6.4: On the us of Involutions

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:

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

or

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

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.