Derrick scaling (${\varphi }_{\lambda }(\mathbf{x})=\varphi (\lambda \mathbf{x})$)

Under Derrick scaling, an energy term containing $m$ spatial derivatives and homogeneous of degree $m$ in those derivatives scales as ${\lambda }^{m-D}$ in $D$ spatial dimensions.

For a smooth gauge field on ${\mathbb{R}}^3$ with finite positive magnetic energy, dilate its connection one-form by $A_i^{(s)}(x)=s{A}_i(sx)$. Its gauge curvature scales as $F_{ij}^{(s)}(x)=s^2{F}_{ij}(sx)$, so the energy scales as $s^4{s}^{-3}E=sE$. 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.

In three space dimensions, rescale $A_i(x)$ to $\lambda A_i(\lambda x)$ and $\Phi (x)$ to $\Phi (\lambda x)$. The gauge curvature then scales with two derivatives, and the adjoint covariant derivative of the Higgs field with one. Magnetic, Higgs-gradient and potential energies give the displayed formula. Stationarity requires $V_B=V_D+3V_U$, which permits nonzero energies. In the Bogomolny-Prasad-Sommerfield monopole limit $V_U=0$, the Bogomolny equations give $V_B=V_D$.