Is "self-interaction" allowed in quantum mechanics?

Scalar fields can be Lorentz covariant. We were wrong to suspect that scalar fields break special relativity. See the update which we wrote to the July 3, 2022 blog post.

However, we found another problem in the Higgs quartic potential. The φ⁴ term in the Higgs potential means that the wave equation is not the massive Klein-Gordon equation.

The Klein-Gordon equation is linear. There is no "self-interaction" of a particle.

In the update to our June 29, 2022 post we asked if a particle can be scattered by itself.

Does self-interaction break basic principles of quantum mechanics when it disperses a wave packet?

Suppose that we prepare a wave packet for a particle. We are able to set a certain probability distribution for the position and the momentum of the particle.

Let us have a wave equation with a self-interaction. The self-interaction may spread the wave packet very quickly, faster than the packet would spread under a linear wave equation.

That is, the uncertainty in the position and the momentum of the particle grows faster than it should.

Is this allowed in quantum mechanics?

There is no external interaction to the wave packet. Conservation of momentum means that we know the state of the particle better than what the self-interacting wave equation claims.

https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma

In our blog we do not like Baron Munchausen tricks. If a particle can disturb itself, that constitutes such a trick.