Vector representation of Lorentz group

[kelly0303]

Homework Statement

In this problem, we'll construct the $(\frac{1}{2},\frac{1}{2})$ representation which acts on "bi-spinors" $V_{\alpha\dot{\alpha}}$ with $\alpha=1,2$ and $\dot{\alpha}=1,2$. It is convential, and convenient, to define these bi-spinors so that the first index transforms under $\Lambda_R$ and the second under the inverse of $\Lambda_L$. That is, we define: $V \to \Lambda_R V \Lambda_L^{-1}$ .
a) Let's write a generic bi-spinor as $V_{\alpha \dot{\alpha}}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ What is the change in a, b, c and d due to $V \to \Lambda_R V \Lambda_L^{-1}$ by an infinitesimal rotation by an angle $\theta$ around the z axis.
b) Write $J_z$ from part a) as a matrix acting on $(a, b, c, d)$. What are its eigenvalues?

Homework Equations

$\Lambda_R=e^{\frac{1}{2}(i\theta_j\sigma_j+\beta_j\sigma_j)}$
$\Lambda_L=e^{\frac{1}{2}(i\theta_j\sigma_j-\beta_j\sigma_j)}$
with $\sigma_j$ being the Pauli matrices.

The Attempt at a Solution

a) For an infinitesimal transformation we have $$\Lambda_R V \Lambda_L^{-1}=(1+\frac{1}{2}i\theta_z\sigma_z)
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} (1-\frac{1}{2}i\theta_z\sigma_z)$$
which, after some calculations gives $$
\begin{pmatrix}
a(1+\frac{\theta_z^2}{4}) & b(1+i\theta_z-\frac{\theta_z^2}{4}) \\
c(1-i\theta_z-\frac{\theta_z^2}{4}) & d(1+\frac{\theta_z^2}{4})
\end{pmatrix}
$$
b) Here I tried this: $$(1+\frac{1}{2}i\theta_zJ_z)
\begin{pmatrix}
a\\
b\\
c\\
d
\end{pmatrix}
=
\begin{pmatrix}
a(1+\frac{\theta_z^2}{4})\\
b(1+i\theta_z-\frac{\theta_z^2}{4}) \\
c(1-i\theta_z-\frac{\theta_z^2}{4})\\
d(1+\frac{\theta_z^2}{4})
\end{pmatrix}
$$
But I am not sure what to do now. I expected to get for $J_z$, something like $diag(0, -1, 0, 1)$ being the combination between a spin 0 and a spin 1 particle. But what I get doesn't look right. Can someone tell me what am I doing wrong? Thank you!

 

[mark726]

Your approach for part a) looks correct, but for part b) you need to isolate the matrix $J_z$ and then find its eigenvalues. To do this, you can equate the two matrices you have written in part a) and solve for $J_z$. This should give you the desired matrix, and then you can find its eigenvalues by solving the characteristic equation.