{"id":1387,"date":"2008-12-21T18:47:54","date_gmt":"2008-12-21T23:47:54","guid":{"rendered":"http:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=1387"},"modified":"2008-12-21T18:47:54","modified_gmt":"2008-12-21T23:47:54","slug":"notes-on-brst-viii-clifford-algebras","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=1387","title":{"rendered":"Notes on BRST VIII: Clifford Algebras"},"content":{"rendered":"

Clifford Algebras<\/strong><\/p>\n

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. <\/p>\n

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<\/p>\n

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

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]<\/p>\n

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<\/p>\n

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

One can show that these complex Clifford algebras are isomorphic to matrix algebras, more precisely<\/p>\n

[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]<\/p>\n

Clifford Algebras and Exterior Algebras<\/strong><\/p>\n

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).<\/p>\n

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 <\/p>\n

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

[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 <\/p>\n

[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]<\/p>\n

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<\/p>\n

[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]<\/p>\n

The inverse [tex]\\sigma=q^{-1}:Cliff(V)\\rightarrow \\Lambda^*(V)[\/tex] is sometime called the “symbol map”.<\/p>\n

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.<\/p>\n

Clifford Modules and Spinors<\/strong><\/p>\n

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<\/p>\n

[tex]\\pi: Cliff(V)\\rightarrow End(M)[\/tex]<\/p>\n

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

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

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

[tex]\\pi(v)\\omega=v\\wedge\\omega – i_v\\omega[\/tex]<\/p>\n

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 <\/p>\n

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

[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.<\/p>\n

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<\/p>\n

[tex](v_1,v_2)=(Jv_1,Jv_2)=(iv_1,iv_2)=-(v_1,v_2)[\/tex]<\/p>\n

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].<\/p>\n

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<\/p>\n

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

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

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

[tex]e_{2j-1}=a_j^+-a_j[\/tex]<\/p>\n

[tex]e_{2j}=-i(a^+_j+a_j)[\/tex]<\/p>\n

The Spin Representation<\/strong><\/p>\n

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. <\/p>\n

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

[tex]\\frac{1}{2}e_{2j-1}e_{2j}[\/tex]<\/p>\n

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

[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]<\/p>\n

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].<\/p>\n

With a standard choice of positive roots, the highest weight of [tex]S^+[\/tex] is<\/p>\n

[tex](+\\frac{1}{2},+\\frac{1}{2}\\cdots,+\\frac{1}{2},+\\frac{1}{2})[\/tex]<\/p>\n

and that of [tex]S^-[\/tex] is<\/p>\n

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

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<\/p>\n

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

So, projectively, the spin representation is just [tex]\\Lambda^*(\\mathbf C^n)[\/tex], but the projective factor is a crucial part of the story. <\/p>\n

The above has been a rather quick sketch of a long story. For more details, a good reference is the book Spin Geometry<\/em> by Lawson and Michelsohn. Chapter 12 of Segal and Pressley’s Loop Groups<\/em> 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<\/a>.<\/p>\n

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]<\/p>\n","protected":false},"excerpt":{"rendered":"

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. … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[12],"tags":[],"class_list":["post-1387","post","type-post","status-publish","format-standard","hentry","category-brst"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1387","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1387"}],"version-history":[{"count":35,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1387\/revisions"}],"predecessor-version":[{"id":1438,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1387\/revisions\/1438"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1387"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1387"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1387"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}