Nonperturbative

Nonperturbative refers to approaches, effects, or phenomena in physics and mathematics that cannot be fully captured by a perturbative expansion in a small parameter, such as a coupling constant. In quantum field theory, perturbation theory expands observables in a power series of the coupling, producing contributions from successive loop orders. Nonperturbative effects are those that either vanish or are invisible at any finite order of perturbation theory, often contributing terms that behave like exp(-const/g) or exp(-const/g^2) at weak coupling. These effects become significant at strong coupling or in the presence of topological structures.

Common nonperturbative phenomena include instantons and solitons, which arise as nontrivial solutions to classical field equations and can mediate tunneling events; confinement of quarks in quantum chromodynamics; mass gaps; and the existence of distinct topological sectors in gauge theories. In statistical mechanics and condensed matter physics, nonperturbative methods describe phase transitions and nontrivial ground states that perturbation theory cannot resolve.

Techniques to study nonperturbative physics include lattice gauge theory, where spacetime is discretized and observables are computed numerically; semiclassical methods that incorporate instantons and other nonperturbative configurations; resummation and resurgence approaches that seek to relate perturbative series to nonperturbative contributions; and dualities or holographic methods that map strong coupling problems to weakly coupled dual theories. Nonperturbative definitions and computations are essential for understanding phenomena in strong coupling regimes, topological effects, and the full structure of quantum field theories.