- Spin representation
In
mathematics , the spin representations are particularprojective representation s of the orthogonal orspecial orthogonal group s in arbitrarydimension and signature (i.e., includingindefinite orthogonal group s). More precisely, they are representations of thespin group s, which aredouble cover s of the special orthogonal groups. They are usually studied over the real orcomplex number s, but they can be defined over otherfield s.Elements of a spin representation are called
spinor s. They play an important role in the physical description offermion s such as theelectron .The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive
real representation s by introducingreal structure s.The properties of the spin representations depend in a subtle way on the dimension and the signature of the orthogonal group. In particular, spin representations often admit
invariant bilinear form s which can be used to embed the spin groups intoclassical Lie group s. In low dimensions, these embeddings aresurjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.et up
Let "V" be a finite dimensional real or complex
vector space with a nondegeneratequadratic form "Q". The (real or complex)linear map s preserving "Q" form theorthogonal group O("V","Q"). The identity component of the group will be called the special orthogonal group SO("V","Q"). (For "V" real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up togroup isomorphism , SO("V","Q") has a unique connecteddouble cover , the spin group Spin("V","Q"). There is thus agroup homomorphism Spin("V","Q") → SO("V","Q") whose kernel has two elements denoted {1, −1}, where 1 is theidentity element .O("V","Q"), SO("V","Q") and Spin("V","Q") are all
Lie groups , and for fixed ("V","Q") they have the sameLie algebra , so("V","Q"). If "V" is real, then "V" is a real vector subspace of itscomplexification "V"C := "V" ⊗R C, and the quadratic form "Q" extends naturally to a quadratic form "Q"C on "V"C. This embeds SO("V","Q") as asubgroup of SO("V"C, "Q"C), and hence we may realise Spin("V","Q") as a subgroup of Spin("V"C, "Q"C). Furthermore, so("V"C, "Q"C) is the just the complexification of so("V","Q").In the complex case, quadratic forms are determined up to isomorphism by the dimension "n" of "V". Concretely, we may assume "V"=C"n" and:Q(z_1,ldots z_n) = z_1^2+ z_2^2+cdots+z_n^2.The corresponding Lie groups and Lie algebra are denoted O("n",C), SO("n",C), Spin("n",C) and so("n",C).
In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers ("p","q") where "n":="p"+"q" is the dimension of "V", and "p"-"q" is the signature. Concretely, we may assume "V"=R"n" and:q(x_1,ldots x_n) = x_1^2+ x_2^2+cdots+x_p^2-(x_{p+1}^2+cdots +x_n^2).The corresponding Lie groups and Lie algebra are denoted O("p","q"), SO("p","q"), Spin("p","q") and so("p","q"). We write R"p","q" in place of R"n" to make the signature explicit.
The spin representations are in some sense the simplest representations of Spin("n",C) and Spin("p","q") which do not come from representations of SO("n",C) and SO("p","q"). A spin representation is therefore, in particular, a real or complex vector space "S" together with a group homomorphism "ρ" from Spin("n",C) or Spin("p","q") to the
general linear group GL("S") such that the element −1 is "not" in the kernel of "ρ".If "S" is such a representation, then according to the relation between Lie groups and Lie algebras, it induces a
Lie algebra representation , i.e., aLie algebra homomorphism from so("n","C") or so("p","q") to the Lie algebra gl("S") of endomorphisms of "S" with the commutator bracket.Spin representations can be analysed according to the following strategy: if "S" is a real spin representation of Spin("p","q"), then its complexification is a complex spin representation of Spin("p","q"); as a representation of so("p","q"), it therefore extends to a complex representation of so("n",C). Proceeding in reverse, we therefore "first" construct complex spin representations of Spin("n",C) and so("n",C), then restrict them to complex spin representations of so("p","q") and Spin("p","q"), then finally analyse possible reductions to real spin representations.
Complex spin representations
Let "V"=C"n" with the standard quadratic form "Q" so that :mathfrak{so}(V,Q) = mathfrak{so}(n,mathbb C).The
symmetric bilinear form on "V" associated to "Q" by polarization will be denoted <.,.>.Isotropic subspaces and root systems
A standard construction of the spin representations of so("n",C) begins with a choice of a pair ("W", "W"∗)of maximal
isotropic subspace s of "V" with "W" ∩ "W"∗ = 0. Let us make such a choice. If "n" = 2"m" or "n" = 2"m"+1, then "W" and "W"∗ both have dimension "m". If "n" = 2"m", then "V" = "W" ⊕ "W"∗, whereas if "n" = 2"m"+1, then "V" = "W" ⊕ "U" ⊕ "W"∗, where "U" is the 1-dimensional orthogonal complement to "W" ⊕ "W"∗. The bilinear form <.,.> induces a pairing between "W" and "W"∗ which must be nondegenerate, because "W" and "W"∗ areisotropic subspace s and "Q" is nondegenerate. Hence "W" and "W"∗ aredual vector space s.More concretely, let "a"1, ... "a""m" be a basis for "W". Then there is a unique basis "α"1, ... "α""m" of "W"∗ such that:langle alpha_i,a_j angle = delta_{ij}.If "A" is a "m" × "m" matrix, then "A" induces a endomorphism of "W" with respect to this basis and the transpose "A"T induces a transformation of "W"∗ with:langle Aw, w^* angle = langle w,A^T w^* anglefor all "w" in "W" and "w"* in "W"∗. It follows that the endomorphism "ρ""A" of "V", equal to "A" on "W", − "A"T on "W"∗ and zero on "U" (if "n" is odd), is skew:langle ho_A v, w angle = -langle v, ho_A w angleand hence an element of so("n",C).
Using the diagonal matrices in this construction defines a
Cartan subalgebra h of so("n",C): the rank of so("n",C) is "m", and the diagonal "m" × "m" matrices determine an "m"-dimensional abelian subalgebra.Let "ε"1, ... "ε""m" be the basis of h∗ such that, for a diagonal matrix "A", "ε""k"("ρ""A") is the "k"th diagonal entry of "A". Clearly this is a basis for h∗. Since the bilinear form identifies so("n",C) with wedge^2 V, it is now easy to construct the
root system associated to h. Theroot space s (simultaneous eigenspaces for the action of h) are spanned by the following elements::a_iwedge a_j,; i eq j, with root (simultaneous eigenvalue) varepsilon_i + varepsilon_j:a_iwedge alpha_j (which is in h if "i" = "j") with root varepsilon_i - varepsilon_j:alpha_iwedge alpha_j,; i eq j, with root varepsilon_i - varepsilon_j,and, if "n" is odd, and "u" is a nonzero element of "U",:a_iwedge u, with root varepsilon_i :alpha_iwedge u, with root varepsilon_i.Thus, with respect the basis "ε"1, ... "ε""m", the roots are the vectors in h∗ which are permutations of:pm 1,pm 1, 0, 0, dots, 0)together with the permutations of:pm 1, 0, 0, dots, 0)if "n" = 2"m"+1 is odd.A system of
positive root s is given by "ε""i"+"ε""j" ("i"≠"j"), "ε""i"−"ε""j" ("i"<"j") and (for "n" odd) "ε""i". The correspondingsimple root s are:varepsilon_1-varepsilon_2, varepsilon_2-varepsilon_3, ldots, varepsilon_{m-1}-varepsilon_m, left{egin{matrix}varepsilon_{m-1}+varepsilon_m& n=2m\varepsilon_m & n=2m+1.end{matrix} ight.The positive roots are nonnegative integer linear combinations of the simple roots.pin representations and their weights
One construction of the spin representations of so("n",C) uses the
exterior algebra (s):S=wedge^ullet W and/or S'=wedge^ullet W^*.There is an action of "V" on "S" such that for any element "v" = "w"+"w"* in "W" ⊕ "W"∗ and any "ψ" in "S" the action is given by::vcdot psi = wwedgepsi+iota(w^*)psi, where the second term is a contraction (interior multiplication ) defined using the bilinear form which pairs "W" and "W"∗. This action respects theClifford relation s "v"2 = "Q"("v")1, and so induces a homomorphism from theClifford algebra Cl"n"C of "V" to End("S"). A similar action can be defined on "S"′, so that both "S" and "S"′ areClifford module s.The Lie algebra so("n",C) is a Lie subalgebra of Cl"n"C (under the commutator bracket) via the embedding:v wedge w mapsto frac12(vw-wv).It follows that both "S" and "S"′ are representations of so("n",C). They are actually equivalent representations, so we focus on "S".
The explicit description shows that the elements "α""i"∧"a""i" of the Cartan subalgebra h act on "S" by:alpha_iwedge a_i) cdot psi = frac12 iota(alpha_i)(a_iwedgepsi)- frac12a_iwedge(iota(alpha_i)psi)= frac12 psi - a_iwedge(iota(alpha_i)psi).A basis for "S" is given by elements of the form:a_{i_1}wedge a_{i_2}wedgecdotswedge a_{i_k}for 0 ≤ "k" ≤ "m" and "i"1 < ... < "i""k". These clearly span
weight space s for the action of h: "α""i"∧"a""i" has eigenvalue -1/2 on the given basis vector if "i" = "i""j" for some "j", and has eigenvalue 1/2 otherwise.It follows that the weights of "S" are all possible combinations of:igl(pm frac12,pm frac12, ldots pm frac12igr)and each
weight space is one dimensional. Elements of "S" are calledDirac spinor s.When "n" is even, "S" is not an
irreducible representation : S_+=wedge^{mathrm{even W and S_-=wedge^{mathrm{odd W are invariant subspaces. The weights are divided into those which have an even number of minus signs, and those which have an odd number of minus signs. Both "S"+ and "S"− are irreducible representations of dimension 2"m"−1 whose elements are calledWeyl spinor s. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, thehighest weight s of "S"+ and "S"− are:igl( frac12, frac12, ldots frac12, frac12igr) and igl( frac12, frac12, ldots frac12, - frac12igr)respectively. The Clifford action identifies Cl"n"C with End("S") and the even subalgebra is identified with the endomorphisms preserving "S"+ and "S"−. The otherClifford module "S"′ is isomorphic to "S" in this case.When "n" is odd, "S" is an irreducible representation of so("n",C) of dimension 2"m": the Clifford action of a unit vector "u" ∈ "U" is given by :ucdot psi = left{egin{matrix}psi&hbox{if } psiin wedge^{mathrm{even W\-psi&hbox{if } psiin wedge^{mathrm{odd Wend{matrix} ight.and so elements of so("n",C) of the form "u"∧"w" or "u"∧"w"* do not preserve the even and odd parts of the exterior algebra of "W". The highest weight of "S" is:igl( frac12, frac12, ldots frac12igr).The Clifford action is not faithful on "S": Cl"n"C can be identified with End("S") ⊕ End("S"′), where "u" acts with the opposite sign on "S"′. More precisely, the two representations are related by the parity involution "α" of Cl"n"C (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl"n"C. In other words, there is a
linear isomorphism from "S" to "S"′, which identifies the action of "A" in Cl"n"C on "S" with the action of "α"("A") on "S"′.Bilinear forms
if "λ" is a weight of "S", so is −"λ". It follows that "S" is isomorphic to the
dual representation "S"∗.When "n" = 2"m"+1 is odd, the isomorphism "B": "S" → "S"∗ is unique up to scale by
Schur's lemma , since "S" is irreducible, and it defines a nondegenerate invariant bilinear form "β" on "S" via:eta(varphi,psi) = B(varphi)(psi).Here invariance means that: eta(xicdotvarphi,psi) + eta(varphi,xicdotpsi) = 0for all "ξ" in so("n",C) and "φ", "ψ" in "S" — in other words the action of "ξ" is skew with respect to "β". In fact, more is true: "S"∗ is a representation of the opposite Clifford algebra, and therefore, since Cl"n"C only has two nontrivialsimple module s "S" and "S"′, related by the parity involution "α", there is anantiautomorphism "τ" of Cl"n"C such that: quadeta(Acdotvarphi,psi) = eta(varphi, au(A)cdotpsi)qquad (1)for any "A" in Cl"n"C. In fact "τ" is reversion (the antiautomorphism induced by the identity on "V") for "m" even, and conjugation (the antiautomorphism induced by minus the identity on "V") for "m" odd. These two antiautomorphisms are related by parity involution "α", which is the automorphism induced by minus the identity on "V". Both satisfy "τ"("ξ") = −"ξ" for "ξ" in so("n",C).When "n" = 2"m", the situation depends more sensitively upon the parity of "m". For "m" even, a weight "λ" has an even number of minus signs if and only if −"λ" does; it follows that there are separate isomorphisms "B"±: "S"± → "S"±∗ of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism "B": "S" → "S"∗. For "m" odd, "λ" is a weight of "S"+ if and only if −"λ" is a weight of "S"−; thus there is an isomorphism from "S"+ to "S"−∗, again unique up to scale, and its transpose provides an isomorphism from "S"− to "S"+∗. These may again be combined into an isomorphism "B": "S" → "S"∗.
For both "m" even and "m" odd, the freedom in the choice of "B" may be restricted to an overall scale by insisting that the bilinear form "β" corresponding to "B" satisfies (1), where "τ" is a fixed antiautomorphism (either reversion or conjugation).
ymmetry and the tensor square
The symmetry properties of "β": "S" ⊗ "S" → C can be determined using Clifford algebras or representation theory. In fact much more can be said: the tensor square "S" ⊗ "S" must decompose into a direct sum of "k"-forms on "V" for various "k", because its weights are all elements in h∗ whose components belong to {−1,0,1}. Now
equivariant linear maps "S" ⊗ "S" → ∧"k""V"∗ correspond bijectively to invariant maps ∧"k""V" ⊗ "S" ⊗ "S" → C and nonzero such maps can be constructed via the inclusion of ∧"k""V" into the Clifford algebra. Furthermore if "β"("φ","ψ") = "ε" "β"("ψ","φ") and "τ" has sign "ε""k" on ∧"k""V" then:eta(Acdotvarphi,psi) = varepsilonvarepsilon_k eta(Acdotpsi,varphi)for "A" in ∧"k""V".If "n" = 2"m"+1 is odd then it follows from Schur's Lemma that:Sotimes S cong igoplus_{j=0}^{m} wedge^{2j} V^*(both sides have dimension 22"m" and the representations on the right are inequivalent). Because the symmetries are governed by an involution "τ" which is either conjugation or reversion, the symmetry of the ∧"2j""V"∗ component alternates with "j". Elementary combinatorics gives:sum_{j=0}^m (-1)^j dim wedge^{2j} C^{2m+1} = (-1)^{frac12 m(m+1)} 2^m = (-1)^{frac12 m(m+1)}(dim mathrm S^2S-dim wedge^2 S)and the sign determines which representations occur in S2"S" and which occur in ∧2"S". [This sign can also be determined from the observation that if "φ" is a highest weight vector for "S" then "φ"⊗"φ" is a highest weight vector for ∧"m""V" ≅ ∧"m"+1"V", so this summand must occur in S2"S".] In particular:eta(phi,psi)=(-1)^{frac12 m(m+1)}eta(psi,phi), and:eta(vcdotphi,psi) = (-1)^m(-1)^{frac12 m(m+1)}eta(vcdotpsi,phi) = (-1)^m eta(phi,vcdotpsi)for "v" ∈ "V" (which is isomorphic to ∧2"m""V"), confirming that "τ" is reversion for "m" even, and conjugation for "m" odd.
If "n"=2"m" is even, then the analysis is more involved, but the result is a more refined decomposition: S"2"S"±, ∧"2"S"± and "S"+ ⊗ "S"− can each be decomposed as a direct sum of "k"-forms (where for "k"="m" there is a further decomposition into selfdual and antiselfdual "m"-forms).
The main outcome is a realisation of so("n",C) as a subalgebra of a classical Lie algebra on "S", depending upon "n" modulo 8, according to the following table:(†) "N" is even for "n">3 and for "n"=3, this is sp(1).
The even dimensional case is similar. For "n">2, the complex half-spin representations are even dimensional. We have additionally to deal with hermitian structures and the real forms of sl(2"N",C), which are sl(2"N",R), su("K","L") with "K" + "L" = 2"N", and sl("N",H). The resulting even spin representations are summarized as follows.(*) For "pq"=0, we have instead so(2"N")+so(2"N")
(†) "N" is even for "n">4 and for "pq"=0 (which includes "n"=4 with "N"=1), we have instead sp("N")+sp("N")
The low dimensional isomorphisms in the complex case have the following real forms.The only special isomorphism of real Lie algebras missing from this table ismathfrak{so}^*(3,mathbb H) cong mathfrak{su}(3,1).
References
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*. See also [http://www.math.ias.edu/QFT the programme website] for a preliminary version.
* citation| last1=Fulton | first1=William| author1-link=William Fulton
last2=Harris| first2=Joe | author2-link=Joe Harris (mathematician)
title=Representation theory. A first course| publisher=Springer-Verlag | location=New York| series=Graduate Texts in Mathematics , Readings in Mathematics| isbn=0-387-97495-4| id=MathSciNet | id = 1153249, ISBN 0-387-97527-6| year=1991| volume=129.
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