Every field equation with two interacting fields is nonlinear: how do we know in physics that singularities do not form?

Suppose that we have two fields φ and ψ. As free fields (no interaction) both fields are governed by a linear differential equation.

The fields might be the electromagnetic field and the Dirac field.

Theorem. If we add an interaction, then the system of φ and ψ is no longer linear.

Proof. Assume first that ψ_0 is identically zero. Pick a solution of φ_0 which is not zero.

The pair (φ_0, ψ_0) is a solution of the combined system.

Then pick a solution (φ_1, ψ_1) where φ_1 is identically zero but ψ_1 not.

But the sum these two solutions is not a solution because of the interaction. QED.

Thus, most of the field equations in quantum physics are nonlinear.

Mathematically, it is very hard to prove that a nonlinear differential equation does not develop singularities. Christodoulou, Klainerman, and Tao have tried to solve this problem for the Einstein equations or the Navier-Stokes equations.

We have the same problem of smoothness for most equations in quantum physics.

http://philsci-archive.pitt.edu/13432/1/PhD%2520Bacelar%2520Valente%2520redux.pdf

Freeman Dyson gave an argument for the non-convergence of the perturbation series in QED. The argument is explained in the above link, the Ph.D. thesis of Mario Bacelar Valente.

The convergence of a perturbation series is related to the general problem if a system has any solutions which do not develop singularities. Thus, the convergence problem of QED is really a ubiquitous problem in quantum physics.

What about renormalization? Infinities may appear because:

1. the field equation itself develops singularities, or

2. our approximation method is bad and creates singularities.

Suppose that we have a nonlinear field equation and send a 1 eV photon to a system governed by that field equation. How do we know that:

1. The system does not output a 1 GeV photon?

2. The system does not create a singularity and collapse into a black hole?

3. There is any solution to the problem at all?

4. The perturbation series which we use to approximate the process converges and does not have huge errors?

For nonlinear field equations, we generally do not know the answer to any of the above questions.

When doing physics we assume that the system is well-behaved. If we use Green's functions to construct a solution, we assume that high momenta p in the "spike" sources are canceled by destructive interference.

The renormalization problem and the convergence problem of QED are no isolated cases. Similar problems appear for all nonlinear field equations.