The Path to Perturbation Theory

In physics, most problems—from the anharmonic oscillator to Quantum Electrodynamics (QED)— cannot be solved exactly. We rely on Perturbation Theory, expanding the solution in powers of a small coupling constant (often denoted λ or α).

A classic example is the quantum anharmonic oscillator, where the Hamiltonian is

(with the quartic term treated as a small perturbation λ ≪ 1). Another famous case is QED, where the fine-structure constant α ≈ 1/137 serves as the small parameter for expanding processes involving electrons and photons.

The challenge? These perturbative expansions are almost always asymptotic series with a null radius of convergence. The coefficients typically grow factorially as (or even faster), making the series eventually diverge regardless of how small the coupling constant is.

For instance, in the anharmonic oscillator, the perturbative corrections to the energy levels involve terms whose coefficients grow like , leading to a divergent series for any nonzero λ. Similarly, in QED, the perturbative series for quantities like the electron’s anomalous magnetic moment (g−2) is asymptotic: it gives spectacular agreement with experiment up to high orders, but diverges if pushed too far.

This is where our journey pays off. By applying Borel resummation and Padé approximants to these divergent series, we can extract non-perturbative information and obtain remarkably accurate physical predictions. In quantum field theory, Borel summation helps recover meaningful results from perturbative expansions (for example, in certain Euclidean field theories or by handling contributions from instantons and renormalons). Padé approximants are routinely used to accelerate convergence and estimate the behavior of series in QCD and other models.

In our upcoming posts, we will see how this mathematical framework becomes the essential language of modern theoretical physics.