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A Derrick scaling variation changes a configuration's size without changing its topological sector. Use $U_L(\mathbf{x})=U(\mathbf{x}/L)$, so $L>1$ expands the configuration. An integrated nonnegative term homogeneous in $s$ spatial derivatives scales as $L^{d-s}{E}_s$. For a scalar gradient-plus-potential energy, for example,A static solution is a stationary point of the full energy functional, so it must in particular satisfy $E^{\prime }(1)=0$. In this example the Derrick virial identity is $(d-2)E_2+d{E}_0=0$. If both energies are nonnegative and the coefficients cannot balance, a nontrivial localized static solution is impossible. If balance is possible, it fixes relations among energy contributions and may select a characteristic size. A negative $E^{\prime \prime }(1)$ proves an instability under size changes; a positive second derivative gives stability only against that one deformation. Passing the scale test is a necessary condition, not an existence theorem or a proof of full stability. It does not directly rule out time-dependent or charge-constrained solitons outside the static variational problem.

For the specified model, each spatial current is anti-Hermitian: differentiating $U{U}^{†}=1$ gives $R_i^{†}=-R_i$. Thus $-\mathrm{Tr}(R_i^2)$ is nonnegative, and the static energy of the nonlinear sigma model isThe boundary condition $U\to 1_2$ identifies spatial infinity to one point, so one-point compactification turns the field into a based map $S^d\to SU(2)$. By SU(2) as the three-sphere, its sectors are classified by the homotopy group ${\pi }_d(S^3)$. In particular,There is no corresponding topological protection in one or two spatial dimensions. In three dimensions there is an integer degree of a map between oriented manifolds, the usual topological baryon number in the Skyrme model when the same field is interpreted in that theory. With a compatible orientation convention it can be writtenConjugation from the left current to the right current preserves the trace in this formula. Higher dimensions are governed by their corresponding homotopy groups, which need not all be trivial; no assumption of their triviality is needed in the scaling obstruction below.

For the pure sigma model, $E(L)=L^{d-2}{E}_2$, so stationarity requires $(d-2)E_2=0$. In every $d\ne 2$ a finite-energy nonconstant static solution is excluded by Derrick scaling. In $d=1$ spreading lowers its energy; in $d>2$ shrinking lowers it. In particular, a degree-carrying three-dimensional sigma configuration can collapse toward zero size and energy within its sector, with a singular limiting field. The topological charge prevents smooth unwinding, but not this singular collapse.

In $d=2$, the sigma energy is scale invariant. The scaling identity supplies no obstruction, but ${\pi }_2(S^3)=0$ gives no topological soliton sector either. One must not infer nonexistence of every nonconstant stationary harmonic map from these two facts: scale-invariant sigma models can have non-topological harmonic-map saddles, for example using a totally geodesic two-sphere in the target three-sphere. The conclusion is absence of a topologically protected sigma soliton here, with no positive scale-restoring force supplied by the quadratic term alone.

For the Skyrme model, the added commutator term gives the nonnegative static energyAnti-Hermiticity of the commutators fixes its sign. The target and boundary condition are unchanged, so the topological classification is unchanged. The Derrick dimension test for the Skyrme model instead becomesIn $d=3$, the opposite exponents permit a balance:For an arbitrary reference shape with both energies nonzero, its preferred scale is $L_{\ast }=(E_4/E_2{)}^{1/2}$ and its optimized energy is $2\sqrt{E_2{E}_4}$. Expansion costs quadratic-gradient energy and contraction costs quartic-gradient energy. Integer-degree Skyrmions are therefore topologically possible and evade the pure sigma collapse. This mechanism permits finite-size stable solitons, although existence and stability in all field directions require more than the scaling argument itself.

The other dimensions behave differently. In $d=1$, all spatial commutators vanish identically, so the added term changes nothing and nonconstant static finite-energy fields remain excluded. In $d=2$, stationarity forces $E_4=0$; any configuration with nonzero commutator energy can lower it by spreading. In fact a smooth finite-energy static field of this particular potential-free model must be constant. To prove this commuting-current obstruction in a two-dimensional Skyrme model, $E_4=0$ forces $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma equation ${\partial }_i{R}_i=0$. The right-current Maurer-Cartan equation isDivergence-free and curl-free currents have componentwise $\Delta R_i=0$. Their components belong to L2 space by finite $E_2$. An entire square-integrable harmonic function on the plane vanishes: the mean-value formula and Cauchy–Schwarz bound its value by $(\pi R^2{)}^{-1/2}$ times its full L2 norm, tending to zero as the disk radius $R$ grows. Thus $R_i=0$, and the boundary condition fixes $U=1_2$. This additional argument uses the vanishing commutator; it does not apply to the pure sigma model's general noncommuting currents.

In $d=4$, the virial relation gives $2E_2=0$, hence a constant field and zero $E_4$. In $d>4$ both coefficients are positive and the same conclusion follows. Only three spatial dimensions allow a nontrivial finite-size balance of these two positive derivative energies. Higher-dimensional topological classes, where present, do not evade this obstruction.