Translation-invariant relativistic Langevin equation derived from first principles

Abstract: The relativistic Langevin equation poses a number of technical and conceptual problems related to its derivation and underlying physical assumptions. Recently, a method has been proposed in [A. Petrosyan and A. Zaccone, J. Phys. A: Math. Theor. 55 015001 (2022)] to derive the relativistic Langevin equation from a first-principles particle-bath Lagrangian. As a result of the particle-bath coupling, a new ``restoring force'' term appeared, which breaks translation symmetry. Here we revisit this problem aiming at deriving a fully translation-invariant relativistic Langevin equation. We successfully do this by adopting the renormalization potential protocol originally suggested by Caldeira and Leggett. The relativistic renormalization potential is derived here and shown to reduce to Caldeira and Leggett's form in the non-relativistic limit. The introduction of this renormalization potential successfully removes the restoring force and a fully translation-invariant relativistic Langevin equation is derived for the first time. The physically necessary character of the renormalization potential is discussed in analogy with non-relativistic systems, where it emerges due to the renormalization of the tagged particle dynamics due to its interaction with the bath oscillators (a phenomenon akin to level-repulsion or avoided-crossing in condensed matter). We discuss the properties that the corresponding non-Markovian friction kernel has to satisfy, with implications ranging from transport models of the quark-gluon plasma, to relativistic viscous hydrodynamic simulations, and to electrons in graphene.