[Paper Review] A chaotic dynamical reduction model for the quantum mechanical state vector

Henrik Brusheim-Johansson, Johan Hansson|ArXiv.org|Oct 31, 2006

Statistical Mechanics and Entropy2 references3 citations

TL;DR

This paper proposes a nonlinear, chaotic dynamical model for quantum state vector reduction, where phase instability drives irreversible collapse to a classical eigenstate. By introducing a non-Hermitian, nonlinear Hamiltonian that induces chaotic phase evolution, the model reproduces Born's statistical rule without ensemble averaging, offering a deterministic, continuous collapse mechanism compatible with individual quantum systems and nonlocality via phase-mediated 'spooky action at a distance'.

ABSTRACT

A new model is proposed for the purpose of modelling the ``wave function collapse'' of a two-state quantum system. The collapse to a classical state is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the apparent indeterminacy we are experiencing for a single quantum system. Through this randomness, the statistical ``ensemble'' behaviour, due to Born, to describe a single quantum system, is no longer needed.

Motivation & Objective

• To resolve the measurement problem in quantum mechanics by replacing the postulated collapse with a continuous, dynamical process.
• To explain the statistical outcomes of quantum measurements not through ensemble averaging, but through deterministic chaos in the absolute phase.
• To provide a mechanism for irreversibility and nonlocality in quantum measurement using a single nonlinear evolution equation.
• To eliminate the need for the standard collapse postulate by deriving it from a chaotic, nonlinear dynamics of the quantum phase.
• To connect the reduction time to physical interaction strengths, particularly in gauge fields, for potential experimental testability.

Proposed method

• Formulates a nonlinear evolution equation for the quantum state vector using time-dependent amplitudes $ x_n(t) $ and phases $ \theta_n(t) $, with $ c_n(t) = \sqrt{x_n(t)} e^{i\theta_n(t)} $.
• Derives the time evolution of $ x_n $ and $ \theta_n $ via a nonlinear interaction Hamiltonian $ H_I^{NL} $, introducing phase-dependent coupling terms.
• Introduces a non-Hermitian Hamiltonian that drives all $ x_n $ to zero except one, which is driven to unity, resulting in state vector collapse.
• Models the collapse process using a characteristic reduction time $ \tau_r $, inversely proportional to the strength of the nonlinear interaction.
• Uses Heaviside functions $ \Theta_+ $ and coupling functions $ f_n(\alpha_k) $ to encode nonlocal correlations and phase-mediated entanglement.
• Demonstrates that the collapse is continuous and smooth, not instantaneous, with a transient phase potentially detectable in high-resolution interference experiments.

Experimental results

Research questions

• RQ1 Can the apparent indeterminacy in single-system quantum measurements be explained by deterministic chaos in the quantum phase rather than intrinsic randomness?
• RQ2 How can a nonlinear, nonunitary, and irreversible evolution law be derived from a fundamental physical principle to replace the standard collapse postulate?
• RQ3 What role does the absolute phase play in mediating nonlocal correlations, such as those in EPR-type entanglement, within a dynamical reduction framework?
• RQ4 Can the characteristic reduction time $ \tau_r $ be linked to measurable physical parameters like interaction energy or gauge field nonlinearity?
• RQ5 Is the continuous, chaotic collapse process experimentally distinguishable from the instantaneous collapse of standard quantum mechanics?

Key findings

• The model reproduces Born's rule for probability distribution through chaotic, indeterminable phase evolution, eliminating the need for statistical ensembles.
• The collapse is continuous and irreversible, with a characteristic time $ \tau_r $ that depends on the strength of the nonlinear interaction, which is inversely proportional to the interaction energy.
• The collapse process is governed by a single nonlinear evolution equation, unifying unitary and nonunitary dynamics under one framework.
• The phase acts as a hidden variable that mediates nonlocality, providing a physical mechanism for 'spooky action at a distance' through phase coupling.
• The transient phase of the collapse is potentially detectable in high-resolution interference experiments, offering a possible experimental signature.
• The model suggests that the reduction time $ \tau_r $ is small for macroscopic systems and large for microscopic ones, consistent with the classical limit.