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

TL;DR

This paper reviews foundational issues with nonlinear modifications of the Schr"odinger equation, highlighting anomalies like superluminal communication and discussing their implications for quantum mechanics and relativity.

Contribution

It summarizes key anomalies arising from nonlinear Schr"odinger equations, emphasizing their foundational significance and exploring less-known but impactful issues.

Findings

Nonlinear Schr"odinger equations can enable superluminal communication.

Anomalies challenge the compatibility of nonlinear quantum mechanics with relativity.

The paper discusses foundational implications of these anomalies.

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.