Characterization of the D'Alembertian by the Poincar\'e Invariance

TL;DR

This paper proves that the d'Alembertian operator is uniquely characterized by its invariance under Poincaré transformations and dilations in Minkowski space-time, highlighting its fundamental role in physics.

Contribution

It demonstrates that the d'Alembertian is the only second-order linear PDE invariant under Poincaré group and dilations, providing a mathematical characterization based on symmetry.

Findings

The d'Alembertian is uniquely invariant under Poincaré and dilation transformations.

The proof involves analyzing polynomial invariance under reflections and rotations.

This characterization emphasizes the fundamental symmetry properties of the d'Alembertian.

Abstract

Many physical models are described by partial differential equations and the most important mathematical structure of the equations is governed by the corresponding linear partial differential operators. Those linear partial differential operators are sometimes determined by the symmetry under the group of motion. In this paper, the d'Alembertian is shown to be characterized as the only linear partial differential operator of the second order that is invariant under the Poincar\'e group and dilations in the Minkowski space-time $\mathbb R\times\mathbb R^n$. The method of proof depends on the analysis of the invariance of the corresponding polynomial in space-time under the time reflections and space rotations.