Schrödinger’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 ρ, and a scalar field S which defines a probability current ~j = ρ∇S/m. A first equation for ρ 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 χ. Using these assumptions and the simplest possible Ansatz χ(ρ, S), for the relation between χ and ρ, S, Schrödinger’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 χ(ρ, S, q, t), which allows for an explicit q, t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ 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 nonunique external q, t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials ~ A, φ 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.