Is a "spinor" an element of the Spin group, or an object that transforms under the Spin group—or both?

Question

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]

Answer 1

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.)

2. 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.