Quaternionic Madelung Transformation and Non-Abelian Fluid Dynamics

TL;DR

This paper extends Madelung's transformation to quaternionic quantum mechanics, deriving non-abelian fluid equations that facilitate new simulation algorithms for non-abelian fluids.

Contribution

It introduces a quaternionic Madelung transformation and derives Hamiltonian equations for non-abelian fluid dynamics from quaternionic Schrödinger equations.

Findings

Derived non-abelian fluid equations from quaternionic Schrödinger equation.

Developed Hamiltonian and Poisson brackets for quaternionic quantum mechanics.

Proposed algorithms for simulating non-abelian fluids based on this framework.

Abstract

In the 1920's, Madelung noticed that if the complex Schroedinger wavefunction is expressed in polar form, then its modulus squared and the gradient of its phase may be interpreted as the hydrodynamic density and velocity, respectively, of a compressible fluid. In this paper, we generalize Madelung's transformation to the quaternionic Schroedinger equation. The non-abelian nature of the full SU(2) gauge group of this equation leads to a richer, more intricate set of fluid equations than those arising from complex quantum mechanics. We begin by describing the quaternionic version of Madelung's transformation, and identifying its ``hydrodynamic'' variables. In order to find Hamiltonian equations of motion for these, we first develop the canonical Poisson bracket and Hamiltonian for the quaternionic Schroedinger equation, and then apply Madelung's transformation to derive non-canonical Poisson brackets yielding the desired equations of motion. These are a particularly natural set of equations for a non-abelian fluid, and differ from those obtained by Bistrovic et al. only by a global gauge transformation. Because we have obtained these equations by a transformation of the quaternionic Schroedinger equation, and because many techniques for simulating complex quantum mechanics generalize straightforwardly to the quaternionic case, our observation leads to simple algorithms for the computer simulation of non-abelian fluids.