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arXiv:1211.0983

Symmetries and conservation laws in the Lagrangian picture of quantum hydrodynamics

Peter Holland

Published Nov 5, 2012• 1 citation
arXivHTMLPDF

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 Schrödinger 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.

DOI
10.48550/arxiv.1211.0983

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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 Schrödinger 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.
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We discuss the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler-Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie-Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie-Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.
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The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ħ $\hbar $ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ħ ≠ 0 ℏ≠0\hbar \ne 0ℏ=0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.