Finite Yang-Mills action forbids nontrivial translation symmetry

A smooth gauge field of finite positive Euclidean Yang-Mills action cannot be invariant under a nonzero translation, even up to gauge transformation. The positive gauge-invariant action density would be periodic along that translation. A ball on which the density is bounded below by a positive constant has infinitely many disjoint translated copies, making the action infinite. Continuous translation invariance is a special case. Consequently a nonflat translation-invariant field satisfying the Anti-self-dual Yang-Mills equations on ${\mathbb{R}}^3\times \mathbb{R}$ is not a finite-action Yang-Mills instanton. Flat connections are the zero-action exception. Translations still act on instanton moduli, sending a localized instanton to a distinct centered configuration.