[QUE/ME-02008] ME-PROBLEM

For a four vector \(x=(\vec{x},x_4)\equiv(x_1,x_2,x_3,x_4)\), define a \(2\times2\) matrix \(M\) by \[M = x_\mu\sigma_\mu= x_4 + \vec{x}\cdot\vec{\sigma}\] where \(\sigma_4\) is \(2\times2\) identity matrix and \(\vec{\sigma}\) are Pauli matrices given by \begin{equation*} \sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \sigma_1=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad \sigma_1=\begin{pmatrix}1&0\\0&-1\end{pmatrix} \end{equation*} Also define a matrix \(U\) by \begin{equation*} U = \cos\frac{\alpha}{2} + i\sin\frac{\alpha}{2}(\hat{n}\cdot\vec{\sigma}) \end{equation*} where \(\hat{n}=(n_1,n_2,n_3)\) is a unit vector.

• Show that \(U\) is unitary and \(\det U=1\).
• Prove that \(x_\mu = \frac{1}{2} Tr(\sigma_\mu M)\)
• Let \(M{'}= U M U^\dagger\) Compute \(x_\mu{'} \) and show that \begin{equation} x_4{'}=x_4; \vec{x}{'}=\vec{x}-\sin\alpha(\hat{n}\times\vec{x}) +(1-\cos\alpha)\hat{n}\times(\hat{n} \times\vec{x}) \end{equation}

The last equation shows that to every rotation in three dimensions there are two \(SU(2)\) matrices given by \(\pm U\).