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 ${L}_{f}=(\frac{1}{8\ensuremath{\pi}}){\frac{1}{2}{F}_{\ensuremath{\alpha}{\ensuremath{\beta}}^{2}}+{a}^{2}{(\frac{\ensuremath{\partial}{F}_{\ensuremath{\alpha}\ensuremath{\beta}}}{\ensuremath{\partial}{x}_{\ensuremath{\beta}}})}^{2}}$. This leads to a theory that contains the Land\'e-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.