6.4: On the us of Involutions

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The existence of the three involutions ( see Equations A.1.1 above), provides a great deal of flexilbity. However, the most efficient use of these concepts calls for some care.

For any matrix of \(\mathcal{A}_{2}\),

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

In the case of Hermitian matrices we have two alternatives:

\[k_{0} r_{0}-\vec{k} \cdot \vec{r}=\frac{1}{2} \operatorname{Tr}(K \tilde{R})\label{2}\]

\[k_{0} r_{0}-\vec{k} \cdot \vec{r}=\frac{1}{2} \operatorname{Tr}(K \bar{R})\label{3}\]

It will appear, however from later discussions, that the complex reflection of Equation \ref{3} is more appropriate to describe the transition from contravariant to covariant entities.

A case in point is the formal representation of the mirroring of a four-vector in a plane with the normal along \(\hat{x}_{1}\). We have

\[\begin{equation}
\begin{aligned}
K^{\prime} &=\sigma_{1} \bar{K} \sigma_{1}=\sigma_{1}\left(k_{0} 1-k_{1} \sigma_{1}-k_{2} \sigma_{2}-k_{3} \sigma_{3}\right) \sigma_{1} \\
&=\sigma_{1}^{2}\left(k_{0} 1-k_{1} \sigma_{1}+k_{2} \sigma_{2}+k_{3} \sigma_{3}\right) \\
&=k_{0} 1-k_{1} \sigma_{1}+k_{2} \sigma_{2}+k_{3} \sigma_{3}
\end{aligned}
\end{equation}\label{4}\]

More generally the mirroring in a plane with normal x is achieve by means of the operation

\[K^{\prime}=\hat{a} \cdot \vec{\sigma} \bar{K} \hat{a} \cdot \vec{\sigma}\label{5}\]

Again, we could have chosen \(\tilde{K} \text { instead of } \bar{K}\).

However, Eq (22) generalizes to the inversion of the electromagnetic six-vector \(\vec{f}=\vec{E}+i \vec{B}\):

\[\left(\vec{E}^{\prime}+i \vec{B}^{\prime}\right) \cdot \vec{\sigma}=\overline{(\vec{E}+i \vec{B}) \cdot \vec{\sigma}}=(-\vec{E}+i \vec{B}) \cdot \vec{\sigma}\label{6}\]

This relation takes into account the fact that \(\vec{E}\) is a polar and \(\vec{B}\) an axial vector.