Electrons in metals interact via the Coulomb interaction. The size of this effect is not negligible: it is of order e2/a where a is the mean distance between electrons. For a metal this ≈ 3 eV, that is, comparable to EF. An approach to the problem would be to treat the interaction as a perturbation. However, the perturbation is strong and long-ranged, and this gives rise to many difficulties, such as divergences in finite orders of perturbation theory. The problems can be overcome by an application of the methods of quantum field theory to the problem, and is treated in books on many-body theory. We will not pursue this approach here, but try to get some of the central conclusions with physical reasoning.

At first glance, we might be tempted to conclude that the sharp Fermi surface that we have discussed, and which is readily observed, should be wiped out by the interaction. The much smaller energy kBT does lead to smearing of the Fermi distribution; it is natural to conclude that this is true for interactions as well. This natural conclusion is not correct in three dimensions. The situation in one dimension is complicated, and the Fermi surface does disappear, leading to a state called the Luttinger liquid. In three dimensions the sharp Fermi surface survives the presence of interactions. This is an important phenomenon which demands explanation.