Tensors for Physics

260 14 Rotation of Tensors

The rotation by an finite angleφ=nδφis given by( 1 +δφH)n. Withδφ=φ/n,
the limitn→∞leads to

a′μ=(exp[φH])μνaν≡Rμν(φ)aν. (14.3)

In principal, the rotation tensorRcan be expressed in terms of the power series

Rμν(φ)=δμν+φHμν+

1

2

φ^2 HμκHκν+.... (14.4)

Due to the special properties ofH, to be discussed next,Rcan be represented in a
more compact form.

14.1.2 Hamilton Cayley and Projection Tensors

Due toHμκHκν=hμhν−δμνandhσHσν=0, the tensorHobeys the relation

H^3 +H= 0. (14.5)

This corresponds to a Hamilton-Cayley equation forHwith the eigenvaluesim,
wherem=0,±1. Second rank projection tensorsP(m)are defined by

P(m)=

∏

m′=m

H−im′ 1

im−im′

, m,m′= 0 ,± 1. (14.6)

In (14.6), the symbol 1 stands for the second rank unit tensor, viz. forδμν. These
projectors are explicitly given by

Pμν(^0 )=hμhν, Pμν(±^1 )=

1

2

(δμν−hμhν∓iεμλνhλ). (14.7)

The projection tensors possess the following properties:

Pμκ(m)P(m

′)

κν =δmm′P

(m)

μν, (14.8)

(Pμν(m))∗=Pμν(−m)=Pνμ(m), (14.9)

∑^1

m=− 1

Pμν(m)=δμν, Pμμ(m)= 1. (14.10)

The eigenvalue equation

Pμκ(m)Hκν=HμκPκν(m)=imPμν(m), (14.11)