6.4: On the us of Involutions
- Page ID
- 31981
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.