# Necessity, significance of Spinors

This is an area I am researching at my own pace, general rotations in 3D. I've known about the plate trick for a while as well, and have a very rough understanding of the concept of simple-connectedness and where it may be important.

I'm trying to understand where the importance is with having the ability to spatially orient an abstract entity without 'tangling', in the context of having a topology free from 'handles'.

I can appreciate the utility with this, if there was a need to describe an infinitude of connected, orientable points free from isolated spaces. I'm at a loss to picture where this could be utilized and for what reason, above some other approach.

For instance, fermions are spinors of a sort, so that tells me that a fermion field is such an infinitude of simply-connected orientations and magnitudes in space. Assuming this approach is on the right track, I'm at a loss to see the necessity in the connectedness of orientations. More bluntly, since space-time does not appear to be 'foamy' or 'grainy', I'm not sure how or why it needs to be connected.

Is it as simple as, to put crudely, a requirement as to be able to define what is and is not 'local' to a point in space(time)? If so why does orientation matter or even affect connectedness?

## 1 Answer

Classically, spinors are insignificant, and should not be expected to fulfill any physical role, since they are not proper representations of the rotation group $\mathrm{SO}(3)$ or the Lorentz group $\mathrm{SO}(1,3)$. The importance of spinors does not arise from any classically intuitive thinking about orientations.

Instead, the reason why spinors appear is fully quantum: Quantumly, the space that has to carry the representations of our symmetry group is the Hilbert space of states, but only rays in this space are different states - the "actual" space of states is the projective Hilbert space, and it are thus the projective representations of our symmetry group that are relevant to quantum physics.

It turns out that, for a large class of Lie groups - the semi-simple ones, or rather those whose complexified Lie algbera is semi-simple - their projective representations are in bijection to linear representations of their universal cover - which is by definition the simply-connected group covering the group. Therefore, spinors appear naturally in quantum physics as allowed representations of the rotation or Lorentz group, and since they are the fundamental, i.e. "simplest", representations of the universal cover, it also seems natural that they should appear, which one experimentally easily verifies by observing fermionic behaviour.