Dispersion relations alone cannot guarantee causality

TL;DR

Dispersion relations alone do not ensure causality; causality emerges only from superpositions of modes in stable local theories, highlighting the importance of mode interactions in relativistic physics.

Contribution

This paper demonstrates that individual dispersion relations cannot guarantee causality and that causality arises from superimposing all excitation branches in stable local theories.

Findings

Single dispersion relations allow superluminal propagation.

Causality depends on mode superpositions, not individual relations.

Local covariantly stable theories automatically ensure causality through mode cancellation.

Abstract

We show that linear superpositions of plane waves involving a single-valued, covariantly stable dispersion relation $\omega(k)$ always propagate outside the lightcone, unless $\omega(k) =a+b k$. This implies that there is no notion of causality for individual dispersion relations, since no mathematical condition on the function $\omega(k)$ (such as the front velocity or the asymptotic group velocity conditions) can serve as a sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emerge from a careful cancellation that occurs when one superimposes all the excitation branches of a physical model. This is shown to happen automatically in local theories of matter that are covariantly stable. Hence, we find that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need for antiparticles in relativistic quantum mechanics.