Renormalization

To describe a quantum field theory, one typically starts with a Lagrangian or an action, which contains terms that describe the fundamental fields and their interactions. The Lagrangian contains "bare" coupling constants -- the charge of an electron, for instance -- that determine the strengths of the interactions.

The couplings in the Lagrangian, however, are not the same as the couplings we actually measure. The physical interactions receive quantum corrections. The charge of an electron, for example, is screened by "vacuum polarization." This effect occurs because the vacuum is full of virtual pairs of particles and antiparticles (see the entry for Hawking radiation), which act, roughly, as a sort of dielectric medium. An electron in the vacuum attracts the positively charged members of virtual particle pairs and repels the negatively charged members, so the effective charge of the electron, as seen from a distance, is reduced. The amount of screening depends on distance (or energy) -- the closer you can get to an electron, the fewer virtual pairs lie between you and the electron, and the less screening occurs. Other kinds of interactions can lead to "antiscreening," which occurs in quantum chromodynamics.

This means that the effective value of a coupling constant will normally depend on the energy at which you probe the interaction. At short distances/high energies, the charge of an electron, for example, becomes higher -- it is less screened by virtual particle-antiparticle pairs. The color charge of a quark, in contrast, becomes lower, approaching zero at infinite energy in a phenomenon called "asymptotic freedom." This variation of coupling constants with energy or scale is called the "renormalization group flow."

The important thing to keep in mind is that the observed coupling constants are not the same as the bare ones that occur in the Lagrangian. In fact, even if a coupling constant is zero in the Lagrangian, its effective value can be nonzero once quantum corrections are accounted for. In general relativity, for example, a cosmological constant will be induced by quantum fluctuations even if the "bare" cosmological constant is set to zero.

A renormalizable theory is one in which the number of undetermined coupling constants is finite, even after quantum corrections are taken into account. A nonrenormalizable theory is one in which the number of undetermined effective coupling constants is infinite. General relativity, when treated as an ordinary quantum field theory, is nonrenormalizable.

Now, even a nonrenormalizable theory can have some predictive power -- for general relativity, see for example, work by Donoghue and the review by Burgess on "effective field theory." But given its infinite number of free parameters, a nonrenormalizable theory probably cannot serve as a complete description of physics.

(One possible loophole, suggested by Steven Weinberg, might make certain nonrenormalizable theories more acceptable. As I noted earlier, the coupling constants in any theory "flow" with energy. It could be that even if a theory has infinitely many coupling constants, they flow to a finite-dimensional surface at high energies. In such an "asymptotically safe" theory, the coupling constants, although infinite in number, would be determined by a finite number of parameters of the high-energy theory. It is an open question whether general relativity is asymptotically safe; here is a sympathetic review.)

For a much more detailed, but entertaining and not-too-technical, description, see John Baez's Renormalization Made Easy.

Contributors and Attributions

• Steve Carlip (Physics, UC Davis)