Spin Representations

Abstract

In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields. Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures. The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions. Set up Let V be a finite dimensional real or complex vector space with a nondegenerate quadratic form Q. The (real or complex) linear maps preserving Q form the orthogonal group O(V,Q). The identity component of the group is called the special orthogonal group SO(V,Q). (For V real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, SO(V,Q) has a unique connected double cover, the spin group Spin(V,Q). There is thus a group homomorphism Spin(V,Q) → SO(V,Q) whose kernel has two elements denoted {1, −1}, where 1 is the identity element. O(V,Q), SO(V,Q) and Spin(V,Q) are all Lie groups, and for fixed (V,Q) they have the same Lie algebra, so(V,Q). If V is real, then V is a real vector subspace of its complexification V C := V ⊗ R C, and the quadratic form Q extends naturally to a quadratic form Q C on V C . This embeds SO(V,Q) as a subgroup of SO(V C , Q C), and hence we may realise Spin(V,Q) as a subgroup of Spin(V C , Q C). Furthermore, so(V C , Q C) is the complexification of so(V,Q). In the complex case, quadratic forms are determined up to isomorphism by the dimension n of V. Concretely, we may assume V=C n and The corresponding Lie groups and Lie algebra are denoted O(n,C), SO(n,C), Spin(n,C) and so(n,C). In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers (p,q) where n:=p+q is the dimension of V, and p-q is the signature. Concretely, we may assume V=R n and The corresponding Lie groups and Lie algebra are denoted O(p,q), SO(p,q), Spin(p,q) and so(p,q). We write R p,q in place of R n to make the signature explicit. The spin representations are, in a sense, the simplest representations of Spin(n,C) and Spin(p,q) that do not come from representations of SO(n,C) and SO(p,q). A spin representation is, therefore, a real or complex vector space S together with a group homomorphism ρ from Spin(n,C) or Spin(p,q) to the general linear group GL(S) such that the element −1 is not in the kernel of ρ.