Definition
Renormalization is a mathematical procedure that removes the infinities arising in quantum field theory calculations by absorbing them into the definitions of measurable parameters — masses, charges, couplings — that are then matched to experiment. In its modern interpretation, due largely to Kenneth Wilson, it is also a framework for understanding how physical laws change with the energy or distance scale at which we probe them.
It is the reason quantum electrodynamics can predict the electron's magnetic moment to twelve decimal places.
Why it matters
How it works
When you calculate a process like electron-photon scattering to higher orders in quantum electrodynamics, integrals over the momentum of virtual particles can diverge — the contribution grows without bound as you include arbitrarily high-momentum (short-distance) modes. Naively, the prediction is infinity.
The cure, worked out by Tomonaga, Schwinger, Feynman, and Dyson in the late 1940s, is to recognize that the "bare" parameters of the theory — the bare electron mass m₀, the bare charge e₀ — were never observable. What we measure is always the dressed quantity: the electron together with its cloud of virtual photons and electron-positron pairs. Define the physical mass and charge in terms of measured quantities. The infinities of the bare theory cancel against the redefinition, leaving finite, well-defined predictions for any other observable.
Wilson's reinterpretation in the 1960s-70s changed the meaning. Imagine integrating out the high-momentum modes of a field theory step by step. The result is a new effective theory at lower energies with shifted parameters. As you change the energy scale you "flow" through a space of theories. The infinities are not pathological — they are the recognition that we never had access to arbitrarily short distances. Different starting parameters at high energies lead to the same physics at low energies, sorted by relevant, marginal, and irrelevant couplings.
That picture explained universality in critical phenomena (why disparate physical systems have the same critical exponents) and asymptotic freedom in QCD (why the strong coupling decreases at high energy, allowing perturbative calculations of high-energy collisions). It also explained why the Standard Model's renormalizable interactions are so important: only those couplings remain significant down to low energies.