  The existence of precise particle trajectories in any quantum state is
accounted for in a consistent way by allowing delocalization of the particle
charge. The relativistic mass of the particle remains within a small volume
surrounding a singularity moving along the particle trajectory. The singularity
is the source of an electric displacement field. The field induces a
polarization charge in the vacuum and this charge is equated with the charge of
the particle. Under dynamic conditions a distributed charge density rho(x,t) is
induced in the vacuum. The volume integral of the charge density is equal to
the charge of the particle and is rigorously conserved. The charge density is
derived from a complex-valued physical field psi(x,t) such that rho(x,t) =
|psi(x,t)|^2. The position probability density is equated with the mean charge
density. The mean field psi(x,t) for many sample realizations with a given
energy E and potential V(x) is the sum of the individual fields. In order for
the sum to be non-zero, the components in the spectral decompositions of the
individual fields must be spatially coherent. The particle has a spin frequency
given by Planck's relation hv = T - V + mc^2, where T is the kinetic energy,
determined from the momentum p and V is a quantum potential such that E = T + V
is conserved. The instantaneous phase of the spin is given by the phase of
exp(ikx) in the spectral decomposition a(k) of psi(x,t). It is spatially
coherent, due to the dependence on x. The momentum probability distribution is
given by the squared magnitude of the coefficients a(k). The Schrodinger
equation is derived by requiring local conservation of mean energy.
