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arXiv:2502.02698

Chaos in a Nonlinear Wavefunction Model: An Alternative to Born's Probability Hypothesis

W. David Wick

Published Feb 4, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2502.02698

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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.
Paper I of this series introduced a nonlinear version of quantum mechanics that blocks cats, and paper II postulated a random part of the wavefunction to explain outcomes in experiments such as Stern-Gerlach or EPRB. However, an ad hoc extra parameter was assumed for the randomness. Here I provide some analytic and simulation evidence that the nonlinear theory exhibits sensitive dependence on initial conditions in measurement scenarios, perhaps implying that the magnitude of randomness required is determined by structural features of the model, and does not require a free parameter.
In two previous papers the author described ``Islands of Instability" that may appear in wavefunction models with nonlinear evolution (of a type proposed originally in the context of the Measurement Problem). Such ``IsoI" represent a new scenario for Hamiltonian systems implying so-called ``chaos". Criteria was derived for, and shown to be fulfilled in, some finite-dimensional (multi-qubit) models, and generalized in the second paper to continuum models. But the only example produced of the latter was a model whose linear Schrodinger equation was exactly-solvable. As exact solutions of many-body problems are rare, here I show that the instability criteria can be verified by plugging test-functions into certain computable expressions, bypassing the solvability blockade. The method can accommodate realistic inter-molecular potentials and so may be relevant to instabilities in fluids and gasses.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
We analyze the consequences of iterative measurement-induced nonlinearity on the dynamical behavior of qubits. We present a one-qubit scheme where the equation governing the time evolution is a complex-valued nonlinear map with one complex parameter. In contrast to the usual notion of quantum chaos, exponential sensitivity to the initial state occurs here. We calculate analytically the Lyapunov exponent based on the overlap of quantum states, and find that it is positive. We present a few illustrative examples of the emerging dynamics.
There has been proven that mathematical origins of randomness in quantum and Newtonian physics are coming from the same source that is dynamical instability. However in Newtonian physics this instability is measured by positive finite Liapunov exponents averaged over infinite time period, while in quantum physics the instability is accompanied by a loss of the Lipchitz condition and represented by an infinite divergence of trajectories in a singular point. Although from a mathematical viewpoint such a difference is significant, from physical viewpoint it does not justify division of randomness into “deterministic “(chaos) and “true” (quantum physics). The common origin of randomness in Newtonian and quantum physics presents a support of the correspondence principle that is being searched by quantum chaos theory.
How classical chaos emerges from quantum mechanics remains a central open question, as the unitary evolution of isolated quantum systems forbids exponential sensitivity to initial conditions. A key insight is that this quantum-classical link is provided by measurement processes. In this work, we identify gain competition in a chaotic photon gas as an operational quantum measurement that selects single motional modes from an initial superposition through stochastic, nonlinear amplification. We show that this mechanism naturally gives rise to classical chaotic behavior, most notably sensitivity to initial conditions. Our results provide a concrete physical mechanism for the quantum-classical transition in a chaotic system and demonstrate that essential aspects of quantum measurement-state projection, Born-rule-like selection, and irreversibility-can naturally emerge from intrinsic gain dynamics.