[QUE/QFT-03002] QFT-PROBLEM

Write an infinitesimal Lorentz transformation as
\begin{equation} \widetilde{X{'}} = \widetilde{\Lambda} \widetilde{X} \end{equation} where \(\widetilde{\Lambda}\) is a four by four matrix. Taking \(Lambda\) to differ infinitesimal transformation of the form \[\Lambda = I + \Omega \] and find restrictions on the matrix \(\Omega\) using the fact that under Lorentz transformations \(x^\mu x_\mu \) remains invariant. Count independent number of parameters in \(\Omega\) and write its most general form. 

 The most general form of \(\Omega\) is \begin{equation*} \Omega = \begin{pmatrix} a & \alpha_1 & \alpha_2 & \alpha_3 \\ \alpha_1 & b & \alpha_4 & \alpha_5 \\ \alpha_2 & -\alpha_4 & c & \alpha_6\\ \alpha_3 & -\alpha_5 & -\alpha_6 & d \end{pmatrix} \end{equation*} }\end{Answer} \begin{Solution} The Lorentz invariance of \(x^\mu x_\mu= x^\mu g_{\mu\nu} x^\nu\) implies \[ \widetilde{X}^{{'} \,T} \tilde{g} \widetilde{X}{'} = {X}^{{'} \,T} \tilde{g} {X}{'}\] where \(\tilde{g}\) is the diagonal matrix with diagonal entries (1,-1,-1,-1). The requirement of Lorentz invariance of the four scalar product gives the following condition on \(\Lambda\). \begin{equation} \widetilde{\Lambda} \tilde{g} \widetilde{\Lambda} = \tilde{g} \end{equation} Using \(\widetilde{X}^{{'} \,T} g \widetilde{X}{'}\) We rewrite the above equation in the form \begin{equation} (I + \Omega^T) \tilde{g} (1+ \Omega) = \tilde{g} \end{equation} Simplifying and keeping only first order terms in \(\Omega\), gives \begin{equation}\label{EQ04} \Omega^T \tilde{g} + \tilde{g} \Omega = 0. \end{equation}
Writing and multiplying out the matrices gives
\begin{eqnarray} \Omega \tilde{g} &=& \begin{pmatrix} a & -\alpha_1 & -\alpha_2 & -\alpha_3 \\ \alpha_1 & b & \alpha_4 & \alpha_5 \\ \alpha_2 & -\alpha_4 & c & \alpha_6\\ \alpha_3 & -\alpha_5 & -\alpha_6 & d \end{pmatrix}\\ 
\tilde{g}\Omega &=& \begin{pmatrix} a & \alpha_1 & \alpha_2 & \alpha_3 \\ \alpha_1{'} & b & \alpha_4 & \alpha_5 \\ \alpha_2{'} & -\alpha_4{'} & c & \alpha_6\\ \alpha_3{'} & -\alpha_5{'} & -\alpha_6{'} & d \end{pmatrix} \end{eqnarray}