Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion and conservation laws

TL;DR

This paper develops equations of motion and conservation laws for quantum many-body systems in a co-moving Lagrangian frame, linking them to a space with a time-dependent metric and effective magnetic field, with applications to harmonic potentials and Fermi systems.

Contribution

It introduces a formalism for quantum many-body dynamics in a Lagrangian frame using a time-dependent metric and vorticity tensor, and applies it to harmonic potentials and Fermi gases.

Findings

Equations of motion are equivalent to a quantum system in a space with a dynamic metric and magnetic field.

Explicit solution of the kinetic equation for a Fermi system in the Hartree approximation.

Formulation of a nonlinear elasticity theory for Fermi gases.

Abstract

We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor $g_{\mu\nu}(\bm\xi,t)$ and a skew-symmetric vorticity tensor $\widetilde{F}_{\mu\nu}(\bm\xi,t)$, where $\bm\xi$ in the Lagrangian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with time-dependent metric $g_{\mu\nu}(\bm\xi,t)$ in the presence of an effective magnetic field $\widetilde{F}_{\mu\nu}(\bm\xi,t)$. To illustrate the general formalism we apply it to the proof of the harmonic potential theorem. As another example of application we consider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In this case the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate the description of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of our results to time-dependent density functional theory.