Reconstructing a nonlinear dynamical framework for testing quantum mechanics
type: Item
instance of: scholarly article
title: Reconstructing a nonlinear dynamical framework for testing quantum mechanics
zbMATH Open document ID: 0783.58091
author: Thomas F. Jordan
published in: Annals of Physics
publication date: 1993-05-25T00:00:00Z
review text: 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 ca…
Mathematics Subject Classification ID: 58Z05
Mathematics Subject Classification ID: 82C10
zbMATH DE Number: 178236
zbMATH Keywords: quantum dynamics
zbMATH Keywords: density matrix
zbMATH Keywords: Hamiltonian
zbMATH Keywords: scaling
zbMATH Keywords: time dependence
MaRDI profile type: Publication
full work available at URL: aphy.1993.1053
OpenAlex ID: W2011184092
DOI: 10.1006/APHY.1993.1053
description: scientific article; zbMATH DE number 178236