nLab
spin representation

Context

Representation theory

representation theory

geometric representation theory

Ingredients

Definitions

representation, 2-representation, ∞-representation

Geometric representation theory

Theorems

Representation theory

Higher spin geometry

Contents

Idea

A representation of the spin group.

Definition

Definition

A quadratic vector space (V,,)(V, \langle -,-\rangle) is a vector space VV over finite dimension over a field kk of characteristic 0, and equipped with a symmetric bilinear form ,:VVk\langle -,-\rangle \colon V \otimes V \to k.

Conventions as in (Varadarajan 04, section 5.3).

We write q:vv,vq\colon v \mapsto \langle v ,v \rangle for the corresponding quadratic form.

Definition

The Clifford algebra CL(V,q)CL(V,q) of a quadratic vector space, def. 1, is the associative algebra over kk which is the quotient

Cl(V,q)T(V)/I(V,q) Cl(V,q) \coloneqq T(V)/I(V,q)

of the tensor algebra of VV by the ideal generated by the elements vvq(v)v \otimes v - q(v).

Since the tensor algebra T(V)T(V) is naturally \mathbb{Z}-graded, the Clifford algebra Cl(V,q)Cl(V,q) is naturally /2\mathbb{Z}/2\mathbb{Z}-graded.

Let ( n,q=||)(\mathbb{R}^n, q = {\vert \vert}) be the nn-dimensional Cartesian space with its canonical scalar product. Write Cl ( n)Cl^\mathbb{C}(\mathbb{R}^n) for the complexification of its Clifford algebra.

Proposition

There exists a unique complex representation

Cl ( n)End(Δ n) Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)

of the algebra Cl ( n)Cl^\mathbb{C}(\mathbb{R}^n) of smallest dimension

dim (Δ n)=2 [n/2]. dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]} \,.
Definition

The Spin group Spin(V,q)Spin(V,q) of a quadratic vector space, def. 1, is the subgroup of the group of units in the Clifford algebra Cl(V,q)Cl(V,q)

Spin(V,q)GL 1(Cl(V,q)) Spin(V,q)\hookrightarrow GL_1(Cl(V,q))

on those elements which are even number multiples v 1v 2kv_1 \cdots v_{2k} of elements v iVv_i \in V with q(V)=1q(V) = 1.

Specifically, “the” Spin group is

Spin(n)Spin( n). Spin(n) \coloneqq Spin(\mathbb{R}^n) \,.

A spin representation is a linear representation of the spin group, def. 3.

Properties

Complex representations

Complex representations of the spin group follow a mod-2 Bott periodicity.

In even d=2nd = 2n there are two inequivalent complex-linear irreducible representations of Spin(d1,1)Spin(d-1,1), each of complex dimension 2 d/212^{d/2-1}, called the two chiral representations, or the two Weyl spinor representations.

For instance for d=10d = 10 one often writes these as 16\mathbf{16} and 16\mathbf{16}'.

The direct sum of the two chiral representation is called the Dirac spinor representation, for instance 16+16\mathbf{16} + \mathbf{16}'.

In odd d=2n+1d = 2n+1 there is a single complex irreducible representation of complex dimension 2 (d1)/22^{(d-1)/2}. For instance for d=11d = 11 one often writes this as 32\mathbf{32}. This is called the Dirac spinor representation in this odd dimension.

For d=2nd = 2n, if {Γ 1,,Γ n}\{\Gamma^1, \cdots, \Gamma^n\} denote the generators of the Clifford algebra Cl d1,1Cl_{d-1,1} then there is the chirality operator

Γ d+1Γ 1Γ 2Γ d \Gamma^{d+1} \coloneqq \Gamma^1 \cdot \Gamma^2 \cdots \Gamma^d

on the Dirac representation, whose eigenspaces induce its decomposition into the two chiral summands.

The unique irreducible Dirac representation in the odd dimension d+1d+1 is, as a complex vector space, the sum of the two chiral representations in dimension dd, with the Clifford algebra represented by Γ 1\Gamma^1 through Γ d\Gamma^d acting diagonally on the two chiral representations, and the chirality operator Γ d+1\Gamma^{d+1} in dimension dd acting on their sum, now being the representation of the (d+1)(d+1)st Clifford algebra generator.

Real representations (Majorana representations)

One may ask in which dimensions dd the above complex representations admit a real structure

Real spinor representations are also called Majorana representations (with variants such as “symplectic Majorana”), and an element of a real/Majorana spin representation is also called a Majorana spinor. On a Majorana representation SS there is a non-vanishing symmetric and Spin(d1,1)Spin(d-1,1)-invariant bilinear form SS dS \otimes S \longrightarrow \mathbb{R}^d, projectively unique if SS is irreducible. This serves as the odd-odd Lie bracket in the super Lie algebra called the super Poincaré Lie algebra extension of the ordinary Poincaré Lie algebra induced by SS. This is “supersymmetry” in physics.

The above irreducible complex representations admit a real structure for d=1,2,3mod8d = 1,2,3 \, mod \, 8. Therefore in dimension d=2mod8d = 2 \, mod \, 8 there exist Majorana-Weyl spinor representations.

The above irreducible complex representations admit a quaternionic structure for d=5,6,7mod8d = 5,6,7 \, mod \, 8.

Let VV be a quadratic vector space, def. 1, over the real numbers with bilinear form of Lorentzian signature, hence V= d1,1V = \mathbb{R}^{d-1,1} is Minkowski spacetime of some dimension dd.

The following table lists the irreducible real representations of Spin(V)Spin(V) (Freed 99, page 48).

ddSpin(d1,1)Spin(d-1,1)minimal real spin representation SSdim Sdim_{\mathbb{R}} S\;\;VV in terms of S *S^\astsupergravity
1 2\mathbb{Z}_2SS real1V(S *) 2V \simeq (S^\ast)^{\otimes}^2
2 >0× 2\mathbb{R}^{\gt 0} \times \mathbb{Z}_2S +,S S^+, S^- real1V(S + *) 2(S *) 2V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}
3SL(2,)SL(2,\mathbb{R})SS real2VSym 2S *V \simeq Sym^2 S^\ast
4SL(2,)SL(2,\mathbb{C})S SSS_{\mathbb{C}} \simeq S' \oplus S''4V S *S *V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\astd=4 N=1 supergravity
5Sp(1,1)Sp(1,1)S S 0 WS_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W8 2S 0 *V \wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}
6SL(2,)SL(2,\mathbb{H})S ±S 0 ± WS^\pm_{\mathbb{C}} \simeq S_0^\pm \otimes_{\mathbb{C}} W8V 2S 0 + *( 2S 0 *) *V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast
7S S 0 WS_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W16 2S 0 *V 2V \wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}
8S S S S_{\mathbb{C}} \simeq S^\prime \oplus S^{\prime\prime}16S *S *V 3V {S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}
9SS real16Sym 2S *V 4VSym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V
10S +,S S^+ , S^- real16Sym 2(S ±) *V ± 5VSym^2(S^\pm)^\ast \simeq V \oplus \wedge_\pm^5 Vtype II supergravity
11SS real32Sym 2S *V 2V 5VSym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V11-dimensional supergravity

Here WW is the 2-dimensional complex vector space on which the quaternions naturally act.

Remark

The last column implies that in each dimension there exists a linear map

Γ:S *S * d1,1 \Gamma \;\colon\; S^\ast \otimes S^\ast \longrightarrow \mathbb{R}^{d-1,1}

which is

  1. symmetric;

  2. Spin(V)Spin(V)-equivariant.

This allows to form the super Poincaré Lie algebra in each of these cases. See there and see Spinor bilinear forms below for more.

Spinor bilinear forms

Let (V,,)(V, \langle -,-\rangle) be a quadratic vector space, def. 1. For pp \in \mathbb{R} write pV\wedge^p V for its ppth skew-symmetrized tensor power, regarded naturally as a representation of the spin group Spin(V)Spin(V).

For S 1,S 2Rep(Spin(V))S_1, S_2 \in Rep(Spin(V)) two irreducible representations of Spin(V)Spin(V), we discuss here homomorphisms of representations (hence kk-linear maps respecting the Spin(V)Spin(V)-action) of the form

S 1S 2 pV. S_1 \otimes S_2 \longrightarrow \wedge^p V \,.

These appear notably in the following applications:

p=0p = 0 – spinor metric

We discuss spinor bilinear pairings to scalars.

Over the complex numbers
Proposition

Let VV be a quadratic vector space, def. 1 over the complex numbers of dimension dd. Then there exists in dimensions d2,6mod8d \neq 2,6 \; mod \, 8, up to rescaling, a unique Spin(V)Spin(V)-invariant bilinear form

C:SS C \;\colon\; S \otimes S \longrightarrow \mathbb{C}

on a complex irreducible representation SS of Spin(V)Spin(V), or in dimension 2 and 6 a bilinear pairing

C:S +S C \;\colon\; S^+ \otimes S^- \longrightarrow \mathbb{C}

which is non-degenerate and whose symmetry is given by the following table:

dmod8d \, mod\, 8C
0symmetric
1symmetric
2S ±S^\pm dual to each other
3skew-symmetric
4skew-symmetric
5skew-symmetric
6S ±S^\pm dual to each other
7symmetric

This appears for instance as (Varadarajan 04, theorem 6.5.7).

Remark

The matrix representation of the bilinear form in prop. 2 is known in the physics literature as the charge conjugation matrix. In matrix calculus the symmetry property means that the transpose matrix C TC^T satisfies

C T=ϵC C^T = - \epsilon C

with ϵ{1,1}\epsilon \in \{-1,1\} given in dimension dd by the following table

dmod8d \, mod \, 8CC
0-1
1-1
2either
3+1
4+1
5+1
6either
7-1

For instance (van Proeyen 99, table 1).

Over the real numbers (for Majorana spinors)
Proposition

Let VV be a quadratic vector space, def. 1 over the real numbers of dimension dd with Loentzian signature. Then there exists, up to rescaling, a unique Spin(V)Spin(V)-invariant bilinear form

C:SS C \;\colon\; S \otimes S \longrightarrow \mathbb{R}

on a real irreducible representation SS of Spin(V)Spin(V), and its symmetry is given by the following table

dmod8d \, mod \, 8CC
0symmetric
1symmetric
2S ±S^{\pm} dual to each other
3skew symmetric
4skew symmetric
5symmetric
6S ±S^{\pm} dual to each other
7symmetric

This appears for instance as (Freed 99, around (3.4), Varadarajan 04, theorem 6.5.10).

p=1p = 1 – super Poincaré bracket (supersymmetry)

We discuss spinor bilinear pairings to vectors.

Over the complex numbers
Proposition

Let VV be a quadratic vector space, def. 1 over the complex numbers of dimension dd.

Then there exists unique Spin(V)Spin(V)-representation morphisms

Γ:SS \Gamma \;\colon\; S\otimes S \longrightarrow \mathbb{C}

for odd dd and SS the unique irreducible representation, and

Γ:S ±S \Gamma \;\colon\; S^{\pm} \otimes S^{\mp} \longrightarrow \mathbb{C}

for even dd and S ±S^\pm the two inequivalent irreducible representations.

This is (Varadarajan 04, theorem 6.6.3).

Over the real numbers (for Majorana spinors)
Proposition

Let VV be a quadratic vector space, def. 1 over the real numbers of dimension dd.

Then there exists unique Spin(V)Spin(V)-representation morphisms

dmod8d \,mod \, 8
0S ±S VS^\pm \otimes S^\mp \to V
1SSVS \otimes S \to V
2S ±S ±VS^\pm \otimes S^\pm \to V
3SSVS \otimes S \to V
4S ±S VS^\pm \otimes S^\mp \to V
5SSVS \otimes S \to V
6S ±S ±VS^\pm \otimes S^\pm \to V
7SSVS \otimes S \to V

This is (Varadarajan 04, theorem 6.5.10).

For more see (Varadarajan 04, section 6.7).

Pairing to a vector in terms of the charge conjugation matrix
Remark

In terms of a matrix representation with respect to a chosen basis as in remark 2 the pairing of prop. 5 is given by the matrices Γ a={(Γ a) α β}\Gamma^a = \{(\Gamma^a)^\alpha{}_\beta\} that represent the Clifford algebra by raising and lowering indices with the charge conjugation matrix of remark 2 (e.g Freed 99 (3.5)).

In such a notation if ϕ=(ϕ α)\phi = (\phi^\alpha) denotes the component-vector of a spinor, then the result of “lowering its index” is given by acting with the metric in form of the charge conjugation matrix. The result is traditionally denoted

ϕ¯ϕ TC \overline{\phi} \coloneqq \phi^T C

hence

ϕ¯ αϕ βC βα. \overline{\phi}_\alpha \coloneqq \phi^\beta C_{\beta \alpha} \,.

This yields the component formula for the pairings to scalars and to vectors which is traditional in the physics literature as follows:

C(ϕ,ψ) =ϕ αC αβψ β =ϕ¯ αψ α =ϕ¯ψ \begin{aligned} C(\phi,\psi) &= \phi^\alpha C_{\alpha \beta} \,\psi^\beta \\ & = \overline{\phi}_\alpha \psi^\alpha \\ & = \overline{\phi} \psi \end{aligned}

and

Γ a(ϕ,ψ) =ϕ αΓ a αβψ β =ϕ αC ακΓ aκ βψ β =ϕ TCΓ aψ ϕ¯Γ aψ. \begin{aligned} \Gamma^a(\phi, \psi) &= \phi^\alpha \Gamma^a{}_{\alpha \beta} \psi^\beta \\ &= \phi^\alpha C_{\alpha \kappa} \Gamma^{a \kappa}{}_\beta \psi^\beta \\ & = \phi^T C \Gamma^a \psi \\ & \coloneqq \overline{\phi} \Gamma^a \psi \end{aligned} \,.

(Recall that all this is here for Majorana spinors, as in the previous prop. 5.)

This yields the component expressions for the bilinear pairings as familiar from the physics supersymmetry literature, for instance (Polchinski 01, (B.2.1), (B.5.1))

Counting numbers of supersymmetries

A spinor bilinear pairing to a vector Γ:SSV\Gamma \;\colon\; S \otimes S \to V as above serves as the odd-odd bracket in a super Poincaré Lie algebra extension of VV. Since this is also called a “supersymmetrysuper Lie algebra, with the spinors being the supersymmetry generators, the decomposition of SS into minimal/irreducible representations is also called the number of supersymmetries. This is traditionally denoted by a capital NN and in even dimensions and over the complex numbers it is traditional to write

N=(N +,N ) N = (N_+, N_-)

to indicate that there are N +N_+ copies of the irreducible Spin(V)Spin(V)-representation of one chirality, and N N_- of those of the other chirality (i.e. left and right handed Weyl spinors).

This counting however is more subtle over the real numbers (Majorana spinors) and the notation in this case (which happens to be the more important case) is not entirely consistent through the literature.

There is no issue in those dimensions in which the complex Weyl representation already admits a real structure itself, hence when there are Majorana-Weyl spinors. In this case one just counts them with N +N_+ and N N_- as in the case over the complex numbers.

However, in some dimensions it is only the direct sum of two Weyl spinor representations which carries a real structure. For instance for d=4d = 4 and d=8d = 8 in Lorentzian signature (see the above table) it is the complex representations 22\mathbf{2} \oplus \mathbf{2}' and 1616\mathbf{16} \oplus \mathbf{16}', respectively, which carry a real structure. Hence the real representation underlying this parameterizes N=1N = 1 supersymmetry in terms Majorana spinors, even though its complexification would be N=(1,1)N = (1,1). See for instance (Freed 99, p. 53).

Similarly in dimensions 5,6 and 7 mod 8, the minimal real representation is obatained from tensoring the complex spinors with the complex 2-dimensional canonical quaternionic representation WW (as in the above table). These are also called symplectic Majorana representations. For instance in in 6d one typically speaks of the 6d (2,0)-superconformal QFT to refer to that with a single “symplectic Majorana-Weyl” supersymmetry (e.g. Figueroa-OFarrill, p. 9), which might therefore be counted as (1,0)(1,0) real supersymmetric, but which involves two complex irreps and is hence often denoted counted as (2,0)(2,0).

p=2p = 2 – superconformal bracket

For the moment see at supersymmetry – Superconformal and super anti de Sitter symmetry.

Expression of real representations via real normed division algebras

We discuss a close relation between real spin representations and division algebras, due to Kugo-Townsend 82, Sudbery 84 and others: The real spinor representations in dimensions 3,4,6,103,4,6, 10 happen to have a particularly simple expression in terms of Hermitian matrices over the four real normed division algebras: the real numbers \mathbb{R} themselves, the complex numbers \mathbb{C}, the quaternions \mathbb{H} and the octonions 𝕆\mathbb{O}. Derived from this also the real spinor representations in dimensions 4,5,7,114,5,7,11 have a fairly simple corresponding expression. We follow the streamlined discussion in Baez-Huerta 09 and Baez-Huerta 10.

Real normed division algebras

To amplify the following pattern and to fix our notation for algebra generators, recall these definitions:

Definition

The complex numbers \mathbb{C} is the commutative algebra over the real numbers \mathbb{R} which is generated from one generators {e 1}\{e_1\} subject to the relation

  • (e 1) 2=1(e_1)^2 = -1.
Definition

The quaternions \mathbb{H} is the associative algebra over the real numbers which is generated from three generators {e 1,e 2,e 3}\{e_1, e_2, e_3\} subject to the relations

quaternion multiplication table
  1. for all ii

    (e i) 2=1(e_i)^2 = -1

  2. for (i,j,k)(i,j,k) a cyclic permutation of (1,2,3)(1,2,3) then

    1. e ie j=e ke_i e_j = e_k

    2. e je i=e ke_j e_i = -e_k

(graphics grabbed from Baez 02)

Definition

The octonions 𝕆\mathbb{O} is the nonassociative algebra over the real numbers which is generated from seven generators {e 1,,e 7}\{e_1, \cdots, e_7\} subject to the relations

octonion multiplication table
  1. for all ii

    (e i) 2=1(e_i)^2 = -1

  2. for e ie je ke_i \to e_j \to e_k an edge or circle in the diagram shown (a labeled version of the Fano plane) then

    1. e ie j=e ke_i e_j = e_k

    2. e je i=e ke_j e_i = -e_k

    and all relations obtained by cyclic permutation of the indices in these equations.

(graphics grabbed from Baez 02)

One defines the following operations on these real algebras:

Definition

For 𝕂{,,,𝕆}\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}, let

() *:𝕂𝕂 (-)^\ast \;\colon\; \mathbb{K} \longrightarrow \mathbb{K}

be the antihomomorphism of real algebras

(ra) *=ra * ,forr,a𝕂 (ab) *=b *a * ,fora,b𝕂 \begin{aligned} (r a)^\ast = r a^\ast &, \text{for}\;\; r \in \mathbb{R}, a \in \mathbb{K} \\ (a b)^\ast = b^\ast a^\ast &,\text{for}\;\; a,b \in \mathbb{K} \end{aligned}

given on the generators of def. 4, def. 5 and def. 6 by

(e i) *=e i. (e_i)^\ast = - e_i \,.

This operation makes 𝕂\mathbb{K} into a star algebra. For the complex numbers \mathbb{C} this is called complex conjugation, and in general we call it conjugation.

Let then

Re:𝕂 Re \;\colon\; \mathbb{K} \longrightarrow \mathbb{R}

be the function

Re(a)12(a+a *) Re(a) \;\coloneqq\; \tfrac{1}{2}(a + a^\ast)

(“real part”) and

Im:𝕂 Im \;\colon\; \mathbb{K} \longrightarrow \mathbb{R}

be the function

Im(a)12(aa *) Im(a) \;\coloneqq \; \tfrac{1}{2}(a - a^\ast)

(“imaginary part”).

It follows that for all a𝕂a \in \mathbb{K} then the product of a with its conjugate is in the real center of 𝕂\mathbb{K}

aa *=a *a𝕂 a a^\ast = a^\ast a \;\in \mathbb{R} \hookrightarrow \mathbb{K}

and we write the square root of this expression as

|a|aa * {\vert a\vert} \;\coloneqq\; \sqrt{a a^\ast}

called the norm or absolute value function

||:𝕂. {\vert -\vert} \;\colon\; \mathbb{K} \longrightarrow \mathbb{R} \,.

This norm operation clearly satisfies the following properties (for all a,b𝕂a,b \in \mathbb{K})

  1. |a|0\vert a \vert \geq 0;

  2. |a|=0a=0{\vert a \vert } = 0 \;\;\;\;\; \Leftrightarrow\;\;\;\;\;\; a = 0;

  3. |ab|=|a||b|{\vert a b \vert } = {\vert a \vert} {\vert b \vert}

and hence makes 𝕂\mathbb{K} a normed algebra.

Since \mathbb{R} is a division algebra, these relations immediately imply that each 𝕂\mathbb{K} is a division algebra, in that

ab=0a=0orb=0. a b = 0 \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; a = 0 \;\; \text{or} \;\; b = 0 \,.

Hence the conjugation operation makes 𝕂\mathbb{K} a real normed division algebra.

Remark

Sending each generator in def. 4, def. 5 and def. 6 to the generator of the same name in the next larger algebra constitutes a sequence of real star-algebra homomorphisms

𝕆. \mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O} \,.
Proposition

(Hurwitz theorem)

The four algebras of real numbers \mathbb{R}, complex numbers \mathbb{C}, quaternions \mathbb{H} and octonions 𝕆\mathbb{O} from def. 4, def. 5 and def. 6 respectively, which are real normed division algebras via def. 7, are, up to isomorphism, the only real normed division algebras that exist.

Remark

While hence the sequence from remark 4

𝕆 \mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}

is maximal in the category of real normed non-associative division algebras, there is a pattern that does continue if one disregards the division algebra property. Namely each step in this sequence is given by a construction called forming the Cayley-Dickson double algebra. This continues to an unbounded sequence of real nonassociative star-algebras

𝕆𝕊 \mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O} \hookrightarrow \mathbb{S} \hookrightarrow \cdots

where the next algebra 𝕊\mathbb{S} is called the sedenions.

What actually matters for the following relation of the real normed division algebras to real spin representations is that they are also alternative algebras:

Definition

Given any non-associative algebra AA, then the trilinear map

[,,]AAAA [-,-,-] \;-\; A \otimes A \otimes A \longrightarrow A

given on any elements a,b,cAa,b,c \in A by

[a,b,c](ab)ca(bc) [a,b,c] \coloneqq (a b) c - a (b c)

is called the associator (in analogy with the commutator [a,b]abba[a,b] \coloneqq a b - b a ).

If the associator is completely antisymmetric (in that for any permutation σ\sigma of three elements then [a σ 1,a σ 2,a σ 3]=(1) |σ|[a 1,a 2,a 3][a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3] for |σ|\vert \sigma \vert the signature of the permutation) then AA is called an alternative algebra.

If the characteristic of the ground field is different from 2, then alternativity is readily seen to be equivalent to the conditions that for all a,bAa,b \in A then

(aa)b=a(ab)and(ab)b=a(bb). (a a)b = a (a b) \;\;\;\;\; \text{and} \;\;\;\;\; (a b) b = a (b b) \,.

We record some basic properties of associators in alternative star-algebras that we need below:

Proposition

Let AA be an alternative algebra (def. 8) which is also a star algebra. Then

  1. the associator vanishes when at least one argument is real

    [Re(a),b,c] [Re(a),b,c]
  2. the associator changes sign when one of its arguments is conjugated

    [a,b,c]=[a *,b,c]; [a,b,c] = -[a^\ast,b,c] \,;
  3. the associator vanishes when one of its arguments is the conjugate of another:

    [a,a *,b]=0; [a,a^\ast, b] = 0 \,;
  4. the associator is purely imaginary

    Re([a,b,c])=0. Re([a,b,c]) = 0 \,.
Proof

That the associator vanishes as soon as one argument is real is just the linearity of an algebra product over the ground ring.

Hence in fact

[a,b,c]=[Im(a),Im(b),Im(c)]. [a,b,c] = [Im(a), Im(b), Im(c)] \,.

This implies the second statement by linearity. And so follows the third statement by skew-symmetry:

[a,a *,b]=[a,a,b]=0. [a,a^\ast,b] = -[a,a,b] = 0 \,.

The fourth statement finally follows by this computation:

[a,b,c] * =[c *,b *,a *] =[c,b,a] =[a,b,c]. \begin{aligned} [a,b,c]^\ast & = -[c^\ast, b^\ast, a^\ast] \\ & = -[c,b,a] \\ & = -[a,b,c] \end{aligned} \,.

Here the first equation follows by inspection and using that (ab) *=b *a *(a b)^\ast = b^\ast a^\ast, the second follows from the first statement above, and the third is the ant-symmetry of the associator.

It is immediate to check that:

Proposition

The real algebras of real numbers, complex numbers, def. 4,quaternions def. 5 and octonions def. 6 are alternative algebras (def. 8).

Proof

Since the real numbers, complex numbers and quaternions are associative algebras, their associator vanishes identically. It only remains to see that the associator of the octonions is skew-symmetric. By linearity it is sufficient to check this on generators. So let e ie je ke_i \to e_j \to e_k be a circle or a cyclic permutation of an edge in the Fano plane. Then by definition of the octonion multiplication we have

(e ie j)e j =e ke j =e je k =e i =e i(e je j) \begin{aligned} (e_i e_j) e_j &= e_k e_j \\ &= - e_j e_k \\ & = -e_i \\ & = e_i (e_j e_j) \end{aligned}

and similarly

(e ie i)e j =e j =e ke i =e ie k =e i(e ie j). \begin{aligned} (e_i e_i ) e_j &= - e_j \\ &= - e_k e_i \\ &= e_i e_k \\ &= e_i (e_i e_j) \end{aligned} \,.

The analog of the Hurwitz theorem (prop. 6) is now this:

Proposition

The only division algebras over the real numbers which are also alternative algebras (def. 8) are the real numbers themselves, the complex numbers, the quaternions and the octonions.

This is due to (Zorn 30).

For the following, the key point of alternative algebras is this equivalent characterization:

Proposition

A nonassociative algebra is alternative, def. 8, precisely if the subalgebra generated by any two elements is an associative algebra.

This is due to Emil Artin, see for instance (Schafer 95, p. 18).

Proposition 10 is what allows to carry over a minimum of linear algebra also to the octonions such as to yield a representation of the Clifford algebra on 9,1\mathbb{R}^{9,1}. This happens in the proof of prop. 13 below.

So we will be looking at a fragment of linear algebra over these four normed division algebras. To that end, fix the following notation and terminology:

Definition

Let 𝕂\mathbb{K} be one of the four real normed division algebras from prop. 6, hence one of the four real alternative division algebras from prop. 9.

Say that an n×nn \times n matrix with coefficients in 𝕂\mathbb{K}, AMat n×n(𝕂)A\in Mat_{n\times n}(\mathbb{K}) is a hermitian matrix if the transpose matrix (A t) ijA ji(A^t)_{i j} \coloneqq A_{j i} equals the componentwise conjugated matrix (def. 7):

A t=A *. A^t = A^\ast \,.

Hence with the notation

() (() t) * (-)^\dagger \coloneqq ((-)^t)^\ast

then AA is a hermitian matrix precisely if

A=A . A = A^\dagger \,.

We write Mat 2×2 her(𝕂)Mat_{2 \times 2}^{her}(\mathbb{K}) for the real vector space of hermitian matrices.

Definition

(trace reversal)

Let AMat 2×2 her(𝕂)A \in Mat_{2 \times 2}^{her}(\mathbb{K}) be a hermitian 2×22 \times 2 matrix as in def. 9. Its trace reversal is the result of subtracting its trace times the identity matrix:

A˜A(trA)1 n×n. \tilde A \;\coloneqq\; A - (tr A) 1_{n\times n} \,.

Spacetime in dimensions 3,4,6 and 10

We discuss how Minkowski spacetime of dimension 3,4,6 and 10 is naturally expressed in terms of the real normed division algebras 𝕂\mathbb{K} from prop. 6, equivalently the real alternative division algebras from prop. 9.

Proposition

Let 𝕂\mathbb{K} be one of the four real normed division algebras from prop. 6, hence one of the four real alternative division algebras from prop. 9.

There is a isomorphism (of real inner product spaces) between Minkowski spacetime (def. \ref{MinkowskiSpacetime}) of dimension

d=2+dim (𝕂) d = 2 + dim_{\mathbb{R}}(\mathbb{K})

hence

  1. 2,1\mathbb{R}^{2,1} for 𝕂=\mathbb{K} = \mathbb{R};

  2. 3,1\mathbb{R}^{3,1} for 𝕂=\mathbb{K} = \mathbb{C};

  3. 5,1\mathbb{R}^{5,1} for 𝕂=\mathbb{K} = \mathbb{H};

  4. 9,1\mathbb{R}^{9,1} for 𝕂=𝕆\mathbb{K} = \mathbb{O}.

and the real vector space of 2×22 \times 2 hermitian matrices over 𝕂\mathbb{K} (def. 9) equipped with the inner product whose norm-square is the negative of the determinant operation on matrices:

dim (𝕂)+1,1(Mat 2×2 her(𝕂),det). \mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} \;\simeq\; \left(Mat_{2 \times 2}^{her}(\mathbb{K}), -det \right) \,.

As a linear map this is given by

(x 0,x 1,,x d1)(x 0+x 1 y y * x 0x 1)withyx 21+x 3e 1+x 4e 2++x 2+dim (𝕂)e dim (𝕂)1. (x_0, x_1, \cdots, x_{d-1}) \;\mapsto\; \left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) \;\;\; \text{with}\; y \coloneqq x_2 1 + x_3 e_1 + x_4 e_2 + \cdots + x_{2 + dim_{\mathbb{R}(\mathbb{K})}} \,e_{dim_{\mathbb{R}}(\mathbb{K})-1} \,.

Under this identification the operation of trace reversal from def. 10 corresponds to time reversal in that

widetlde(x 0+x 1 y y * x 0x 1)=(x 0+x 1 y y * x 0x 1). \widetlde{ \left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) } \;=\; \left( \array{ -x_0 + x_1 & y \\ y^\ast & -x_0 - x_1 } \right) \,.
Proof

This is immediate from the nature of the conjugation operation () *(-)^\ast from def. 7:

det(x 0+x 1 y y * x 0x 1) =(x 0+x 1)(x 0x 1)+yy * =(x 0) 2+a=1d1(x a) 2. \begin{aligned} - det \left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) & = -(x_0 + x_1)(x_0 - x_1) + y y^\ast \\ & = -(x_0)^2 + \underoverset{a = 1}{d-1}{\sum} (x_a)^2 \end{aligned} \,.

By direct computation one finds:

Proposition

In terms of the trace reversal operation ()˜\widetilde{(-)} from def. 10, the determinant operation on hermitian matrices (def. 9) has the following alternative expression

det(A) =AA˜ =A˜A. \begin{aligned} -det(A) & = A \tilde A \\ & = \tilde A A \end{aligned} \,.

and the Minkowski inner product has the alternative expression

η(A,B)=12Re(tr(AB˜))=12Re(tr(A˜B)). \eta(A,B) = \tfrac{1}{2}Re(tr(A \tilde B)) = \tfrac{1}{2} Re(tr(\tilde A B)) \,.

(Baez-Huerta 09, prop. 5)

Real spinors in dimensions 3, 4, 6 and 10

We now discuss how real spin representations in dimensions 3,4, 6 and 10 are naturally induced from linear algebra over the four real alternative division algebras.

In particular we establish the following table of exceptional isomorphisms of spin groups:

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)"SL(2,𝕆)"Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}𝕆\mathbb{O} the octonionsheterotic/type II string
Remark

Prop .11 immediately implies that for 𝕂{,,}\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\} then there is a monomorphism from the special linear group SL(2,𝕂)SL(2,\mathbb{K}) to the spin group in the given dimension:

SL(2,𝕂)Spin(dim (𝕂)+1,1) SL(2,\mathbb{K}) \hookrightarrow Spin(dim_{\mathbb{R}(\mathbb{K} )} +1 ,1)

given by

AA()A . A \mapsto A(-)A^\dagger \,.

This preserves the determinant, and hence the Lorentz form, by the multiplicative property of the determinant:

det(A()A )=det(A)=1det()det(A)=1 *=det(). det(A(-)A^\dagger) = \underset{=1}{\underbrace{det(A)}} det(-) \underset{= 1}{\underbrace{det(A)}}^\ast = det(-) \,.

Hence it remains to show that this is surjective, and to define this action also for 𝕂\mathbb{K} being the octonions, where general matrix calculus does not apply, due to non-associativity.

Definition

Let 𝕂\mathbb{K} be one of the four real normed division algebras from prop. 6, hence one of the four real alternative division algebras from prop. 9.

Define a real linear map

Γ: dim (𝕂)+1,1End (𝕂 4) \Gamma \;\colon\; \mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} \simeq End_{\mathbb{R}}(\mathbb{K}^4)

from (the real vector space underlying) Minkowski spacetime to real linear maps on 𝕂 4\mathbb{K}^4

Γ(A)(ψ ϕ)(A˜ϕ Aψ). \Gamma(A) \left( \array{ \psi \\ \phi } \right) \;\coloneqq\; \left( \array{ \tilde A \phi \\ A \psi } \right) \,.

Here on the right we are using the isomorphism from prop. 11 for identifying a spacetime vector with a 2×22 \times 2-matrix, and we are using the trace reversal idetilde()\idetilde(-) from def. 10.

Remark

Each operation of Γ(A)\Gamma(A) in def. 11 is clearly a linear map, even for 𝕂\mathbb{K} being the non-associative octonions. The only point to beware of is that for 𝕂\mathbb{K} the octonions, then the composition of two such linear maps is not in general given by the usual matrix product.

Proposition

The map Γ\Gamma in def. 11 gives a representation of the Clifford algebra Cl( dim (𝕂+1,1))Cl(\mathbb{R}^{dim_{\mathbb{R}}}(\mathbb{K}+1,1) ) (def. \ref{CliffordAlgebra}), i.e of

  1. Cl( 2,1)Cl(\mathbb{R}^{2,1}) for 𝕂=\mathbb{K} = \mathbb{R};

  2. Cl( 3,1)Cl(\mathbb{R}^{3,1}) for 𝕂=\mathbb{K} = \mathbb{C};

  3. Cl( 5,1)Cl(\mathbb{R}^{5,1}) for 𝕂=\mathbb{K} = \mathbb{H};

  4. Cl( 9,1)Cl(\mathbb{R}^{9,1}) for 𝕂=𝕆\mathbb{K} = \mathbb{O}.

Hence this Clifford representation induces representations of the spin group Spin(dim (𝕂)+1,1)Spin(dim_{\mathbb{R}}(\mathbb{K})+1,1) on the real vector spaces

S ±𝕂 2. S_{\pm } \coloneqq \mathbb{K}^2 \,.

(Baez-Huerta 09, p. 6)

Proof

We need to check that the Clifford relation

(Γ(A)) 2=η(A,A)1 (\Gamma(A))^2 = -\eta(A,A)1

is satisfied. Now by definition, for any (ϕ,ψ)𝕂 4(\phi,\psi) \in \mathbb{K}^4 then

(Γ(A)) 2(ϕ ψ)=(A˜(Aϕ) A(A˜ψ)), (\Gamma(A))^2 \left( \array{ \phi \\ \psi } \right) \;=\; \left( \array{ \tilde A(A \phi) \\ A(\tilde A \psi) } \right) \,,

where on the right we have in each component ordinary matrix product expressions.

Now observe that both expressions on the right are sums of triple products that involve either one real factor or two factors that are conjugate to each other:

A(A˜ψ) =(x 0+x 1 y y * x 0x 1) =((x 0+x 1)ϕ 1+yϕ 2 y *ϕ 1(x 0+x 1)ϕ 2) =((x 0 2+x 1 2)ϕ 1+(x 0+x 1)(yϕ 2)+y(y *ϕ 1)y((x 0+x 1)ϕ 2) ). \begin{aligned} A (\tilde A \psi) & = \left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) \\ & = \left( \array{ (-x_0 + x_1) \phi_1 + y \phi_2 \\ y^\ast \phi_1 - (x_0 + x_1)\phi_2 } \right) \\ & = \left( \array{ (-x_0^2 + x_1^2) \phi_1 + (x_0 + x_1)(y \phi_2) + y (y^\ast \phi_1) - y( (x_0 + x_1) \phi_2 ) \\ \cdots } \right) \end{aligned} \,.

Since the associators of triple products that involve a real factor and those involving both an element and its conjugate vanish by prop. 7 (hence ultimately by Artin’s theorem, prop. 10). In conclusion all associators involved vanish, so that we may rebracket to obtain

(Γ(A)) 2(ϕ ψ)=((A˜A)ϕ (AA˜)ψ). (\Gamma(A))^2 \left( \array{ \phi \\ \psi } \right) \;=\; \left( \array{ (\tilde A A) \phi \\ (A \tilde A) \psi } \right) \,.

This implies the statement via the equality AA˜=A˜A=det(A)A \tilde A = \tilde A A = -det(A) (prop. 12).

Remark

Prop. 13 says that the isomorphism of prop. 11 is that given by forming generalized Pauli matrices. In standard physics notation these matrices are written as

Γ(x a)=(γ αα˙ a). \Gamma(x^a) = (\gamma^a_{\alpha \dot \alpha}) \,.
Proposition

The spin representations given via prop. 13 by the Clifford representation of def. 11 are the following:

  1. for 𝕂=\mathbb{K} = \mathbb{R} the Majorana representation of Spin(2,1)Spin(2,1) on S +S S_+ \simeq S_-;

  2. for 𝕂=\mathbb{K} = \mathbb{C} the Majorana representation of Spin(3,1)Spin(3,1) on S +S S_+ \simeq S_-;

  3. for 𝕂=\mathbb{K} = \mathbb{H} the Weyl representation of Spin(5,1)Spin(5,1) on S +S_+ and on S S_-;

  4. for 𝕂=𝕆\mathbb{K} = \mathbb{O} the Majorana-Weyl representation of Spin(9,1)Spin(9,1) on S +S_+ and on S S_-.

Proposition

Under the identification of prop. 13 the bilinear pairings

()¯():S +S \overline{(-)}(-) \;\colon\; S_+ \otimes S_-\longrightarrow \mathbb{R}

and

()¯Γ():S ±S ±V \overline{(-)}\Gamma (-) \;\colon\; S_\pm \otimes S_{\pm}\longrightarrow V

from above are given, respectively, by forming the real part of the canonical 𝕂\mathbb{K}-inner product

()¯():S +S \overline{(-)}(-) \colon S_+\otimes S_- \longrightarrow \mathbb{R}
(ψ,ϕ)ψ¯ϕRe(ψ ϕ) (\psi,\phi)\mapsto \overline{\psi} \phi \coloneqq Re(\psi^\dagger \cdot \phi)

and by forming the product of a column vector with a row vector to produce a matrix, possibly up to trace reversal (def. 10):

S +S +V S_+ \otimes S_+ \longrightarrow V
(ψ,ϕ)ψ¯Γϕψϕ +ϕψ ˜ (\psi , \phi) \mapsto \overline{\psi}\Gamma \phi \coloneqq \widetilde{\psi \phi^\dagger + \phi \psi^\dagger}

and

S S V S_- \otimes S_- \longrightarrow V
(ψ,ϕ)ψϕ +ϕψ (\psi , \phi) \mapsto {\psi \phi^\dagger + \phi \psi^\dagger}

For AVA \in V the AA-component of this map is

η(ψ¯Γϕ,A)=Re(ψ (Aϕ)). \eta(\overline{\psi}\Gamma \phi, A) = Re (\psi^\dagger (A\phi)) \,.

(Baez-Huerta 09, prop. 8, prop. 9).

Example

Consider the case 𝕂=\mathbb{K} = \mathbb{R} of real numbers.

Now V=Mat 2×2 symm()V= Mat^{symm}_{2\times 2}(\mathbb{R}) is the space of symmetric 2x2-matrices with real numbers.

V={(t+x y y tx)|t,x,y} V = \left\{ \left. \left( \array{ t + x & y \\ y & t - x } \right) \right\vert t,x,y\in \mathbb{R} \right\}

The “light-cone”-basis for this space would be

{v +(1 0 0 0),v (0 0 0 1),v y(0 1 1 0)} \left\{ v^+ \coloneqq \left( \array{ 1 & 0 \\ 0 & 0 } \right) \,, \; v^- \coloneqq \left( \array{ 0 & 0 \\ 0 & 1 } \right) \,, \; v^y \coloneqq \left( \array{ 0 & 1 \\ 1 & 0 } \right) \right\}

Its trace reversal (def. 10) is

{v˜ +(0 0 0 1),v˜ (1 0 0 0),v˜ y(0 1 1 0)} \left\{ \tilde{v}^+ \coloneqq \left( \array{ 0 & 0 \\ 0 & -1 } \right) \,, \; \tilde v^- \coloneqq \left( \array{ -1 & 0 \\ 0 & 0 } \right) \,, \; \tilde v^y \coloneqq \left( \array{ 0 & 1 \\ 1 & 0 } \right) \right\}

Hence the Minkowski metric of prop. 11 in this basis has the components

η=(0 1 0 1 0 0 0 0 2). \eta = \left( \array{ 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 2 } \right) \,.

As vector spaces S ±= 2S_{\pm} = \mathbb{R}^2.

The bilinear spinor pairing ()¯():S +S \overline{(-)}(-) \colon S_+ \otimes S_- \to \mathbb{R} is given by

ψ¯ϕ =ψ tϕ =ψ 1ϕ 1+ψ 2ϕ 2. \begin{aligned} \overline{\psi}\phi &= \psi^t \cdot \phi \\ & = \psi_1 \phi_1 + \psi_2 \phi_2 \end{aligned} \,.

The spinor pairing S +S +V *S_+ \otimes S_+ \otimes V^\ast \to \mathbb{R} from prop. 15 is given on an A=A +v ++A v +A yv yA = A_+ v^+ + A_- v^- + A_y v^y (A +,A ,A yA_+, A_-, A_y \in \mathbb{R}) by the components

η(ψ¯Γϕ,A) =ψ tAϕ =ψ 1ϕ 1A ++ψ 2ϕ 2A +(ψ 1ϕ 2+ψ 2ϕ 1)A y \begin{aligned} \eta(\overline{\psi}\Gamma\phi,A) &= \psi^t \cdot A \cdot \phi \\ & = \psi_1 \phi_1 A_+ + \psi_2 \phi_2 A_- + (\psi_1 \phi_2 + \psi_2 \phi_1) A_y \end{aligned}

and S S V *S_- \otimes S_- \otimes V^\ast \to \mathbb{R} is given by the components

η(ψ¯Γϕ,A) =ψ tA˜ϕ =ψ 1ϕ 1A +ψ 2ϕ 2A +(ψ 1ϕ 2+ψ 2ϕ 1)A y. \begin{aligned} \eta(\overline{\psi}\Gamma\phi,A) &= \psi^t \cdot \tilde A \cdot \phi \\ &= -\psi_1 \phi_1 A_+ - \psi_2 \phi_2 A_- + (\psi_1 \phi_2 + \psi_2 \phi_1) A_y \end{aligned} \,.

and so, in view of the above metric components, in terms of dual bases {ψ i}\{\psi^i\} this is

μ=ψ 1ψ 1A ψ 2ψ 2A ++12(ψ 1ψ 2ψ 2ψ 1)A y \mu = - \psi^1 \otimes \psi^1 \otimes A_- - \psi^2 \otimes \psi^2 \otimes A_+ + \frac{1}{2} (\psi^1 \otimes \psi^2 \oplus \psi^2 \otimes\psi^1) \otimes A_y

So there is in particular the 2-dimensional space of isomorphisms of super Minkowski spacetime super translation Lie algebras

2,1|2 2,1|2¯ \mathbb{R}^{2,1|\mathbf{2}} \stackrel{\simeq}{\longrightarrow} \mathbb{R}^{2,1|\bar\mathbf{2}}

(not though of the corresponding super Poincaré Lie algebras, because for them the difference in the Spin-representation does matter) spanned by

(θ 1,θ 2,e)(θ 1,θ 2,e) (\theta_1,\theta_2, \vec e) \mapsto (\theta_1, -\theta_2, -\vec e)

and by

(θ 1,θ 2,e)(θ 1,θ 2,e). (\theta_1,\theta_2, \vec e) \mapsto (-\theta_1, \theta_2, -\vec e) \,.

Hence there is a 1-dimensional space of non-trivial automorphism

2,1|2 2,1|2 \mathbb{R}^{2,1|\mathbf{2}} \stackrel{\simeq}{\longrightarrow} \mathbb{R}^{2,1|\mathbf{2}}

spanned by

(θ 1,θ 2,e)(θ 1,θ 2,e). (\theta_1,\theta_2, \vec e) \mapsto (-\theta_1, -\theta_2, \vec e) \,.

Real spinors in dimensions 4,5,7 and 11

Definition

Write VMat 2×2 hermitian(𝕂)V \coloneqq Mat^{hermitian}_{2\times 2}(\mathbb{K}) \oplus \mathbb{R}.

Write S𝕂 4S \coloneqq \mathbb{K}^4. Define a real linear map

Γ:VEnd(S) \Gamma \;\colon\; V\longrightarrow End(S)

given by left matrix multiplication

Γ(A,a)(a1 2×2 A˜ A a1 2×2). \Gamma(A,a) \coloneqq \left( \array{ a \cdot 1_{2\times 2} & \tilde A \\ A & -a \cdot 1_{2\times 2} } \right) \,.
Remark

The real vector space VV in def. 12 equipped with the inner product η(,)\eta(-,-) given by

η((A,a),(A,a))=det(A)+a 2 \eta((A,a), (A,a)) = det(A) + a^2

is Minkowski spacetime

  1. 3,1\mathbb{R}^{3,1} for 𝕂=\mathbb{K} = \mathbb{R};

  2. 4,1\mathbb{R}^{4,1} for 𝕂=\mathbb{K} = \mathbb{C};

  3. 6,1\mathbb{R}^{6,1} for 𝕂=\mathbb{K} = \mathbb{H};

  4. 10,1\mathbb{R}^{10,1} for 𝕂=𝕆\mathbb{K} = \mathbb{O}.

Proposition

The map Γ\Gamma in def. 12 gives a representation of the Clifford algebra of

  1. 3,1\mathbb{R}^{3,1} for 𝕂=\mathbb{K} = \mathbb{R};

  2. 4,1\mathbb{R}^{4,1} for 𝕂=\mathbb{K} = \mathbb{C};

  3. 6,1\mathbb{R}^{6,1} for 𝕂=\mathbb{K} = \mathbb{H};

  4. 10,1\mathbb{R}^{10,1} for 𝕂=𝕆\mathbb{K} = \mathbb{O}.

Under restriction along Spin(n+2,1)Cl(n+2,1)Spin(n+2,1) \hookrightarrow Cl(n+2,1) this is isomorphic to

  1. for 𝕂=\mathbb{K} = \mathbb{R} the Majorana representation of Spin(3,1)Spin(3,1) on SS;

  2. for 𝕂=\mathbb{K} = \mathbb{C} the Dirac representation of Spin(4,1)Spin(4,1) on SS;

  3. for 𝕂=\mathbb{K} = \mathbb{H} the Dirac representation of Spin(6,1)Spin(6,1) on SS;

  4. for 𝕂=𝕆\mathbb{K} = \mathbb{O} the Majorana representation of Spin(10,1)Spin(10,1) on SS.

(Baez-Huerta 10, p. 10, prop. 8, prop. 9)

Write

Γ 0(0 1 2x2 1 2×2 0). \Gamma^0 \coloneqq \left( \array{ 0 & - 1_{2x2} \\ 1_{2\times 2} & 0 } \right) \,.
Proposition

Under the identification of prop. 16 of the bilinear pairings

()¯():SS \overline{(-)}(-) \;\colon\; S \otimes S \longrightarrow \mathbb{R}

and

()¯Γ():SSV \overline{(-)}\Gamma (-) \;\colon\; S \otimes S \longrightarrow V

of remark 1, the first is given by

(ψ,ϕ)ψ¯ϕRe(ψ Γ 0ϕ) (\psi,\phi) \mapsto \overline\psi\phi \coloneqq Re(\psi^\dagger \Gamma^0 \phi)

and the second is characterized by

η(ψ¯Γϕ,A) =ψ¯(Γ(A)ϕ) =Re(ψ Γ 0Γ(A)ϕ). \begin{aligned} \eta \left( \overline{\psi}\Gamma\phi, A \right) &= \overline{\psi}(\Gamma(A)\phi) \\ & = Re( \psi^\dagger \Gamma^0 \Gamma(A)\phi) \end{aligned} \,.

(Baez-Huerta 10, prop. 10, prop. 11).

References

Accounts in the mathematical literature include

Specifically for Lorentzian signature and with an eye towards supersymmetry in QFT, see

For the component notation traditionally used in physics see for instance

For good math/physics discussion with special emphasis on the symplectic Majorana spinors and their role in the 6d (2,0)-superconformal QFT see

A clean summary of the relation of the real representation to Hermitian forms over the real normed division algebras is in

Revised on January 18, 2017 11:29:21 by Urs Schreiber (194.210.233.24)