Tensors for Physics

14.3 Rotation of Tensors of Rank 265

14.3 Rotation of Tensors of Rank‘.......................

The obvious generalization of the generator for the infinitesimal rotation of second
rank tensors, viz. (14.24), to tensors of rank,isthe2th rank tensorH()defined by

Hμ() 1 μ 2 ···μ(),ν 1 ν 2 ···ν (14.36)

≡εμ 1 λν 1 hλδμ 2 ν 2 ···δμν+...+δμ 1 ν 1 ···δμ− 1 ν− 1 εμλνhλ

=Hμ 1 ν 1 δμ 2 ν 2 ···δμν+δμ 1 ν 1 Hμ 2 ν 2 ···δμν+...+δμ 1 ν 1 ···δμ− 1 ν− 1 Hμν.

The infinitesimal rotation of a symmetric tensorSis also described byH(),viz.

Sμ′ 1 μ 2 ···μ=Sμ 1 μ 2 ···μ+δφHμ() 1 μ 2 ···μ(),ν 1 ν 2 ···νSν 1 ν 2 ···ν. (14.37)

In the following, it is assumed thatSis also traceless, i.e. it is an irreducible tensor.
The rotated tensorS′is also irreducible. Then one has

H() S=H() S,

where the symbol indicates the-fold contraction as occurring in the equations
above, andH()is equivalent to

Hμ() 1 μ 2 ···μ(),ν 1 ν 2 ···ν=hλ()μ 1 μ 2 ···μ(),λ,ν 1 ν 2 ···ν, (14.38)

for...see (11.16).
Projection tensorsP(m), wherem=−,−+ 1 ,..., 0 ,...,− 1 ,, are defined
by analogy to (14.31). These projectors are eigen-tensors ofH(),viz.

H() P(m)=P(m) H()=imP(m). (14.39)

In terms of these projectors, the spectral decomposition ofH()reads

Hμ() 1 μ 2 ···μ(),ν 1 ν 2 ···ν=

∑

m=−

imPμ(m 1 μ) 2 ···μ(),ν 1 ν 2 ···ν. (14.40)

The obvious generalizations of equations (14.34) and (14.35) describing the rotation
of second rank tensors to those ofth rank tensors is

R()μ 1 μ 2 ···μ(),ν 1 ν 2 ···ν(φ)=

∑

m=−

exp[im]Pμ(m 1 μ) 2 ···μ(),ν 1 ν 2 ···ν, (14.41)