Traveling Wave Form Description for Dirac Field and Its Deduction To Pauli Equation Type Forms in Quantum Mechanics

TL;DR

This paper develops a traveling wave form description for the Dirac field and derives related non-relativistic equations like the Schrödinger and Pauli equations, extending to higher-spin fields using Bargmann-Wigner formalism.

Contribution

It introduces a novel traveling wave form framework for Dirac and higher-spin fields, connecting relativistic and non-relativistic quantum equations.

Findings

Derived a traveling wave form for Dirac field.

Reduced to Schrödinger-type equations in the non-relativistic limit.

Extended the formalism to higher-spin fields like Rarita-Schwinger and gravitational fields.

Abstract

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 $s\geq 1/2$, 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.