How acausal equations emerge from causal dynamics

TL;DR

This paper constructs a causal kinetic model that can reproduce any stable dispersion relation at real wavenumbers, demonstrating that acausal equations can emerge from causal microscopic dynamics.

Contribution

It provides a counterexample showing microscopic causality does not uniquely determine the form of dispersion relations at real wavenumbers.

Findings

The model reproduces arbitrary stable dispersion relations.

Macroscopic observables can follow acausal-looking equations.

Causality at microscopic level does not constrain the analytic form of dispersion relations.

Abstract

We construct a causal and covariantly stable kinetic model whose spectrum at real wavenumbers $k$ reproduces any rest-frame stable dissipative dispersion relation $\omega(k)$ via suitable initialization of the microscopic degrees of freedom. Macroscopic observables can therefore obey arbitrary linear evolution equations (including forms that would be acausal if taken as fundamental), while the underlying dynamics remains causal, and all apparent propagation is encoded in the initial data. This provides an explicit counterexample to the idea that microscopic causality alone constrains the analytic form of dispersion relations at real $k$. In particular, bounds on transport coefficients based solely on the analytic structure of $\omega(k)$, such as the hydrohedron bounds, require additional assumptions about the region in the complex $k$-plane where $\omega(k)$ corresponds to physical modes.