Clifford Algebra - Spinors - Real Spinors

Real Spinors

For more details on this topic, see spinor.

To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pin_p,q is the set of invertible elements in Cℓ_{p, q} which can be written as a product of unit vectors:

Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group O(p, q). The Spin group consists of those elements of Pin_{p, q} which are products of an even number of unit vectors. Thus by the Cartan-Dieudonné theorem Spin is a cover of the group of proper rotations SO(p,q).

Let α : Cℓ → Cℓ be the automorphism which is given by the mapping v ↦ −v acting on pure vectors. Then in particular, Spin_p,q is the subgroup of Pin_{p, q} whose elements are fixed by α. Let

(These are precisely the elements of even degree in Cℓ_{p, q}.) Then the spin group lies within Cℓ0_{p, q}.

The irreducible representations of Cℓ_{p, q} restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of Cℓ0_{p, q}

To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)

Cℓ0_p,q ≈ Cℓ_p,q−1, for q > 0

Cℓ0_p,q ≈ Cℓ_q,p−1, for p > 0

and realize a spin representation in signature (p,q) as a pin representation in either signature (p,q−1) or (q,p−1).