I asked this question on Math.SE as I thought the perspective of representation theory might be enlightening.

But since the question was provoked by a description of Spinors describing the spin of electrons in Dr Tongs notes where he described that 'one has to walk around an electron twice' for it to return to the same position; I thought I'd also ask it here.

Consider a Möbius strip; draw on one side of it an arrow aligned vertically; now take it for a trip by around the strip; then when it comes back to the same position it has flipped direction; another circumnavigation of the strip returns it to the right way up.

Now Spinors have to be rotated twice to return it to the same position.

Can these two pictures be connected in some way?

There is also this plate and belt trick; which might or might not be connected.

  • $\begingroup$ On an orientable manifold, to have spinors one has to find a lifting of the principle bundle associated with $SO(n)$ to the $Spin(n)$ (i.e. spin structure). For non-orientable manifold, the frames now lie in $O(n)$ and the lifting problem is $O(n)$ to $Pin^\pm(n)$. One can show that there is no obstruction in doing so for 2D Riemann surfaces. For a physicist-friendly discussion, you may want to take a look at projecteuclid.org/download/pdf_1/euclid.cmp/1104159727. $\endgroup$ – Meng Cheng May 7 '15 at 16:22
  • $\begingroup$ @cheng: a section of a principal bundle whose structure group is Spin(n) is essentially a spinor field? $\endgroup$ – Mozibur Ullah May 7 '15 at 16:55
  • $\begingroup$ Except we can't have global sections, so I should qualify that as a local section. $\endgroup$ – Mozibur Ullah May 7 '15 at 16:56
  • $\begingroup$ Essentially, yes, although as you already pointed out there is no section. Spin(n) bundle really just tells you how to rotate the Dirac matrices together with the local framing. $\endgroup$ – Meng Cheng May 7 '15 at 16:59
  • $\begingroup$ How does one connect Dirac matrices to Spin(n); I just think of it as the universal cover of the rotation group SO(n), which happens to be a double cover; is it a coordinatisation? In the sense matrices coordinatisation SO(n)? $\endgroup$ – Mozibur Ullah May 7 '15 at 17:03

Can these two pictures be connected in some way?

Yes, that's why the Wikipedia spinor article features a picture of a Möbius strip: enter image description here GNUFDL image by Slawekb, see Wikipedia

The Mobius strip also features in the Mathspages Dirac's belt article where you can read that it's "reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state". Dirac's belt is your "belt and plate trick".

You might want to look at the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". Also read about Goudsmit and the discovery of electron spin. The Leiden article is offline at the moment, but the key sentence is this: "It means that the electron has a spin, that it rotates". If you also look at an old version of the Wikipedia Stern-Gerlach article you can spot the non-sequitur which I will paraphrase as: the electron can't be rotating like a planet, so it can't be rotating at all. Well duh, of course it isn't rotating like a planet. It's a spin ½ particle. It's a bispinor. It rotates round the major axis, AND round the minor axis. The AND serves as a multiplier. Note that in atomic orbitals electrons "exist as standing waves", and that standing waves look motionless, even though they're not. We can diffract electrons. The wave nature of matter is not in doubt. So what sort of wave are we talking about? One that moves in a straight line at c? Methinks not.


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