Re: Has a moebius strip spin 1/2?

Dear Ewen!

Excellent observation! Although the connection between the rotational properties of spin and a Moebius strip is not as deep as you might have imagined, your question touches some of the most underlying concepts in mathematical physics. I will try to elaborate a bit on the mathematical background, assuming that you are not too well grounded in group theory and topology.

Spin 1/2 is a mathematical concept which is often misunderstood. Sometimes people imagine spin as a little `arrow' indicating some axis of `rotation' for an elementary particle. While it is correct that rotation has something to do with it, the connection is more subtle. Physicists describe rotations in space as a certain group, called SO(3). Now you don't really have to understand what a group actually is. Suffice it to say here that it is a set of elements (rotations in our case) and there is an operation which produces out of two group elements another one. This would be two successive rotations about two axes, which can be described as a single rotation about another axis.

Now imagine a vector (an arrow) pointing somewhere in space. The vector has three components, i.e. we have to give three numbers to tell its length and direction. Now mathematics tells us how to treat such an entity under rotations: We take an element of SO(3), represented in this case by a 3x3 matrix, and multiply the matrix with the vector to get the new, rotated vector. So far so good. Now physicists have found out that spin 1/2 particles also change under rotations, just like vectors. The difference is that spin 1/2 is not represented by three numbers like a vector, but only by two numbers. Now don't make the mistake to imagine this as a vector that can only have certain directions in space; this entity (called a spinor, btw) does not live in ordinary three-dimensional space as we know it. Nevertheless, mathematics knows how to rotate a spinor (or better, how it chenges when viewed from different directions). In this case we need a 2x2 matrix to do the trick.

But there is something I haven't told you yet: Although the 2x2 matrices I have just talked about describe the same type of rotations as the 3x3 ones from the previous paragraph, the group they belong to is slightly `larger' than SO(3); it contains more elements, and for the sake of completeness I will tell you that it's called SU(2). This makes hardly any difference when the rotations are about small angles, but when the angle becomes large the SU(2) rotations show some peculiarity, especially when the angle is 2 times pi (a full rotation). While a full rotation in SO(3) is the identity, the same rotation in SU(2) is the inversion, i.e. the rotated spinor gets a minus sign. This is unbelievable: One might ask why electrons, which have spin 1/2, don't have exceptional properties because of that strange behaviour. The answer is that the minus sign is unobservable in practice (I will not elaborate on the reason for that). While the mathematical structure I have outlined is very important for the physics of spin 1/2 particles in general, going `once around an electron' will not change our perception of it in an way.

So let me summarize why spin 1/2 behaves in such a special way under rotations: Spin 1/2 particles transform under rotations according to a different representation of the rotation group, actually another group, which is slightly larger because it contains inversions (which SO(3) does not). This fact expresses itself as the minus sign when rotating an electron about a full angle.

Now what about the Moebius strip? Imagine wandering on the surface of such a strip, being able to see just a small portion of your environment, i.e. not the whole strip at once. This is now your universe. Go once around it: You will stand headlong! But you will not be able to deduce this behaviour from the start, just by watching your surroundings; this `rotation' is a global property of the strip. Locally, your world looks flat and uninteresting. See the connection to the spin example? Although locally, i.e. for small rotations SU(2) and SO(3) look the same (and a flat surface and your Moebius strip look the same), there is a global difference. If someone had cut the strip and glued it together again without the `twist' (and without your knowledge), you would actually have to make a full round trip do see what happened.

So there are mathematical structures which describe changes like rotations and translations, and it is possible to represent those structures in different ways which are identical locally, i.e. for small changes, but not globally. The physical situation dictates which representation one has to choose. Your observation is a consequence of this fact. The connection between spin 1/2 and the Moebius strip lives on that mathematical level; there is no deep physical insight.

It is entirely possible that my description was too mathematical for you. If that is so, feel free to send another question to the MSN.

Bye,
Georg.