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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!

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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.

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You might equally well ask, "How does the physical belt in the Dirac trick sense the topology?" This question is, when you think about it, no less mysterious than yours. The answer, by experiment, is that it simply does.

And ultimately, if something transforms "compatibly" with the Lorentz group, or with $SO(3)$, then there is really only a one-bit question to ask: are we talking about a representation of the original group (i.e. $SO(3)$ or $SO^+(1,\,3)$) or its double cover ($SU(2)$ or $SL(2,\,\mathbb{C})$)? There are no other possibilities (as you probably know). The question is pretty much the same for an electron or a Dirac ribbon.

The Dirac belt is only a physical analogy - albeit a pretty good one - for a homotopy class for a $C^1$ path through $SO(3)$ linking the identity to a given $\gamma\in SO(3)$. If you idealise the physical belt to a set of mathematical idealisations that intuitively seem pretty reasonable (i.e. in keeping with our experimental intuition gleaned from playing with ribbons and Dirac belts), then yes, the analogy becomes exact, as I discuss in Example 14.23 of my article "Lie Group Homotopy and Global Topology" on my website.

But then your question is tantamount to asking why does the real, physical object (Dirac belt) behave in the way described by the mathematical idealisations I talk about in my article. The answer is simply a wholly experimental one, to wit: it simply does, by experimental induction! You can't dig any deeper than this.

Now, you could ask a somewhat similar question along the lines of "Can we say spinors behave as though they are linked with little ribbons to spacetime, only in the sense of being mathematically analogous to the mathematical idealisation of a Dirac belt that, e.g. I discuss?" then the answer is of course yes. But this is subtly, but most assuredly, different from your question.

You can then say that any experimentally spinorial object: electron, any spin $\frac{1}{2}$ particle or indeed Dirac ribbon is experimentally seen to seem to somehow keep an "imprint" of how they have been rotated (path dependence/memory). Things that keep this imprint (i.e. recall the homotopy class) are by definition experimentally analogous to a member of the universal cover of $SO(3)$ or $SO^+(1,\,3)$.

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  • $\begingroup$ Thank you so much for your answer! I find it easier to "accept" that the physical belt in the Dirac trick senses the topology of $SU(2)$ for the very fact that it is connected to the surrounding! So the number of $2\pi$ rotations can be counted by the number of twists in the belt. Anybody - even a person who did not whitness the act of rotating the belt - can see the number $n$ of turns imprinted on the belt with their own eyes. In case of a spinor there is no such obvious thing that could keep track of the number of twistings - yet it magically does distinguish between odd and even $n$... $\endgroup$ – quantumorsch Feb 6 '15 at 13:31
  • $\begingroup$ @quantumorsch As I said, you can make a pretty reasonable mathematical idealisation, which maybe you can look at on my website. But the fact that the idealisation models the real device well is still experimental. But I think I'm at last getting you: you seem to be focussing on the "memory" aspect: the twists are like a state machine: is this what you're thinking? $\endgroup$ – WetSavannaAnimal Feb 6 '15 at 13:36
  • $\begingroup$ Yes, the memory aspect is what I find mysterious. In case of the physical belt, the number of twists $n$ is clearly visible for everybody to count. The imprint of the path is in plain sight! 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 anyting. $\endgroup$ – quantumorsch Feb 6 '15 at 13:46
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    $\begingroup$ @quantumorsch Good question. I am definitely still thinking so I shall be talking to you. Yours is one of those nagging questions (no criticism meant) that I can't work out whether the answer is deep, whether it is a tautological, or trivial or maybe all three! It is the kind of question a child asks (again, no criticism meant - quite the opposite) because we grownups are too contaminated with prejudice to see the question. Incidentally, a child of seven asked me the same kind of question some time ago, so you might like to see the answer I gave in the section "Amber Particles" .... $\endgroup$ – WetSavannaAnimal Feb 7 '15 at 2:40
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    $\begingroup$ @quantumorsch ... to my late primary schoolchild activity "Twist and Wonder" on my website that has been a bit hit at my daughter's school. Maybe you could read this whilst QMechanic and I are busy thinking! PS: I don't see why you are getting close votes just because it's a question we can't answer. $\endgroup$ – WetSavannaAnimal Feb 7 '15 at 2:42

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