Fluid Lagrangian approach to the classical-quantum transition

With a two-field Lagrangian of the Hamilton-Jacobi equation, a coupling bet ween the two fields can be introduced to yield an appropriate quantum Lagra ngian. The quantum effect is nothing more than a quantum pressure force act ing 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 rem ains valid. Similar, but not identical, to the classical collisionless flui ds, the flow circulation and magnetic but can remain locked together even f or a spatially and temporally varying magnetic field. Extensions beyond the present scope of quantum formulation are also discussed.