Schrodinger's Equation with Gauge Coupling Derived from a Continuity Equation

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density rho, and a scalar field S which defines a probability current j=rho a double dagger S/m. A first equation for rho and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable chi. Using these assumptions and the simplest possible Ansatz chi(rho,S), for the relation between chi and rho,S, Schrodinger's equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz chi(rho,S,q,t), which allows for an explicit q,t-dependence of chi, one obtains a generalized Schrodinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrodinger' equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of chi, one obtains Schrodinger's equation with electrodynamic potentials A,phi in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out.