Spinor representation

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In mathematics, a spinor representation is a particular kind of projective representation of a special orthogonal group, or orthogonal group. The group acts in a particular way on spinors, which are a geometrically-twisted type of vector. The point of the approach through representation theory is to describe the geometry involved by algebraic means.

[edit] Description

Spinor representations of \mathfrak{so}(p,q) are modules of the Clifford algebra

Clp,q,

which are generated as a ring by a vector space

Cp+q.

Weyl spinors exist when p+q is even. Weyl spinors which are annihilated by a maximum dimension ((p+q)/2) subspace of this vector space are called pure spinors.

[edit] Complex representation of the Lie algebra

Let's focus on complex representations first. It's convenient to work with the complexified Lie algebra

\mathfrak{so}(p,q)

of the special orthogonal group

SO(p,q).

Since the complexification of \mathfrak{so}(p,q) is the same as the complexification of \mathfrak{so}(p+q), we can focus upon the latter, at least for complex representations.

The rank of \mathfrak{so}(2n) is

n = p + q,

and its roots are the vectors that are permutations of

(\pm 1,\pm 1, 0, 0, \dots, 0)

where: there are n coordinates; and all but two are zero ;and the absolute values of the nonzero coordinates are 1. (This does not apply to \mathfrak{so}(2), which isn't semisimple.)

The rank of \mathfrak{so}(2n+1) is n and its roots are the permutations of

(\pm 1, \pm 1, 0, 0, \dots, 0)

and the permutations of

(\pm 1, 0, 0, \dots, 0).

For \mathfrak{so}(2n), there is an irreducible representation whose weights are all possible combinations of

(\pm {1\over 2},\pm {1\over 2}, \dots, \pm{1\over 2})

with an even number of minuses and each weight has multiplicity 1. This is a Weyl spinor and it is 2n-1 dimensional.

There is also another irreducible representation whose weights are all possible combinations of

(\pm{1\over 2},\pm{1\over 2},\dots,\pm{1\over 2})

with an odd number of minuses and each weight has multiplicity 1. This is an inequivalent spinor and it is 2n-1 dimensional.

The direct sum of both Weyl spinors is a Dirac spinor.

Let's now go over to \mathfrak{so}(2n+1). Here, there's an irreducible representation whose weights are all possible combinations of

(\pm {1\over 2},\pm {1\over 2},\dots,\pm{1\over 2})

and each weight has multiplicity 1. This is a Dirac spinor and it is 2n dimensional.

In both even and odd dimensions, the tensor product of the Dirac representation with itself contains the trivial representation, the vector representation and the adjoint representation. The first means the Dirac representation is self-dual. The second means there is a nonzero intertwiner from the tensor product of the vector representation and the Dirac representation to the dual of the Dirac representation. This is represented by the γ matrices, γi.

In 4n dimensions, each Weyl representation is self-dual. In 4n+2 dimensions, both Weyl representations are duals of each other.

One thing to note, though, is these spinors are not unitary except in Euclidean space. This means complex conjugate representations and dual representations do not coincide for \mathfrak{so}(p,q) unless either p or q is zero.

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