Generalized continuity equations from two-field Schr\"odinger Lagrangians

TL;DR

This paper develops a variational framework using two-field Lagrangians to derive generalized continuity equations in quantum systems, unifying Hermitian, non-Hermitian, and $ ext{PT}$ symmetric cases.

Contribution

It introduces a novel Lagrangian approach involving two complex wave fields to generate symmetry-induced continuity equations and bilocal conservation laws in quantum mechanics.

Findings

Derives mixed continuity equations from a two-field Lagrangian.

Shows bilocal conservation laws arise from discrete symmetries.

Reproduces $ ext{PT}$ symmetric quantum mechanics results.

Abstract

A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary currents, and to divergenceless bilocal currents for stationary states. The formalism reproduces the bilocal continuity equation obtained in the special case of $\mathcal{PT}$ symmetric quantum mechanics and paraxial optics.