Reconstructing a nonlinear dynamical framework for testing quantum mechanics
The nonlinar generalization of quantum dynamics constructed by Weinberg as a basis for experimental tests is reconstructed in terms of density- matrix elements, as suggested by Polchinski, to allow independent dynamics for subsystems. Dynamics is generated with a Lie bracket and a nonlinear Hamiltonian function. It takes density matrices to density matrices and pure states to pure states. Each density matrix has a Hamiltonian operator that makes its evolution for an infinitesimal time the same as in ordinary quantum mechanics, but the Hamiltonian operator may be different for different density matrices and may change in time as the density matrix changes. A Hamiltonian function for a subsystem serves also for the entire system. Independence of separate subsystems is confirmed both by seeing that brackets are for functions from different subsystems and by looking at the Hamiltonian operator for each density matrix. Scaling properties of Hamiltonian functions are found to be of little consequence by themselves but important in connection with locality. An example of all this, including subsystem independence, is obtained from every one of the local nonlinear Schrödinger equations described by Bialynicki-Birula and Mycielski; the logarithmic nonlinearity they favor is distinguished by a scaling property. Examples are worked out for spins coupled together or to fields, demonstrating Hamiltonian functions and equations of motion written directly in terms of physical mean values. Observables and states are taken to be the same as in ordinary quantum mechanics. An attempt to find nonlinear representations of observables by characterizing propositions as functions equal to their squares yields a negative result: for two-by-two matrices, no nontrivial propositions are found to be significantly different from those of ordinary quantum mechanics. Sharper interpretation of mixed states is proposed. In a mixture of parts that are prepared separately, time dependence has to be calculated separately for each part. Then different mixtures that yield the same density matrix can be distinguished. It is argued that no criticism has shown that a consistent interpretation can not be made this way. Nonlinearity remains a viable hypothesis for experimental tests.
scientific article; zbMATH DE number 178236