2.6 Parity 27
2.6.3 Consequences for Linear Relations
The electromagnetic interaction underlying all relevant interactions encountered in
every days life, i.e. in gases, liquids and solids, is invariant under the parity operation.
The equations governing physical properties and phenomena must not violate this
parity invariance. This means, for example, when the vectorbin the relationbμ=
Cμνaνhas the parity−1 (polar vector), the vectoraand the tensorCmust have the
parities−1 and 1 (polar vector and proper tensor) or 1 and−1 (axial vector and
pseudo tensor). More general, letPa,Pb,PCthe values of the parities of the tensors
a,b,Cin the linear relation (2.51). Parity invariance requires
Pb=PCPa. (2.57)Likewise, when the parities ofaandbare given by their physical meaning, the
coefficient tensorCmust have the parity
PC=PaPb, (2.58)in order that the linear relation (2.51) does not violate parity.
2.6.4 Application: Linear and Nonlinear Susceptibility Tensors
The electric fieldE, the electric displacement fieldDand the electric polarizationP
used inelectrodynamicsare polar vectors. They are linked by the general relation
D=ε 0 E+P,whereε 0 is the electric permeability coefficient of the vacuum. In a material, called
linear medium, the electric polarization is linearly related to the electric field, accord-
ing to
Pμ=ε 0 χμνEν,whereχμνis thelinear susceptibility tensor. In the special case of a linear medium,
one has
Dμ=ε 0 εμνEν,εμν=ε 0 (δμν+χμν),with the dimensionless dielectric tensorεμν. In general, in particular for strong
electric fields as, e.g. encountered in a (focussed) laser beam, terms nonlinear in the
electric field give significant contributions to the electric polarization. Up to third
order in the electric field, the electric polarization is given by