A Generalized Electrodynamics Part I—Non-Quantum

If one wishes to derive generalized field equations from a Lagrangian, at the same time preserving the linear character of the equations, one must admit terms involving derivatives of the field quantities. It turns out that the only non-trivial generalization of this kind, leading to differential equations of order below eighth, is obtained by taking $L_f=\left(\frac{1}{8\pi }\right)\{\frac{1}{2}{F}_{\alpha {\beta }^2}+a^2{\left(\frac{\partial F_{\alpha \beta }}{\partial x_{\beta }}\right)}^2\}$. This leads to a theory that contains the Landé-Thomas theory and accounts for the choice of sign required when one wishes to consider the total field as consisting of the Maxwell-Lorentz and the Yukawa fields.