Fluid Lagrangian approach to the classical-quantum transition

With a two-field Lagrangian of the Hamilton-Jacobi equation, a coupling between the two fields can be introduced to yield an appropriate quantum Lagrangian. The quantum effect is nothing more than a quantum pressure force acting on the otherwise classical pressureless fluid when one of the fields is properly renormalized. Inclusion of the electromagnetic coupling under the present formulation shows that the collisionless quantum fluid picture remains valid. Similar, but not identical, to the classical collisionless fluids, the flow circulation and magnetic flux can remain locked together even for a spatially and temporally varying magnetic field. Extensions beyond the present scope of quantum formulation are also discussed.