Relativistic hydrodynamic fluctuations from an effective action: causality, stability, and the information current

TL;DR

This paper develops a covariant effective action framework for relativistic hydrodynamic fluctuations, ensuring causality and stability, and introduces a new symmetry to connect different formalisms and improve understanding of fluctuation dynamics.

Contribution

It formulates a covariant effective action for relativistic hydrodynamics incorporating causality, stability, and a novel symmetry, linking Schwinger-Keldysh and Martin-Siggia-Rose approaches.

Findings

Correlation functions exhibit physical properties within the valid regime.

A new $ ext{Z}_2$ symmetry analogous to KMS symmetry is identified.

The effective action framework clarifies the connection between different fluctuation formalisms.

Abstract

Causality is necessary for retarded Green's functions to remain retarded in all inertial frames in relativity, which ensures that dissipation of fluctuations is a Lorentz invariant concept. For first-order BDNK theories with stochastic fluctuations, introduced via the Schwinger-Keldysh formalism, we show that imposing causality and stability leads to correlation functions of hydrodynamic fluctuations that only display the expected physical properties at small frequencies and wavenumber, i.e., within the expected regime of validity of the first-order approach. For second-order theories of Israel and Stewart type, constructed using the information current such that entropy production is always non-negative, a stochastic formulation is presented using the Martin-Siggia-Rose approach where imposing causality and stability leads to correlators with the desired properties. We also show how Green's functions can be determined from such an action. We identify a $\mathbb{Z}_2$ symmetry, analogous to the Kubo-Martin-Schwinger symmetry, under which this Martin-Siggia-Rose action is invariant. This modified Kubo-Martin-Schwinger symmetry provides a new guide for the effective action formulation of hydrodynamic systems with dynamics not solely governed by conservation laws. Furthermore, this symmetry ensures that the principle of detailed balance is valid in a covariant manner. We employ the new symmetry to further clarify the connection between the Schwinger-Keldysh and Martin-Siggia-Rose approaches, establishing a precise link between these descriptions in second-order theories of relativistic hydrodynamics. Finally, the modified Kubo-Martin-Schwinger symmetry is used to determine the corresponding action describing diffusion in Israel-Stewart theories in a general hydrodynamic frame.