Hello I'm a quantum mechanics newbie.

I learned about spinors, and how they are different from vectors because unlike vectors, rotating them once does not give the original spinor, but the negative of it.(Although while proving this, we only learned the definition of rotation operators and didn't learn how they were derived or what they meant (the professor said we will learn it in our senior years).

Anyways, this made me wonder.... are there things that, instead of spinors, has to be rotated 3 times, or 4 times, ... n times to come back to what it was? (will it be something like, if it comes back to what it was after n rotations, something like $ e^{2\pi/n} $ multiplied after one rotation? (So that it comes back to its position after n rotations))


In more than two spatial dimensions, there are no such objects.

In two dimensions, there are such objects, known as anyons, which can have not only the half-integer spin of spinors, but any fractional spin.

I'm afraid the underlying reason is rather technical:

The reason why the spatial dimension is important is because the rotation group has a different topological structure in two dimensions than in more dimensions. The rotation group $\mathrm{SO}(2)$ is the circle and has infinite fundamental group, while the higher orthogonal groups $\mathrm{SO}(n)$ have only the two-element group as their fundamental group. So passing to the universal cover when seeking the quantum representations of the rotation group only "doubles" the available representations from the vectorial ones for higher dimensions, but yields infinitely many anyonic representations between two vectorial representations in two dimensions.

For why the topology of the rotation group and universal covers are relevant to quantum mechanics to begin with, see this Q&A of mine.


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