Solving the Impossible in Quantum Field Theory

Quantum field theory can’t be solved exactly for even simple particle interactions because there are infinitely many ways events can unfold in intermediate states. The practical breakthrough is to treat “impossible” equations as a starting point for approximations: use perturbation theory to rank which interactions matter most, organize the bookkeeping with Feynman diagrams, and then tame the infinities that appear in loop corrections through renormalization. Together, these tools turn an uncomputable problem into predictions that match experiments.

Electron scattering in quantum electrodynamics (QED) illustrates the challenge. In classical electrodynamics, two electrons repel through a straightforward Coulomb force mediated by an electromagnetic field. In QED, the electromagnetic field exists everywhere, and its excitations are photons. An electron itself is an excitation of the electron field, and the two fields interact: one electron can emit a virtual photon that transfers momentum to the other. A single-photon exchange can be drawn as a simple Feynman diagram—two incoming electron lines, a squiggly virtual-photon line between them, and two outgoing electron lines—with vertices marking emission and absorption. The math associated with that diagram sums the ways the two electrons can scatter using only one virtual photon.

But real scattering is more complicated because the event “two electrons go in, two electrons come out” hides an enormous number of possible intermediate processes. The exchanged photon might be accompanied by additional photon exchanges; an electron can emit and reabsorb a virtual photon; or a virtual photon can momentarily create a virtual electron–positron pair. In principle, all these possibilities contribute, and some interpretations even treat all compatible intermediate histories as occurring. Exact calculation would require summing an infinite set of contributions, so physicists instead expand around the most likely interactions. Perturbation theory does this by treating the interaction strength as a small correction: each extra vertex in a Feynman diagram suppresses the contribution by roughly a factor of 100. That means one-photon exchange dominates, while four-vertex processes—like two-photon exchange, electron self-emission/reabsorption, or virtual pair excitation—enter at about 1% of the main contribution, with more complex diagrams shrinking rapidly.

The remaining obstacle is that some corrections come from loop interactions, where virtual particles appear and disappear. These loops generate self-energy effects: an electron’s constant interaction with virtual photons effectively increases its mass. When the self-energy is computed directly in QED, the correction diverges—integrals over arbitrarily large photon energies drive the “extra mass” to infinity. Renormalization is the fix. Instead of starting from an unmeasurable “bare” mass, the theory folds the self-energy into the measured mass and rewrites parameters so that infinities cancel against terms absorbed into experimental inputs. The trade-off is that the theory can’t predict the renormalized value of every quantity from scratch; it predicts other observables relative to what’s measured.

With these methods—diagram-driven perturbative expansions and renormalization—QFT becomes predictive across scattering, self-energy, particle creation and annihilation, and decay processes. The same diagrammatic rules that make calculations manageable also helped shape the Standard Model, the most complete description of known subatomic physics so far.