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 two equivalent 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 be a finite-dimensional vector space over or , equipped with a nondegenerate quadratic form . The orthogonal group and special orthogonal group of are
Thus denotes the determinant-one subgroup, rather than necessarily the identity component of . If is real and is indefinite, then generally has two connected components; its identity component is denoted by or .
Over , every nondegenerate quadratic form is determined up to isomorphism by the dimension of . One may therefore take and
The corresponding groups and Lie algebra are denoted by , , , and . The group is connected.
Over , Sylvester's law of inertia shows that a nondegenerate quadratic form is determined up to isomorphism by its signature , where . One may take and
The corresponding groups and Lie algebra are denoted by , , , and .
The spin group is defined naturally through the Clifford algebra of . With the convention used here,
so that in . Let be its even subalgebra, and let denote reversion, the anti-automorphism determined by
The spin group may then be defined by
Conjugation in the Clifford algebra gives a homomorphism
In the complex case this gives an exact sequence
In real signature , the corresponding exact sequence is
Thus . The element here is the scalar element of the Clifford algebra and should not be confused with the orthogonal transformation .. Although is a twofold covering homomorphism, is not in general the universal covering group of in indefinite signature. The analogous construction for the full orthogonal group uses a pin group.
The differential of identifies the Lie algebra of the spin group with the orthogonal Lie algebra:
If is real, its complexification carries the complexified quadratic form , and
The inclusion of the real Clifford algebra into its complexification realizes as a real subgroup of .
A complex spinor module is an irreducible module for the complex Clifford algebra. Restricting its Clifford action to the spin group gives the complex spin representation. Its form depends on the parity of :
If is odd, the complex Clifford algebra has two inequivalent irreducible modules of dimension . Their restrictions to are equivalent and give a single irreducible spin representation, denoted by .
If is even, the complex Clifford algebra has one irreducible module of dimension . Its restriction to decomposes as
where and are inequivalent irreducible representations of dimension . They are called the positive and negative half-spin, chiral, or Weyl representations; is called the Dirac spin representation...
The central element acts on every spinor module as . Consequently, a spin representation does not descend to a linear representation of the corresponding special orthogonal group, but its projectivization does descend to a projective representation. The condition , however, does not characterize the spin representations: it merely says that an arbitrary representation of the spin group does not factor through the special orthogonal group.
For a real quadratic space of signature , the complex spin representations are obtained by restricting the complex spin modules to . The existence of invariant real, quaternionic, or Hermitian structures on these modules depends on modulo . It is therefore convenient first to construct the complex spin representations and then to analyze their restrictions and possible real structures.
Complex spin representations
Let with the standard quadratic form so that
The symmetric bilinear form on associated to by polarization is denoted .
Isotropic subspaces and root systems
Suppose that for . Then . We say that is isotropic and that it generates a one-dimensional totally isotropic subspace . Similarly, is isotropic and generates a one-dimensional totally isotropic subspace.
A standard construction of the spin representations of begins with a choice of a pair of maximal totally isotropic subspaces (with respect to ) of with . Let us make such a choice. If or , then and both have dimension . If , then , whereas if , then , where is the 1-dimensional orthogonal complement to . The bilinear form associated to induces a pairing between and , which must be nondegenerate, because and are totally isotropic subspaces and is nondegenerate. Hence and are dual vector spaces.
More concretely, let be a basis for . Then there is a unique basis of such that
If is an matrix, then induces an endomorphism of with respect to this basis and the transpose induces a transformation of with
for all in and in . It follows that the endomorphism of , equal to on , on and zero on (if is odd), is skew,
for all in , and hence (see classical group) an element of .
Using the diagonal matrices in this construction defines a Cartan subalgebra of : the rank of is , and the diagonal matrices determine an -dimensional abelian subalgebra.
Let be the basis of such that, for a diagonal matrix is the th diagonal entry of . Clearly this is a basis for . Since the bilinear form identifies with , explicitly, Chapter I.6, p.41. If we follow the convention of Chapter 20, p.303, then a factor 2 appears and the following formulas have to be changed accordingly it is now easy to construct the root system associated to . The root spaces (simultaneous eigenspaces for the action of ) are spanned by the following elements: with root (simultaneous eigenvalue) (which is in if with root with root and, if is odd, and is a nonzero element of , with root with root Thus, with respect to the basis , the roots are the vectors in that are permutations of
together with the permutations of
if is odd.
A system of positive roots is given by and (for odd) . The corresponding simple roots are
The positive roots are nonnegative integer linear combinations of the simple roots.
Spin representations and their weights
One construction of the spin representations of uses the exterior algebra(s) and/or There is an action of on such that for any element in and any in the action is given by:
where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairs and . This action respects the Clifford relations , and so induces a homomorphism from the Clifford algebra of to . A similar action can be defined on , so that both and are Clifford modules.
The Lie algebra is isomorphic to the complexified Lie algebra in via the mapping induced by the covering since if is the covering, then , so and since is a scalar, we get
It follows that both and are representations of . They are actually equivalent representations, so we focus on S.
The explicit description shows that the elements of the Cartan subalgebra act on by
A basis for is given by elements of the form
for and . These clearly span weight spaces for the action of : has eigenvalue −1/2 on the given basis vector if for some , and has eigenvalue otherwise.
It follows that the weights of are all possible combinations of
and each weight space is one-dimensional. Elements of are called Dirac spinors.
When is even, is not an irreducible representation: and are invariant subspaces. The weights divide into those with an even number of minus signs, and those with an odd number of minus signs. Both S+ and S− are irreducible representations of dimension 2m−1 whose elements are called Weyl spinors. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, the highest weights of S+ and S− are and respectively. The Clifford action identifies ClnC with End(S) and the even subalgebra is identified with the endomorphisms preserving S+ and S−. The other Clifford module S′ is isomorphic to S in this case.
When n is odd, S is an irreducible representation of so(n,C) of dimension 2m: the Clifford action of a unit vector u ∈ U is given by
and so elements of so(n,C) of the form u∧w or u∧w∗ do not preserve the even and odd parts of the exterior algebra of W. The highest weight of S is
The Clifford action is not faithful on S: ClnC can be identified with End(S) ⊕ End(S′), where u acts with the opposite sign on S′. More precisely, the two representations are related by the parity involution α of ClnC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of ClnC. In other words, there is a linear isomorphism from S to S′, which identifies the action of A in ClnC on S with the action of α(A) on S′.
Bilinear forms
if λ is a weight of S, so is −λ. It follows that S is isomorphic to the dual representation S∗.
When n = 2m + 1 is odd, the isomorphism B: S → S∗ is unique up to scale by Schur's lemma, since S is irreducible, and it defines a nondegenerate invariant bilinear form β on S via
Here invariance means that
for all ξ in so(n,C) and φ, ψ in S — in other words the action of ξ is skew with respect to β. In fact, more is true: S∗ is a representation of the opposite Clifford algebra, and therefore, since ClnC only has two nontrivial simple modules S and S′, related by the parity involution α, there is an antiautomorphism τ of ClnC such that
for any A in ClnC. In fact τ is reversion (the antiautomorphism induced by the identity on V) for m even, and conjugation (the antiautomorphism induced by minus the identity on V) for m odd. These two antiautomorphisms are related by parity involution α, which is the automorphism induced by minus the identity on V. Both satisfy τ(ξ) = −ξ for ξ in so(n,C).
When n = 2m, the situation depends more sensitively upon the parity of m. For m even, a weight λ has an even number of minus signs if and only if −λ does; it follows that there are separate isomorphisms B±: S± → S±∗ of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism B: S → S∗. For m odd, λ is a weight of S+ if and only if −λ is a weight of S−; thus there is an isomorphism from S+ to S−∗, again unique up to scale, and its transpose provides an isomorphism from S− to S+∗. These may again be combined into an isomorphism B: S → S∗.
For both m even and m odd, the freedom in the choice of B may be restricted to an overall scale by insisting that the bilinear form β corresponding to B satisfies (1), where τ is a fixed antiautomorphism (either reversion or conjugation).
Symmetry and the tensor square