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I've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presented as "vector-like" objects that transform under the Spin group (the latter seeming to be more common in physics settings).

Which description is more accurate? Is this a case of "overloaded" terminology referring to two different (albeit related) objects? Or are the two equivalent, and the distinction is irrelevant?

Lastly, if spinors are in fact elements of the Spin group—and thus also of the relevant Special Unitary group—how does one reconcile this with the way spinors are typically depicted in physics? In 3 dimensions, for example, how would one reconcile the concept of spinors as elements of SU(2) with their usual depiction as two-component complex column "vectors"?

Thanks!

[Apologies if this question is too "physics-y"—I'm mainly interested in the pure math perspective here, but I'll repost to the Physics SE if it's not appropriate for this forum]

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    $\begingroup$ Where have you seen the first definition? I'm only familiar with the second one. $\endgroup$ – pregunton Jul 17 at 6:24
  • $\begingroup$ I've bounced around a bunch of different resources lately, so I can't immediately recall a specific example off the top of my head; let me look back and see if I can find one for you. I'm fairly certain it's come up a few times, but there's also a very decent chance I misunderstood something along the way. $\endgroup$ – TheMac Jul 17 at 6:35
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    $\begingroup$ I never seen the first one either, but in physics literature people routinely conflate elements of a Lie groups with elements of its Lie algebra, so it would not be surprising if they also conflate elements of a vector space on which $Spin(n)$ is acting with elements of $Spin(n)$. But, in addition, one frequently uses the word spinor to denote sections of a vector bundles associated with $Spin$-representations, i.e. spinor fields are frequently called spinors. $\endgroup$ – Moishe Kohan Jul 17 at 6:47
  • $\begingroup$ en.wikipedia.org/wiki/Spinor as well as math.stackexchange.com/questions/444457/…. $\endgroup$ – Moishe Kohan Jul 17 at 6:54
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    $\begingroup$ Oof... I skimmed around a bit and must have missed that sentence—that really ought to have tipped me off (though granted, I was already pretty confused even before I found that reference). $\endgroup$ – TheMac Jul 17 at 7:33
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Just to wrap up the discussion. Let $Spin(n)$ denote the spin-group (the 2-fold covering group) of $SO(n)$. Then spinors (with respect to the group $Spin(n)$) are elements of a vector space $V$ on which $Spin(n)$ acts via a (usually) irreducible finite-dimensional linear representation $(V,\rho)$ which does not descend to a representation of $SO(n)$. (In this generality, the notion of a spinor depends on the particular choice of a linear representation.) In most cases, $V$ is taken to be a fundamental representation of $Spin(n)$ constructed via the Clifford algebra $Cl(n)$ (associated with $R^n$ equipped with its inner product, used to define the orthogonal group $O(n)$). Thus, spinors are not elements of the spin-group. Regarding spinors as such is an abuse of notation which one should avoid.

One frequently generalizes this definition of spinors to include not just vectors (in a suitable vector space $V$) but spinor fields which are (smooth) sections of some vector bundle over an $n$-dimensional manifold $M$. Such a vector bundle is derived from a spinor representation $(V,\rho)$ via the associated vector bundle construction. With this definition, in local coordinates, spinors appear as (smooth) maps $$ U_\alpha\subset R^n\to V, $$ which transform, under local change of coordinates, according to the spinor representation $(V,\rho)$.

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  • $\begingroup$ Thanks for clearing things up! Just to ask for one final clarification—would it be accurate, in that case, to say that spinors don't really "exist" outside of a given choice of representation, while Spin(n) has meaningful representation-independent significance through its connection to SO(n)? $\endgroup$ – TheMac Jul 17 at 7:56
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    $\begingroup$ @TheMac: Right. However, in the literature people frequently abuse the terminology and refer to spinors as elements of the fundamental representation (unique up to isomorphism). One can say that this representation is the most important one... $\endgroup$ – Moishe Kohan Jul 17 at 8:37
  • $\begingroup$ Got it, thanks so much! $\endgroup$ – TheMac Jul 17 at 8:45
  • $\begingroup$ I accepted your earlier answer -- see also math.stackexchange.com/q/3299960/141334 $\endgroup$ – annie heart Jul 22 at 0:07

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