Nonlinear Schr\"odinger equation in foundations: summary of 4 catches

Abstract

Fundamental modifications of the standard Schr\"odinger equation by additional nonlinear terms have been considered for various purposes over the recent decades. It came as a surprise when, inverting Abner Shimony's observation of "peaceful coexistence" between standard quantum mechanics and relativity, N. Gisin proved in 1990 that any (deterministic) nonlinear Schr\"odinger equation would allow for superluminal communication. This is by now the most spectacular and best known anomaly. We discuss further anomalies, simple but foundational, less spectacular but not less dramatic.