Question 13.6: Construct the matrix α · p + mβ that occurs in the above equ...
Construct the matrix α · p + mβ that occurs in the above equation.
In the Dirac-Pauli representation of the α and β matrices (13.42), the matrix α^{i} is
\begin{matrix} α^{i}=\left[\begin{matrix} 0 &σ^{i}\\ σ^{i}& 0 \end{matrix} \right], & β =\left[\begin{matrix} I & 0 \\0 & I \end{matrix} \right] .\end{matrix} (13.42)
\alpha ^{i}=\left[\begin{matrix}0& σ^{i}\\σ^{i}& 0 \end{matrix} \right] , (13.57)
where σ^{i} is a two-dimensional Pauli matrix defined by Eq. (13.43). Multiplying the above α-matrix with p^{i} gives
\begin{matrix} σ^{1}=\left[\begin{matrix}0 &1\\1& 0 \end{matrix} \right],& σ^{2}= \left[\begin{matrix}0 &-i\\i& 0\end{matrix} \right], &σ^{3}= \left[\begin{matrix}1 &0\\0& −1 \end{matrix} \right].\end{matrix} (13.43)
\alpha ^{i}p^{i} =\left[\begin{matrix}0& σ^{i}p^{i}\\σ^{i}p^{i}& 0 \end{matrix} \right].
The dot product is accomplished by summing over the i indices to obtain
\alpha · p =\left[\begin{matrix}0& σ·p\\σ·p& 0 \end{matrix} \right]. (13.58)
Similarly, the β matrix is
β =\left[\begin{matrix}I &0\\0& −I \end{matrix} \right],
where I denotes the unit 2 × 2 matrix. Multiplying β by m gives
mβ =\left[\begin{matrix}mI &0\\0& −mI \end{matrix} \right].
The 2 × 2 unit matrices I that occur in this last equation can be understood writing simply
mβ =\left[\begin{matrix}m &0\\0& −m \end{matrix} \right]. (13.59)
Combining Eqs. (13.58) and (13.59), we obtain
α · p + mβ =\left[\begin{matrix}m& σ · p\\σ · p& −m \end{matrix} \right]. (13.60)