In the framework of Bohmian quantum mechanics, the Klein--Gordon equation can be seen as representing a particle with mass m which is guided by a guiding wave $\phi(x)$ in a causal manner. Here a relevant question is whether Bohmian quantum mechanics is applicable to a non-linear Klein--Gordon equation? We examine this approach for $\phi_{4}(x)$ and sine-Gordon potentials. It turns out that this method leads to equations for quantum states which are identical to those derived by field theoretical methods used for quantum solitons. Moreover, the quantum force exerted on the particle can be determined. This method can be used for other non-linear potentials as well.

There is a famous criterion for objective wave function reduction which is derived by using the Shrödinger–Newton equation [L Diosi, Phys. Lett. A 105(4–5), 199 (1984)]. In this regard, a critical mass for the transition from quantum world to the classical world is determined for a particle or an object. In this paper, we shall derive that criterion by using the concept of Bohmian trajectories. This study has two consequences. The first is, it provides a geometric framework for the problem of wave function reduction. The second is, it represents the role of quantum and gravitational forces in the reduction process.