They play an important role in the physical description of fermions such as the electron.
The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group.
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).
(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.)
In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimension n of V. Concretely, we may assume V = Cn and The corresponding Lie groups are denoted O(n, C), SO(n, C), Spin(n, C) and their Lie algebra as 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.
We write Rp,q in place of Rn to make the signature explicit.
A standard construction of the spin representations of so(n, C) begins with a choice of a pair (W, W∗) of maximal totally isotropic subspaces (with respect to Q) of V with W ∩ W∗ = 0.
It follows that the endomorphism ρA of V, equal to A on W, −AT on W∗ and zero on U (if n is odd), is skew, for all u, v in V, and hence (see classical group) an element of so(n, C) ⊂ End(V).
One construction of the spin representations of so(n, C) uses the exterior algebra(s) There is an action of V on S such that for any element v = w + w∗ in W ⊕ W∗ and any ψ in S the action is given by: where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairs W and W∗.
Both S+ and S− are irreducible representations of dimension 2m−1 whose elements are called Weyl spinors.
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.
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.
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.
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).
The symmetry properties of β: S ⊗ S → C can be determined using Clifford algebras or representation theory.
In fact much more can be said: the tensor square S ⊗ S must decompose into a direct sum of k-forms on V for various k, because its weights are all elements in h∗ whose components belong to {−1,0,1}.
If n = 2m+1 is odd then it follows from Schur's Lemma that (both sides have dimension 22m and the representations on the right are inequivalent).
The main outcome is a realisation of so(n,C) as a subalgebra of a classical Lie algebra on S, depending upon n modulo 8, according to the following table: For n ≤ 6, these embeddings are isomorphisms (onto sl rather than gl for n = 6): The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras.
However, there are additional "reality" structures that are invariant under the action of the real Lie algebras.
The type of structure invariant under so(p,q) depends only on the signature p − q modulo 8, and is given by the following table.
To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms.
The odd case is simpler, there is only one complex spin representation S, and hermitian structures do not occur.
The only special isomorphisms of real Lie algebras missing from this table are