We are finally ready to face up to the fact that most of the expressions we write in the Feynman-diagram solution of quantum field theory are nonsensical, which is to say infinite. The purpose of this introductory section is to demonstrate the simple and general counting rules which prove that this is so, and to give the reader a conceptual orientation to the general theory of renormalization, which we will discuss in the rest of this chapter.

We will begin with the conceptual. The formalism of quantum field theory makes statements about arbitrarily short distances in space-time and arbitrarily high energies. At a given moment in history experiment will never probe arbitrarily short distances or high energies. We should expect that the formalism will have to change to fit the data as we uncover more empirical facts about short distances. This has happened before. Hydrodynamics is a field theory, but we know that at short enough distance scales it is not a good description of water. A better one is given (in principle) by the Schrödinger equation for hydrogen and oxygen nuclei interacting with electrons, and that in turn is superseded by the standard model of particle physics as we go to even shorter distances and higher energies. Yet only a fool would imagine that one should try to understand the properties of waves in the ocean in terms of Feynman-diagram calculations in the standard model, even if the latter understanding is possible “in principle.”