Spinors and Möbius strips

Question

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.

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Mozibur Ullah

Comments

I do not know, but it reminds me of some of John Baez's writings. The punchline is this paper. The story starts here, and to my knowledge, spinorial stuff first get mentioned here.

This post imported from StackExchange Mathematics at 2015-06-03 19:35 (UTC), posted by SE-user pjs36

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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.

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Meng Cheng

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@cheng: a section of a principal bundle whose structure group is Spin(n) is essentially a spinor field?

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Mozibur Ullah

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Except we can't have global sections, so I should qualify that as a local section.

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Mozibur Ullah

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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.

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Meng Cheng

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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)?

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Mozibur Ullah

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Spinors are more closely related to the Plate trick - the fact that a 360-degree rotation can't be 'undone' continuously, but a 720-degree rotation can.

This post imported from StackExchange Mathematics at 2015-06-03 19:35 (UTC), posted by SE-user Steven Stadnicki

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More on belt trick: physics.stackexchange.com/q/163642/2451

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user Qmechanic

Answer 1

The two are related but not as closely as you might hope. The first thing happens because Möbius strips are not orientable. One way to describe what it means for a manifold to be orientable is that its tangent bundle admits a reduction of the structure group from $O(n)$ to $SO(n)$. There is a $\mathbb{Z}_2$ floating around here, one manifestation of which is the short exact sequence

$$1 \to SO(n) \to O(n) \xrightarrow{\det} \mathbb{Z}_2 \to 1$$

and another of which is the first Stiefel-Whitney class $w_1 \in H^1(BO(n), \mathbb{Z}_2)$.

Spinors instead happen on manifolds which are not only orientable but which also have a spin structure, which means a reduction of the structure group to a group called $Spin(n)$. There is a different $\mathbb{Z}_2$ floating around here, one manifestation of which is the short exact sequence

$$1 \to \mathbb{Z}_2 \to Spin(n) \to SO(n) \to 1$$

and another of which is the second Stiefel-Whitney class $w_2 \in H^2(BSO(n), \mathbb{Z}_2)$.

One thing to say here is that when people say spinors have to be rotated twice to be returned to the same position, this rotation is not happening in space. It is a "gauge transformation," or an "internal" symmetry. Mathematically this has to do with the reduction of the structure group to $Spin(n)$ and then the construction of associated vector bundles of the corresponding principal $Spin(n)$-bundle; spinors are sections of these.

Spin isn't called spin because anything is spinning in space, it's called spin because it behaves the way angular momentum behaves in quantum mechanics. Mathematically this is because both things are controlled by the representation theory of $Spin(3)$.

This post imported from StackExchange Mathematics at 2015-06-03 19:35 (UTC), posted by SE-user Qiaochu Yuan

Answer 2

Can these two pictures be connected in some way?

Yes, that's why the Wikipedia spinor article features a picture of a Möbius strip: 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.

This post imported from StackExchange Physics at 2015-06-03 19:33 (UTC), posted by SE-user John Duffield