Spin representation

Template:Short description In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.

Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron.

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 representations by introducing real structures.

The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.

Set-up

Let Template:Math be a finite-dimensional real or complex vector space with a nondegenerate quadratic form Template:Math. The (real or complex) linear maps preserving Template:Math form the orthogonal group Template:Math. The identity component of the group is called the special orthogonal group Template:Math. (For Template:Math 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 to group isomorphism, Template:Math has a unique connected double cover, the spin group Template:Math. There is thus a group homomorphism Template:Math whose kernel has two elements denoted Template:Math, where Template:Math is the identity element. Thus, the group elements Template:Math and Template:Math of Template:Math are equivalent after the homomorphism to Template:Math; that is, Template:Math for any Template:Math in Template:Math.

The groups Template:Math and Template:Math are all Lie groups, and for fixed Template:Math they have the same Lie algebra, Template:Math. If Template:Math is real, then Template:Math is a real vector subspace of its complexification Template:Math, and the quadratic form Template:Math extends naturally to a quadratic form Template:Math on Template:Math. This embeds Template:Math as a subgroup of Template:Math, and hence we may realise Template:Math as a subgroup of Template:Math. Furthermore, Template:Math is the complexification of Template:Math.

In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimension Template:Math of Template:Math. Concretely, we may assume Template:Math and

$Q(z_1,\dots ,z_n)=z_1^2+z_2^2+\cdots +z_n^2.$

The corresponding Lie groups are denoted Template:Math and their Lie algebra as Template:Math.

In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers Template:Math where Template:Math is the dimension of Template:Math, and Template:Math is the signature. Concretely, we may assume Template:Math and

$Q(x_1,\dots ,x_n)=x_1^2+x_2^2+\cdots +x_p^2-(x_{p+1}^2+\cdots +x_{p+q}^2).$

The corresponding Lie groups and Lie algebra are denoted Template:Math and Template:Math. We write Template:Math in place of Template:Math to make the signature explicit.

The spin representations are, in a sense, the simplest representations of Template:Math and Template:Math that do not come from representations of Template:Math and Template:Math. A spin representation is, therefore, a real or complex vector space Template:Math together with a group homomorphism Template:Math from Template:Math or Template:Math to the general linear group Template:Math such that the element Template:Math is not in the kernel of Template:Math.

If Template:Math is such a representation, then according to the relation between Lie groups and Lie algebras, it induces a Lie algebra representation, i.e., a Lie algebra homomorphism from Template:Math or Template:Math to the Lie algebra Template:Math of endomorphisms of Template:Math with the commutator bracket.

Spin representations can be analysed according to the following strategy: if Template:Math is a real spin representation of Template:Math, then its complexification is a complex spin representation of Template:Math; as a representation of Template:Math, it therefore extends to a complex representation of Template:Math. Proceeding in reverse, we therefore first construct complex spin representations of Template:Math and Template:Math, then restrict them to complex spin representations of Template:Math and Template:Math, then finally analyse possible reductions to real spin representations.

Complex spin representations

Let Template:Math with the standard quadratic form Template:Math so that

$𝔰𝔬(V,Q)=𝔰𝔬(n,ℂ).$

The symmetric bilinear form on Template:Math associated to Template:Math by polarization is denoted Template:Math.

Isotropic subspaces and root systems

A standard construction of the spin representations of Template:Math begins with a choice of a pair Template:Math of maximal totally isotropic subspaces (with respect to Template:Math) of Template:Math with Template:Math. Let us make such a choice. If Template:Math or Template:Math, then Template:Math and Template:Math both have dimension Template:Math. If Template:Math, then Template:Math, whereas if Template:Math, then Template:Math, where Template:Math is the 1-dimensional orthogonal complement to Template:Math. The bilinear form Template:Math associated to Template:Math induces a pairing between Template:Math and Template:Math, which must be nondegenerate, because Template:Math and Template:Math are totally isotropic subspaces and Template:Math is nondegenerate. Hence Template:Math and Template:Math are dual vector spaces.

More concretely, let Template:Math be a basis for Template:Math. Then there is a unique basis Template:Math of Template:Math such that

$⟨{\alpha }_i,a_j⟩={\delta }_{ij}.$

If Template:Math is an Template:Math matrix, then Template:Math induces an endomorphism of Template:Math with respect to this basis and the transpose Template:Math induces a transformation of Template:Math with

$⟨Aw,w^{*}⟩=⟨w,A^{\mathrm{T}}{w}^{*}⟩$

for all Template:Math in Template:Math and Template:Math in Template:Math. It follows that the endomorphism Template:Math of Template:Math, equal to Template:Math on Template:Math, Template:Math on Template:Math and zero on Template:Math (if Template:Math is odd), is skew,

$⟨{\rho }_{A}u,v⟩=-⟨u,{\rho }_{A}v⟩$

for all Template:Math in Template:Math, and hence (see classical group) an element of Template:Math.

Using the diagonal matrices in this construction defines a Cartan subalgebra Template:Math of Template:Math: the rank of Template:Math is Template:Math, and the diagonal Template:Math matrices determine an Template:Math-dimensional abelian subalgebra.

Let Template:Math be the basis of Template:Math such that, for a diagonal matrix Template:Math is the Template:Mathth diagonal entry of Template:Math. Clearly this is a basis for Template:Math. Since the bilinear form identifies Template:Math with ${\wedge }^{2}V$, explicitly,

$x\wedge y\mapsto {\phi }_{x\wedge y},{\phi }_{x\wedge y}(v)=⟨y,v⟩x-⟨x,v⟩y,x\wedge y\in {\wedge }^{2}V,x,y,v\in V,{\phi }_{x\wedge y}\in 𝔰𝔬(n,ℂ),$

it is now easy to construct the root system associated to Template:Math. The root spaces (simultaneous eigenspaces for the action of Template:Math) are spanned by the following elements:

$a_i\wedge a_j,i\ne j,$ with root (simultaneous eigenvalue) ${\epsilon }_i+{\epsilon }_j$

$a_i\wedge {\alpha }_j$ (which is in Template:Math if Template:Math with root ${\epsilon }_i-{\epsilon }_j$

${\alpha }_i\wedge {\alpha }_j,i\ne j,$ with root $-{\epsilon }_i-{\epsilon }_j,$

and, if Template:Math is odd, and Template:Math is a nonzero element of Template:Math,

$a_i\wedge u,$ with root ${\epsilon }_i$

${\alpha }_i\wedge u,$ with root $-{\epsilon }_i.$

Thus, with respect to the basis Template:Math, the roots are the vectors in Template:Math that are permutations of

$(\pm 1,\pm 1,0,0,\dots ,0)$

together with the permutations of

$(\pm 1,0,0,\dots ,0)$

if Template:Math is odd.

A system of positive roots is given by Template:Math and (for Template:Math odd) Template:Math. The corresponding simple roots are

${\epsilon }_1-{\epsilon }_2,{\epsilon }_2-{\epsilon }_3,\dots ,{\epsilon }_{m-1}-{\epsilon }_m,\{\begin{array}{cc}{\epsilon }_{m-1}+{\epsilon }_m& n=2m\\ {\epsilon }_m& n=2m+1.\end{array}$

The positive roots are nonnegative integer linear combinations of the simple roots.

Spin representations and their weights

One construction of the spin representations of Template:Math uses the exterior algebra(s)

$S={\wedge }^{\bullet }W$ and/or $S^{\prime }={\wedge }^{\bullet }{W}^{*}.$

There is an action of Template:Math on Template:Math such that for any element Template:Math in Template:Math and any Template:Math in Template:Math the action is given by:

$v\cdot \psi =2^{\frac{1}{2}}(w\wedge \psi +\iota (w^{*})\psi ),$

where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairs Template:Math and Template:Math. This action respects the Clifford relations Template:Math, and so induces a homomorphism from the Clifford algebra Template:Math of Template:Math to Template:Math. A similar action can be defined on Template:Math, so that both Template:Math and Template:Math are Clifford modules.

The Lie algebra Template:Math is isomorphic to the complexified Lie algebra Template:Math in Template:Math via the mapping induced by the covering Template:Math

$v\wedge w\mapsto \frac{1}{4}[v,w].$

It follows that both Template:Math and Template:Math are representations of Template:Math. They are actually equivalent representations, so we focus on S.

The explicit description shows that the elements Template:Math of the Cartan subalgebra Template:Math act on Template:Math by

$({\alpha }_i\wedge a_i)\cdot \psi =\frac{1}{4}(2^{\frac{1}{2}}{)}^2(\iota ({\alpha }_i)(a_i\wedge \psi )-a_i\wedge (\iota ({\alpha }_i)\psi ))=\frac{1}{2}\psi -a_i\wedge (\iota ({\alpha }_i)\psi ).$

A basis for Template:Math is given by elements of the form

$a_{i_1}\wedge a_{i_2}\wedge \cdots \wedge a_{i_k}$

for Template:Math and Template:Math. These clearly span weight spaces for the action of Template:Math: Template:Math has eigenvalue −1/2 on the given basis vector if Template:Math for some Template:Math, and has eigenvalue Template:Math otherwise.

It follows that the weights of Template:Math are all possible combinations of

$(\pm \frac{1}{2},\pm \frac{1}{2},\dots \pm \frac{1}{2})$

and each weight space is one-dimensional. Elements of Template:Math are called Dirac spinors.

When Template:Math is even, Template:Math is not an irreducible representation: $S_{+}={\wedge }^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}W$ and $S_{-}={\wedge }^{\mathrm{o}\mathrm{d}\mathrm{d}}W$ are invariant subspaces. The weights divide into those with an even number of minus signs, and those with an odd number of minus signs. Both S₊ and S₋ are irreducible representations of dimension 2^m−1 whose elements are called Weyl spinors. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, the highest weights of S₊ and S₋ are

$(\frac{1}{2},\frac{1}{2},\dots \frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},\frac{1}{2},\dots \frac{1}{2},-\frac{1}{2})$

respectively. The Clifford action identifies Cl_nC with End(S) and the even subalgebra is identified with the endomorphisms preserving S₊ and S₋. The other Clifford 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

$u\cdot \psi =\{\begin{array}{cc}\psi & \text{if }\psi \in {\wedge }^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}W\\ -\psi & \text{if }\psi \in {\wedge }^{\mathrm{o}\mathrm{d}\mathrm{d}}W\end{array}$

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

$(\frac{1}{2},\frac{1}{2},\dots \frac{1}{2}).$

The Clifford action is not faithful on S: Cl_nC 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_nC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl_nC. In other words, there is a linear isomorphism from S to S′, which identifies the action of A in Cl_nC 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 = 2m + 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

$\beta (\phi ,\psi )=B(\phi )(\psi ).$

Here invariance means that

$\beta (\xi \cdot \phi ,\psi )+\beta (\phi ,\xi \cdot \psi )=0$

for 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_nC only has two nontrivial simple modules S and S′, related by the parity involution α, there is an antiautomorphism τ of Cl_nC such that

$\beta (A\cdot \phi ,\psi )=\beta (\phi ,\tau (A)\cdot \psi )(1)$

for any A in Cl_nC. 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 = 2m, 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).

Symmetry 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 → ∧^kV^∗ correspond bijectively to invariant maps ∧^kV ⊗ S ⊗ S → C and nonzero such maps can be constructed via the inclusion of ∧^kV into the Clifford algebra. Furthermore, if β(φ,ψ) = ε β(ψ,φ) and τ has sign ε_k on ∧^kV then

$\beta (A\cdot \phi ,\psi )=\epsilon {\epsilon }_k\beta (A\cdot \psi ,\phi )$

for A in ∧^kV.

If n = 2m+1 is odd then it follows from Schur's Lemma that

$S\otimes S\cong \underset{j=0}{\overset{m}{⨁}}{\wedge }^{2j}{V}^{*}$

(both sides have dimension 2^2m and the representations on the right are inequivalent). Because the symmetries are governed by an involution τ that is either conjugation or reversion, the symmetry of the ∧^2jV^∗ component alternates with j. Elementary combinatorics gives

$\sum _{j=0}^m(-1{)}^j\dim {\wedge }^{2j}{ℂ}^{2m+1}=(-1{)}^{\frac{1}{2}m(m+1)}{2}^m=(-1{)}^{\frac{1}{2}m(m+1)}(\dim {\mathrm{S}}^{2}S-\dim {\wedge }^{2}S)$

and the sign determines which representations occur in S²S and which occur in ∧²S. In particular

$\beta (\varphi ,\psi )=(-1{)}^{\frac{1}{2}m(m+1)}\beta (\psi ,\varphi ),$ and

$\beta (v\cdot \varphi ,\psi )=(-1{)}^m(-1{)}^{\frac{1}{2}m(m+1)}\beta (v\cdot \psi ,\varphi )=(-1{)}^m\beta (\varphi ,v\cdot \psi )$

for v ∈ V (which is isomorphic to ∧^2mV), confirming that τ is reversion for m even, and conjugation for m odd.

If n = 2m is even, then the analysis is more involved, but the result is a more refined decomposition: S²S_±, ∧²S_± 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 mod 8	0	1	2	3	4	5	6	7
Spinor algebra	$𝔰𝔬(S_{+})\oplus 𝔰𝔬(S_{-})$	$𝔰𝔬(S)$	$𝔤𝔩(S_{\pm })$	$𝔰𝔭(S)$	$𝔰𝔭(S_{+})\oplus 𝔰𝔭(S_{-})$	$𝔰𝔭(S)$	$𝔤𝔩(S_{\pm })$	$𝔰𝔬(S)$

For n ≤ 6, these embeddings are isomorphisms (onto sl rather than gl for n = 6):

$𝔰𝔬(2,ℂ)\cong 𝔤𝔩(1,ℂ)(=ℂ)$

$𝔰𝔬(3,ℂ)\cong 𝔰𝔭(2,ℂ)(=𝔰𝔩(2,ℂ))$

$𝔰𝔬(4,ℂ)\cong 𝔰𝔭(2,ℂ)\oplus 𝔰𝔭(2,ℂ)$

$𝔰𝔬(5,ℂ)\cong 𝔰𝔭(4,ℂ)$

$𝔰𝔬(6,ℂ)\cong 𝔰𝔩(4,ℂ).$

Real representations

The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types.

1. There is an invariant complex antilinear map r: S → S with r² = id_S. The fixed point set of r is then a real vector subspace S_R of S with S_R ⊗ C = S. This is called a real structure.
2. There is an invariant complex antilinear map j: S → S with j² = −id_S. It follows that the triple i, j and k:=ij make S into a quaternionic vector space S_H. This is called a quaternionic structure.
3. There is an invariant complex antilinear map b: S → S^∗ that is invertible. This defines a pseudohermitian bilinear form on S and is called a hermitian structure.

The type of structure invariant under so(p,q) depends only on the signature p − q modulo 8, and is given by the following table.

p−q mod 8	0	1	2	3	4	5	6	7
Structure	R + R	R	C	H	H + H	H	C	R

Here R, C and H denote real, hermitian and quaternionic structures respectively, and R + R and H + H indicate that the half-spin representations both admit real or quaternionic structures respectively.

Description and tables

To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms. Since n = p + q ≅ p − q mod 2, there are two cases: the dimension and signature are both even, and the dimension and signature are both odd.

The odd case is simpler, there is only one complex spin representation S, and hermitian structures do not occur. Apart from the trivial case n = 1, S is always even-dimensional, say dim S = 2N. The real forms of so(2N,C) are so(K,L) with K + L = 2N and so^∗(N,H), while the real forms of sp(2N,C) are sp(2N,R) and sp(K,L) with K + L = N. The presence of a Clifford action of V on S forces K = L in both cases unless pq = 0, in which case KL=0, which is denoted simply so(2N) or sp(N). Hence the odd spin representations may be summarized in the following table.

	n mod 8	1, 7	3, 5
p−q mod 8		so(2N,C)	sp(2N,C)
1, 7	R	so(N,N) or so(2N)	sp(2N,R)
3, 5	H	so∗(N,H)	sp(N/2,N/2)† or sp(N)

(†) Template:Math is even for Template:Math and for Template:Math, this is Template:Math.

The even-dimensional case is similar. For Template:Math, the complex half-spin representations are even-dimensional. We have additionally to deal with hermitian structures and the real forms of Template:Math, which are Template:Math, Template:Math with Template:Math, and Template:Math. The resulting even spin representations are summarized as follows.

	n mod 8	0	2, 6	4
p-q mod 8		so(2N,C)+so(2N,C)	sl(2N,C)	sp(2N,C)+sp(2N,C)
0	R+R	so(N,N)+so(N,N)∗	sl(2N,R)	sp(2N,R)+sp(2N,R)
2, 6	C	so(2N,C)	su(N,N)	sp(2N,C)
4	H+H	so∗(N,H)+so∗(N,H)	sl(N,H)	sp(N/2,N/2)+sp(N/2,N/2)†

(*) For Template:Math, we have instead Template:Math

(†) Template:Math is even for Template:Math and for Template:Math (which includes Template:Math with Template:Math), we have instead Template:Math

The low-dimensional isomorphisms in the complex case have the following real forms.

Euclidean signature	Minkowskian signature	Other signatures
$𝔰𝔬(2)\cong 𝔲(1)$	$𝔰𝔬(1,1)\cong ℝ$
$𝔰𝔬(3)\cong 𝔰𝔭(1)$	$𝔰𝔬(2,1)\cong 𝔰𝔩(2,ℝ)$
$𝔰𝔬(4)\cong 𝔰𝔭(1)\oplus 𝔰𝔭(1)$	$𝔰𝔬(3,1)\cong 𝔰𝔩(2,ℂ)$	$𝔰𝔬(2,2)\cong 𝔰𝔩(2,ℝ)\oplus 𝔰𝔩(2,ℝ)$
$𝔰𝔬(5)\cong 𝔰𝔭(2)$	$𝔰𝔬(4,1)\cong 𝔰𝔭(1,1)$	$𝔰𝔬(3,2)\cong 𝔰𝔭(4,ℝ)$
$𝔰𝔬(6)\cong 𝔰𝔲(4)$	$𝔰𝔬(5,1)\cong 𝔰𝔩(2,ℍ)$	$𝔰𝔬(4,2)\cong 𝔰𝔲(2,2)$	$𝔰𝔬(3,3)\cong 𝔰𝔩(4,ℝ)$

The only special isomorphisms of real Lie algebras missing from this table are ${𝔰𝔬}^{*}(3,ℍ)\cong 𝔰𝔲(3,1)$ and ${𝔰𝔬}^{*}(4,ℍ)\cong 𝔰𝔬(6,2).$

References

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