6.3: Typical Examples

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Example 1

\[\begin{equation}
\begin{aligned}
K^{\prime}=H K H, \quad H=& \exp \left(\frac{\mu}{2} \hat{h} \cdot \vec{\sigma}\right) \\
\vec{k}=\vec{k}_{\|}+\vec{k}_{\perp} \quad \vec{k}_{\|}=(\vec{k} \cdot \hat{h}) \hat{h}
\end{aligned}
\end{equation}\label{1}\]

By using (6a) and (7b):

\[\begin{equation}
\begin{array}{r}
\vec{k}_{\|} \cdot \vec{\sigma} H=H \vec{k}_{\|} \cdot \vec{\sigma}, \quad \vec{k}_{\perp} \cdot \vec{\sigma} H=H^{-1} \vec{k}_{\perp} \cdot \vec{\sigma} \\
\vec{k}_{\|}^{\prime}=\vec{k}_{\|}=k \hat{h}
\end{array}
\end{equation}\label{2}\]

\[\begin{equation}
\begin{aligned}
\left(k_{0}^{\prime}+\vec{k}_{\|}^{\prime} \cdot \vec{\sigma}\right) &=H^{2}\left(k_{0}+\vec{k}_{\|} \cdot \vec{\sigma}\right) \\
&=(\cosh \mu+\sinh \mu \hat{h} \cdot \vec{\sigma})\left(k_{0}+\vec{k}_{\|} \cdot \vec{\sigma}\right)
\end{aligned}
\end{equation}\label{3}\]

\[\begin{equation}
\begin{array}{l}
k_{0}^{\prime}=k_{0} \cosh \mu+k \sinh \mu \\
k^{\prime}=k_{0} \sinh \mu+k \cosh \mu
\end{array}
\end{equation}\label{4}\]

Example 2

\[\begin{equation}
\begin{aligned}
K^{\prime}=U K U^{-1}, & U=\exp \left(-i \frac{\phi}{2} \hat{u} \cdot \vec{\sigma}\right) \\
\vec{k}=\vec{k}_{\|}+\vec{k}_{\perp} \quad \vec{k}_{\|}=(\vec{k} \cdot \hat{u}) \hat{u}
\end{aligned}
\end{equation}\label{5}\]

\[\vec{k}_{\|} \cdot \vec{\sigma} U^{-1}=U^{-1} \vec{k}_{\|} \cdot \vec{\sigma}, \quad \vec{k}_{\perp} \cdot \vec{\sigma} U^{-1}=U \vec{k}_{\perp} \cdot \vec{\sigma}\label{6}\]

\[\vec{k}_{\|}^{\prime}=\vec{k}_{\|}\label{7}\]

\[\begin{equation}
\begin{aligned}
\vec{k}_{\perp}^{\prime} \cdot \vec{\sigma} &=\left(\cos \frac{\phi}{2} 1-i \sin \frac{\phi}{2} \hat{u} \cdot \vec{\sigma}\right)^{2} \vec{k}_{\perp} \cdot \vec{\sigma} \\
&=(\cos \phi 1-i \sin \phi \hat{u} \cdot \vec{\sigma}) \vec{k}_{\perp} \cdot \vec{\sigma} \\
\vec{k}_{\perp}^{\prime} &=\cos \phi \vec{k}_{\perp}+\sin \phi \hat{u} \times \vec{k}_{\perp}
\end{aligned}
\end{equation}\label{8}\]