First-Order Field Equations in Spin 1/2 Form

TL;DR

This paper derives first-order covariant field equations for spin 1/2 particles based on transformation properties, emphasizing that these equations are consequences of the underlying symmetries and operator transformations.

Contribution

It introduces a method to derive covariant first-order field equations directly from transformation properties of quantum operators and fields.

Findings

Derived covariant first-order field equations for spin 1/2 fields.

Showed equations follow from transformation properties of operators.

Connected field equations to symmetry and transformation principles.

Abstract

From one point of view in the quantum theory of fields, free quantum fields are uniquely determined, not by field equations, but by the transformations of the field and the annihilation and creation operators from which the field is constructed. One says that a free field equation merely records the fact that some field components are superfluous. Here, free field equations that are first order and covariant are derived so that the already determined field is one solution. The unknowns are the vector matrices that combine with the known gradient of the field to make an invariant equation: the scalar product of the vector matrices and the gradient are proportional to the field. Thus these free field equations are direct consequences of the transformation properties of the annihilation and creation operators and the transformation properties of the field.