  The Madelung transformation of the space in which a quantum wave function
takes its values is generalized from complex numbers to include field spaces
that contain orbits of groups that are diffeomorphic to spheres. The general
form for the resulting real wave equations then involves structure constants
for the matrix algebra that is associated with the group action. The particular
cases of the algebras of complex numbers, quaternions, and complex quaternions,
which pertain to the Klein-Gordon equation, the relativistic Pauli equation,
and the bi-Dirac equation, resp., are then discussed.
