5.4: On the us of Involutions
- Page ID
- 41021
The existence of the three involutions ( see Equations A.1.1 above), provides a great deal of flexil bity. However, the most efficient use of these concepts calls for some care.
For any matrix of \(\mathcal{A}_{2}\)
\[\begin{array}{c} {A^{-1} = \frac{\tilde{A}}{|A|} |A| = \frac{1}{2} Tr(A \tilde{A})} \end{array}\]
In the case of Hermitian matrices we have two alternatives:
\[\begin{array}{c} {k_{0}r_{0}-\vec{k} \cdot \vec{r} = \frac{1}{2} Tr(K \tilde{R})} \end{array}\]
or
\[\begin{array}{c} {k_{0}r_{0}-\vec{k} \cdot \vec{r} = \frac{1}{2} Tr(K \bar{R})} \end{array}\]
It will appear, however from later discussions, that the complex reflection of Equation A.4.3 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
\[\begin{array}{c} {K' = \sigma_{1} \bar{K} \sigma_{1} = \sigma_{1} (k_{0}1-k_{1} \sigma_{1}-k_{2} \sigma_{2}-k_{3} \sigma_{3}) \sigma_{1}}\\ {= \sigma_{1}^{2} (k_{0}1-k_{1} \sigma_{1}+k_{2} \sigma_{2}+k_{3} \sigma_{3})}\\ {= k_{0}1-k_{1} \sigma_{1}+k_{2} \sigma_{2}+k_{3} \sigma_{3}} \end{array}\]
More generally the mirroring in a plane with normal x is achieve by means of the operation
\[\begin{array}{c} {K' = \hat{a} \cdot \vec{\sigma} \bar{K} \hat{a} \cdot \vec{\sigma}} \end{array}\]
Again, we could have chosen \(\tilde{K}\) instead of \(\bar{K}\).
However, Eq (22) generalizes to the inversion of the electromagnetic six-vector \(\vec{f} = \vec{E}+i \vec{B}\):
\[\begin{array}{c} {(\vec{E}'+i \vec{B}') \cdot \vec{\sigma} = \overline{(\vec{E}+i \vec{B}) \cdot \vec{\sigma}} = (-\vec{E}+i \vec{B}) \cdot \vec{\sigma}} \end{array}\]
This relation takes into account the fact that \(\vec{E}\) is a polar and \(\vec{B}\) an axial vector.