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Correlation experiments in nonlinear quantum mechanics

Marek Czachor, H. -d. Doebner

Published Oct 1, 2001• 27 citations
OpenAlex

Abstract

We show how one can compute multiple-time multi-particle correlation functions in nonlinear quantum mechanics (NLQM) in a way which guarantees locality of the formalism. On this basis we derive a version of the projection postulate which is appropriate for correlation experiments in NLQM. I.

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The well known interpretational difficulties with nonlinear Schrödinger and von Neumann equations can be reduced to the problem of computing multiple-time correlation functions in the absence of Heisenberg picture. Having no Heisenberg picture one often resorts to Zeno-type reasoning which explicitly involves the projection postulate as a means of computing conditional and joint probabilities. Although the method works well in linear quantum mechanics, it completely fails for nonlinear evolutions. We propose an alternative way of performing the same task in linear quantum mechanics and show that the method smoothly extends to the nonlinear domain. The trick is to use appropriate time-dependent Hamiltonians which involve "switching-off functions". We apply the technique to the EPR problem in nonlinear quantum mechanics and show that paradoxes of Gisin and Polchinski disappear.
The problem of the quantization of an elementary model with the second class constraints, O(n) rigid rotator model, is investigated by using the Dirac method, the projection operator method (POM) and the nonlinear quantum mechanics (NLQM). It is proved that the quantum effects, which are equivalent to those of NLQM, appear in the equation of motion of POM. Furthermore, it is shown that POM as well as NLQM preserves the isometric symmetry, which the system possesses in the classical level, through the quantization process.
Sequential measurements on a single particle play an important role in fundamental tests of quantum mechanics. We provide a general method to analyze temporal quantum correlations, which allows us to compute the maximal correlations for sequential measurements in quantum mechanics. As an application, we present the full characterization of temporal correlations in the simplest Leggett-Garg scenario and in the sequential measurement scenario associated with the most fundamental proof of the Kochen-Specker theorem.
Linear Quantum Mechanics (LQM): Its Successes and Problems Macroscopic Quantum Effects and Properties of Motion of Quasiparticles The Fundamental Principles and Theories of Nonlinear Quantum Mechanics Wave-Corpuscle Duality of Microscopic Particles (MIP) in the NLQM Nonlinear Interaction and Localization of the MIP Nonlinear Versus Linear Quantum Mechanics The Problems Solving in the NLQM Properties and States of the MIP in Different Nonlinear Systems Nonlinearly Quantum-Mechanical Properties of the Excitons and Phonons Properties of Nonlinear Excitations and Motions for Protons, Polarons and Magnons in Different Systems.