268 14 Rotation of Tensors
with the longitudinal, perpendicular and transverse conductivity coefficients
determined by
σ‖=σ 0 ≡ne^2 τ/m,σ⊥=σ 0 ( 1 +φ^2 )−^1 ,σtrans=σ 0 φ( 1 +φ^2 )−^1 .(14.53)The magnetic field is an axial vector, the same applies forh. The constitutive law
(14.48), with the conductivity tensor given here conserves parity. Notice that the
conductivity tensor has the symmetry property
σμν(h)=σνμ(−h). (14.54)Sinceτ>0, one hasσ‖>0 andσ⊥>0. The relation of the associated longitudinal
and perpendicular parts of the current density with the electric field violate time-
reversal invariance, typical for an irreversible process. The transverse coefficient
which underlies the Hall-effect, is of reversible character, the coefficientσtransmay
have either sign.
14.5 Additional Formulas Involving Projectors
The application of the fourth rank projection tensor on the symmetric traceless tensor
aμνis explicitly given by
Pμν,μ(^0 ) ′ν′aμ′ν′=3
2
hμhνhμ′hν′aμ′ν′, (14.55)Pμν,μ(±^1 )′ν′aμ′ν′=1
2
(hμhκaκν+hνhκaκμ)−hμhνaμ′ν′hμ′hν′∓i
2(hμHντaτκhκ+hνHμτaτκhκ),Pμν,μ(±^2 )′ν′aμ′ν′=1
2
aμν−hμhκaκν+1
4
hμhνhμ′hν′aμ′ν′ (14.56)∓
i
4(Hμτaτν−hμHντaτκhκ+Hντaτμ−hνHμτaτκhκ).Multiplicationoftheexpressionsabovebythesymmetrictracelesstensorbμνleadsto
bμνPμν,μ(^0 ) ′ν′aμ′ν′=
3
2
(hμbμνhν)(hμ′aμ′ν′hν′), (14.57)bμνPμν,μ(±^1 )′ν′aμ′ν′=hμbμνaνκhκ−(hμbμνhν)(hμ′aμ′ν′hν′)∓ihμbμνHντaτκhκ,