A very brief introduction is given to all that is needed to appreciate the formal structure of the Dirac equation and why–without destroying this structure–it cannot be reduced to a Pauli-Schrödinger type equation.
We derive an equivalent traveling wave form description for Dirac field. In the non-relativistic limit, such form can reduce to inverse-Galilean transformed Schrodinger-type equation. We find that, the resulting two-component Schrodinger-type equation from the reduction of traveling wave form description of Dirac field is different to the naive Galilean transformed Schrodinger equation. Taking into account the interactions of the system to electromagnetic field by adding proper forms of covariant derivative, the traveling wave form description for Pauli equation can be similarly obtained in the non-relativistic limit. Such descriptions allow one to choose arbitrary convenient reference frame for quantum system involving spins. Using Bargmann-Wigner formalism for field with arbitrary spin s≥1/2, which satisfy Dirac-type equations in all its indices, the traveling wave description for such a field can be similarly obtained from the traveling wave form description of Dirac field, for example, for the spin-3/2 Rarita-Schwinger field and spin-2 gravitational field.
The stochastic theory of quantum mechanics is further developed to include the problem of extended rigid particles, thus allowing the introduction of spin. It is demonstrated that the stochastic equations for the system's center of mass give rise to a generalized Schrödinger equation for integral or half-integral spin; in the particular case of spin ½, upon elimination of the internal variables, Pauli's equation is obtained. A formal simplified relativistic extension of the theory is worked out and shown to lead to Dirac's equation in the case of spin ½ and for a gyromagnetic ratio equal to 2; in the case of arbitrary spin, the theory gives an equation of the Feynman-Gell-Mann type.