= No static finite-energy lump in three-dimensional pure Yang-Mills theory
{title2=$E[A^{(s)}]=sE[A]$}
For a smooth <gauge field> on $\mathbb R^3$ with finite positive magnetic <energy>, dilate its <connection one-form> by $A_i^{(s)}(x)=sA_i(sx)$. Its <gauge curvature> scales as $F_{ij}^{(s)}(x)=s^2F_{ij}(sx)$, so the energy scales as $s^4s^{-3}E=sE$. A static stationary configuration must have zero derivative at $s=1$, forcing $E=0$ and hence $F=0$. This <Derrick scaling> argument assumes the dilation preserves the boundary conditions and that boundary terms vanish. It does not exclude four-dimensional <Yang-Mills instantons>, whose Euclidean action is scale invariant.
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