Nonlocal-looking equations can make nonlinear quantum dynamics local

A general method for extending a nondissipative nonlinear Schrodinger and Liouville-von Neumann one-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is completely separable, which is the strongest condition one can impose on dynamics of composite systems. It requires that for all initial states (entangled or not) not only can a subsystem not be influenced by any action undertaken by an observer in a separated system (strong separability), but additionally the self-consistency condition Tr(2 degrees)phi(1+2)(t)=phi(1 degrees)(t)Tr(2) is fulfilled. It is shown that a correct extension to N particles involves integro-differential equations, which, in spite of their nonlocal appearance, make the theory fully local. As a consequence, a much larger class of nonlinearities satisfying the complete separability condition is allowed than has been assumed so far. In particular all nonlinearities of the form F(\psi(x)\) are acceptable. This shows that the locality condition does not single out logarithmic or one-homogeneous nonlinearities.