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Dirac Matrices

Define the $4\times 4$ matrices

$\displaystyle \sigma_i$ $\textstyle =$ $\displaystyle {\hbox{\sf I}}\otimes\sigma_{i,{\rm\ Pauli}}$ (1)
$\displaystyle \rho_i$ $\textstyle =$ $\displaystyle \sigma_{i,{\rm\ Pauli}}\otimes{\hbox{\sf I}},$ (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,
$\displaystyle {\hbox{\sf I}}$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0\\  0 & 1 & 0 & 0\\  0 & 0 & 1 & 0\\  0 & 0 & 0 & 1\end{array}\right]$ (3)
$\displaystyle \sigma_1$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}0 & 1 & 0 & 0\\  1 & 0 & 0 & 0\\  0 & 0 & 0 & 1\\  0 & 0 & 1 & 0\end{array}\right]$ (4)
$\displaystyle \sigma_2$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}0 & -i & 0 & 0\\  i & 0 & 0 & 0\\  0 & 0 & 0 & -i\\  0 & 0 & i & 0\end{array}\right]$ (5)
$\displaystyle \sigma_3$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0\\  0 & -1 & 0 & 0\\  0 & 0 & 1 & 0\\  0 & 0 & 0 & -1\end{array}\right]$ (6)
$\displaystyle \rho_1$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}0 & 0 & 1 & 0\\  0 & 0 & 0 & 1\\  1 & 0 & 0 & 0\\  0 & 1 & 0 & 0\end{array}\right]$ (7)
$\displaystyle \rho_2$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}0 & 0 & -i & 0\\  0 & 0 & 0 & -i\\  i & 0 & 0 & 0\\  0 & i & 0 & 0\end{array}\right]$ (8)
$\displaystyle \rho_3$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0\\  0 & 1 & 0 & 0\\  0 & 0 & -1 & 0\\  0 & 0 & 0 & -1\end{array}\right].$ (9)

These matrices satisfy the anticommutation identities
\begin{displaymath}
\sigma_i\sigma_j+\sigma_j\sigma_i=2\delta_{ij}{\hbox{\sf I}}
\end{displaymath} (10)


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

where $\delta_{ij}$ is the Kronecker Delta, the commutation identity
\begin{displaymath}[\sigma_i, \rho_j]=\sigma_i\rho_j-\rho_j\sigma_i=0,
\end{displaymath} (12)

and are cyclic under permutations of indices
\begin{displaymath}
\sigma_i\sigma_j=i\sigma_k
\end{displaymath} (13)


\begin{displaymath}
\rho_i\rho_j=i\rho_k.
\end{displaymath} (14)


A total of 16 Dirac matrices can be defined via

\begin{displaymath}
{\hbox{\sf E}}_{ij}=\rho_i\sigma_j
\end{displaymath} (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
\begin{displaymath}
{\hbox{\sf A}}=\sum_{i,j=0}^3 c_{ij}{\hbox{\sf E}}_{ij},
\end{displaymath} (16)

where the $c_{ij}$ are real or complex and are given by
\begin{displaymath}
c_{mn}={\textstyle{1\over 4}}\mathop{\rm tr}({\hbox{\sf A}}{\hbox{\sf E}}_{mn})
\end{displaymath} (17)

(Arfken 1985).


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

$\displaystyle \alpha_i$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{1i}=\rho_1\sigma_i$ (18)
$\displaystyle \alpha_4$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{30}=\rho_3,$ (19)

for $i=1$, 2, 3, giving
$\displaystyle \alpha_1$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{11}=\left[\begin{array}{cccc}0 & 0 & 0 & 1\\  0 & 0 & 1 & 0\\  0 & 1 & 0 & 0\\  1 & 0 & 0 & 0\end{array}\right]$ (20)
$\displaystyle \alpha_2$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{12}=\left[\begin{array}{cccc}0 & 0 & 0 & -i\\  0 & 0 & i & 0\\  0 & -i & 0 & 0\\  i & 0 & 0 & 0\end{array}\right]$ (21)
$\displaystyle \alpha_3$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{13}=\left[\begin{array}{cccc}0 & 0 & 1 & 0\\  0 & 0 & 0 & -1\\  1 & 0 & 0 & 0\\  0 & -1 & 0 & 0\end{array}\right]$ (22)
$\displaystyle \alpha_4$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{30}=\left[\begin{array}{cccc}1 & 0 & 0 & 0\\  0 & 1 & 0 & 0\\  0 & 0 & -1 & 0\\  0 & 0 & 0 & -1\end{array}\right].$ (23)

The additional matrix
\begin{displaymath}
\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]
\end{displaymath} (24)

is sometimes defined. Other sets of Dirac matrices are sometimes defined as
$\displaystyle y_i$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{2i}$ (25)
$\displaystyle y_4$ $\textstyle =$ $\displaystyle {\hbox{\sf E}}_{30}$ (26)
$\displaystyle y_5$ $\textstyle =$ $\displaystyle -{\hbox{\sf E}}_{10}$ (27)

and
\begin{displaymath}
\delta_i={\hbox{\sf E}}_{3i}
\end{displaymath} (28)

for $i=1$, 2, 3 (Arfken 1985) and
$\displaystyle \gamma_i$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}{\hbox{\sf0}} & \sigma_i\\  -\sigma_i & {\hbox{\sf0}}\end{array}\right]$ (29)
$\displaystyle \gamma_4$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}{\hbox{\sf I}}& {\hbox{\sf0}}\\  2{\hbox{\sf I}}& -{\hbox{\sf I}}\end{array}\right]$ (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

\begin{displaymath}
{\hbox{\sf M}}^2={\hbox{\sf M}}.
\end{displaymath} (31)

In addition
\begin{displaymath}
\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.
\end{displaymath} (32)

The products of $\alpha_i$ and $y_i$ satisfy
\begin{displaymath}
\alpha_1\alpha_2\alpha_3\alpha_4\alpha_5=1
\end{displaymath} (33)


\begin{displaymath}
y_1y_2y_3y_4y_5=1.
\end{displaymath} (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.



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© 1996-9 Eric W. Weisstein
1999-05-24