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

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.