Someone here recently noted that "The spin-statistics thing isn't a problem, it is a theorem (a demonstrably valid proposition), and it shouldn't be addressed, it should be understood and celebrated."

"Spin statistics" is of course the shorthand way of referencing a most curious fact about the universe, which is this: If a particle of any type has a "spin" measured in whole units of quantized angular momentum, it will be a boson, a group that includes energy-like force particles such as photons. However, if its spin is (rather strangely) off-set by half a unit, it will instead be a fermion, which includes the particles that occupy space and that make up most ordinary matter.

The rule is very simple. The explanation of it is arguably a bit more complicated.

I find it fascinating that Nobel Laureate Richard Feynman worried over this simple theorem for decades, yet he never seemed to find an explanation for it that truly satisfied him. It was not a lack of mathematical explanations, I should note. It was because Feynman deeply believed in a rather simple search heuristic: very simple relationships should in general also have simple, easily-conveyed explanations.

Alas, Feynman's last attempt to explain spin statistics, in his Dirac Lecture, always seemed to me one of his least clear bits of exposition ever. I am fairly confident Feynman would have assessed his Lecture that way himself, as he tended to be quite brutal in self-critiques on anything related to clarity of explanation.

(I think there is an interesting family insight in that observation, incidentally: Richard Feynman's scientifically inclined father always hoped that his son, who had received the education he was never able to have, would someday explain all those little physics mysteries to him. The young Richard took that duty very seriously, and never really abandoned it, even towards the end of his own life.)

So, my question and challenge: How is everyone doing on Feynman's spin-statistics challenge these days?

Do you, fair reader, have in your hands some truly simple explanation for why whole-spin particles always seems to be bosons, and ones with half-spin offsets always seem to be fermions?

I am not asking for twisty belts and wine glasses (please, no!), nor am I asking for something math free... though I do think anyone trying to answer this question should first look at how Feynman handled even complex numbers in his book QED. What I am asking for is insight, the kind of explanation that makes the reader stop and think wow, of course that's it, why didn't I see it what way before?

So, anyone? I probably will not put an explicit bonus on this one, but if someone can provide an explanation that knocks everyone's +1 socks off, I guarantee I'll contribute at least a couple of hundred points to that overall consensus.

  • $\begingroup$ For those who wanted to have a link to Feynman's Dirac lecture. youtu.be/cKzzG5DS6V8 $\endgroup$
    – aignas
    Mar 24, 2013 at 22:00
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    $\begingroup$ A playlist with all parts of Feynman's Dirac lecture is here: youtu.be/cKzzG5DS6V8?list=PLC3D8F5EA631EBA02 $\endgroup$
    – aignas
    Mar 24, 2013 at 22:12
  • $\begingroup$ @gns-ank, thanks, I didn't know it was available online! $\endgroup$ Mar 24, 2013 at 22:48
  • $\begingroup$ Honestly, your question is not clear for me, "why whole-spin particles always seems to be bosons", do you mean why Bosons behaves like Bosons? or why there is two types of fundamental particles? or how spin affects there behavior? $\endgroup$
    – TMS
    Mar 24, 2013 at 22:59
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    $\begingroup$ @lcv Any "beautiful proof" that doesn't use relativity is wrong, because the theorem requires relativity as a hypothesis. It's not true for nonrelativistic theories. $\endgroup$
    – knzhou
    Jul 17, 2019 at 0:08

1 Answer 1


The simplest proof of the spin-statistics theorem I know of is shown in the two videos on http://www.motionmountain.net/videos.html One video shows spin 1/2 behavior, the other video shows that fermion behavior follows automatically.

  • $\begingroup$ +1 for a really nice pair of belt videos, despite my very sincere "please no!" plea on that point :). "Exchanging buckles twice = 720$^{\circ}$" is going to confuse folks who see only a 360$^{\circ}$ rotation, but Feynman and others do explain why that is. Spin 1 becomes "no belts, just buckles", which alas is not mentioned in the description. The shared underlying math is correctly noted. But alas: I'm guessing that most folks who see even a good video like that will go away wondering why belts are "just like spin 1/2" (spinors) and buckles "just like spin 1". So: Good Feynman capture. $\endgroup$ Mar 28, 2013 at 18:18
  • $\begingroup$ The belt trick video is a classic, but it's just wrong. It tells us that phases acquired by rotation (related to spin) are related to phases acquired by physically exchanging particles. This is not the same thing as the phase associated with particle statistics, which is what the spin-statistics theorem is about, so the video doesn't prove it. $\endgroup$
    – knzhou
    Jul 17, 2019 at 0:06
  • $\begingroup$ To say it another way: the spin-statistics theorem only holds for relativistic theories. The belt trick proof doesn't mention relativity anywhere; it proves too much, so it must be wrong. $\endgroup$
    – knzhou
    Jul 17, 2019 at 0:07
  • $\begingroup$ @knzhou do not be too quick too dismiss the relevance of the belt trick: in the usual belt trick we take for granted the fact that the location of the belt is unphysical (i.e. we don't 'count' steps where we only move the belt and not the buckles). If these belts are certain strings sticking out of fermions, then relativity could be used to argue that these strings are topological (having to do with the arbitrariness of spatial slices) and can be moved around at will. $\endgroup$ Jun 30, 2021 at 19:34
  • $\begingroup$ @RubenVerresen Very interesting! Sounds deep, I'll have to think about it some more... $\endgroup$
    – knzhou
    Jun 30, 2021 at 20:15

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