Clifford algebra representations

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

Matrix representations of real Clifford algebras

We will have to study anticommutative matrices (AB = -BA) because in Clifford algebras orthogonal vectors anticommutate

A⋅ ⋅ B=12(AB+BA)=0{displaystyle Acdot B={frac {1}{2}}(AB+BA)=0}

For the real Clifford algebras R_{p, q} we need p + q mutually anticommuting matrices, of which p have +1 as a square and q have −1 as a square.

γ γ a2=+1Yeah.1≤ ≤ a≤ ≤ pγ γ a2=− − 1Yeah.p+1≤ ≤ a≤ ≤ p+qγ γ aγ γ b=− − γ γ bγ γ aYeah.aI was. I was. b{displaystyle {begin{matrix}gamma _{a}{a}{7}{2}{mbox{si}}}{1leq aleq pgamma _{a}{a}{a}{2}{mbox{a}{a{a#}{a{a}{a #1⁄4}{ax}{ax}{ax}{a

The "K" to name arrays

First we present a convenient method for naming 2ⁿ x 2ⁿ matrices

K0=(1001),K1=(0110),K2=(0− − 110),K3=(100− − 1).{displaystyle K_{0}={begin{pmatrix}1 fake0 fake1end{pmatrix}},K_{1}={begin{pmatrix}{1}{11st}{ptrix}{,K_{2}={begin{pmatrix}{x1}{x1striend}{x{x1}{x{x1111x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1 !

Note that K₀ is the identity matrix. The names were chosen in such a way that there is a simple rule to remember the products:

K₁ K₂ = K₃

K₁ K₃ = K₂

K₂ K₃ = K₁

K₂ K₁ = K₃

K₃ K₁ = K₂

K₃ K₂ = K₁.

The increase in indices gives a positive result. Indices that decrease give negative results.

Attention! These are not the same relationships that hold for the standard basis of quaternions. If we named i = i₁, j = i₂ and k = i₃, we would get

i₁ i₂ =₃

i₂ i₃ =₁

i₃ i₁ =₂

The last rule is different. We will see later that the pure quaternions i, j and k can be represented by K₁₂, K₂₀ and K₃₂

It is emphasized that

K02=K12=K32=K0{displaystyle K_{0}{2}=K_{1}{2}=K_{3⁄2}{2}{2}=K_{0}}}

K22=− − K0{displaystyle K_{2}{2}=-K_{0}}}

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

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

Kab=Ka Kb{displaystyle K_{ab}=K_{a}otimes K_{b}}

Kabc=Ka Kbc=Ka Kb Kc{displaystyle K_{abc}=K_{a}otimes K_{bc}=K_{a}otimes K_{b}otimes K_{c}}}}

Some examples

K30=(1000010000− − 10000− − 1),K11=(0001001001001000){displaystyle K_{30}={begin{pmatrix}1 fake0 fake0 fake0}{ fake0 fake0 fake0 fake0 fake0 fake0-1end{pmatrix}}},K_{11}={begin{pmatrix}0{pmatrix}0{1{1{111{}{}{}{1{1{111{1st}{1st}{}{1st}{}{}{}{1st}{}{1st}{1st}{1st}{1st}{}{}{1st}{}{1st}{1st}{}{pmatrix}{1st}{}{}{1st}{1st}{}{}{

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 it is very easy to multiply large square matrices since

(A B)(C D)=AB CD{displaystyle (Aotimes B)(Cotimes D)=ABotimes CD}

Let's solve an example

K ₁₂₃ K ₂₂₂ = K ₃₀₁

level-8x8 1 by 2 da 3

level-4x4 2 by 2 da 0 but remember the sign less

level-2x2 3 by 2 da 1 but again a lesser sign

(there is cancellation of the two minus signs so the result is K₃₀₁)

We can now begin to build sets of mutually anticommuting orthogonal matrices, sometimes called Dirac matrices. It is obvious that two such matrices anticommute in an odd number of indices (index 0 commutes with all other indices).

K₁₃ for example anticommutes with

K₀₁, K₀₂, K₁₁, K₁₂, K₂₀, K₂₃, K₃₀, K₃₃

and toggles with

K₀₀, K₁₀, K₁₃, K₂₁, K₂₂, K₃₁, K₃₂.

If the index 2 occurs 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 occurs 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

Now we have a very simple way to build the largest possible sets of anticommutant matrices.

Start with an existing set {K₁, K₂, K₃}

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

Then add two more matrices that anticommutate at the new level and commute at the old level (via zero index 0)

Getting {K₁₁, K₁₂, K₁₃, K₂₀, K ₃₀}

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 more Kplus than Kminus.

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

Real Clifford algebra R2,0

p = 2 and q = 0 therefore we need 2 Kplus as basis vectors

degree 0 (scalar)

1=K0{displaystyle {begin{matrix}1=K_{0}end{matrix}}}}

grade 1 (the vectors)

γ γ 1=K1⇒ ⇒ γ γ 12=K0=1{displaystyle gamma _{1}=K_{1}Rightarrow gamma _{1⁄2}{2}=K_{0}=1}

γ γ 2=K3⇒ ⇒ γ γ 22=K0=1{displaystyle gamma _{2}=K_{3}Rightarrow gamma _{2}{2}{2}=K_{0}=1}

degree 2 (the pseudoscalar)

γ γ 1∧ ∧ γ γ 2=γ γ 1γ γ 2=K2⇒ ⇒ (γ γ 1∧ ∧ γ γ 2)2=(γ γ 1γ γ 2)2=K22=− − 1{displaystyle gamma _{1}land gamma _{2}=gamma _{1gamma _{2}=K_{2}Rightarrow (gamma _{1}land gamma _{2})^{2}={2}=(gamma _{1}gamma _{2}){2}

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

Real Clifford algebra R1,1

p = 1 and q = 1 we need a Kplus and 1 Kminus as basis vectors

degree 0 (scalar)

1=K0{displaystyle {begin{matrix}1=K_{0}end{matrix}}}}

grade 1 (the vectors)

γ γ 1=K1⇒ ⇒ γ γ 12=K0=1{displaystyle gamma _{1}=K_{1}Rightarrow gamma _{1⁄2}{2}=K_{0}=1}

γ γ 2=K2⇒ ⇒ γ γ 22=− − K0=− − 1{displaystyle gamma _{2}=K_{2}Rightarrow gamma _{2}{2}{2}=-K_{0}=-1}

degree 2 (the pseudoscalar)

γ γ 1∧ ∧ γ γ 2=γ γ 1γ γ 2=K3⇒ ⇒ (γ γ 1∧ ∧ γ γ 2)2=(γ γ 1γ γ 2)2=K32=K0=1{displaystyle gamma _{1}land gamma _{2}=gamma _{1gamma _{2}={2}=K_{3}Rightarrow (gamma _{1}land gamma _{2}){2}{2}{2}={2}{1⁄2}{1}{2}{2}}{2}{2}{1⁄2}}}{1⁄2}}}{1⁄2}{1⁄2}}}{1⁄2}{1⁄2}}{1⁄2}}}{1⁄2}{1⁄2}}{1⁄2}{1⁄2}}{1⁄2}}{1⁄2}}}{1⁄2}{1}}{1}}{1⁄2}}}{1⁄2}}{1⁄2}}}{1⁄2}{1⁄2}{1⁄2}{1⁄2}}}{1⁄2}}}}{1⁄2}}}}{gamma

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

Real Clifford algebra R2,1

p = 2 and q = 1 we need two Kplus and 1 Kminus as basis vectors

degree 0 (scalar)

1=K0{displaystyle {begin{matrix}1=K_{0}end{matrix}}}}

grade 1 (the vectors)

γ γ 1=K1⇒ ⇒ γ γ 12=K0=1{displaystyle gamma _{1}=K_{1}Rightarrow gamma _{1⁄2}{2}=K_{0}=1}

γ γ 2=K3⇒ ⇒ γ γ 22=K0=1{displaystyle gamma _{2}=K_{3}Rightarrow gamma _{2}{2}{2}=K_{0}=1}

γ γ 3=K2⇒ ⇒ γ γ 32=− − K0=− − 1{displaystyle gamma _{3}=K_{2}Rightarrow gamma _{3^}{2}=-K_{0}=-1}

The signature is (+ + -)

degree 2 (bivectors)

γ γ 1∧ ∧ γ γ 2=γ γ 3=K2⇒ ⇒ (γ γ 1∧ ∧ γ γ 2)2=− − 1{displaystyle gamma _{1}land gamma _{2}=gamma _{3}=K_{2}Rightarrow (gamma _{1}land gamma _{2}){2}{2} = 1}

γ γ 1∧ ∧ γ γ 3=γ γ 2=K3⇒ ⇒ (γ γ 1∧ ∧ γ γ 3)2=+1{displaystyle gamma _{1}land gamma _{3}=gamma _{2}={2}=K_{3}Rightarrow (gamma _{1}land gamma _{3})^{2}=+1}

γ γ 2∧ ∧ γ γ 3=− − γ γ 1=− − K1⇒ ⇒ (γ γ 2∧ ∧ γ γ 3)2=+1{displaystyle gamma _{2}land gamma _{3}=-gamma _{1=}-K_{1Rightarrow (gamma _{2}land gamma _{3})^{2}=+1}

degree 3 (the pseudoscalar)

γ γ 1∧ ∧ γ γ 2∧ ∧ γ γ 3=− − 1⇒ ⇒ (γ γ 1∧ ∧ γ γ 2∧ ∧ γ γ 3)2=(− − 1)2=+1{displaystyle gamma _{1}land gamma _{2}land gamma _{3=}-1Rightarrow (gamma _{1}land gamma _{2}land gamma _{3})^{2}=(-1){2}=+1}

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

(the Hodge dual of each element is simply minus the original)

↓ ↓ A=− − A{displaystyle}A=-A

Real Clifford algebra R0,2

p = 0 and q = 2 we need two Kminus as basis vectors, this is not possible with real 2x2 matrices so we need to use 4x4 matrices, we have many possibilities. This algebra is isomorphic with H (the quaternions)

degree 0 (scalar)

1=K00{displaystyle {begin{matrix}1=K_{00}end{matrix}}}}

grade 1 (the vectors)

γ γ 1=K12⇒ ⇒ γ γ 12=− − K00=− − 1{displaystyle gamma _{1}=K_{12}Rightarrow gamma _{1^}{2}=-K_{00}=-1}

γ γ 2=K20⇒ ⇒ γ γ 22=− − K00=− − 1{displaystyle gamma _{2}=K_{20}Rightarrow gamma _{2}{2}{2}=-K_{00}=-1}

The signature is (- -)

degree 2 (the pseudoscalar)

γ γ 1∧ ∧ γ γ 2=K12K20=K32⇒ ⇒ (γ γ 1∧ ∧ γ γ 2)2=K322=− − K00=− − 1{displaystyle gamma _{1}land gamma _{2}=K_{12}K_{20}=K_{32}Rightarrow (gamma _{1}land gamma _{2})^{2}=K_{32}{2}{2}=-K_{00}=-1}

The isomorphism with the quaternions is as follows

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

A Clifford number is a linear combination of the 4 elements 1 i j and k.

1=K00,i=K12,j=K20k=K32{displaystyle {begin{matrix}1=K_{00}, strangeri=K_{12}, fakej=K_{20}{matrix}}}}}{matrix}}}}}}}

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

Real Clifford algebra R0,3

p = 0 and q = 3 we need 3 Kminus as basis vectors, this is the usual way to work with quaternions i, j and k but now they are basis vectors and the ijk = -1 is the pseudoscalar. This algebra is again isomorphic with H (the quaternions)

degree 0 (scalar)

1=K0{displaystyle {begin{matrix}1=K_{0}end{matrix}}}}

degree 1 (the vectors)

γ γ 1=K12=i⇒ ⇒ γ γ 12=− − K00=− − 1{displaystyle gamma _{1}=K_{12}=iRightarrow gamma _{1}^{2}=-K_{00}=-1}

γ γ 2=K20=j⇒ ⇒ γ γ 22=− − K00=− − 1{displaystyle gamma _{2}=K_{20}=jRightarrow gamma _{2}^{2}=-K_{00}=-1}

γ γ 3=K32=k⇒ ⇒ γ γ 32=− − K00=− − 1{displaystyle gamma _{3}=K_{32}=kRightarrow gamma _{3}^{2}=-K_{00}=-1}

The signature is (- - -)

degree 2 (bivectors)

γ γ 1∧ ∧ γ γ 2=K12K20=K32=γ γ 3{displaystyle gamma _{1}land gamma _{2}=K_{12}K_{20}=K_{32}=gamma _{3}}}

γ γ 3∧ ∧ γ γ 1=K32K12=K20=γ γ 2{displaystyle gamma _{3}land gamma _{1}=K_{32}K_{12}=K_{20}=gamma _{2}}}

γ γ 2∧ ∧ γ γ 3=K20K32=K12=γ γ 1{displaystyle gamma _{2}land gamma _{3}=K_{20}K_{32}=K_{12}=gamma _{1}}}

degree 3 (the pseudoscalar)

γ γ 1∧ ∧ γ γ 2∧ ∧ γ γ 3=K12K20K32=− − K00=− − 1{displaystyle gamma _{1}land gamma _{2}land gamma _{3=}=K_{12}K_{20}K_{32}=-K_{00}=-1}

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

Real Clifford algebra R3,0

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

degree 0 (scalar)

1=K00{displaystyle {begin{matrix}1=K_{00}end{matrix}}}}

grade 1 (the vectors)

γ γ 1=K10=σ σ 1⇒ ⇒ γ γ 12=K00=+1{displaystyle gamma _{1}=K_{10}=sigma _{1}Rightarrow gamma _{1}{2}=K_{00}=+1}

γ γ 2=K22=σ σ 2⇒ ⇒ γ γ 22=K00=+1{displaystyle gamma _{2}=K_{22}=sigma _{2}Rightarrow gamma _{2}{2}{2}=K_{00}=+1}

γ γ 3=K30=σ σ 3⇒ ⇒ γ γ 32=K00=+1{displaystyle gamma _{3}=K_{30}=sigma _{3}Rightarrow gamma _{3}{2}=K_{00}=+1}

The signature is (+ + +)

degree 2 (bivectors)

σ σ 1∧ ∧ σ σ 2=K10K22=K32=K02K30=iσ σ 3{displaystyle sigma _{1}land sigma _{2}=K_{10}K_{22}=K_{32}=K_{02}K_{30}=isigma _{3}}}}

σ σ 3∧ ∧ σ σ 1=K30K10=− − K20=K02K22=iσ σ 2{displaystyle sigma _{3}land sigma _{1}=K_{30K_{10}=-K_{20}=K_{02}K_{22}=isigma _{2}}}}

σ σ 2∧ ∧ σ σ 3=K22K30=K12=K02K10=iσ σ 1{displaystyle sigma _{2}land sigma _{3}=K_{22K_{30}=K_{12}=K_{02}K_{10}=isigma _{1}}}

degree 3 (the pseudoscalar)

σ σ 1∧ ∧ σ σ 2∧ ∧ σ σ 3=K10K22K30=K02=i{displaystyle sigma _{1}land sigma _{2}land sigma _{3}=K_{10}K_{22}K_{30}=K_{02}=i}

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

Real Clifford algebra R3,1

This is perhaps the most interesting real Clifford algebra because it allows the construction of Dirac-type equations without complex numbers. Majorana discovered her. The actual spinors of are called Majorana spinors. The algebra is also known as the Majorana algebra. It makes use of all 16 real 4x4 matrices. The four basis vectors are in fact the three Pauli matrices (Kplus) completed with a fourth anti-Hermitian matrix (Kmin). The signature is (+ + + -) For the signature (+ - - -) or (- - - +) commonly used in physics, complex 4x4 matrices or real 8x8 matrices are needed because 3 Kmin anticommutant 4x4 matrices cannot be formed.

see R_1,3 for some renderings.

degree 0 (scalar)

1=K0{displaystyle {begin{matrix}1=K_{0}end{matrix}}}}

degree 1 (the vectors)

γ γ 1=K10⇒ ⇒ γ γ 12=K00=+1{displaystyle gamma _{1}=K_{10}Rightarrow gamma _{1^}{2}=K_{00}=+1}

γ γ 2=K22⇒ ⇒ γ γ 22=K00=+1{displaystyle gamma _{2}=K_{22}Rightarrow gamma _{2}{2}{2}=K_{00}=+1}

γ γ 3=K30⇒ ⇒ γ γ 32=K00=+1{displaystyle gamma _{3}=K_{30}Rightarrow gamma _{3^}{2}=K_{00}=+1}

γ γ 4=K23⇒ ⇒ γ γ 32=− − K00=− − 1{displaystyle gamma _{4}=K_{23}Rightarrow gamma _{3^}{2}=-K_{00}=-1}

The signature is (+ + + -)

degree 2 (bivectors, "tree" rotations, and "boosts" of tree)

γ γ 1γ γ 2=K10K22=K32⇒ ⇒ (γ γ 1γ γ 2)2=− − K00=− − 1{displaystyle gamma _{1}gamma _{2}=K_{10}K_{22}=K_{32Rightarrow (gamma _{1}gamma _{2}){2}{2}=-K_{00}=-1}

γ γ 1γ γ 3=K10K30=K20⇒ ⇒ (γ γ 1γ γ 3)2=− − K00=− − 1{displaystyle gamma _{1}gamma _{3}=K_{10}K_{30}=K_{20}Rightarrow (gamma _{1}gamma _{3}){2}=-K_{00}=-1}

γ γ 2γ γ 3=K22K30=K12⇒ ⇒ (γ γ 2γ γ 3)2=− − K00=− − 1{displaystyle gamma _{2}gamma _{3}=K_{22}K_{30}=K_{12}Rightarrow (gamma _{2}gamma _{3}){2}{2}=-K_{00}=-1}

γ γ 1γ γ 4=K10K23=K33⇒ ⇒ (γ γ 1γ γ 4)2=K00=+1{displaystyle gamma _{1}gamma _{4}=K_{10}K_{23=}=K_{33}Rightarrow (gamma _{1}gamma _{4}){2}=K_{00}=+1}

γ γ 2γ γ 4=K22K23=− − K01⇒ ⇒ (γ γ 1γ γ 2)2=K00=+1{displaystyle gamma _{2}gamma _{4}=K_{22}K_{23}=}-K_{01Rightarrow (gamma _{1}gamma _{2}){2}{2}=K_{00}=+1}

γ γ 3γ γ 4=K30K23=− − K13⇒ ⇒ (γ γ 1γ γ 2)2=K00=+1{displaystyle gamma _{3}gamma _{4}=K_{30}K_{23=}-K_{13Rightarrow (gamma _{1}gamma _{2}){2}{2}=K_{00}=+1}

degree 3 (pseudovectors, Hodge duals of vectors)

γ γ 2γ γ 3γ γ 4=K22K30K23=K31⇒ ⇒ (γ γ 2γ γ 3γ γ 4)2=K00=+1{displaystyle gamma _{2}gamma _{3}gamma _{4}=K_{22}K_{30}K_{23}=K_{31}Rightarrow (gamma _{2}gamma _{3}gamma _{4}{4})^{2}=K_{00}=+1}

γ γ 1γ γ 3γ γ 4=K10K30K23=− − K03⇒ ⇒ (γ γ 1γ γ 3γ γ 4)2=K00=+1{displaystyle gamma _{1}gamma _{3}gamma _{4}=K_{10}K_{30}K_{23}=-K_{03}Rightarrow (gamma _{1}gamma _{3}gamma _{4}{4})^{2}=K_{00}=+1}

γ γ 1γ γ 2γ γ 4=K10K22K23=− − K11⇒ ⇒ (γ γ 1γ γ 2γ γ 4)2=K00=+1{displaystyle gamma _{1}gamma _{2}gamma _{4}=K_{10}K_{22}K_{23}=-K_{11}Rightarrow (gamma _{1}gamma _{2}gamma _{4}{4})^{2}=K_{00}=+1}

γ γ 1γ γ 2γ γ 3=K10K22K30=K02=i⇒ ⇒ (γ γ 1γ γ 2γ γ 3)2=− − K00=− − 1{displaystyle gamma _{1}gamma _{2}gamma _{3}=K_{10}K_{22}K_{30}=K_{02}=iRightarrow (gamma _{1}gamma _{2}gamma _{3})^{2}=-K_{00}=-K_{00}=-1}

the last one was the pseudoscalar in R_3,0

degree 4 (the pseudoscalar)

γ γ 1γ γ 2γ γ 3γ γ 4=K10K22K30K23=K21⇒ ⇒ (γ γ 1γ γ 2γ γ 3γ γ 4)2=− − K00=− − 1{displaystyle 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}={4}{2}{2}