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= Derrick theorem
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Derrick's scaling argument tests whether a static localized field configuration can be an energy extremum under a change of size. For $\phi_\lambda(x)=\phi(\lambda x)$ in $D$ spatial dimensions, a term homogeneous in $m$ spatial derivatives scales as $E_m(\lambda)=\lambda^{m-D}E_m$.

Stationarity requires $\sum_m(m-D)E_m=0$. Nonzero scaling contributions must balance, while terms with $m=D$ have zero scaling weight and need no opposite-power partner. This qualification matters for scale-invariant two-dimensional gradient models. In an unconstrained scalar model with positive ordinary gradient and potential energies in more than two dimensions, both scale in the same direction, obstructing a nontrivial static finite-energy minimum. Higher derivatives, gauge fields, constraints, time dependence or conserved-charge conditions can change the argument. The scaling identity is necessary, not a complete proof of stability.


 Derrick's theorem (source code)