Define the
matrices
where
are the Pauli Matrices, I is the Identity Matrix,
, 2, 3, and
is the matrix Direct Product. Explicitly,
![$\displaystyle {\hbox{\sf I}}$](d2_10.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{array}\right]$](d2_11.gif) |
(3) |
![$\displaystyle \sigma_1$](d2_12.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right]$](d2_13.gif) |
(4) |
![$\displaystyle \sigma_2$](d2_14.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}0 & -i & 0 & 0\\ i & 0 & 0 & 0\\ 0 & 0 & 0 & -i\\ 0 & 0 & i & 0\end{array}\right]$](d2_15.gif) |
(5) |
![$\displaystyle \sigma_3$](d2_16.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\end{array}\right]$](d2_17.gif) |
(6) |
![$\displaystyle \rho_1$](d2_18.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\end{array}\right]$](d2_19.gif) |
(7) |
![$\displaystyle \rho_2$](d2_20.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}0 & 0 & -i & 0\\ 0 & 0 & 0 & -i\\ i & 0 & 0 & 0\\ 0 & i & 0 & 0\end{array}\right]$](d2_21.gif) |
(8) |
![$\displaystyle \rho_3$](d2_22.gif) |
![$\textstyle =$](d2_3.gif) |
![$\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\end{array}\right].$](d2_23.gif) |
(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}](d2_24.gif) |
(10) |
![\begin{displaymath}
\rho_i\rho_j+\rho_j\rho_i=2\delta_{ij}{\hbox{\sf I}},
\end{displaymath}](d2_25.gif) |
(11) |
where
is the Kronecker Delta, the commutation identity
![\begin{displaymath}[\sigma_i, \rho_j]=\sigma_i\rho_j-\rho_j\sigma_i=0,
\end{displaymath}](d2_27.gif) |
(12) |
and are cyclic under permutations of indices
![\begin{displaymath}
\sigma_i\sigma_j=i\sigma_k
\end{displaymath}](d2_28.gif) |
(13) |
![\begin{displaymath}
\rho_i\rho_j=i\rho_k.
\end{displaymath}](d2_29.gif) |
(14) |
A total of 16 Dirac matrices can be defined via
![\begin{displaymath}
{\hbox{\sf E}}_{ij}=\rho_i\sigma_j
\end{displaymath}](d2_30.gif) |
(15) |
for
, 1, 2, 3 and where
. These matrices satisfy
- 1.
, where
is the Determinant,
- 2.
,
- 3.
, making them Hermitian, and therefore unitary,
- 4.
, except
,
- 5. Any two
multiplied together yield a Dirac matrix to within a multiplicative factor of
or
,
- 6. The
are linearly independent,
- 7. The
form a complete set, i.e., any
constant matrix may be written as
![\begin{displaymath}
{\hbox{\sf A}}=\sum_{i,j=0}^3 c_{ij}{\hbox{\sf E}}_{ij},
\end{displaymath}](d2_42.gif) |
(16) |
where the
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}](d2_44.gif) |
(17) |
(Arfken 1985).
Dirac's original matrices were written
and were defined by
for
, 2, 3, giving
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}](d2_57.gif) |
(24) |
is sometimes defined. Other sets of Dirac matrices are sometimes defined as
and
![\begin{displaymath}
\delta_i={\hbox{\sf E}}_{3i}
\end{displaymath}](d2_64.gif) |
(28) |
for
, 2, 3 (Arfken 1985) and
for
, 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
, then
![\begin{displaymath}
{\hbox{\sf M}}^2={\hbox{\sf M}}.
\end{displaymath}](d2_70.gif) |
(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}](d2_71.gif) |
(32) |
The products of
and
satisfy
![\begin{displaymath}
\alpha_1\alpha_2\alpha_3\alpha_4\alpha_5=1
\end{displaymath}](d2_73.gif) |
(33) |
![\begin{displaymath}
y_1y_2y_3y_4y_5=1.
\end{displaymath}](d2_74.gif) |
(34) |
The 16 Dirac matrices form six anticommuting sets of five matrices each:
- 1.
,
,
,
,
,
- 2.
,
,
,
,
,
- 3.
,
,
,
,
,
- 4.
,
,
,
,
,
- 5.
,
,
,
,
,
- 6.
,
,
,
,
.
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.
© 1996-9 Eric W. Weisstein
1999-05-24