6.1: Useful formulas

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\(A=a_{0} 1+\vec{a} \cdot \vec{\sigma} \tilde{A}=a_{0} 1-\vec{a} \cdot \vec{\sigma} A^{\dagger}=a_{0}^{*} 1+\vec{a}^{*} \cdot \vec{\sigma} \bar{A}=\tilde{A}^{\dagger}=a_{0}^{*} 1-\vec{a}^{*} \cdot \vec{\sigma}\)

\[\frac{1}{2} \operatorname{Tr}(A)=a_{0}, \quad|A|=a_{0}^{2}-\vec{a}^{2} 1 \frac{1}{2} \operatorname{Tr}(A \tilde{A})\label{1}\]

\[\frac{1}{2} \operatorname{Tr}(A \tilde{B})=a_{0} b_{0}-\vec{a} \cdot \vec{b}\label{2}\]

\[A^{-1}=\frac{\tilde{A}}{|A|} \quad \text { for } \quad|A|=1: A^{-1}=\tilde{A}\label{3}\]

\[(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})=\vec{a} \cdot \vec{b} 1+i(\vec{a} \times \vec{b}) \cdot \vec{\sigma}\label{4}\]

\[\text { For } \vec{a} \| \vec{b} \quad \frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}} \quad \vec{a} \times \vec{b}=0\label{5}\]

\[(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})-(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma})=[(\vec{a} \cdot \vec{\sigma}),(\vec{b} \cdot \vec{\sigma})]=0\label{6}\]

\[\text { For } \quad A=a_{0} 1+\vec{a} \cdot \vec{\sigma}, \quad B=b_{0} 1+\vec{b} \cdot \vec{\sigma}\label{7}\]

\[[A, B]=0 \quad \text { iff } \quad \vec{a} \| \vec{b}\label{8}\]

\[\begin{equation}
\begin{array}{ll}
\text { For } \quad & \vec{a} \perp \vec{b}, \quad \vec{a} \cdot \vec{b} \\
& \{\vec{a} \cdot \vec{\sigma}, \vec{b} \cdot \vec{\sigma}\} \quad \equiv \quad(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma})+(\vec{b} \cdot \vec{\sigma})(\vec{a} \cdot \vec{\sigma})=0
\end{array}
\end{equation}\label{9}\]

\[A(\vec{b} \cdot \vec{\sigma})=(\vec{b} \cdot \vec{\sigma}) \tilde{A}\label{10}\]

\[U=U\left(\hat{u}, \frac{\phi}{2}\right)=\cos \frac{\phi}{2} 1-i \sin \frac{\phi}{2} \hat{n} \cdot \vec{\sigma}=\exp \left(-i \frac{\phi}{2} \hat{n} \cdot \vec{\sigma}\right)\label{11}\]

\[H=H\left(\hat{h}, \frac{\mu}{2}\right)=\cosh \frac{\mu}{2} 1+\sinh \frac{\mu}{2} \hat{h} \cdot \vec{\sigma}=\exp \left(\frac{\mu}{2} \hat{h} \cdot \vec{\sigma}\right)\label{12}\]

\(U\) unitary unimodular, H Hermitian and positive.