis to quantum theory as a branch of physics?

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Dec 13, 2019 at 1:27 comment added Aaron Bergman @TimothyChow Actually, I was referring specifically to functional analysis. In other areas, particularly geometry, I would say that the subtle issues brought to the fore by mathematicians have proved to be very important — understanding global effects, for example. For whatever reason, to my knowledge, this just hasn’t been the case for analysis. I don’t know why. Back when I did this for a living, I would worry from time to time that QFT has not been made rigorous, and maybe that means there is something important we’re missing there.
Dec 12, 2019 at 17:54 history made wiki Post Made Community Wiki by Todd Trimble
Dec 12, 2019 at 16:02 comment added Timothy Chow Perhaps a good example to consider is general relativity. In Einstein the Stubborn, Galina Weinstein makes the case that the rigorous mathematics of Levi-Civita and Hilbert played a crucial role in correcting an error in Einstein's development of the theory. If we accept Weinstein's narrative, it supports Francois Ziegler's point and goes against Aaron Bergman's claim that problems don't arise when physicists just barrel forward with non-rigorous arguments. (Though maybe some will argue that Einstein would have given up his incorrect theory anyway.)
Dec 12, 2019 at 14:40 comment added Francois Ziegler @AaronBergman We may be talking past each other. Sure the envelope may need pushed into the not-yet stress-tested, as with unbounded operators or infinite-dimensional representations, but it’s (almost always?) there to begin with. Apollonius’ conics, Lagrange and Gauss’ divergence theorem, Riemann and Ricci/Levi-Civita’s geometry, Hilbert’s spectra, Sturm and Liouville’s eigenfunctions, Frobenius’ representations, Cartan’s spinors, etc., were there for the taking by Kepler, Maxwell, Einstein, Born et al., Schrödinger, Wigner, Pauli, etc. (I’d call path integrals or string theory atypical.)
Dec 12, 2019 at 13:42 comment added Paul Siegel @AaronBergman In my undergraduate QM class we used "An Introduction to Quantum Theory" by Hannabuss, who writes on page 75: "Given a self-adjoint operator, $A$, in an infinite-dimensional space, it is useful to define its spectrum to be the set of complex numbers $\alpha$ for which $A - \alpha$ is not invertible. (The notion of invertible is not quite straightforward either, but our discussion will not be sensitive to the precise details.)" Now this book is in the "Oxford Graduate Texts in Mathematics" series, but my class was in the physics department, so maybe it's a push.
Dec 12, 2019 at 13:03 comment added Aaron Bergman @PaulSiegel I would be very impressed if you could find any quantum mechanics book, either undergraduate or graduate, that contains the correct definition of the spectrum outside of a finite dimensional space.
Dec 12, 2019 at 13:00 comment added Aaron Bergman @FrancoisZiegler For better or for worse, it’s exactly the other way around. Physicists used (and use) these concepts well before they have been stress-tested from a mathematics point of view. The math has almost always, to my knowledge, come later. As I said, it’s somewhat impressive that this hasn’t led to more problems.
Dec 12, 2019 at 12:35 comment added Paul Siegel The fact that lots of physicists are confused by the notion of the spectrum in functional analysis stems from the fact that most of them took at least one quantum mechanics course in which the spectrum of a linear operator on Hilbert space was defined (probably correctly, at least in context). That physics professors and textbook authors view this formalism as a foundational topic in the field - and thus most physicists are even aware of its existence - is in my mind evidence of the influence of functional analysis.
Dec 12, 2019 at 6:59 comment added Michael Engelhardt @AaronBergman - a propos, true story: Physicist as Dean's Rep in a math oral final asks for difference between spectrum and eigenvalues. Crickets ...
Dec 12, 2019 at 6:55 comment added Francois Ziegler @AaronBergman I think his point is that math provides a toolbox of concepts that have already been stress-tested. Only then can others afford to use them more sloppily.
Dec 12, 2019 at 3:25 comment added Aaron Bergman This is getting a lot of upvotes, but I’m not sure I agree. The number of physicists who could tell you what a bounded operator is, much less the right definition of the spectrum (which I just had to look up) is a small fraction of the whole in my experience. Most physicists are at least a sloppy with definitions as they are with theorems. Most of the time, for whatever reason, it just doesn’t seem to matter, and the mathematicians can clean everything up later (at least with QM — we haven’t been so lucky with QFT).
Dec 11, 2019 at 20:56 comment added Michael Engelhardt Well - theorems are important too, say, the residue theorem. But I agree, the relative importance of definitions vs. theorems is usually underestimated. Language guides patterns of thinking.
Dec 11, 2019 at 19:18 comment added Noah Schweber +1 for the first two sentences of the second paragraph alone.
Dec 11, 2019 at 18:10 comment added Abdelmalek Abdesselam I agree with the importance of definitions. For a recent example see this article arxiv.org/abs/1911.07895 by Binder and Rychkov where they establish some theorems but perhaps more importantly give definitions of QFT models with $N$ component valued fields even if $N$ is not a positive integer. BTW the big open problem "Is there a measure spece on which path integrals make sense" has a solution: for Euclidean path integrals on flat space, the measure space is the space of temperate distributions $\mathcal{S}'(\mathbb{R}^d)$.
Dec 11, 2019 at 18:00 history answered Paul Siegel CC BY-SA 4.0