Dynamics of the electric current in an ideal electron gas: a sound mode inside the quasi-particles

We study the equation of motion for the Noether current in an electron gas within the framework of the Schwinger-Keldysh Closed-Time-Path formalism. The equation is shown to be highly non-linear and irreversible even for a non-interacting, ideal gas of electrons at non-zero density. We truncate the linearised equation of motion, written as the Laurent series in Fourier space, so that the resulting expressions are local in time, both at zero and at small finite temperatures. Furthermore, we show that the one-loop Coulomb interactions only alter the physical picture quantitatively, while preserving the characteristics of the dynamics that the electric current exhibits in the absence of interactions. As a result of the composite nature of the Noether current, {\it composite sound} waves are found to be the dominant IR collective excitations at length scales between the inverse Fermi momentum and the mean free path that would exist in an interacting electron gas. We also discuss the difference and the transition between the hydrodynamical regime of an ideal gas, defined in this work, and the hydrodynamical regime in phenomenological hydrodynamics, which is normally used for the description of interacting gases.