Aspects of Nonlinearity in Quantum Mechanics

Abstract:
The possibility of nonlinear modifications of quantum mechanics is connected with a dichotomy for the interpretation of wave functions formulated by Einstein at the 1927 Solvay congress. Nonlinear extensions are excluded when quantum states as given by density matrices are understood as statistical ensembles, but they seem natural when wave functions are seen as mechanical variables for individual systems. The latter view was refuted by Einstein Podolsky and Rosen in 1935. So quantum mechanics is rigidly linear. Nevertheless nonlinear equations of exactly the kind suggested by the view of wave functions as mechanical variables reappear in mean field quantum dynamics. Examples include nonlinear dynamics on the Poincaré sphere, as well as Gross-Pitaevsky equations for Bose-Einstein condensates.