Relativistically Extended Modification of the Schroedinger Equation

Abstract

We propose a nonlinear modification of the Schr\"{o}dinger equation that possesses the main properties of this equation such as the Galilean invariance, the weak separability of composite systems, and the homogeneity in the wave function. The modification is derived from the relativistic relation between the energy and momentum of free particle and, as such, it is the best relativistic extension of the Schr\"{o}dinger equation that preserves the properties in question. The only change it effectively entails in the Schr\"{o}dinger equation involves the conserved probability current. It is pointed out that it partially retains the linear superposition principle and that it can be used to model the process of decoherence.