Quaternionic Madelung transformation and non-Abelian fluid dynamics

We find the quaternionic generalization of the Madelung transformation of the Schroedinger equation. The non-Abelian nature of the full SU(2) gauge group of the quaternionic Schroedinger equation leads to a more complex set of equations for the fluid variables. We identify the “hydrodynamic” variables for the quaternionic Madelung transformation. In order to find equations of motion for these quantities we cast the Madelung transformation in the form of a non-canonical transformation of the classical Hamiltonian field theory for the quaternionic Schroedinger equation. We first define the canonical bracket and Hamiltonian for the quaternionic Schroedinger equation and then obtain the non-canonical brackets yielding the equations of motion for the quaternionic Madelung variables. These equations 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 (hep-th/0210143, 2002). Obtaining these equations by a transformation of the quaternionic Schroedinger equation makes their simulation by a variety of methods particularly simple.