Natural higher-derivatives generalization for the Klein–Gordon equation

Abstract

We propose a natural family of higher-order partial differential equations generalizing the second-order Klein–Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing higher-derivative terms. The limit obtained by considering arbitrarily higher-order powers of the d’Alembertian operator leading to a formal infinite-order partial differential equation is discussed. The general model is constructed using the exponential of the d’Alembertian differential operator. The canonical energy–momentum tensor densities and field propagators are explicitly computed. We consider both homogeneous and non-homogeneous situations. The classical solutions are obtained for all cases.