Dirac Matrices

Define the $4\times 4$ matrices

$$\sigma_i = {\hbox{\sf I}}\otimes\sigma_{i,{\rm\ Pauli}}$$ (1)

$$\rho_i = \sigma_{i,{\rm\ Pauli}}\otimes{\hbox{\sf I}},$$ (2)

where $\sigma_{i,{\rm\ Pauli}}$ are the Pauli Matrices, I is the Identity Matrix, $i=1$, 2, 3, and ${\hbox{\sf A}}\otimes{\hbox{\sf B}}$ is the matrix Direct Product. Explicitly,

$${\hbox{\sf I}} = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{array}\right]$$ (3)

$$\sigma_1 = \left[\begin{array}{cccc}0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right]$$ (4)

$$\sigma_2 = \left[\begin{array}{cccc}0 & -i & 0 & 0\\ i & 0 & 0 & 0\\ 0 & 0 & 0 & -i\\ 0 & 0 & i & 0\end{array}\right]$$ (5)

$$\sigma_3 = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\end{array}\right]$$ (6)

$$\rho_1 = \left[\begin{array}{cccc}0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\end{array}\right]$$ (7)

$$\rho_2 = \left[\begin{array}{cccc}0 & 0 & -i & 0\\ 0 & 0 & 0 & -i\\ i & 0 & 0 & 0\\ 0 & i & 0 & 0\end{array}\right]$$ (8)

$$\rho_3 = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\end{array}\right].$$ (9)

These matrices satisfy the anticommutation identities

$$\sigma_i\sigma_j+\sigma_j\sigma_i=2\delta_{ij}{\hbox{\sf I}}$$ (10)

$$\rho_i\rho_j+\rho_j\rho_i=2\delta_{ij}{\hbox{\sf I}},$$ (11)

where $\delta_{ij}$ is the Kronecker Delta, the commutation identity

$$[\sigma_i, \rho_j]=\sigma_i\rho_j-\rho_j\sigma_i=0,$$ (12)

and are cyclic under permutations of indices

$$\sigma_i\sigma_j=i\sigma_k$$ (13)

$$\rho_i\rho_j=i\rho_k.$$ (14)

A total of 16 Dirac matrices can be defined via

$${\hbox{\sf E}}_{ij}=\rho_i\sigma_j$$ (15)

for $i,j=0$, 1, 2, 3 and where $\sigma_0=\rho_0\equiv {\hbox{\sf I}}$. These matrices satisfy

1. $\vert{\hbox{\sf E}}_{ij}\vert=1$, where $\vert{\hbox{\sf A}}\vert$ is the Determinant,

2. ${\hbox{\sf E}}_{ij}^2={\hbox{\sf I}}$,

3. ${\hbox{\sf E}}_{ij}={\hbox{\sf E}}_{ij}^\dagger$, making them Hermitian, and therefore unitary,

4. $\mathop{\rm tr}({\hbox{\sf E}}_{ij})=0$, except $\mathop{\rm tr}({\hbox{\sf E}}_{00})=4$,

5. Any two ${\hbox{\sf E}}_{ij}$ multiplied together yield a Dirac matrix to within a multiplicative factor of $-1$ or $\pm i$,

6. The ${\hbox{\sf E}}_{ij}$ are linearly independent,

7. The ${\hbox{\sf E}}_{ij}$ form a complete set, i.e., any $4\times 4$ constant matrix may be written as

$${\hbox{\sf A}}=\sum_{i,j=0}^3 c_{ij}{\hbox{\sf E}}_{ij},$$ (16)

where the $c_{ij}$ are real or complex and are given by

$$c_{mn}={\textstyle{1\over 4}}\mathop{\rm tr}({\hbox{\sf A}}{\hbox{\sf E}}_{mn})$$ (17)

(Arfken 1985).

Dirac's original matrices were written $\alpha_i$ and were defined by

$$\alpha_i = {\hbox{\sf E}}_{1i}=\rho_1\sigma_i$$ (18)

$$\alpha_4 = {\hbox{\sf E}}_{30}=\rho_3,$$ (19)

for $i=1$, 2, 3, giving

$$\alpha_1 = {\hbox{\sf E}}_{11}=\left[\begin{array}{cccc}0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\end{array}\right]$$ (20)

$$\alpha_2 = {\hbox{\sf E}}_{12}=\left[\begin{array}{cccc}0 & 0 & 0 & -i\\ 0 & 0 & i & 0\\ 0 & -i & 0 & 0\\ i & 0 & 0 & 0\end{array}\right]$$ (21)

$$\alpha_3 = {\hbox{\sf E}}_{13}=\left[\begin{array}{cccc}0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\end{array}\right]$$ (22)

$$\alpha_4 = {\hbox{\sf E}}_{30}=\left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\end{array}\right].$$ (23)

The additional matrix

$$\alpha_5={\hbox{\sf E}}_{20}=\rho_2=\left[{\matrix{0 & 0 & -... ...r 0 & 0 & 0 & -i\cr i & 0 & 0 & 0\cr 0 & i & 0 & 0\cr}}\right]$$ (24)

is sometimes defined. Other sets of Dirac matrices are sometimes defined as

$$y_i = {\hbox{\sf E}}_{2i}$$ (25)

$$y_4 = {\hbox{\sf E}}_{30}$$ (26)

$$y_5 = -{\hbox{\sf E}}_{10}$$ (27)

and

$$\delta_i={\hbox{\sf E}}_{3i}$$ (28)

for $i=1$, 2, 3 (Arfken 1985) and

$$\gamma_i = \left[\begin{array}{cc}{\hbox{\sf0}} & \sigma_i\\ -\sigma_i & {\hbox{\sf0}}\end{array}\right]$$ (29)

$$\gamma_4 = \left[\begin{array}{cc}{\hbox{\sf I}}& {\hbox{\sf0}}\\ 2{\hbox{\sf I}}& -{\hbox{\sf I}}\end{array}\right]$$ (30)

for $i=1$, 2, 3 (Goldstein 1980).

Any of the 15 Dirac matrices (excluding the identity matrix) commute with eight Dirac matrices and anticommute with the other eight. Let ${\hbox{\sf M}}\equiv {\textstyle{1\over 2}}(1+{\hbox{\sf E}}_{ij})$, then

$${\hbox{\sf M}}^2={\hbox{\sf M}}.$$ (31)

In addition

$$\left[{\matrix{\alpha_1\cr \alpha_2\cr \alpha_3\cr}}\right]\... ...\matrix{\alpha_1\cr \alpha_2\cr \alpha_3\cr}}\right]=2i\sigma.$$ (32)

The products of $\alpha_i$ and $y_i$ satisfy

$$\alpha_1\alpha_2\alpha_3\alpha_4\alpha_5=1$$ (33)

$$y_1y_2y_3y_4y_5=1.$$ (34)

The 16 Dirac matrices form six anticommuting sets of five matrices each:

1. $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$, $\alpha_5$,

2. $y_1$, $y_2$, $y_3$, $y_4$, $y_5$,

3. $\delta_1$, $\delta_2$, $\delta_3$, $\rho_1$, $\rho_2$,

4. $\alpha_1$, $y_1$, $\delta_1$, $\sigma_2$, $\sigma_3$,

5. $\alpha_2$, $y_2$, $\delta_2$, $\sigma_1$, $\sigma_3$,

6. $\alpha_3$, $y_3$, $\delta_3$, $\sigma_1$, $\sigma_2$.

See also Pauli Matrices

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211-213, 1985.

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980.