nonperturbatively

Nonperturbatively refers to approaches, effects, or phenomena that cannot be captured by a perturbative expansion in a small parameter, typically a coupling constant. In quantum field theory and statistical mechanics, perturbation theory expresses quantities as a power series in a small coupling; nonperturbative physics is required when the coupling is not small, or when the system exhibits effects that are invisible to any finite truncation of the series. Such effects often scale nonanalytically with the coupling, for example as exp(-const/g) or exp(-const/g^2), so they do not appear in ordinary perturbation theory.

Examples include confinement of quarks in quantum chromodynamics, dynamical mass generation, instantons and other solitonic configurations, monopoles, and topological defects. Many phenomena in strong coupling regimes are inherently nonperturbative.

Common nonperturbative techniques are lattice gauge theory, which formulates field theories on a spacetime lattice and relies on numerical simulations; strong-coupling expansions; and functional methods such as Dyson–Schwinger equations or variational and resummation schemes. In some theories, exact nonperturbative results exist, as in certain two-dimensional models or integrable systems, while in others one relies on numerical or approximate nonperturbative frameworks.

In mathematics, nonperturbative analysis studies problems without assuming a small parameter, often involving global or qualitative properties of solutions, singularities, or nonanalytic dependence on parameters. The term highlights the distinction from perturbative methods that rely on series expansions around a known, simple limit.