# A media in which the electrical displacement vector is not causal

I recently did an electrodynamics homework problem in which we showed that in a certain model (Lorentz-Drude), where the permittivity of free space $\epsilon$ was dependent upon the angular frequency of the electromagnetic waves travelling through it, it is possible to show that the electrical displacement vector $\mathbf{D}$ only depends upon the values of the electric field $\mathbf{E}$ in the past and present i.e. for $t' \leq t$.

During the calculation it became evident that the angular dependence of the media was the reason for the causal relationship between $\mathbf{D}$ and $\mathbf{E}$, in light of this does there exist some (perhaps fictitious) media in which "future" values of the electric field affect the electrical displacement vector? If there do exist such media, then what are the experimental consequences? Intuitively it seems like this cannot be, since this would imply that there are media in which perfect knowledge of the electric field as a function of time would not correctly predict the electrical displacement vector at a given time.

## 1 Answer

The causality of the response, along with the fact that it would never introduce imaginary-valued D as a response to real-valued E, stipulates that the spectra of permittivity conform to the Kramers-Kronig relations.

These form a very tight relation between the real and imaginary parts of permittivity. In particular, each part is a Hilbert transform of the other one multiplied by i. This can be relatively easily proven even without the use of complex integration (see, e.g., https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations#/media/File:KramersKronig.svg).

With broad enough spectral data, the K-K relations enable one to deduce one part from another one, so with some limitations one can measure spectra of refractive index from the material's absorption.

Removing the requirement of causality enables you to obtain virtually any shape of the permittivity.