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stub for spin-statistics theorem. Just recording a first few references so far.
Added an idea paragraph and the reference to the Guido/Longo paper.
There’s a really nice discussion of this from a conceptual POV in Tony Zee’s Quantum Field Theory in a Nutshell.
Sure, why not. Some people would say that handwaving of this kind does more harm than good, but I’m not one of them. (Where is the argument for spin 1/2 ? :-)
I’m not quite happy with the situation in the Haag-Kastler either, as far as I understand it, I think there should be an argument that is easier…
@Tim: Are you referring to Zee’s conceptual description? Wasn’t sure.
Yes, I mean the chapter II.4 Spin-Statistics Connection, subchapter “The price of perversity”: The problem with the kind of argument presented here is that there are several no-go theorems that say that the objects manipulated here cannot be made precise in certain ways, one of the best review papers in AQFT, Hans Halvorson, Michael Müger, “Algebraic Quantum Field Theory” (reference is on the AQFT page) lists some of these.
This leads some people working in AQFT to the conclusion that arguments like this should be avoided at all costs, even at an undergraduate level.
This reminds me of an anecdote about Heisenberg: First, he intended to study mathematics, but when he told the math professor whom he consulted in Göttingen about this (I forgot his name) that he had read “Raum, Zeit, Materie” by Weyl, the professor said: “Then you have been spoilt for mathematics anyway”. Heisenberg then went to Arnold Sommerfeld and decided to study physics instead…
Hmmm. Interesting. I’ve heard of the Halvorson/Müger paper but have never read it. I would be curious to know what kind of arguments we should be using with undergraduates. Personally, I prefer correctness over simplicity, but, on the other hand, I also prefer that the student have some level of actual understanding and not simply be able to parrot what I say. It’s a tough line to straddle (I sincerely hope people appreciate how hard it can be to teach undergraduates well - sometimes I feel as if undergraduate education gets treated as the “ugly duckling” as it were).
I like the Heisenberg anecdote, by the way. I went hunting for Weyl’s grave in Zürich in January but wasn’t able to find it (I found Pauli’s, Hopf’s, James Joyce’s, and Fritz Zwicky’s).
I would be curious to know what kind of arguments we should be using with undergraduates.
My outdated answer is: When I was a student, theoretical QFT was an optional class on the way to the Diplom, so the professors would simply say “we will do it our way and anyone who does not like it does not have to attend” (which means that there were introductory classes in QFT starting with the definition of a $C^*$ algebra).
There has been a big reform recently, the Bologna process, which changed that. I do not know how they do it now.
Bologna is good change for bad universities and disaster for good, as every democracy-like change in a structured society is: detrimental for elite and good for the masses.
Bologna is good change for bad universities and disaster for good, as every democracy-like change in a structured society is: detrimental for elite and good for the masses.
From my experience, this is quite right.
There has been endless discussion about this in the press here. One simple fact seemed to be implicit in each and every contribution that I have read, but was never made explicit:
nowadays there are simply two different kind of people going to universities, for two different kinds of reasons: one group wants to enter academia, the other group wants a high-level professional education. The whole problem is that these two groups are not being distinguished.
The whole problem is that these two groups are not being distinguished.
I would like to mention that, after my question at mathoverflow here about the PCT and the spin-statistics theorem did not get an answer (was worth a try anyway), I asked Professor Rehren from the Göttinger AQFT group the same question, and he confirmed that the references cited on the nLab are pretty much the state of the art. (He also mentions some papers about CFT that I don’t know, I’ll have a look).
Thanks, Tim!
By the way, I just read in a new QM text, that Feynman once challenged the physics/mathematics community to develop an elementary proof of the spin-statistics theorem. In 2002 the editor of the American Journal of Physics brought up Feynman’s challenge again as having gone unfulfilled. Apparently this is still the case.
“Elementary” is in the eye of the beholder. I think Feynman originally said (in response to a challenge) that he would would prepare a “freshman lecture” to explain the theorem, but eventually gave up. (“Freshman lecture” meaning something on the order of his famous lecture notes on physics.)
“Elementary” is in the eye of the beholder.
My concept of “simple” in this case is:
a) axioms used: Can I understand the physical motivation of those?
c) how easy is it to spot when and where and how which of the axioms are used in the proof?
For example, the Reeh-Schlieder theorem needs heavy math machinery like the SNAG-theorem and the edge-of-the-wedge theorem, but on the other hand it is very easy to see which axioms are used, and that e.g. the causality axiom is not (which is a little bit disappointing for anyone who thinks that the Reeh-Schlieder theorem violates causality and that we should change the causality axiom to dodge it).
…the editor of the American Journal of Physics brought up Feynman’s challenge again as having gone unfulfilled.
Ok, but is there any money in it?
@Tim: What happened to point b)? ;)
Ok, but is there any money in it?
Probably not, but if my nanotech startup takes off and I get rich (as opposed to simply delusional), I’ll sponsor a prize for it. :-)
finally polished up this bibitem
and added pointer to this one:
added (here) a bunch of references treating the spin-statistics theorem for non-relativistic particles via the topology of their configuration space of points.
Will put that list now into its on entry spin-statistics via configuration spaces – references, to be !include
-ed here and at configuration space of points
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