Q:

Is the quantum dynamics of a system of interacting particles linear or non-linear?

As I understand it, the linearity of quantum mechanics is considered to be an inviolable principle - e.g., this paper - because (among other things) causality would be violated or and/or superluminal communication would be possible if quantum mechanics were nonlinear.  However, even in the simple case of multiple electrons, the path (or evolution of the wavefunction) of an individual electron would influence the paths of the electrons, which would in turn influence the path of the individual electron.  This looks a lot like self-interaction which would imply nonlinearity. 
I think it is possible to represent the multiple-electron system by a set of coupled quasi-linear equations; but I also think that an equation in terms of the wavefunction of the whole multiple-electron system would be nonlinear. {By "quasi-linear" I mean linear in some of the unknowns, in such a way that the other unknowns resemble constants.)  This SE answer may be relevant, but seems not to directly address my problem: Does separability into coupled quasi-linear equations somehow  meet the overall linearity requirement?

A:

Depends on what you mean by "linear" - it is certainly not linear in the sense that the full state can be composed of a product of many single-particle wave functions + (anti)symmetrization. For an interacting theory, the many-body wave function contains many correlations that do not allow for the decomposition into separable independent single-particle wave functions.
It is linear in the sense that if $|\psi(t)\rangle, |\varphi(t)\rangle$ are two solutions of the Schrödinger equation then $|\psi(t)\rangle + |\varphi(t)\rangle$ will also be a solution. This is relevant also for a many-body wave function $\langle \vec{r}_1,\ldots \vec{r}_N | \psi(t) \rangle = \psi(\vec{r}_1,\ldots,\vec{r}_N, t)$. It doesn't mean that there is a way to write $\psi(\vec{r}_1,\ldots,\vec{r}_N, t) = \prod_j \psi_j(\vec{r}_j)$ with each of these functions a solution.
The non-linear Schrödinger equation, for example one that involves $|\psi|^2$, is an approximation. Usually (in the contexts that I am familiar with) a mean-field approximation where the effects of the other particles are taken as an external potential, and then solved self-consistently.

