Rigorous Finite-Dimensional Perturbation Theory

In quantum field theory, a crucial role is played by renormalization. Let us now study this phenomenon in a very simplified manner.

• We want to show how mathematical difficulties arise if nonlinear equations are linearized in the incorrect place.

• Furthermore, we will discuss how to overcome these difficulties by using the methods of bifurcation theory.

The main trick is to replace the original problem by an equivalent one by introducing so-called regularizing terms. We have to distinguish between

• the non-resonance case (N) (or regular case), and

• the resonance case (R) (or singular case).

In celestial mechanics, it is well-known that resonance may cause highly complicated motions of asteroids.

In rough terms, the complexity of phenomena in quantum field theory is caused by resonances.

In Sect. 7.16, the non-resonance case and the resonance case were studied for linear operator equations. We now want to generalize this to nonlinear problems.