A Generalized Electrodynamics Part II—Quantum

When the Lagrangian from which the field equations are derived contains second and higher derivatives of the generalized field coordinates, the method of quantizing the field equations developed by Heisenberg and Pauli cannot be immediately applied. By generalizing a method due to Ostrogradsky for converting Lagrange's equations of motion of a particle, when higher derivatives are present, into canonical Hamiltonian form, it becomes possible to perform a similar transformation of the field equations. Applying this method to Podolsky's generalized electrodynamics, we obtain the Hamiltonian of the field and double the usual number of generalized coordinates and momenta. The quantization of the field follows without any special assumptions. The last two sections are devoted to the discussion of the auxiliary conditions and some of their consequences.