Notes on BRST VIII: Clifford Algebras

Clifford Algebras

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

v\otimes w + w\otimes v = -2(v,w)1

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

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

e_i^2=-1,\ \ e_ie_j=-e_je_i\ \  (i\neq j)

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

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)

Clifford Algebras and Exterior Algebras

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

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

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

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

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)}

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

q(e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_p})=e_{i_1}e_{i_2}\cdots e_{i_p}

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

\pi: Cliff(V)\rightarrow End(M)

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

\pi(v)\pi(w) +\pi(w)\pi(v)= -2(v,w)Id

One such Clifford module is M=\Lambda^*V, with

\pi(v)\omega=v\wedge\omega – i_v\omega

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

\sigma: a\in Cliff(V)\rightarrow \pi(a)1\in \Lambda^*(V)

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

(v_1,v_2)=(Jv_1,Jv_2)=(iv_1,iv_2)=-(v_1,v_2)

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

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

\{a_i,a_j\}=\{a^+_i,a^+_j\}=0

\{a_i,a^+_j\}=\delta_{ij}

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

e_{2j-1}=a_j^+-a_j

e_{2j}=-i(a^+_j+a_j)

The Spin Representation

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

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

\frac{1}{2}e_{2j-1}e_{2j}

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

\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^+]

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

With a standard choice of positive roots, the highest weight of S^+ is

(+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},+\frac{1}{2})

and that of S^- is

(+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},-\frac{1}{2})

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

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

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

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6 Responses to Notes on BRST VIII: Clifford Algebras

  1. John says:

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

    I’m wondering what their primary applications are.

  2. Peter Woit says:

    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.

  3. John says:

    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

  4. Peter Woit says:

    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.

  5. John says:

    Thanks.

  6. andy.s says:

    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.

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