# Causality and dielectric functions for linear media with spatial dispersion

## TL;DR

This paper extends Kramers-Kronig relations to dielectric functions dependent on both frequency and wave number, incorporating causality constraints related to signal propagation speed, with applications to microscopic models and linear response theories.

## Contribution

It introduces a generalized form of Kramers-Kronig relations accounting for spatial dispersion and causality constraints in dielectric functions.

## Key findings

- Extended Kramers-Kronig relations for spatially dispersive media.
- Comparison with previous generalizations of dielectric response.
- Applicability to microscopic models and isotropic linear response theories.

## Abstract

We extend Kramers-Kronig relations beyond the optical approximation to dielectric functions that depend not only on frequency but on the wave number as well. This implies extending the notion of causality commonly used in the theory of Kramers-Kronig relations to include the fact that signals cannot propagate faster than light in vacuo. The extension is applied to some microscopic models for the dielectric function and is compared with previous generalizations. The results derived here also apply to general theories of isotropic linear response in which the response function depends on both wave number and frequency.