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.
Proof of Theorem 1 For ξ = (τ, ξ) ∈ R × Rn, we introduce eξ ∈ C∞(R × Rn; R) by eξ(x) = eξ(t, x) = exp(τ t+ ξ · x) for x = (t, x) ∈ R × Rn
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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.