Formation of Singularirties in Madelung fluid : a nonconventional application of Ito calculus to foundations of quantum Mechanics

Stochastic Quantization is a procedure which provides the equation of motion of a Quantum System starting from its classical description and incorporating quantum effects into a stochastic kinematics. After the pioneering work by E.Nelson in 1966 the method has been developed in the eighties in various different ways. In this communication I revisit an approach based on a lagrangian variational principle were 3/2 order contributions in It\^o calculus are needed. The result is a generalization of Madelung fluid equations where in particular velocity fields with vorticity are allowed. Such a vorticity induces dissipation of the energy so that the irrotational solutions, corresponding to the usual conservative solutions of Schroedinger equation, act as an attracting set.Recent numerical experiments show generation of zeroes of the density with concentration of vorticity and formation of isolated central vortex lines.