On the connection between the Wigner and the Bohm quantum formalism

In this paper, we discuss the connection between two alternative descriptions of the quantum motion proposed by Wigner in 1932 and by Bohm in 1952. Such formalism provides different representations of the quantum dynamics. The Wigner description is based on the definition of a non positive quasi distribution function in the quantum phase space and the Bohm formalism on the introduction of the pilot wave field associated to the particle motion along trajectories. We discuss the representation of the Bohm dynamics in the phase space in terms of a suitable Wigner measure. We propose an extension of the Wigner transformation and we derive a family of models where the quantum dynamics is described by equivalent mathematical formulations. Similarly to the classical limit, we show the asymptotic convergence in distributional sense of the extended Wigner distribution to a measure. We prove that such a limit is the solution of a classical Liouville equation containing as a quantum correction the Bohm potential