First-Order Field Equations in Spin 1/2 Form
  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. 
Richard Shurtleff