Finite-energy field configuration ($E[\Phi ]<\infty$)

A finite-energy field configuration has integrable energy density after the vacuum energy has been subtracted. The boundary conditions used for a localized classical field-theory soliton require approaching a vacuum at spatial infinity. In one dimension the two ends may approach distinct scalar-field vacua; in a gauge theory the asymptotic field can have nontrivial winding even when its local energy density tends to zero.

For arbitrary maps $\varphi :{\mathbb{R}}^2\to S^2$, integrability of the Dirichlet energy does not imply continuous extension to the one-point compactification. For large $r$, takeand interpolate smoothly to a constant on a bounded disk. Its exterior energy isThe field has no limit because $a(r)$ keeps winding. A continuous extension at the point at infinity instead requires $\varphi (x)\to {\varphi }_{\infty }$ uniformly outside sufficiently large disks. This fixed-limit boundary condition is standard for a sigma-model lump, but is extra information beyond finite energy for arbitrary fields.