Inverse kinetic theory for quantum hydrodynamic equations

A remarkable feature of standard quantum mechanics is its analogy with classical fluid dynamics. This has motivated in the past efforts to formulate phase-space techniques based on various statistical models of quantum hydrodynamic equations. In this work an inverse kinetic theory for the Schrödinger equation has been constructed in order to formally describe the standard quantum dynamics by means of a classical dynamical system (to be denoted as phase-space Schrödinger dynamical system). It is shown that the inverse kinetic theory can be (non)uniquely determined under suitable mathematical prescriptions. In particular, when the quantum linear momentum is identified with a suitable linear kinetic momentum, it follows that the fluctuations of the position vector and the kinetic linear momentum satisfy identically the Heisenberg theorem.