Lorentz transformation (II)

 Problem: Let $k^{\mu}\equiv[1, 1,  0,  0]^T$, find a Lorentz transformation (matrix) $W^{\mu}_{\,\,\,\,\nu}$  such that $ W^{\mu}_{\,\,\,\,\nu}\,k^{\nu}=k^{\mu}$.

(This problem is about the proof of Eq. (2.5.28) in Weinberg's QFT book, Volume I)

Solution:

Let $t^{\mu}\equiv[1, 0,  0,  0]^T$, the relations \begin{equation}\begin{split}(Wt)^{\mu}(Wt)_{\mu}&=&t^{\mu}t_{\mu}=-1\,,\\ (Wt)^{\mu}k_{\mu}&=&t^{\mu}k_{\mu}=-1\,, \end{split}\end{equation} suggest that $(Wt)^{\mu}$ is of the form \begin{equation} (Wt)^{\mu}= [1+\zeta\,,\zeta\,, \alpha\,, \beta]^T\,,\end{equation} with \begin{equation}\zeta = (\alpha^2+\beta^2)/2\,.\tag{1}\end{equation}

As a result, the Lorentz matrix must take the form of \begin{equation}W = \begin{pmatrix} \begin{matrix} 1+\zeta & -\zeta \\ \zeta & 1-\zeta \end{matrix}  & \mathbf{X} \\  \begin{matrix} \alpha & \quad -\alpha \\ \beta & \quad -\beta \end{matrix}  & \mathbf{R} \end{pmatrix}\,.\tag{2}\end{equation} where $\mathbf{X}$ and $\mathbf{R}$ are 2x2 block matrices to be solved.

As a Lorentz matrix, $W$ should satisfy the condition \begin{equation}W^T\eta W =\eta\,,\tag{3}\end{equation} with the metric $\eta \equiv [-1, 1, 1, 1]^T$. Submitting the form (2) into Eq. (3) gives the relations: \begin{eqnarray}\left[\begin{matrix} -1-\zeta & \zeta \\ \zeta & 1-\zeta \end{matrix}\right]\mathbf{X} + \left[\begin{matrix} \alpha & \beta \\ -\alpha & -\beta \end{matrix}\right] \mathbf{R}&=&\mathbf{0}\,,\\ \mathbf{R}^T\mathbf{R}&=&\mathbf{I}_{2\times 2}\,.\end{eqnarray}

Therefore, $\mathbf{R}$ is a two-dimension matrix that can be parametrized by \begin{equation}\mathbf{R}= \left[\begin{matrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix}\right]\,\end{equation} and $\mathbf{X}$ can be expressed in terms of $\mathbf{R}$ as \begin{equation}\mathbf{X} = \left[\begin{matrix} 1+\zeta & -\zeta \\ -\zeta & -1+\zeta \end{matrix}\right]^{-1} \left[\begin{matrix} \alpha & \beta \\ -\alpha & -\beta \end{matrix}\right] \mathbf{R}=\left[\begin{matrix} \alpha & \beta \\ \alpha & \beta \end{matrix}\right] \mathbf{R}\,.\end{equation}

In sum, the general solution of $W$ can be parametrized in three parameters $\alpha,\beta,\theta$ as \begin{equation}W(\alpha, \beta, \theta)=\left[\begin{matrix} 1+\zeta & -\zeta & \alpha & \beta \\ \zeta & 1-\zeta & \alpha & \beta \\ \alpha & -\alpha &  1 & 0 \\ \beta & -\beta & 0 & 1 \end{matrix}\right]\left[\begin{matrix} 1 &  &  & \\  & 1 & & \\ & &  \cos\theta & -\sin\theta  \\ & & \sin\theta & \cos\theta \end{matrix}\right]\,.\tag{4}\end{equation}