Madelung Structure of the Dirac Equation

We consider the Dirac equations in polar form proving that they can equivalently be re-configured into a system of equations consisting of derivatives of the velocity density plus the Hamilton-Jacobi equation, giving the momentum in terms of relativistic quantum potentials (i.e. displaying first-order derivatives of the two degrees of freedom of the spinor field): this system is said to have Madelung structure. Conservation laws, second-order equations and multi-valuedness are also discussed.