**Clifford Algebras**

Clifford algebras are well-known to physicists, in the guise of matrix algebras generated by the [tex]\gamma[/tex] -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 [tex]V[/tex] with a symmetric bilinear form [tex](\cdot,\cdot)[/tex], the associated Clifford algebra [tex]Cliff (V,(\cdot,\cdot))[/tex] can be defined by starting with the tensor algebra [tex]T^*(V)[/tex] ([tex]T^k(V)[/tex] is the k-th tensor power of [tex]V[/tex]), and imposing the relations

[tex]v\otimes w + w\otimes v = -2(v,w)1[/tex]

where [tex]v,w\in V=T^1(V),\ 1\in T^0(V)[/tex]. Note that many authors use a plus instead of a minus sign in this relation. The case of most interest in physics is [tex]V=\mathbf R^4, (\cdot,\cdot)[/tex] the Minkowski inner product of signature (3,1). The theory of Clifford algebras for real vector spaces [tex]V[/tex] is rather complicated. Here we’ll stick to complex vector spaces [tex]V[/tex], where the theory is much simpler, partially because over [tex]\mathbf C[/tex] there is, up to equivalence, only one non-degenerate symmetric bilinear form. We will suppress mention of the bilinear form in the notation, writing [tex]Cliff(V)[/tex] for [tex]Cliff(V,(\cdot,\cdot)).[/tex]

For a more concrete definition, one can choose an orthonormal basis [tex]e_i[/tex] of [tex]V[/tex]. Then [tex]Cliff(V)[/tex] is the algebra generated by the [tex]e_i[/tex], with multiplication satisfying the relations

[tex]e_i^2=-1,\ \ e_ie_j=-e_je_i\ \ (i\neq j)[/tex]

One can show that these complex Clifford algebras are isomorphic to matrix algebras, more precisely

[tex]Cliff(\mathbf C^{2n})\simeq M(\mathbf C, 2^n),\ \ \ Cliff(\mathbf C^{2n+1})\simeq M(\mathbf C, 2^n)\oplus M(\mathbf C, 2^n)[/tex]

**Clifford Algebras and Exterior Algebras**

The exterior algebra [tex]\Lambda^*(V)[/tex] is the algebra of anti-symmetric tensors, with product the wedge product [tex]\wedge[/tex]. This is also exactly what one gets if one takes the Clifford algebra [tex]Cliff(V)[/tex], with zero bilinear form. Multiplying a non-degenerate symmetric bilinear form [tex](\cdot,\cdot)[/tex] by a parameter [tex]t[/tex] gives for non-zero [tex]t[/tex] a Clifford algebra [tex]Cliff(V, t(\cdot,\cdot))[/tex] that can be thought of as a deformation of the exterior algebra [tex]\Lambda^*(V)[/tex]. Thinking of the exterior algebra on [tex]V[/tex] 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 [tex]\Lambda^*(V)[/tex]) to operators (elements of [tex]Cliff(V)[/tex], matrices in this case).

While [tex]\Lambda^*(V)[/tex] is a [tex]\mathbf Z[/tex] graded algebra, [tex]Cliff(V)=Cliff^{even}(V)\oplus Cliff^{odd}(V)[/tex] is only [tex]\mathbf Z_2[/tex]-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 [tex]\mathbf Z[/tex] degree, taking [tex]F_p(Cliff(V))\subset Cliff(V)[/tex] to be the subspace of elements that can be written as sums of [tex]\leq p[/tex] generators. The exterior algebra is naturally isomorphic to the associated graded algebra for this filtration

[tex]\Lambda^p(V)\simeq F_p(Cliff(V))/F_{p-1}(Cliff(V))[/tex]

[tex]\Lambda^*(V)[/tex] and [tex]Cliff(V)[/tex] are isomorphic as vector spaces. One choice of such an isomorphism is given by composing the skew-symmetrization map

[tex]v_1\wedge v_2\wedge\cdots\wedge v_p=\frac{1}{p!}\sum_{s\in S_p}sgn(s)v_{s(1)}\otimes v_{s(2)}\otimes\cdots\otimes v_{s(p)}[/tex]

with the projection [tex]T^*(V)\rightarrow Cliff(V)[/tex]. Denoting this map by q, it is sometimes called the “quantization map”. Using an orthonormal basis [tex]e_i[/tex], [tex]q[/tex] acts as

[tex]q(e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_p})=e_{i_1}e_{i_2}\cdots e_{i_p}[/tex]

The inverse [tex]\sigma=q^{-1}:Cliff(V)\rightarrow \Lambda^*(V)[/tex] 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 [tex]M[/tex] and an algebra homomorphism

[tex]\pi: Cliff(V)\rightarrow End(M)[/tex]

To specify [tex]\pi[/tex], we just need to know it on generators, and see that it satisfies

[tex]\pi(v)\pi(w) +\pi(w)\pi(v)= -2(v,w)Id[/tex]

One such Clifford module is [tex]M=\Lambda^*V[/tex], with

[tex]\pi(v)\omega=v\wedge\omega – i_v\omega[/tex]

where [tex]i_v[/tex] is contraction by [tex]v[/tex]. This gives the inverse to the quantization map (the symbol map [tex]\sigma[/tex]) as

[tex]\sigma: a\in Cliff(V)\rightarrow \pi(a)1\in \Lambda^*(V)[/tex]

[tex]\Lambda^*(V)[/tex] is not an irreducible Clifford module, and we would like to decompose it into irreducibles. For [tex]dim_{\mathbf C}V =2n[/tex] even, there will be a single such irreducible [tex]S[/tex], of dimension [tex]2^n[/tex], and the module map [tex]\pi:Cliff(V)\rightarrow End(S)[/tex] 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 [tex]S\subset \Lambda^*(V)[/tex], one can begin by choosing a linear map [tex]J:V\rightarrow V[/tex] such that [tex]J^2=-1[/tex] and [tex]J[/tex] is orthogonal [tex]((Jv,Jw)=(v,w))[/tex]. Then let [tex]W_J\subset V[/tex] be the subspace on which [tex]J[/tex] acts by [tex]+i[/tex], [tex]\overline W_J[/tex] be the subspace on which [tex]J[/tex] acts by [tex]-i[/tex]. Note that [tex]V[/tex] is a complex vector space, and now has two linear maps on it that square to [tex]-1[/tex], multiplication by [tex]i[/tex], and multiplication by [tex]J[/tex]. [tex]W_J[/tex] is an isotropic subspace of [tex]V[/tex], since

[tex](v_1,v_2)=(Jv_1,Jv_2)=(iv_1,iv_2)=-(v_1,v_2)[/tex]

for any [tex]v_1,v_2\in W_J[/tex]. We now have a decomposition [tex]V=W_j\oplus \overline W_J[/tex] into two isotropic subspaces. Since the bilinear form is zero on these subspaces, we get two subalgebras of the Clifford algebra, [tex]\Lambda^*(W_J)[/tex] and [tex]\Lambda^*(\overline{W_J})[/tex]. It turns out that one can choose [tex]S\simeq \Lambda^*(W_J)[/tex].

One can make this construction very explicit by picking a particular [tex]J[/tex], for instance the one that acts on the element of an orthonormal basis by [tex]Je_{2j-1}=e_{2j},\ Je_{2j}=-e_{2j-1}[/tex] for [tex]j=1,\cdots n[/tex]. Letting [tex]w_j=e_{2j-1}+ie_{2j}[/tex] we get a basis of [tex]W_J[/tex]. To get an explicit representation of [tex]S[/tex] as a [tex]Cliff(V)[/tex] module isomorphic to [tex]\Lambda^*(\mathbf C^n)[/tex], 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 [tex]a_i^+[/tex] and [tex]a_i[/tex] for [tex]i=1,\cdots,n[/tex] satisfying

[tex]\{a_i,a_j\}=\{a^+_i,a^+_j\}=0[/tex]

[tex]\{a_i,a^+_j\}=\delta_{ij}[/tex]

In terms of these operators on [tex]\Lambda^*(\mathbf C^n)[/tex], [tex]Cliff(n)[/tex] acts by

[tex]e_{2j-1}=a_j^+-a_j[/tex]

[tex]e_{2j}=-i(a^+_j+a_j)[/tex]

**The Spin Representation**

The group that preserves [tex](\cdot,\cdot)[/tex] is [tex]O(n,\mathbf C)[/tex], and its connected component of the identity [tex]SO(n,\mathbf C)[/tex] has compact real form [tex]SO(n)[/tex]. [tex]SO(n)[/tex] has a non-trivial double cover, the group [tex]Spin(n)[/tex]. One can construct [tex]Spin(n)[/tex] explicitly as invertible elements in [tex]Cliff(V)[/tex] for [tex]V=\mathbf R^n[/tex], and its Lie algebra using quadratic elements of [tex]Cliff(V)[/tex], with the Lie bracket given by the commutator in the Clifford algebra.

For the even case, a basis for the Cartan subalgebra of [tex]Lie\ Spin(2n)[/tex] is given by the elements

[tex]\frac{1}{2}e_{2j-1}e_{2j}[/tex]

These act on the spinor module [tex]S\simeq\Lambda^*(\mathbf C^n)[/tex] as

[tex]\frac{1}{2}e_{2j-1}e_{2j}=-i\frac{1}{2}(a_j^+-a_j)(a_j^++a_j)=i\frac{1}{2}[a_j,a_j^+][/tex]

with eigenvalues [tex](\pm\frac{1}{2},\cdots,\pm\frac{1}{2})[/tex]. [tex]S[/tex] is not irreducible as a representation of [tex]Spin(2n)[/tex], but decomposes as [tex]S=S^+\oplus S^-[/tex] into two irreducible half-spin representations, corresponding to the even and odd degree elements of [tex]\Lambda^*(\mathbf C^n)[/tex].

With a standard choice of positive roots, the highest weight of [tex]S^+[/tex] is

[tex](+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},+\frac{1}{2})[/tex]

and that of [tex]S^-[/tex] is

[tex](+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},-\frac{1}{2})[/tex]

Note that the spinor representation is not a representation of [tex]SO(2n)[/tex], just of [tex]Spin(2n)[/tex]. However, if one restricts to the [tex]U(n)\subset SO(2n)[/tex] preserving [tex]J[/tex], then the [tex]\Lambda^*(W_J)[/tex] are the fundamental representations of this [tex]U(n)[/tex]. These representations have weights that are 0 or 1, shifted by [tex]+\frac{1}{2}[/tex] from those of the spin representation. One can’t restrict from [tex]Spin(2n)[/tex] to [tex]U(n)[/tex], but one can restrict to [tex]\tilde U(n)[/tex], a double cover of [tex]U(n)[/tex]. On this double cover the notion of [tex]\Lambda^n(\mathbf C^n)^{\frac{1}{2}[/tex] makes sense and one has, as [tex]\tilde U(n)[/tex] representations

[tex]S\otimes \Lambda^n(\mathbf C^n)^{\frac{1}{2}}\simeq\Lambda^*(\mathbf C^n)[/tex]

So, projectively, the spin representation is just [tex]\Lambda^*(\mathbf C^n)[/tex], 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 [tex]V=\mathfrak g[/tex], and studies the Clifford algebra [tex]Cliff(\mathfrak g)[/tex]

Last Updated on

It seems only few years back that Clifford Algebras have been getting more attention…

I’m wondering what their primary applications are.

John,

The primary applications of Clifford algebras have always been to construct spinors (and thus the spin representations of orthogonal groups, from which all other reps of these groups can be constructed) and write down the Dirac equation. Much more speculative has been the idea of trying to use them to understand internal symmetries. Lots of people have tried to do this, with varying results.

In physics, whenever one works with anti-commuting variables, when one quantizes them, what one gets is a a Clifford algebra. So, the things really are everywhere.

In mathematics, the use of Clifford algebras to do representation theory, more generally than just for orthogonal groups, has been getting more attention in recent years, and that’s part of the story I’m trying to write about here.

Peter, I’m sure you know of David Hestenes’ works using Grassmann and (I think also, although I’m not sure) Clifford Algebras…

Aren’t Clifford Algebras then used to reformulate some areas of physics? If they are, I’m wondering which areas and (perhaps) how?

Also, are the Clifford Algebra reps a key component of BRST quantization (Just wondering.)

John

John,

Grassman algebra = exterior algebra

I haven’t looked closely at the work of Hestenes. From what I’ve seen much of it exploits the relationship between the exterior algebra and the Clifford algebra explained in this posting.

Clifford algebras are a crucial part of the approach to BRST I’m explaining in these notes. If you look at standard treatments of BRST, you’ll find that some of them crucially use Clifford algebras, others avoid this.

Thanks.

Hestenes (and also Doran and Lasenby at Cambridge) rewrites basic physics using Clifford algebra to replace vector algebra.

So, for angular momentum L = r /\ p instead of L = r x p, for example.

It’s fun stuff. Doran and Lasenby use this to derive a gauge theory of gravity in flat space.