# What does QFT say about non-linear processes?

In QFT one can use the S matrix theory. We have a IN free system in the far past. It interacts in a black box in the present and there is a free OUT system in the far future. We have OUT = S IN with the S matrix. What if in the black box there are non-linear things like Spontaneous Parametric Down Conversion or non-linear processes? Have we to accept problematic things (like collapse) added to the linear theory?

I have in mind what Susskind wrote in the black hole war He had a controversy with S Hawking about the paradox problem in general relativity. Hawking said the information will be lost in the evaporation of black holes and Susskind did not agree. He wrote that he knew he had won his bet when Maldacena discovered the AdS/CFT correspondence; A problem of gravity (in RG) was equivalent to a quantum mechanical problem. And information is never lost in QM. If you know the state of a system a t, the linearity of the evolution allows you to go back smoothly to any previous state. Several physicists write that the physical processes are the linear processes. Do they think that jumps never occur?

• Consider to spell out acronyms. Dec 19 '20 at 11:27
• You might need to clarify what you mean by nonlinear. In a QFT with linear equations of motion (EoMs), the S-matrix is trivial unless background fields are present. A Poincaré-invariant QFT with a nontrivial S-matrix is always nonlinear, in the sense that the EoMs for the field operators are nonlinear. Your examples suggest that you might be thinking of an effective QFT for the EM field where the dynamic matter fields have been integrated out, and then the same comment applies: the effective EoMs for the EM field are nonlinear, and that's why the S-matrix for photons is nontrivial. Dec 19 '20 at 15:23
• I modified my question to clarify it. Dec 19 '20 at 22:00

You do not need collapse or "problematic things" to describe non-linear processes like spontaneous parametric down-conversion. The OP's observation that the scattering matrix in quantum field theory is a linear operator on Hilbert space does not prohibit non-linear processes in the quantum optical sense.

The point is that you have to ask the question "linear in what?". Processes such as spontaneouus parametric down-conversion are typically non-linear in some observable, but can be obtained as completely normal unitary evolutions of the wavefunction (see also my answer here).

To illustrate this, consider a two-level atom coupled to a single electromagnetic field mode (i.e. the Jaynes-Cummings model). It constitutes a pretty standard quantum theory with the interaction Hamiltonian

$$H_I = g(\hat{a}\hat\sigma^+ + \hat{a}^\dagger\hat\sigma^-) \,.$$

So the wavefunction will evolve in the usual linear way. However you will find that the operator equations of motion are non-linear in some sense, e.g.

$$\frac{d}{dt}\hat{\sigma}^- \propto \hat{a}\hat{\sigma}^z \,.$$

If you look at this from the perspective of correlation functions, you can construct the BBGKY-Hierarchy for these equations. That is, if you take into account an infinite hierarchy of correlation functions, the equations remain linear. However, it is also easy to see how non-linear optical processes occur in this system. As an example, for the case of many photons, mean-field theory is often applicable, such that we can approximate

$$\langle\hat{a}\hat{\sigma}^z\rangle \approx \langle\hat{a}\rangle\langle\hat{\sigma}^z\rangle \,.$$

The resultinig semi-classical equation of motion is then manifestly non-linear due to the product between two operator expectation values and this is precisely what results in the non-linear optical response of this system.

For spontaneous parametric down-conversion, the concept applies identically, but the underlying model is more complicated, which is why I chose a simpler example above.

With regards to the edit of the question, which seems to be about the black hole information paradox, I will just point to this recent popular article, which summarizes recent work proposing a solution within semi-classical quantum gravity. While still under debate, this seems to follow a similar principle: the dynamics of a simple quantum + gravity model lead to complex dynamics of the information flow.