Generalized Madelung transformations for quantum wave equations I: generalized spherical coordinates for field spaces

TL;DR

This paper extends the Madelung transformation to more complex field spaces involving spherical group orbits, leading to new real wave equations linked to various algebraic structures and quantum equations.

Contribution

It introduces a generalized Madelung transformation for field spaces with spherical group orbits, unifying several quantum wave equations under a common framework.

Findings

Derived real wave equations involving structure constants of associated algebras.

Connected algebraic structures to specific quantum equations like Klein-Gordon and Dirac.

Provided a unified geometric approach to different quantum wave equations.

Abstract

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.