Any Poincaré-invariant linear partial differential operator on Minkowski spacetime is a polynomial in the d'Alembertian, and dilation invariance then forces the second order operator to be a scalar multiple of it.
For related subjects, we refer the reader to the references [13–19]
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Characterization of the D'Alembertian by the Poincar\'e Invariance
Any Poincaré-invariant linear partial differential operator on Minkowski spacetime is a polynomial in the d'Alembertian, and dilation invariance then forces the second order operator to be a scalar multiple of it.