Dirac matrices

The Dirac matrices are a set of 16 matrices created from the Pauli matrices by using the Kronecker product. All 16 Dirac matrices square to positive one (i.e.

I
4

{\displaystyle I\_4}

). Any 5 that anticommute can be used as the basis for Cℓ5,0(**R**). Dirac originally used

α
1

,

α
2

,

α
3

,

α
4

,

α
5

{\displaystyle \alpha\_1, \alpha\_2, \alpha\_3, \alpha\_4, \alpha\_5}

 which are shown in blue.

(

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

)

(

0

1

0

0

1

0

0

0

0

0

0

1

0

0

1

0

)

(

0

−
i

0

0

i

0

0

0

0

0

0

−
i

0

0

i

0

)

(

1

0

0

0

0

−
1

0

0

0

0

1

0

0

0

0

−
1

)

(

0

0

1

0

0

0

0

1

1

0

0

0

0

1

0

0

)

(

0

0

0

1

0

0

1

0

0

1

0

0

1

0

0

0

)

(

0

0

0

−
i

0

0

i

0

0

−
i

0

0

i

0

0

0

)

(

0

0

1

0

0

0

0

−
1

1

0

0

0

0

−
1

0

0

)

(

0

0

−
i

0

0

0

0

−
i

i

0

0

0

0

i

0

0

)

(

0

0

0

−
i

0

0

−
i

0

0

i

0

0

i

0

0

0

)

(

0

0

0

−
1

0

0

1

0

0

1

0

0

−
1

0

0

0

)

(

0

0

−
i

0

0

0

0

i

i

0

0

0

0

−
i

0

0

)

(

1

0

0

0

0

1

0

0

0

0

−
1

0

0

0

0

−
1

)

(

0

1

0

0

1

0

0

0

0

0

0

−
1

0

0

−
1

0

)

(

0

−
i

0

0

i

0

0

0

0

0

0

i

0

0

−
i

0

)

(

1

0

0

0

0

−
1

0

0

0

0

−
1

0

0

0

0

1

)

{\displaystyle
\begin{array}{c|c|c|c}
\begin{pmatrix}
 1&0&0&0\\
 0&1&0&0\\
 0&0&1&0\\
 0&0&0&1
\end{pmatrix}
& \begin{pmatrix}
 0&1&0&0\\
 1&0&0&0\\
 0&0&0&1\\
 0&0&1&0
\end{pmatrix}
& \begin{pmatrix}
 0&-i&0&0\\
 i&0&0&0\\
 0&0&0&-i\\
 0&0&i&0
\end{pmatrix}
& \begin{pmatrix}
 1&0&0&0\\
 0&-1&0&0\\
 0&0&1&0\\
 0&0&0&-1
\end{pmatrix} \\\hline
\begin{pmatrix}
 0&0&1&0\\
 0&0&0&1\\
 1&0&0&0\\
 0&1&0&0
\end{pmatrix}
& {\color{blue} \begin{pmatrix}
 0&0&0&1\\
 0&0&1&0\\
 0&1&0&0\\
 1&0&0&0
\end{pmatrix} }
& {\color{blue} \begin{pmatrix}
 0&0&0&-i\\
 0&0&i&0\\
 0&-i&0&0\\
 i&0&0&0
\end{pmatrix} }
& {\color{blue} \begin{pmatrix}
 0&0&1&0\\
 0&0&0&-1\\
 1&0&0&0\\
 0&-1&0&0
\end{pmatrix} } \\\hline
{\color{blue} \begin{pmatrix}
 0&0&-i&0\\
 0&0&0&-i\\
 i&0&0&0\\
 0&i&0&0
\end{pmatrix}}
& \begin{pmatrix}
 0&0&0&-i\\
 0&0&-i&0\\
 0&i&0&0\\
 i&0&0&0
\end{pmatrix}
& \begin{pmatrix}
 0&0&0&-1\\
 0&0&1&0\\
 0&1&0&0\\
 -1&0&0&0
\end{pmatrix}
& \begin{pmatrix}
 0&0&-i&0\\
 0&0&0&i\\
 i&0&0&0\\
 0&-i&0&0
\end{pmatrix} \\\hline
{\color{blue} \begin{pmatrix}
 1&0&0&0\\
 0&1&0&0\\
 0&0&-1&0\\
 0&0&0&-1
\end{pmatrix}}
& \begin{pmatrix}
 0&1&0&0\\
 1&0&0&0\\
 0&0&0&-1\\
 0&0&-1&0
\end{pmatrix}
& \begin{pmatrix}
 0&-i&0&0\\
 i&0&0&0\\
 0&0&0&i\\
 0&0&-i&0
\end{pmatrix}
& \begin{pmatrix}
 1&0&0&0\\
 0&-1&0&0\\
 0&0&-1&0\\
 0&0&0&1
\end{pmatrix}
\end{array}}

Surprisingly the 4 by 4 table above forms a multiplication table even though it is actually created by the following rules:

σ

00

σ

01

σ

02

σ

03

σ

10

σ

11

σ

12

σ

13

σ

20

σ

21

σ

22

σ

23

σ

30

σ

31

σ

32

σ

33

where

{\displaystyle
\begin{array}{c|c|c|c}
\sigma\_{00}
& \sigma\_{01}
& \sigma\_{02}
& \sigma\_{03} \\\hline
\sigma\_{10}
& {\color{blue} \sigma\_{11}}
& {\color{blue} \sigma\_{12}}
& {\color{blue} \sigma\_{13}} \\\hline
{\color{blue} \sigma\_{20}}
& \sigma\_{21}
& \sigma\_{22}
& \sigma\_{23} \\\hline
{\color{blue} \sigma\_{30}}
& \sigma\_{31}
& \sigma\_{32}
& \sigma\_{33}
\end{array} \quad \text{where} \quad }

σ

i
j

=

σ
i

⊗

σ
j

{\displaystyle \quad\quad\begin{align}
\sigma\_{ij} &= \sigma\_i \otimes \sigma\_j
\end{align}}

where

σ
i

{\displaystyle \sigma\_i}

 and

σ
j

{\displaystyle \sigma\_j}

 are the original 2x2 Pauli matrices and

⊗

{\displaystyle \otimes}

 is the Kronecker product (not the tensor product)

σ
0

=

(

1

0

0

1

)

,

σ
1

=

(

0

1

1

0

)

,

σ
2

=

(

0

−
i

i

0

)

,

σ
3

=

(

1

0

0

−
1

)

{\displaystyle
\sigma\_0 =
 \begin{pmatrix}
 1&0\\
 0&1
 \end{pmatrix}, \quad
 \sigma\_1 =
 \begin{pmatrix}
 0&1\\
 1&0
 \end{pmatrix}, \quad
 \sigma\_2 =
 \begin{pmatrix}
 0&-i\\
 i&0
 \end{pmatrix}, \quad
 \sigma\_3 =
 \begin{pmatrix}
 1&0\\
 0&-1
 \end{pmatrix}}

And

(

A

⊗

B

)
(

C

⊗

D

# )
(

A
C

)
⊗
(

B
D

)

{\displaystyle (\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = (\mathbf{AC}) \otimes (\mathbf{BD}) }

The Dirac matrices are commonly referred to by the following name. Note that 4 of the Dirac matrices are denoted

σ
i

{\displaystyle \sigma\_i}

 even though the same symbol can refer to the original Pauli matrices.

σ
0

σ
1

σ
2

σ
3

ρ
1

α
1

α
2

α
3

ρ
2

y
1

y
2

y
3

ρ
3

δ
1

δ
2

δ
3

where

σ
0

=

I
4

−

ρ
1

=

y
5

ρ
2

=

α
5

ρ
3

=

α
4

=

y
4

{\displaystyle
\begin{array}{c|c|c|c}
\sigma\_0
& \sigma\_1
& \sigma\_2
& \sigma\_3 \\\hline
\rho\_1
& {\color{blue} \alpha\_1}
& {\color{blue} \alpha\_2}
& {\color{blue} \alpha\_3} \\\hline
{\color{blue} \rho\_2}
& y\_1
& y\_2
& y\_3 \\\hline
{\color{blue} \rho\_3}
& \delta\_1
& \delta\_2
& \delta\_3
\end{array} \quad \text{where} \quad \begin{align}
 \sigma\_0 &= I\_4 \\
-\rho\_1 &= y\_5 \\
 \rho\_2 &= {\color{blue} \alpha\_5} \\
 \rho\_3 &= {\color{blue} \alpha\_4} = y\_4
\end{align} }

The 16 original Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214).

1. α
1

,

α
2

,

α
3

,

ρ
3

,

ρ
2

(

α
4

,

α
5

)

{\displaystyle {\color{blue} \alpha\_1, \alpha\_2, \alpha\_3,} \quad \rho\_3, \rho\_2 \quad ({\color{blue} \alpha\_4, \alpha\_5})}
2. y
1

,

y
2

,

y
3

,

ρ
3

,
−

ρ
1

(

y
4

,

y
5

)

{\displaystyle y\_1, y\_2, y\_3, \quad \rho\_3, -\rho\_1 \quad (y\_4, y\_5)}
3. δ
1

,

δ
2

,

δ
3

,

ρ
1

,

ρ
2

{\displaystyle \delta\_1, \delta\_2, \delta\_3, \quad \rho\_1, \rho\_2}
4. α
1

,

y
1

,

δ
1

,

σ
2

,

σ
3

{\displaystyle \alpha\_1, y\_1, \delta\_1, \quad \sigma\_2, \sigma\_3}
5. α
2

,

y
2

,

δ
2

,

σ
1

,

σ
3

{\displaystyle \alpha\_2, y\_2, \delta\_2, \quad \sigma\_1, \sigma\_3}
6. α
3

,

y
3

,

δ
3

,

σ
1

,

σ
2

{\displaystyle \alpha\_3, y\_3, \delta\_3, \quad \sigma\_1, \sigma\_2}

Any of the 15 original Dirac matrices (excluding the identity matrix

σ
0

{\displaystyle \sigma\_0}

) anticommute with eight other original Dirac matrices and commute with the remaining eight, including itself and the identity matrix.

Source: Weisstein, Eric W. "Dirac Matrices." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiracMatrices.html
## Alpha multiplication table

Note that unlike the 3 dimensional case with Pauli matrices, the pseudoscalar in 5 dimensions is the negative of the identity matrix.
(

α
1

α
2

α
3

α
4

α
5

=
−

I
4

=

i
2

⊗

i
2

.

{\displaystyle \alpha\_1 \alpha\_2 \alpha\_3 \alpha\_4 \alpha\_5 = -I\_4 = i\_2 \otimes i\_2.}

) In seven dimensions it would be

i
⋅
I

{\displaystyle i \cdot I}

 again. See Classification\_of\_Clifford\_algebras#Unit\_pseudoscalar.

1

2

3

4

5

1

12

13

14

15

2

21

23

24

25

3

31

32

34

35

4

41

42

43

45

5

51

52

53

54

21

2

−
1

54

53

43

31

3

54

−
1

−
52

42

32

−
54

3

−
2

51

−
41

41

4

53

52

−
1

−
32

42

53

4

−
51

−
2

31

43

−
52

51

4

−
3

−
21

51

5

43

−
42

32

−
1

52

−
43

5

41

−
31

−
2

53

42

−
41

5

21

−
3

54

−
32

31

−
21

5

−
4

{\displaystyle
\begin{matrix}
 & {\color{blue}1} & {\color{blue}2} & {\color{blue}3} & {\color{blue}4} & {\color{blue}5} \\
{\color{blue}1} & & 12 & 13 & 14 & 15 \\
{\color{blue}2} & 21 & & 23 & 24 & 25 \\
{\color{blue}3} & 31 & 32 & & 34 & 35 \\
{\color{blue}4} & 41 & 42 & 43 & & 45 \\
{\color{blue}5} & 51 & 52 & 53 & 54 & \\
21 & {\color{blue}2} & {\color{blue}-1} & 54 & 53 & 43 \\
31 & {\color{blue}3} & 54 & {\color{blue}-1} &-52 & 42 \\
32 &-54 & {\color{blue}3} & {\color{blue}-2} & 51 &-41 \\
{\color{green}41} & {\color{blue}4} & 53 & 52 &{\color{blue}-1} &-32 \\
{\color{green}42} & 53 & {\color{blue}4} &-51 &{\color{blue}-2} & 31 \\
{\color{green}43} &-52 & 51 & {\color{blue}4} &{\color{blue}-3} &-21 \\
51 & {\color{blue}5} & 43 &-42 & 32 &{\color{blue}-1} \\
52 &-43 & {\color{blue}5} & 41 &-31 &{\color{blue}-2} \\
53 & 42 &-41 & {\color{blue}5} & 21 &{\color{blue}-3} \\
54 &-32 & 31 &-21 & {\color{blue}5} &{\color{blue}-4}
\end{matrix}
}

## Gamma matrices

The **gamma matrices**,

{

γ
0

,

γ
1

,

γ
2

,

γ
3

}

{\displaystyle \{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \} }

 are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra *C*ℓ1,3(**R**).

α
4

α
0

=

γ
0

=

(

1

0

0

0

0

1

0

0

0

0

−
1

0

0

0

0

−
1

)

,

α
4

α
1

=

γ
1

=

(

0

0

0

1

0

0

1

0

0

−
1

0

0

−
1

0

0

0

)

α
4

α
2

=

γ
2

=

(

0

0

0

−
i

0

0

i

0

0

i

0

0

−
i

0

0

0

)

,

α
4

α
3

=

γ
3

=

(

0

0

1

0

0

0

0

−
1

−
1

0

0

0

0

1

0

0

)

{\displaystyle
\begin{align}
{\color{blue} \alpha\_4 \alpha\_0} = {\color{green} \gamma^0} &= {\color{green} \begin{pmatrix}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & -1 & 0 \\
 0 & 0 & 0 & -1
\end{pmatrix}}, \quad &
{\color{blue} \alpha\_4 \alpha\_1} = {\color{green} \gamma^1} &= {\color{green} \begin{pmatrix}
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0 \\
 0 & -1 & 0 & 0 \\
 -1 & 0 & 0 & 0
\end{pmatrix}} \\

{\color{blue} \alpha\_4 \alpha\_2} = {\color{green} \gamma^2} &= {\color{green} \begin{pmatrix}
 0 & 0 & 0 & -i \\
 0 & 0 & i & 0 \\
 0 & i & 0 & 0 \\
 -i & 0 & 0 & 0
\end{pmatrix}}, \quad &
{\color{blue} \alpha\_4 \alpha\_3} = {\color{green} \gamma^3} &= {\color{green} \begin{pmatrix}
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & -1 \\
 -1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0
\end{pmatrix}}
\end{align}
}

The first squares to one and the rest square to negative one:

(

γ
0

)
2

=

I
4

(

γ
1

)
2

=
−

I
4

(

γ
2

)
2

=
−

I
4

(

γ
3

)
2

=
−

I
4

{\displaystyle
\begin{align}
({\color{green} \gamma^0})^2 &= I\_4 \\
({\color{green} \gamma^1})^2 &= -I\_4 \\
({\color{green} \gamma^2})^2 &= -I\_4 \\
({\color{green} \gamma^3})^2 &= -I\_4
\end{align}
}

For reference:

α
4

α
5

=

(

0

0

−
i

0

0

0

0

−
i

−
i

0

0

0

0

−
i

0

0

)

{\displaystyle
{\color{blue} \alpha\_4 \alpha\_5} = \begin{pmatrix}
 0 & 0 & -i & 0 \\
 0 & 0 & 0 & -i \\
 -i & 0 & 0 & 0 \\
 0 & -i & 0 & 0
\end{pmatrix}}