6.4: On the us of Involutions
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 A2,
A−1=˜A|A||A|=12Tr(A˜A)
In the case of Hermitian matrices we have two alternatives:
k0r0−→k⋅→r=12Tr(K˜R)
or
k0r0−→k⋅→r=12Tr(KˉR)
It will appear, however from later discussions, that the complex reflection of Equation ??? 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 ˆx1. We have
K′=σ1ˉKσ1=σ1(k01−k1σ1−k2σ2−k3σ3)σ1=σ21(k01−k1σ1+k2σ2+k3σ3)=k01−k1σ1+k2σ2+k3σ3
More generally the mirroring in a plane with normal x is achieve by means of the operation
K′=ˆa⋅→σˉKˆa⋅→σ
Again, we could have chosen ˜K instead of ˉK.
However, Eq (22) generalizes to the inversion of the electromagnetic six-vector →f=→E+i→B:
(→E′+i→B′)⋅→σ=¯(→E+i→B)⋅→σ=(−→E+i→B)⋅→σ
This relation takes into account the fact that →E is a polar and →B an axial vector.