differential geometry - Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$.

This is often visualized via the Dirac belt trick, constructing "spinorial objects" with strings attached to the surrounding space. But what does that really mean?

• Are spinors somehow connected to spacetime?


• Spinors maintain an "imprint" of how they have been rotated (path dependence/memory) - how is that possible?


• I understand the topological argument with the simply connectedness of the universal cover vs. the original rotation group - but how can a Dirac particle "sense" the topology?


• 
  

  In the Dirac trick, the imprint of the path (number of rotations) is clearly visible to anybody by the number of twists in the belt! So I dont find its "path memory" as mysterious as for the free fermion. An electron is assumed to be structureless without any inner degrees of freedom except spin - so how can it "keep track" of the number $n$ of twistings just like the belt connected to some fixed background?

  
  

  The distortion/twisting of the belt is in plain sight! I can count it simply looking at the system itself. This distortion is clearly a feature of the system. So it is not so suprising that the two situations (odd or even $n$) are distinguished. But for a spinor, there is no such thing to keep track of $n$ - the free Dirac particle does not interact with anything!

  
  

I am familiar with the usual arguments (homotopy classes etc), but those do not resolve my issue/trouble making sense of spinorial objects - therefore I need further help. Thank you very much!

Answer

I'm not entirely sure what OP's question (v4) is asking, but here are some comments:

I) The Dirac belt trick demonstrates that the Lie group $SO(3)$ of 3D rotations is doubly connected, $$ \pi_1(SO(3))~=~\mathbb{Z}_2. $$

_{(source: naukas.com)}

II) As for the title question Are spinors somehow connected to spacetime? one answer could be: Yes, in the sense that the mere existence of spinors puts topological constraints on possible spacetimes. In detail, the existence of a globally defined (Weyl) spinor on a (spacetime) manifold $M$ has the following topological implications for $M$:

1. 
   
   

   The (spacetime) manifold $M$ should be orientable, i.e. the 1st Stiefel-Whitney class $w_1(M)\in H^1(M,\mathbb{Z}_2)$ should vanish.

   
   


2. 
   

   The 2nd Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb{Z}_2)$ should vanish as well, cf. e.g. Wikipedia.