and the transverse (divergence-free) component must be defined for a given gauge, g. For example, in the Coulomb gauge, PC ⊥ = 0 and in the multi-polar gauge (the dipole gauge beyond the long-wavelength approximation), Pmp ⊥ = j q j rj 1 0 ds δ ⊥ (x -sr j ). To transform between gauges, a more generalized gauge transformation in terms of the polarization fields is needed.