In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. This means that the number of "independent types" of ultraviolet infinities must be finite, and by setting them equal to the measured values, we can predict the results of all other experiments.
On the other hand, Fermi's interaction or general relativity are non-renormalizable. This problem is a hint that these theories should be replaced by a more complete theory at very short distance. These more complete theories are the electroweak theory and string theory, respectively.
The Standard Model of particle physics contains only renormalizable operators, but the interactions of general relativity become nonrenormalizable operators if one attempts to construct a field theory of quantum gravity in the most straightforward manner, suggesting that perturbation theory is useless in application to quantum gravity.
The classic example is the Fermi theory of the weak nuclear force, a nonrenormalizable effective theory whose cutoff is comparable to the mass of the W particle.
It may be that any others that may exist at the GUT or Planck scale simply become too weak to detect in the realm we can observe, with one exception: gravity, whose exceedingly weak interaction is magnified by the presence of the enormous masses of stars and planets.
But the integrals go to infinity very slowly, like the logarithm of momentum, and it turns out that in this case, the theory can be renormalized so that the infinities can be absorbed into a redefinition of a small number of parameters in the theory, such as the mass and charge of the electron.
Quantum electrodynamics was a renormalizable theory, and by the 19402, this was regarded as a solved relativistic quantum theory.
But in 1971, a new type of quantum field theory came on the scene that explained the weak nuclear force by uniting it with electromagnetism into electroweak theory, and it was shown to be renormalizable.
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