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

A reconstruction of quantum theory for nonspinning particles