# Can collective excitations expand faster than the speed of light?

May 8, 2024

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One of the foundational principles of Einstein's theory of relativity is the postulate that information cannot travel faster than the speed of light. This statement, which to date has never been falsified, is called relativistic causality and is built into the mathematical structure of the standard model of particle physics.

While the practical (i.e., experimental) interpretation of relativistic causality is uncontroversial, it is often challenging to translate the general principle into a simple statement about the dynamics of a physical model. For example, it would be tempting to argue that because "information" is transported by particles, relativistic causality should imply that particles must travel in space no faster than the speed of light. However, this statement is not necessarily correct. In principle, a particle may travel faster than light as long as it does not carry any new information that was not already accessible before the detection. Indeed, the Hegerfeldt paradox in quantum field theory describes such a scenario: single-particle wave functions can expand at infinite speed. However, the information contained in this expansion was already accessible before the expansion occurred, being contained in field perturbations that extend outside the region of space covered by the wave function. In other words, single-particle states always carry non-local information, even if their wave function might seem to be localized.

In a recent paper published in *Physical Review Letters*, we generalize Hegerfeldt's result to all collective excitations in matter. Our main result, proven through rigorous mathematical theorems, may be summarized as follows: take an arbitrary chunk of matter in thermodynamic equilibrium and consider its spectrum of collective excitations. Pick a single excitation of interest (i.e., choose one element of the spectrum of collective modes of the system) and use it to build a wave packet. If the system is dispersive (i.e., if the group velocity, or the damping rate, depends on the wavenumber), this wave packet will expand at infinite speed. However, causality is not violated, and no information is propagated faster than the speed of light.

This result can be better understood through a concrete example. Suppose the material is a fluid, and the collective excitation of interest is a sound wave. In the absence of viscosity, the group velocity equals the speed of sound, which does not depend on the wavenumber. Hence, the sound wave indeed propagates at the speed of sound. Now, consider viscous effects. As a result, the wave acquires a damping rate, which depends on the wavenumber. Then, necessarily, the support of the sound-wave packet must expand faster than light. Naively, this result seems to violate relativistic causality, but that is not the case. In fact, we also prove that a single collective excitation can never be truly localized. Hence, like in the Hegerfeldt paradox, one cannot use this infinitely fast expansion to send information, and relativistic causality necessarily holds, as expected.

*Jorge Noronha's research is funded in part by he U.S. Department of Energy, Office of Science, Office for Nuclear Physics under Award No. DE- SC0023861. His co-authors are supported in part by Vanderbilt Seeding Success Grant (Lorenzo Gavassino) and by NSF Grant No. DMS-2107701, a Chancellor’s Faculty Fellowship, DOE Grant No. DE-SC0024711, and a Vanderbilt Seeding Success Grant (Marcelo Disconzi). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the funding agencies.*