Clifford algebras are well-known to physicists, in the guise of matrix algebras generated by the -matrices first used in the Dirac equation. They also have a more abstract formulation, which will be the topic of this posting. One way to think about Clifford algebras is as a “quantization” of the exterior algebra, associated with a symmetric bilinear form.
Given a vector space with a symmetric bilinear form , the associated Clifford algebra can be defined by starting with the tensor algebra ( is the k-th tensor power of ), and imposing the relations
where . Note that many authors use a plus instead of a minus sign in this relation. The case of most interest in physics is the Minkowski inner product of signature (3,1). The theory of Clifford algebras for real vector spaces is rather complicated. Here we’ll stick to complex vector spaces , where the theory is much simpler, partially because over there is, up to equivalence, only one non-degenerate symmetric bilinear form. We will suppress mention of the bilinear form in the notation, writing for
For a more concrete definition, one can choose an orthonormal basis of . Then is the algebra generated by the , with multiplication satisfying the relations
One can show that these complex Clifford algebras are isomorphic to matrix algebras, more precisely
Clifford Algebras and Exterior Algebras
The exterior algebra is the algebra of anti-symmetric tensors, with product the wedge product . This is also exactly what one gets if one takes the Clifford algebra , with zero bilinear form. Multiplying a non-degenerate symmetric bilinear form by a parameter gives for non-zero a Clifford algebra that can be thought of as a deformation of the exterior algebra . Thinking of the exterior algebra on of dimension n as the algebra of functions on n anticommuting coordinates, the Clifford algebra can be thought of as a “quantization” of this, taking functions (elements of ) to operators (elements of , matrices in this case).
While is a graded algebra, is only -graded, since the Clifford product does not preserve degree but can change it by two when multiplying generators. The Clifford algebra is filtered by a degree, taking to be the subspace of elements that can be written as sums of generators. The exterior algebra is naturally isomorphic to the associated graded algebra for this filtration
and are isomorphic as vector spaces. One choice of such an isomorphism is given by composing the skew-symmetrization map
with the projection . Denoting this map by q, it is sometimes called the “quantization map”. Using an orthonormal basis , acts as
The inverse is sometime called the “symbol map”.
This identification as vector spaces is known as the “Chevalley identification”. Using it, one can think of the Clifford algebra as just an exterior algebra with a different product.
Clifford Modules and Spinors
Given a Clifford algebra, one would like to classify the modules over such an algebra, the Clifford modules. Such a module is given by a vector space and an algebra homomorphism
To specify , we just need to know it on generators, and see that it satisfies
One such Clifford module is , with
where is contraction by . This gives the inverse to the quantization map (the symbol map ) as
is not an irreducible Clifford module, and we would like to decompose it into irreducibles. For even, there will be a single such irreducible , of dimension , and the module map is an isomorphism. In the rest of this posting we’ll stick to the this case, for the odd dimensional case see the references mentioned at the end.
To pick out an irreducible module , one can begin by choosing a linear map such that and is orthogonal . Then let be the subspace on which acts by , be the subspace on which acts by . Note that is a complex vector space, and now has two linear maps on it that square to , multiplication by , and multiplication by . is an isotropic subspace of , since
for any . We now have a decomposition into two isotropic subspaces. Since the bilinear form is zero on these subspaces, we get two subalgebras of the Clifford algebra, and . It turns out that one can choose .
One can make this construction very explicit by picking a particular , for instance the one that acts on the element of an orthonormal basis by for . Letting we get a basis of . To get an explicit representation of as a module isomorphic to , we will use the formalism of fermionic annihilation and creation operators. These are the operators on an exterior algebra one gets from wedging by or contracting by an orthonormal vector, operators and for satisfying
In terms of these operators on , acts by
The Spin Representation
The group that preserves is , and its connected component of the identity has compact real form . has a non-trivial double cover, the group . One can construct explicitly as invertible elements in for , and its Lie algebra using quadratic elements of , with the Lie bracket given by the commutator in the Clifford algebra.
For the even case, a basis for the Cartan subalgebra of is given by the elements
These act on the spinor module as
with eigenvalues . is not irreducible as a representation of , but decomposes as into two irreducible half-spin representations, corresponding to the even and odd degree elements of .
With a standard choice of positive roots, the highest weight of is
and that of is
Note that the spinor representation is not a representation of , just of . However, if one restricts to the preserving , then the are the fundamental representations of this . These representations have weights that are 0 or 1, shifted by from those of the spin representation. One can’t restrict from to , but one can restrict to , a double cover of . On this double cover the notion of makes sense and one has, as representations
So, projectively, the spin representation is just , but the projective factor is a crucial part of the story.
The above has been a rather quick sketch of a long story. For more details, a good reference is the book Spin Geometry by Lawson and Michelsohn. Chapter 12 of Segal and Pressley’s Loop Groups contains a very geometric version of the above material, in a form suitable for generalization to infinite dimensions. My notes for my graduate class also have a bit more detail, see here.
In the next posting we’ll see what happens when one chooses , and studies the Clifford algebra