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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,

A1=˜A|A||A|=12Tr(A˜A)

In the case of Hermitian matrices we have two alternatives:

k0r0kr=12Tr(K˜R)

or

k0r0kr=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(k01k1σ1k2σ2k3σ3)σ1=σ21(k01k1σ1+k2σ2+k3σ3)=k01k1σ1+k2σ2+k3σ3

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

K=ˆaσˉKˆaσ

Again, we could have chosen ˜K instead of ˉK.

However, Eq (22) generalizes to the inversion of the electromagnetic six-vector f=E+iB:

(E+iB)σ=¯(E+iB)σ=(E+iB)σ

This relation takes into account the fact that E is a polar and B an axial vector.


This page titled 6.4: On the us of Involutions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by László Tisza (MIT OpenCourseWare) .

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