Do fractional spins like 4/5 exist?

[jk22]

In the case of spin 1/2 the state has to be rotated twice 180° to recover the initial one.

If we consider a square made of arcs of equators on a sphere. The interior angle on the sphere is chosen to be 108°.

Then the sphere is rolled along those arcs on a plane.

The square hence draws segments with a 108° angle between them. On the plane this closes only if we roll the sphere twice on one segment and it draws a pentagon.

Hence the sphere had to be rotate 5/4 times 360°.

Could this in some sense describe a spin 4/5 system ?

 

[kith]

No. The commutation relations for angular momentum operators imply integer or half-integer values. This is a basic result which is proven in probably all textbooks.

 

[PeterDonis]

The commutation relations for angular momentum operators imply integer or half-integer values.

It's perhaps worth noting that this theorem holds in 3 or more dimensions, but not in 2 dimensions; in 2 dimensions there is a continuous range of allowed statistics, corresponding to a continuous range of angles $\theta$ that represent the phase shift on an exchange of particles. (Ordinary fermions and bosons correspond to $\theta = \pi$ and $\theta = 2 \pi$ respectively, which are the only possibilities in 3 or more dimensions.) Quasiparticles with "fractional" statistics are called "anyons", more here:

https://en.wikipedia.org/wiki/Anyon

Note that anyon statistics have been observed in real systems where the effective degrees of freedom are restricted to two dimensions, for example in the fractional Hall effect, as noted in the article.