The representation matrix for alpha and beta in Dirac equation

[Haorong Wu]

Homework Statement

Prove that in the presentation of $\beta$,

$\mathbf \alpha=\begin{pmatrix}\mathbf 0 & \mathbf \sigma \\ \mathbf \sigma & \mathbf 0\end{pmatrix}$ and $\beta=\begin{pmatrix}\mathbf I & \mathbf 0 \\ \mathbf0 & -\mathbf I\end{pmatrix} ,$

where $\mathbf \alpha$ and $\beta$ are in the Dirac equation, in which $H=c \mathbf \alpha \cdot \mathbf p +\beta m c^2$.

Relevant Equations

1. $\mathbf \alpha$ and $\beta$ are Hermitian.
2. $\{ \mathbf \alpha_i, \mathbf \alpha_j \}=0$, if $i\ne j$.
3. $\{ \mathbf \alpha_i, \beta\}=0$.
4. $\alpha_i^2=\beta^2=1$.
5. The traces of $\mathbf \alpha$ and $\beta$ are zero.
6. The eigenvalues of them are $1$ or $-1$.

In the 4-dimensional representation of $\beta$, $\beta=\begin{pmatrix}\mathbf I & \mathbf 0 \\ \mathbf0 & -\mathbf I\end{pmatrix} ,$ and we can suppose $\alpha_i=\begin{pmatrix}\mathbf A_i & \mathbf B_i \\ \mathbf C_i & \mathbf D_i\end{pmatrix}$.

From the anti-commutation relation $\{ \mathbf \alpha_i, \beta\}=0$, I can derive $A_i=D_i=0$.

From $\alpha_i^2=1$, I can have $C_i=B_i^{-1}$. Furthermore, from the Hermiticity, I can have $C_i=B_i^{\dagger}$.

But I could not find a way to prove that $C_i=B_i$. The relation $\{ \mathbf \alpha_i, \mathbf \alpha_j \}=0$ for $i\ne j$ does not help.

Should $\alpha_i$ and $\beta$ always be symmetric? This is not given in the problem. Is there any other properties of $\alpha_i$?

I have looked in some books. These matrices are just given directly without proof.

 

[anuttarasammyak]

Say
$$B=<br /> \begin{pmatrix}<br /> a & b \\<br /> c & -a \\<br /> \end{pmatrix}$$
with trace=0 used.
$$C=B^{-1}=\frac{-1}{det \ B}<br /> \begin{pmatrix}<br /> a & b \\<br /> c & -a \\<br /> \end{pmatrix}<br /> =\frac{-1}{det \ B} B$$
$$det \ B=-a^2-bc=-1$$
because B has eigenvalue 1 and -1 so
$$(a-\lambda)(-a-\lambda)-bc=0$$
for $\lambda=\pm1$. As $\alpha$ is Hermitian we observe a is real and $c=b^*$. so
$$a^2+|b|^2=1$$

 

[Haorong Wu]

@anuttarasammyak Thanks! I forgot the determinant. I will try to generalize it to 4 dimensional case.

 

[anuttarasammyak]

Further to post #2 as general expression
$$B=C=<br /> \begin{pmatrix}<br /> cos\theta & sin\theta e^{-i\phi} \\<br /> sin\theta e^{i\phi} & -cos\theta \\<br /> \end{pmatrix}<br /> =\cos\theta\ \sigma_z+sin\theta cos\phi\ \sigma_x+sin\theta sin\phi\ \sigma_y$$
that seems like unit vector in polar coordinates.