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Representations of Clifford algebras

In mathematics, the representations of Clifford algebras are also known as Clifford modules. In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined.

The abstract algebra theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and A. Shapiro (Clifford Modules, Topology 3 (Suppl. 1) (1964), 3?38). This article gives an explicit theory.

Matrix representations of real Clifford algebras

We will need to study anticommuting matrices (AB = −BA) because in Clifford algebras orthogonal vectors anticommute

<math> A \cdot B = \frac{1}{2}( AB + BA ) = 0<math>

For the real Clifford algebra R_p,q we need p + q mutually anticommuting matrices, of which p have +1 as square and q have −1 as square.

<math> \begin{matrix}

\gamma_a^2 &=& +1 &\mbox{if} &1 \le a \le p \\ \gamma_a^2 &=& -1 &\mbox{if} &p+1 \le a \le p+q\\ \gamma_a \gamma_b &=& -\gamma_b \gamma_a &\mbox{if} &a \ne b \ \\

\end{matrix}<math>

Such a base of gamma matrices is not unique. On can allways obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.

<math> \begin{matrix}

\gamma_{a'} &=& S &\gamma_{a } &S^{-1} \end{matrix} <math>

where S is a non-singular matrix. The sets γ _a' and γ _a belong to the same equivalence class.

Intermezzo: the K-system for naming matrices

We first present a nice method for naming 2ⁿ × 2ⁿ matrices

<math>

K_0 = \begin{pmatrix} 1&0\\0&1 \end{pmatrix}, K_1 = \begin{pmatrix} 0&1\\1&0 \end{pmatrix}, K_2 = \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}, K_3 = \begin{pmatrix} 1&0\\0&-1 \end{pmatrix}. <math>

Notice that K₀ is the identity matrix. The names were so chosen that there is a simple rule for remembering the products:

K₁ K₂ = K₃

K₁ K₃ = K₂

K₂ K₃ = K₁

K₂ K₁ = −K₃

K₃ K₁ = −K₂

K₃ K₂ = −K₁.

Incrementing index is positive result. Decreasing index is negative result.

Attention ! These are NOT the same relations that hold for the standard basis of the quaternions. If you would name i = i₁, j = i₂ and k = i₃ you would get

i₁i₂=i₃

i₂i₃=i₁

i₃i₁=i₂

so the last rule is different. We will later see that the pure quaterions i,j and k can be represented by K₁₂,K₂₀and K₃₂

Remark that

<math> K_0^2 = K_1^2 = K_3^2 = K_0 <math>

<math> K_2^2 = - K_0 <math>

K₂ is the only one with negative square, so it can be regarded as the simplest representation of i

Then we give all possible Kronecker products a name (see matrix multiplication):

<math>

K_{ab} = K_{a} \otimes K_{b} <math>

<math>

K_{abc} = K_{a} \otimes K_{bc}= K_{a} \otimes K_{b} \otimes K_{c} <math>

Some examples

<math>

K_{30} = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix},

K_{11} = \begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{pmatrix} <math>

Each index has its level ( 2x2, 4x4, 8x8, 16x16, ...)

K₁₃ is a K₃ at the 2x2 level and a K₁ at the 4x4 level. With this notation its very easy to multiply large square matrices since

<math> (A \otimes B)(C \otimes D) = AB \otimes CD <math>

Lets work out an example

K₁₂₃ K₂₂₂ = K₃₀₁

8x8-level 1 times 2 gives 3

4x4-level 2 times 2 gives 0 but remember the minus sign

2x2-level 3 times 2 gives 1 but with again a minus sign

( the two minus signs cancel so the result is K₃₀₁

We can now start to construct sets of mutually anticommuting orthogonal matrices, sometimes called Hodge dual of every element is simply minus the original)

<math> *A = -A <math>

Real Clifford algebra R_0,2

Here p = 0 and q = 2 so we need 2 two anti-commuting K₋ matrices as base vectors. This is not possible with real 2×2 matrices so we need to use 4×4 matrices, and there are many possibilities. This algebra is isomorphic with the ring H of quaternions.

grade 0 (the scalar)

<math> \begin{matrix} 1 = K_{00} \end{matrix} <math>

grade 1 (the vectors)

<math> \gamma_1 = K_{12} \Rightarrow \gamma_1^2 = -K_{00} = -1 <math>

<math> \gamma_2 = K_{20} \Rightarrow \gamma_2^2 = -K_{00} = -1 <math>

The signature is (− −)

grade 2 (the pseudoscalar)

<math> \gamma_1 \land \gamma_2 = K_{12}K_{20} = K_{32} \Rightarrow (\gamma_1 :\land \gamma_2)^2 = K_{32}^2 = -K_{00} = -1 <math>

The isomorphism with the quaternions is as follows:

1 is scalar, i and j are vectors and k = ij is the pseudoscalar.

A Clifford number is a linear combination of the four elements 1, i, j and k

<math> \begin{matrix} 1 = K_{00}, &i = K_{12}, &j = K_{20} &k = K_{32} \end{matrix}<math>

The use of k as pseudoscalar ( i times j ) is a bit strange but perfectly sound.

real Clifford algebra R_0,3

p = 0 and q = 3 so we need 3 Kminus basevectors, this is the usual way of working with quaternions i, j and k are now basevectors and ijk = -1 is the pseudoscalar. This algebra is again isomorf with H (the quaternions)

grade 0 (the scalar)

<math> \begin{matrix} 1 = K_0 \end{matrix} <math>

grade 1 (the vectors)

<math> \gamma_1 = K_{12} = i \Rightarrow \gamma_1^2 = -K_{00} = -1 <math>

<math> \gamma_2 = K_{20} = j \Rightarrow \gamma_2^2 = -K_{00} = -1 <math>

<math> \gamma_3 = K_{32} = k \Rightarrow \gamma_3^2 = -K_{00} = -1 <math>

The signature is ( - - - )

grade 2 (the bivectors)

<math> \gamma_1 \land \gamma_2 = K_{12} K_{20} = K_{32} = \gamma_3 <math>

<math> \gamma_3 \land \gamma_1 = K_{32} K_{12} = K_{20} = \gamma_2 <math>

<math> \gamma_2 \land \gamma_3 = K_{20} K_{32} = K_{12} = \gamma_1 <math>

grade 3 (the pseudoscalar)

<math> \gamma_1 \land \gamma_2 \and \gamma_3 = K_{12} K_{20} K_{32} = -K_{00} = -1 <math>

A Clifford number is here again a linear combination of the 4 elements 1 i j and k. The use of -1 as pseudoscalar (ijk)is as we are used to, but it makes the algebra a new example of a non-universal Clifford algebra, since p + q = 3 and we only have 2² elements.

real Clifford algebra R_3,0

This is the famous Pauli algebra, if you think of K₀₂ as i and K₀₀ as 1. We have tree Kplus as basevectors.

grade 0 (the scalar)

<math> \begin{matrix} 1 = K_0 \end{matrix} <math>

grade 1 (the vectors)

<math> \gamma_1 = K_{10} = \sigma_1 \Rightarrow \gamma_1^2 = K_{00} = +1 <math>

<math> \gamma_2 = K_{22} = \sigma_2 \Rightarrow \gamma_2^2 = K_{00} = +1 <math>

<math> \gamma_3 = K_{30} = \sigma_3 \Rightarrow \gamma_3^2 = K_{00} = +1 <math>

The signature is ( + + + )

grade 2 (the bivectors)

<math> \sigma_1 \land \sigma_2 = K_{10} K_{22} = K_{32} = K_{02} K_{30}= i \sigma_3 <math>

<math> \sigma_3 \land \sigma_1 = K_{30} K_{10} = -K_{20} = K_{02}K_{22} = i \sigma_2 <math>

<math> \sigma_2 \land \sigma_3 = K_{22} K_{30} = K_{12} = K_{02} K_{10} = i \sigma_1 <math>

grade 3 (the pseudoscalar)

<math> \sigma_1 \land \sigma_2 \and \sigma_3 = K_{10} K_{22} K_{30} = K_{02} = i <math>

So i is the pseudoscalar and the equations for the bivectors mean in fact that each bivector is the Hodge star of the one vector not part of the bivector.

real Clifford algebra R_3,1

This to me is the most interesting real Clifford algebra because it enables the construction of a Dirac-like equation without complex numbers. Majorana discovered it. Hence real spinors are called Majorana spinors. The algebra is also known as the Majorana-algebra. It makes use of all the 16 4x4 real matrices. The four basevectors are in fact the tree Pauli matrices (Kplus) completed with a fourth antihermitian (Kmin) matrix. The signature is ( + + + - ) See sign convention For the signature ( + - - - ) or ( - - - + ) often used in physics you need 4x4 complex matrices or 8x8 real matrices because you can not form 3 anticommuting Kmin 4x4 matrices. See R_1,3 for several representations.

grade 0 (the scalar)

<math> \begin{matrix} 1 = K_0 \end{matrix} <math>

grade 1 (the vectors)

<math> \gamma_1 = K_{10} \Rightarrow \gamma_1^2 = K_{00} = +1 <math>

<math> \gamma_2 = K_{22} \Rightarrow \gamma_2^2 = K_{00} = +1 <math>

<math> \gamma_3 = K_{30} \Rightarrow \gamma_3^2 = K_{00} = +1 <math>

<math> \gamma_4 = K_{23} \Rightarrow \gamma_4^2 = -K_{00} = -1 <math>

The signature is ( + + + - )

grade 2 (the bivectors, tree rotations and tree boosts)

<math> \gamma_1\gamma_2 = K_{10}K_{22} = K_{32} \Rightarrow (\gamma_1\gamma_2)^2 = -K_{00}= -1 <math>

<math> \gamma_1\gamma_3 = K_{10}K_{30} = K_{20} \Rightarrow (\gamma_1\gamma_3)^2 = -K_{00}= -1 <math>

<math> \gamma_2\gamma_3 = K_{22}K_{30} = K_{12} \Rightarrow (\gamma_2\gamma_3)^2 = -K_{00}= -1 <math>

<math> \gamma_1\gamma_4 = K_{10}K_{23} = K_{33} \Rightarrow (\gamma_1\gamma_4)^2 = K_{00}= +1 <math>

<math> \gamma_2\gamma_4 = K_{22}K_{23} = -K_{01} \Rightarrow (\gamma_1\gamma_2)^2 = K_{00}= +1 <math>

<math> \gamma_3\gamma_4 = K_{30}K_{23} = -K_{13} \Rightarrow (\gamma_1\gamma_2)^2 = K_{00}= +1 <math>

grade 3 (the pseudovectors, the Hodge duals of the vectors)

<math> \gamma_2\gamma_3\gamma_4 = K_{22}K_{30}K_{23} = K_{31} \Rightarrow (\gamma_2\gamma_3\gamma_4)^2 = K_{00} = +1<math>

<math> \gamma_1\gamma_3\gamma_4 = K_{10}K_{30}K_{23} = -K_{03} \Rightarrow (\gamma_1\gamma_3\gamma_4)^2 = K_{00} = +1<math>

<math> \gamma_1\gamma_2\gamma_4 = K_{10}K_{22}K_{23} = -K_{11} \Rightarrow (\gamma_1\gamma_2\gamma_4)^2 = K_{00} = +1<math>

<math> \gamma_1\gamma_2\gamma_3 = K_{10}K_{22}K_{30} = K_{02} = i \Rightarrow (\gamma_1\gamma_2\gamma_3)^2 = -K_{00} = -1 <math>

the last one was the pseudoscalar in R_3,0

grade 4 (the pseudoscalar)

<math> \gamma_1\gamma_2\gamma_3\gamma_4 = K_{10}K_{22}K_{30}K_{23} = K_{21} \Rightarrow (\gamma_1\gamma_2\gamma_3\gamma_4)^2 = -K_{00} = -1 <math>

further representations with real 4x4 matrices

<math> \begin{matrix} R_{2,2} &K_{10} &K_{22} &K_{21} &K_{23} \end{matrix} <math>

<math> \begin{matrix} R_{3,2} &K_{10} &K_{22} &K_{30} &K_{21} &K_{23} \end{matrix} <math>

representations with real 8x8 matrices

<math> \begin{matrix} R_{4,0} &K_{110} &K_{122} &K_{130} &K_{300} \end{matrix} <math>

<math> \begin{matrix} R_{4,1} &K_{110} &K_{122} &K_{130} &K_{300} &K_{200} \end{matrix} <math>

<math> \begin{matrix} R_{4,2} &K_{110} &K_{122} &K_{130} &K_{300} &K_{200} &K_{121} \end{matrix} <math>

<math> \begin{matrix} R_{4,3} &K_{110} &K_{122} &K_{130} &K_{300} &K_{200} &K_{121} &K_{123} \end{matrix} <math>

<math> \begin{matrix} R_{3,3} &K_{110} &K_{122} &K_{130} &K_{200} &K_{121} &K_{123} \end{matrix} <math>

<math> \begin{matrix} R_{2,3} &K_{110} &K_{122} &K_{200} &K_{121} &K_{123} \end{matrix} <math>

<math> \begin{matrix} R_{1,3} &K_{110} &K_{200} &K_{121} &K_{123} \end{matrix} <math>

This last one is very important in physics since it is the most used Clifford algebra for working in Minkowski space-time. Signature ( + - - - ) see sign convention More used representations are

<math> \begin{matrix} R_{1,3} &K_{100} &K_{210} &K_{222} &K_{230} \end{matrix} <math>

Interestingly with 8x8 real matrices one can form 7 anticommuting Kmin matrices. They form a baseset for the non-universal real Clifford algebra R_0,7

<math> \begin{matrix} R_{0,7} &K_{302} &K_{102} &K_{230} &K_{210} &K_{023} &K_{021} &K_{222} \end{matrix} <math>

<math> \begin{matrix} R_{0,6} &K_{302} &K_{102} &K_{230} &K_{210} &K_{023} &K_{021} \end{matrix} <math>

<math> \begin{matrix} R_{0,5} &K_{302} &K_{102} &K_{230} &K_{210} &K_{023} \end{matrix} <math>

<math> \begin{matrix} R_{0,4} &K_{302} &K_{102} &K_{230} &K_{210} \end{matrix} <math>

( For R_0,3 we showed one only needs 4x4 real matrices)

representations with 16x16 real matrices

<math> \begin{matrix} R_{5,0} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000} \end{matrix} <math>

<math> \begin{matrix} R_{5,1} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} \end{matrix} <math>

<math> \begin{matrix} R_{5,2} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} &K_{1123} \end{matrix} <math>

<math> \begin{matrix} R_{5,3} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} &K_{1123} &K_{1200} \end{matrix} <math>

<math> \begin{matrix} R_{5,4} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} &K_{1123} &K_{1200} &K_{2000} \end{matrix} <math>

<math> \begin{matrix} R_{4,4} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{1121} &K_{1123} &K_{1200} &K_{2000} \end{matrix} <math>

<math> \begin{matrix} R_{3,4} &K_{1110} &K_{1122} &K_{1130} &K_{1121} &K_{1123} &K_{1200} &K_{2000} \end{matrix} <math>

<math> \begin{matrix} R_{2,4} &K_{1110} &K_{1122} &K_{1121} &K_{1123} &K_{1200} &K_{2000} \end{matrix} <math>

<math> \begin{matrix} R_{1,4} &K_{1110} &K_{1121} &K_{1123} &K_{1200} &K_{2000} \end{matrix} <math>

<math> \begin{matrix} R_{0,4} &K_{1121} &K_{1123} &K_{1200} &K_{2000} \end{matrix} <math>

<math> \begin{matrix} R_{1,8} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{1222} &K_{2000} &K_{3000} \end{matrix} <math>

<math> \begin{matrix} R_{1,7} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{1222} &K_{3000} \end{matrix} <math>

<math> \begin{matrix} R_{1,6} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{3000} \end{matrix} <math>

<math> \begin{matrix} R_{1,5} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{3000} \end{matrix} <math>

<math> \begin{matrix} R_{1,4} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{3000} \end{matrix} <math>

R_1,3 only needs 4x4 real matrices

<math> \begin{matrix} R_{0,8} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{1222} &K_{2000} \end{matrix} <math>

R_0,7 only needs 8x8 real matrices

<math> \begin{matrix} R_{9,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} &K_{2222} &K_{1000} &K_{3000} \end{matrix} <math>

<math> \begin{matrix} R_{8,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} &K_{2222} &K_{1000} \end{matrix} <math>

<math> \begin{matrix} R_{7,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} &K_{2222} \end{matrix} <math>

<math> \begin{matrix} R_{6,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} \end{matrix} <math>

R_5,0 only needs 8x8 real matrices