Relativistic dispersion spectra across Lorentz boosted frames: spurious modes and the enigma of causality

The analysis of linearized perturbations in relativistic field theories involving derivative corrections frequently leads to pathologies that, although generic, may not always be manifest, like in a local rest frame scenario. Hence, analyzing such theories conclusively in a more general state of equilibrium, like in a Lorentz boosted inertial frame, is imperative, albeit extracting the boosted modes could be quite nontrivial. Motivated by this problem, in this study, we develop a general framework for deriving the dispersion spectra in Lorentz-boosted inertial frames solely from the information of the local rest frame dispersion coefficients. Apart from the well-behaved modes, which are observed to follow an exact mapping across the boost, we show that in some cases, unphysical ``spurious" modes might turn up, which diverge at zero boost limit and thus lack any rest-frame analogue. Finally, we investigate the conflict between the existence of these spurious modes and the causality for a given relativistic theory. The key developments provided here are (i) a convenient way for obtaining the dispersion spectra across inertial frames, bypassing the traditional method of solving the boosted polynomial, and (ii) exploring the direct connection between the mode conservation and the causality of a theory with detailed proof. Due to the general approach, these results can find direct applications for a range of other effective theory formulations, like gauge theory plasmas, strongly coupled systems, and heavy quark effective theory.