Timeline for Any real contribution of functional analysis to quantum theory as a branch of physics?
Current License: CC BY-SA 4.0
31 events
| when toggle format | what | by | license | comment | |
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| Jan 29, 2020 at 2:56 | comment | added | none | Random matrix theory is used in QM a lot, and is apparently now affected quite a bit by free probability, which came out of functional analysis. Maybe that's too indirect though. | |
| Dec 16, 2019 at 9:46 | comment | added | Kostya_I | I actually don't disagree with you. But compare two questions: "what did representation theory contribute to real physics" and similar for FA. While you may answer "definitions" to the first question, a physicist would likely rather say "selection rules, 3j symbols etc." It is not clear (to me) whether FA affected physicist's toolbox in a similar (or any) way - Schroedinger and hordes of undergraduates lived rather happily without the right definitions. I'm not saying yours is not a valid answer, just that it sounds less convincing (to me) than a more concrete one would. | |
| Dec 16, 2019 at 9:29 | comment | added | Kostya_I | @PaulSiegel, I think you answered your own question: if you insist on defining particles as representation theoretical object (as opposed to "something that leaves traces in a cloud chamber"), you are thereby making a prediction (that the irrep of the Poincare group is all that matters). Wigner's theorem then sharpens that to a prediction that mass and spin/helicity is all that matters. | |
| Dec 15, 2019 at 20:21 | comment | added | Timothy Chow | @PaulSiegel : I mostly agree with you, but perhaps you're overstating the case a bit. Consider general relativity. Certainly, Einstein's work in creating the model was the most important step. But there are enough mathematical difficulties that finding solutions to the equations is a hugely important task in its own right. This seems to be a case where mathematical theorems ("such-and-such is a solution to the equations of general relativity") lead directly to predictions. | |
| Dec 15, 2019 at 8:24 | comment | added | Paul Siegel | @Kostya_I Let's take your first example. What does Wigner's theorem predict? Hopefully not the existence of a fundamental particle for every projective irrep of the Poincare group, because there is one of the latter for every positive real mass. The theorem does something better: it shows how to define particles as representation theoretic objects, yielding a framework for doing calculations which make predictions. I'm not making the strawman argument that theorems aren't important to physicists, just that their real purpose is to provide good definitions. | |
| Dec 13, 2019 at 21:01 | comment | added | Francois Ziegler | @Kostya_I Other than maybe Fermi-Pasta, of which I don’t know enough to tell, I believe your examples are actually post-dictions of the exact kind pragmatists (not me!) might (and apparently sometimes did) dismiss as “cocktail party apology”. That is why I prefer the longer-term view of math’s role reflected in Paul’s answer (or my examples, to which one might add Fourier analysis, Clifford algebra,...) | |
| Dec 13, 2019 at 9:04 | answer | added | Kostya_I | timeline score: 7 | |
| Dec 13, 2019 at 8:05 | comment | added | Kostya_I | ... concretely, though, there does exist a sentiment among some physicists (explicitly stated in textbooks) that making QM rigorous does not pay off in terms of physical understanding. You may just look for solutions to stationary Schroedinger equation; if your potential is reasonable, they are either exploding at infinity (unphysical), or bounded but don't go to zero (continuous spectrum), or they go to zero (discrete spectrum). Who needs the general definition of the spectrum of an unbounded operator in a Hilbert space? Would be nice to see concrete examples proving that wrong. | |
| Dec 13, 2019 at 7:41 | comment | added | Kostya_I | @PaulSiegel, examples of theorems that do make predictions are abundant. Wigner's theorem on classification of representations of Poincaré group was proven with the purpose of classifying particles one can expect to discover. The behavior of Fermi-Pasta-Ulam chain was a complete mystery before KAM theorem came along. Spin-statistics theorem... Index theorem... Also, many theorems just underscore and organize general features of calculations, so I do not agree with your countraposing the two. | |
| Dec 13, 2019 at 0:54 | comment | added | Paul Siegel | @Kostya_I Theorems don't make predictions - models do. The hard part about building mathematical models is choosing the right definitions, e.g. Einstein metrics on manifolds or Lie group representations on Hilbert spaces - this is what you need to do calculations. If those calculations consistently agree with experiment then the model is accepted - nobody outside of mathematics could care less whether they are properly justified by theorems. Clarifying and validating definitions (with theorems!) is the primary purpose of mathematical work - that's a boast, not an apology. | |
| Dec 13, 2019 at 0:02 | history | reopened |
Kostya_I Sebastien Palcoux dhy Suvrit Alexandre Eremenko |
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| Dec 12, 2019 at 17:54 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Dec 12, 2019 at 12:25 | review | Reopen votes | |||
| Dec 13, 2019 at 0:05 | |||||
| Dec 12, 2019 at 12:15 | comment | added | Kostya_I | ... most of answers so far are rather weak in this sense, they are advanced versions of "look, you don't need maths IRL but it's helps you to learn to organize your thoughts" cocktail party Maths apology. Closing the question that has chances to get real answers is a pity. | |
| Dec 12, 2019 at 12:09 | comment | added | Kostya_I | I do not agree closing. There is a well-established notion of what real physics is (e. g. "something for which physics Nobel prize can be awarded" or "predicting or explaining experimentally observable phenomena") as well as some common notion of what a real contribution of functional analysis would be. At least, "realness" defines a rather obvious order in both aspects, and hence this question is as good as asking for the strongest known version of a theorem. I would very much like to see answers of the form "a (non-trivial) theorem A informed prediction of phenomenon B", if possible. | |
| Dec 12, 2019 at 8:27 | history | closed |
user6976 Timothy Chow Johannes Hahn Bombyx mori Max Horn |
Opinion-based | |
| Dec 12, 2019 at 5:37 | answer | added | Terry Loring | timeline score: 17 | |
| Dec 11, 2019 at 19:29 | history | became hot network question | |||
| Dec 11, 2019 at 18:57 | comment | added | jjcale | One important branch of functional analysis is the theory of integral equations. And integral equations appear in quantum scattering theory. Another example is the integral quantum hall effect, where the hall conductivity is identified with a Fredholm index. | |
| Dec 11, 2019 at 18:00 | answer | added | Paul Siegel | timeline score: 50 | |
| Dec 11, 2019 at 17:29 | comment | added | Steve Huntsman | physics.stackexchange.com/a/43519/711 | |
| Dec 11, 2019 at 17:22 | answer | added | Michael Engelhardt | timeline score: 10 | |
| Dec 11, 2019 at 15:50 | review | Close votes | |||
| Dec 12, 2019 at 8:27 | |||||
| Dec 11, 2019 at 15:48 | comment | added | Timothy Chow | I've voted to close as "opinion-based". I agree with Konstantinos Kanakoglou that "real" is an opinion-based, or at least unclear, word here. | |
| Dec 11, 2019 at 14:42 | answer | added | Francois Ziegler | timeline score: 12 | |
| Dec 11, 2019 at 12:44 | comment | added | Konstantinos Kanakoglou | Or would you even be interested in insights in the understanding of QM which are supported by the mathematical properties and methods ? | |
| Dec 11, 2019 at 12:42 | comment | added | Konstantinos Kanakoglou | I am not sure i understand the term "real" contribution. Do you mean some development in QM which has been dictated by some result of functional analysis (instead of some demand imposed by experimental data or physical arguments)? | |
| Dec 11, 2019 at 12:12 | comment | added | Nik Weaver | I agree with the entire quote, not just the bolded part. | |
| Dec 11, 2019 at 12:11 | answer | added | Alexandre Eremenko | timeline score: 30 | |
| Dec 11, 2019 at 11:35 | comment | added | lcv | I don't understand the question. You mean real contributions to quantum physics beyond those of von Neumann? Are you alluding to quantum field theory? Otherwise, for what concerns the explanation of atoms and molecules, functional analysis and quantum mechanics are essentially the same thing. Rightly seen from two different perspective (math vs physics). | |
| Dec 11, 2019 at 11:24 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |