Symmetries and conservation laws in the Lagrangian picture of quantum hydrodynamics

TL;DR

This paper explores the symmetries and conservation laws in the Lagrangian formulation of quantum hydrodynamics, revealing new quantum symmetries and their relation to classical invariances through Noether's theorem.

Contribution

It introduces the relabelling symmetry in the Lagrangian quantum hydrodynamics and connects it with known Eulerian symmetries, expanding understanding of quantum symmetries.

Findings

Derived relabelling symmetry as a quantum conservation law.

Connected Lagrangian and Eulerian symmetry structures.

Showed superposition symmetry as a label transformation.

Abstract

Quantum hydrodynamics represents quantum mechanics through two complementary models: the Eulerian picture, a direct transcription of wave mechanics, and the Lagrangian picture, in which the quantum state is represented by the collective motion of a continuum of fluid particles, the latter being obtained by continuously varying the particle label. The Lagrangian picture thus adds variables to the quantum formalism and exhibits a corresponding new quantum symmetry viz. a continuous particle-relabelling covariance group. Using Noether's theorem, the relabelling symmetry is derived as a component of a general investigation of symmetries and conservation laws in the quantum Lagrangian picture. Relations with symmetries and conservation laws in the Eulerian picture are explored in detail. Alongside the infinite relabelling group, the 12-parameter kinematical covariance group of the Schr\"odinger equation is derived. It is stressed that the role of label transformations extends beyond the class corresponding to Eulerian invariance. As an example, it is shown that the linear superposition of waves, a fundamental symmetry of the quantum Eulerian picture, can be generated by a deformation-dependent label transformation in the Lagrangian picture.