Belt trick, spin 1/2 and spin sqrt(3/4)

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frank_k_sheldon@yahoo.co.uk - 06 Jan 2008 16:03 GMT

It is well known that Dirac's belt trick models the behaviour of
spin 1/2 particles. For example, if one turns the end of the
belt (the particle) by 4 pi, the belt can be untwisted even while
keeping the particle fixed. This illustrates that for spin 1/2
particles,
rotations by 4 pi and 0 pi are equivalent, whereas rotations
by 2 pi are not.

But there is another point. Spin 1/2 hbar is in fact the projection
on the z axis; the absolute value of the spin is sqrt(3/4) hbar.

How can one see both the 1/2 and the sqrt(3/4) values
in the belt visualization?
Or:
Is there a way to visualize the sqrt(3/4) value as well,
in the belt trick?

Frank

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pellis - 07 Jan 2008 13:00 GMT

On Jan 6, 4:03=A0pm, frank_k_shel...@yahoo.co.uk wrote:

> It is well known that Dirac's belt trick models the behaviour of
> spin 1/2 particles. For example, if one turns the end of the
[quoted text clipped - 13 lines]
>
> Frank

While I'm no expert, I suspect that the belt trick illustrates the
topology, more than the detailed geometry, and would be pleasantly
surprised if it could encompass both aspects of spin (as someone more
expert than I am may be able to say).

There's a very informative article in Scientific American by Bernstein
and Philips from July 1981, with a title, (from memory) something like
"Vector Bundles in Quantum Theory" that goes into the nature of spin,
and might allow you to think further about your question.

Paul

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Natural Science Forum / Physics / Research / January 2008

Belt trick, spin 1/2 and spin sqrt(3/4)