Natural Higher-Derivatives Generalization for the Klein-Gordon Equation

TL;DR

This paper introduces a family of higher-order PDEs extending the Klein-Gordon equation, using exponential operators and higher derivatives, with explicit solutions and physical quantities computed.

Contribution

It presents a novel generalization of the Klein-Gordon equation through higher-derivative models and infinite-order operators, with explicit formulations and solutions.

Findings

Derived higher-order Klein-Gordon generalizations using exponential operators

Explicitly computed energy-momentum tensors and propagators

Obtained classical solutions for various cases

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.