Spinors and Möbius strips

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.

• 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. – Meng Cheng May 7 '15 at 16:22
• @cheng: a section of a principal bundle whose structure group is Spin(n) is essentially a spinor field? – Mozibur Ullah May 7 '15 at 16:55
• Except we can't have global sections, so I should qualify that as a local section. – Mozibur Ullah May 7 '15 at 16:56
• 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. – Meng Cheng May 7 '15 at 16:59
• 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)? – Mozibur Ullah May 7 '15 at 17:03