Dirac Fields in Hydrodynamic Form and their Thermodynamic Formulation

We consider the theory of spinor fields written in polar form and we re-express it in terms of the so-called 1+1+2 covariant splitting: after this is done for the basic kinematic variables, we proceed to decompose the dynamical equations, both for the case of the Dirac differential field equations and for the case of the energy density tensor. As an explicit example of a real physical application we deal with the hydrogen atom, superconductivity and an analogy with the van der Waals gas.