Perturbative

Perturbative refers to methods that treat a complex problem as a small modification of a simpler, exactly solvable one. In perturbation theory, the quantity of interest is expressed as a series in a small parameter ε that measures the strength of the perturbation: O = O0 + ε O1 + ε^2 O2 + … When ε is small, truncating the series yields an approximate solution whose accuracy improves with more terms. In many cases the series is asymptotic rather than convergent, providing good approximations for small ε but not guaranteed to converge for any finite ε.

Perturbative methods are widely used in physics and applied mathematics. In quantum mechanics, time-independent perturbation theory computes energy shifts and state corrections when a solvable Hamiltonian is subjected to a weak perturbation. In quantum field theory, perturbation theory expands observables in powers of a coupling constant, with calculations organized by Feynman diagrams. Renormalization and regularization are common features of perturbative QFT to handle divergences. In chemistry and molecular physics, perturbation theory underpins methods such as Møller–Plesset MP2 and coupled-cluster theory in the limit of weak inter-electronic interaction. In celestial mechanics and classical mechanics, perturbation theory analyzes small deviations from idealized motions or trajectories.

Limitations include the requirement of a small perturbation parameter; many systems exhibit non-perturbative phenomena that cannot be captured by any finite truncation. Such effects may be treated with non-perturbative techniques, resummation methods, or numerical approaches. The term perturbative is often contrasted with nonperturbative methods.