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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]

TheMac
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    Where have you seen the first definition? I'm only familiar with the second one. – pregunton Jul 17 '19 at 06:24
  • 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. – TheMac Jul 17 '19 at 06:35
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    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. – Moishe Kohan Jul 17 '19 at 06:47
  • https://en.wikipedia.org/wiki/Spinor as well as https://math.stackexchange.com/questions/444457/what-are-spinors-mathematically?rq=1. – Moishe Kohan Jul 17 '19 at 06:54
  • Here's one of the resources I'd stumbled on that seemed to suggest the former definition: Gerrit Coddens - "Spinors for Everyone". See Section 2.1, paragraph 2: "A 2x1 spinor φ can then be shown to be just a stenographic notation for a 2×2 SU(2) rotation matrix R... This is all that spinors in SU(2) are about. Spinors code group elements." – TheMac Jul 17 '19 at 07:04
  • If accurate, the "2x1 column as shorthand for 2x2 matrix" notion here does potentially answer my last question re: reconciling the definitions, but the overall approach seems slightly inconsistent with what I've read elsewhere, hence my confusion. – TheMac Jul 17 '19 at 07:08
  • But in the next paragraph he says "Stating that a spinor in SU(2) is a rotation is actually an abus de langage". My suggestion is to stay away from abusive notations. – Moishe Kohan Jul 17 '19 at 07:17
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    I would agree with @MoisheKohan. The presentation in that reference is quite idiosyncratic, and that's not the only problem, there are some sentences that are just wrong: "The automorphisms of a group G are themselves a group that is isomorphic to G". I guess this all could be explained by this sentence in Section 2.7: "The author has figured out the whole contents of the present paper from scratch because he found the textbook presentations impenetrable. The author has also not studied books on Clifford algebra in depth". – pregunton Jul 17 '19 at 07:23
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    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). – TheMac Jul 17 '19 at 07:33
  • That reference aside, I think some of my confusion was also stemming from the Clifford Algebras/Geometric Algebra characterizations on Wikipedia, where the Spin group is characterized as even products of unit vectors and spinors as elements of the even subalgebra; coming back to it with fresh eyes, I likely conflated the two without noticing. – TheMac Jul 17 '19 at 07:42

1 Answers1

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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 $(V,\rho)$.

Edit. The nicest detailed treatment of spinors that I know, from the mathematical viewpoint, is in the books:

  1. Lawson, H. Blaine jun.; Michelsohn, Marie-Louise, Spin geometry, Princeton Mathematical Series. 38. Princeton, NJ: Princeton University Press. xii, 427 p. (1989). ZBL0688.57001.

Specifically: Chapter I, sections 1-6 (explaining spin groups and their representations); Chapter II, section 1, 3, finally defining spinor fields. (This is before you get to the Dirac operators; Dirac operators are covered in sections 4-7 of Chapter II.)

  1. Friedrich, Thomas, Dirac operators in Riemannian geometry. Transl. from the German by Andreas Nestke, Graduate Studies in Mathematics. 25. Providence, RI: American Mathematical Society (AMS). xvi, 195 p. (2000). ZBL0949.58032.

Specifically, sections 1.1-1.5 of Chapter 1 and 2.1-2.3, 2.5 of Chapter 2. (Dirac operators are discussed in Chapter 3.)

However, both books work with Riemannian metrics and, hence, their spinors are defined with respect to the orthogonal spin group. If one is interested in spinors in the context of General Relativity, their treatment of spinors has to be modified accordingly and one should use pseudo-orthogonal groups and corresponding spin-groups.

Moishe Kohan
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    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)? – TheMac Jul 17 '19 at 07:56
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    @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... – Moishe Kohan Jul 17 '19 at 08:37
  • Got it, thanks so much! – TheMac Jul 17 '19 at 08:45
  • I accepted your earlier answer -- see also https://math.stackexchange.com/q/3299960/141334 – annie marie cœur Jul 22 '19 at 00:07
  • Sorry to comment on an older post but I wanted to make sure I have understood it correctly. So is a spinor an element of the space $(V, \rho)$, where $V$ is a vector space and $\rho$ is a representation of $Spin(n)$ that does not descend that $SO(n)$? In other words, whether an element of a vector space is an "ordinary vector" or a spinor depends on $\rho$ and whether it represents $SO(n)$ or its double cover, which is partly determined by the dimension of $V$. Is that right? – CBBAM Jun 15 '24 at 04:18
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    @CBBAM: Right. If you are familiar with the language of modules, one can say that spinors are elements of certain $\mathbb R Spin(p,q)$-modules. OP was asking about compact spin-group, but one frequently uses spinors for noncompact groups $Spin(p,q)$. – Moishe Kohan Jun 15 '24 at 04:20
  • @MoisheKohan Thank you for the quick answer! – CBBAM Jun 15 '24 at 04:24