Editing Dielectric (section)

===Dispersion and causality===
In general, a material cannot polarise instantaneously in response to an applied field. The more general formulation as a function of time is

<math display="block">\mathbf{P}(t) = \varepsilon_0 \int_{-\infty}^t \chi_e\left(t - t'\right) \mathbf{E}(t')\, dt'.</math>

That is, the polarisation is a [[convolution]] of the electric field at previous times with time-dependent susceptibility given by <math>\chi_e (\Delta t)</math>. The upper limit of this integral can be extended to infinity as well if one defines <math>\chi_e (\Delta t) = 0</math> for <math>\Delta t < 0</math>. An instantaneous response corresponds to [[Dirac delta function]] susceptibility <math>\chi_e (\Delta t) = \chi_e \delta (\Delta t)</math> .

It is more convenient in a linear system to take the [[continuous Fourier transform|Fourier transform]] and write this relationship as a function of frequency. Due to the [[convolution theorem]], the integral becomes a simple product,
<math display="block">\mathbf{P}(\omega) = \varepsilon_0 \chi_e(\omega) \mathbf{E}(\omega).</math>

The susceptibility (or equivalently the permittivity) is frequency dependent. The change of susceptibility with respect to frequency characterises the [[dispersion (optics)|dispersion]] properties of the material.

Moreover, the fact that the polarisation can only depend on the electric field at previous times (i.e., <math>\chi_e (\Delta t) = 0</math> for <math>\Delta t < 0</math>), a consequence of [[causality]], imposes [[Kramers–Kronig relation|Kramers–Kronig constraints]] on the real and imaginary parts of the susceptibility <math>\chi_e (\omega)</math>.