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.

Contents

1 Matrix representations of real Clifford algebras 2 Intermezzo: the K-system for naming matrices 3 Real Clifford algebra R2,0 4 Real Cifford algebra R1,1 5 real Clifford algebra R2,1 6 Real Clifford algebra R0,2 7 real Clifford algebra R0,3 8 real Clifford algebra R3,0 9 real Clifford algebra R3,1 10 further representations with real 4x4 matrices 11 representations with real 8x8 matrices 12 representations with 16x16 real matrices

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. One can always 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 quaternions 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 (see orthogonal matrix), sometimes called Dirac matrices. Its obvious that two such matrices anticommute if they anticommute in an odd number of indexes (index o commutes with all the other indices).

K₁₃ for example anticommutes with

K₀₁,K₀₂,K₁₁,K₁₂,K₂₀,K₂₃,K₃₀,K₃₃

and commutes with

K₀₀,K₁₀,K₁₃,K₂₁,K₂₂,K₃₁,K₃₂.

If the index 2 appears an even number of times in the name then the square of the matrix is plus the identity matrix, let's call this a Kplus

examples are K₁, K₂₂, K₃₁₁, K₂₂₂₂

If the index 2 appears an odd number of times in the name then the square of the matrix is minus the identity matrix, let's call this a Kminus

examples are K₂, K₂₂₂, K₂₁₁, K₁₂₂₂

We have now a very simple way of constructing the largest possible sets of anticommuting matrices.

Start with an existing set {K₁,K₂,K₃}

Insert a constant new index (for example a 1 in first position) and you get {K₁₁,K₁₂,K₁₃}

Then add two more matrices that anticommute in the new level and commute in the old level (by means of the zero index 0)

So you get {K₁₁, K₁₂, K₁₃, K₂₀, K₃₀}

Other examples

{K₂₁, K₂₂, K₂₃, K₁₀, K₃₀}

{K₃₁, K₃₂, K₃₃, K₁₀, K₂₀}

{K₁₁₁, K₁₁₂, K₁₁₃, K₁₂₀, K₁₃₀, K₂₀₀, K₃₀₀}

{K₂₁₁, K₂₁₂, K₂₁₃, K₂₂₀, K₂₃₀, K₁₀₀, K₃₀₀}

{K₃₁₁, K₃₁₂, K₃₁₃, K₃₂₀, K₃₃₀, K₁₀₀, K₂₀₀}

You always get a set with an odd number of matrices and there is always one Kplus more than Kminus.

Each of them can be written as the product of all the other. Example K₁₁ K₁₂ K₁₃ K₂₀ = K₃₀

Real Clifford algebra R_2,0

p = 2 and q =0 so we need 2 Kplus as basevectors

grade 0 (the scalar)

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

grade 1 (the vectors)

<math> \gamma_1 = K_1 \Rightarrow \gamma_1^2 = K_0 = 1 <math>

<math> \gamma_2 = K_3 \Rightarrow \gamma_2^2 = K_0 = 1 <math>

grade 2 (the pseudoscalar)

<math> \gamma_1 \land \gamma_2 = \frac{1}{2}(\gamma_1 \gamma_2 - \gamma_2 \gamma_1 ) = \gamma_1 \gamma_2 = K_2 \Rightarrow (\gamma_1 \land \gamma_2)^2 = (\gamma_1 \gamma_2)^2 = K_2^2 = -1 <math>

n = p + q = 2 and we have 2² = 4 elements so it is what I. Portious calls a universal Clifford algebra.

Real Cifford algebra R_1,1

p=1 and q = 1 so we need one Kplus and 1 Kminus as basevectors

grade 0 (the scalar)

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

grade 1 (the vectors)

<math> \gamma_1 = K_1 \Rightarrow \gamma_1^2 = K_0 = 1 <math>

<math> \gamma_2 = K_2 \Rightarrow \gamma_2^2 = -K_0 = -1 <math>

grade 2 (the pseudoscalar)

<math> \gamma_1 \land \gamma_2 = \gamma_1 \gamma_2 = K_3 \Rightarrow (\gamma_1 \land \gamma_2)^2 = (\gamma_1 \gamma_2)^2 = K_3^2 = K_{0} = 1 <math>

Here again we have 2ⁿ elements in the algebra with n = p+q so it is again a universal Clifford algebra

real Clifford algebra R_2,1

p = 2 and q = 1 so 2 Kplus basevectors and 1 Kminus basevector

grade 0 (the scalar)

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

grade 1 ( the vectors)

<math> \gamma_1 = K_1 \Rightarrow \gamma_1^2 = K_0 = 1<math>

<math> \gamma_2 = K_3 \Rightarrow \gamma_2^2 = K_0 = 1<math>

<math> \gamma_3 = K_2 \Rightarrow \gamma_3^2 = -K_0 = -1<math>

The signature is ( + + - )

grade 2 (the bivectors)

<math> \gamma_1 \land \gamma_2 = \gamma_3 = K_2 \Rightarrow (\gamma_1 \land \gamma_2)^2 = -1<math>

<math> \gamma_1 \land \gamma_3 = \gamma_2 = K_3 \Rightarrow (\gamma_1 \land \gamma_3)^2 = +1<math>

<math> \gamma_2 \land \gamma_3 = -\gamma_1 = -K_1 \Rightarrow (\gamma_2 \land \gamma_3)^2 = +1<math>

grade 3 (the pseudoscalar)

<math> \gamma_1 \land \gamma_2 \land \gamma_3 = -1 \Rightarrow (\gamma_1 \land \gamma_2 \land \gamma_3)^2 = (-1)^2 = +1 <math>

This is the first example of a non-universal Clifford algebra since p+q= 3 and we only have 2² elements and not 2³. The reason is very simple, every matrix is used twice, once as vector and once as bivector. And the pseudoscalar is real just as the scalar.

(The 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 Kminus-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 isomorphic 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 matriceses:Representaciones de álgebras de Clifford