  This work is based on the idea that the classical counterpart of quantum
theory (QT) is not mechanics but probabilistic mechanics. We therefore choose
the theory of statistical ensembles in phase space as starting point for a
reconstruction of QT. These ensembles are described by a probability density
$\rho (q, p, t)$ and an action variable $S (q, p, t)$. Since the state
variables of QT only depend on $q$ and $t$, our first step is to carry out a
projection $p \Rightarrow M (q, t)$ from phase space to configuration space. We
next show that instead of the momentum components $M_{k}$ one must introduce
suitable potentials as dynamic variables. The quasi-quantal theory resulting
from the projection is only locally valid. To correct this failure, we have to
perform as a second step a linearization or randomization, which ultimately
leads to QT. In this work we represent $M$ as an irrotational field, where all
components $M_{k}$ may be derived from a single function $S (q, t)$. We obtain
the usual Schr\"odinger equation for a nonspinning particle. However, space is
three-dimensional and $M$ must be described by $3$ independent functions. In
the fourth work of this series, a complete representation of $M$ will be used,
which explains the origin of spin. We discuss several fundamental questions
that do not depend on the form of $M$ and compare our theory with other recent
reconstructions of QT.
