Chaos in a Nonlinear Wavefunction Model: An Alternative to Born's Probability Hypothesis
W. David Wick
TL;DR
This work investigates whether intrinsic chaos can arise in nonlinear wavefunction dynamics as an alternative to Born's probability hypothesis. It develops a Determinant Criterion for Instability (DCI) based on the Jacobian $M(t)$ with $\det M < 0$ and analyzes a finite-dimensional three-qubit model, complemented by simulations that estimate the maximal Lyapunov exponent $\gamma$ via tangent vectors. The results show a positive $\gamma$ beyond a threshold near $w \approx 1.15$, and an accompanying algebraic framework yields sufficient conditions for instability that extend to continuum settings via operator inequalities and a proposed Proposition. The continuum extension suggests analogous instability criteria, indicating that chaos may delineate the boundary between classical and wavefunction physics and potentially underpin stochastic outcomes without invoking intrinsic randomness in the wavefunction itself. This advances a dynamical interpretation of quantum outcomes and offers concrete criteria and methods for linking nonlinear wavefunction dynamics to observable chaos.
Abstract
In a prior paper, the author described an instability in a nonlinear wavefunction model. Proposed in connection with the Measurement Problem, the model contained an external potential creating a ``classical'' instability. However, it is interesting to ask whether such models possess an intrinsic randomness -- even ``chaos" -- independent of external potentials. In this work, I investigate the criterion analytically and simulate from a small (``3 qubit") model, demonstrating that the Lyapunov exponent -- a standard measure of ``chaos" -- is positive. I also extend the instability criterion to models in the continuum. These results suggest that the boundary between classical and wavefunction physics may also constitute the threshold of chaos, and present an alternative to Max Born's ad hoc probability hypothesis: random outcomes in experiments result not from ``wave-particle duality" or ``the existence of the quantum," but from sensitive dependence on initial conditions, as is common in the other sciences.