Bohmian mechanics without pilot waves

Bing-Ying Li, Jian-Ming Zhang, Yu-Cheng Chang +1 more

E. Deotto, G. C. Ghirardi

Emmanuel E. Haven, Andrei Yu. Khrennikov

The pilot wave function and its uses outside of quantum mechanics Let us recall our discussion on Bohmian mechanics in Chapter 6. The idea of using Bohmian mechanics outside of quantum mechanics is not new anymore. Khrennikov [1] provides for an overview on how this model can contribute in precise terms to areas such as economics and finance.1 Historically, the work by Bohm and Hiley [2] and also Hiley and Pylkkanen [3] brought forward the idea that the pilot wave function could be seen as a wave function containing information. In Khrennikov [4], Choustova [5], and Haven [6], the idea of using pilot wave theory to finance was investigated. Khrennikov [1] (p. 160) remarks that “the force induced by the pilot wave field does not depend on the amplitude of the wave.” He cites the argument made by Bohm and Hiley [2] that because of this property the pilot wave is an information wave. The various interpretations (with applications) of the wave function will be covered in detail in Chapter 13. Here are some of the key features which can be of use in an financial/economics setting. When the wave function is not factorized, then a change in the price of a stock i will affect the prices of stocks, j , with j ≠ i . See Khrennikov [1] (p. 161). The Bohmian theory, as we remarked already in Chapters 1 and 6, is non-local.

K. Berndl, M. Daumer, D. Dürr +2 more

Valeri Frumkin, David Darrow, John W. M. Bush +1 more

In certain instances, the particle paths predicted by Bohmian mechanics are considered to be at odds with classical intuition, and so deemed to be ``surreal.'' Through a combined experimental and numerical study, the authors demonstrate that individual trajectories in the hydrodynamic pilot-wave system exhibit the key features of their surreal Bohmian counterparts.

David Darrow

Abstract Bohmian mechanics supplements the quantum wavefunction with deterministic particle trajectories, offering an alternate, dynamical language for quantum theory. However, the Bohmian wavefunction evolves independently of these trajectories, and is thus unaffected by the observable properties of the system. While this property is widely assumed necessary to ensure agreement with quantum mechanics, much work has recently been dedicated to understanding classical pilot-wave systems, which feature a two-way coupling between particle and wave. These systems—including the “walking droplet” system of Couder and Fort (Couder and Fort (2006) Phys. Rev. Lett. 97:154101) and its various abstractions (Dagan and Bush (2020) CR Mecanique 348:555–571; Durey and Bush (2020) Front. Phys. 8:300; (2021) Chaos 31:033136; Darrow and Bush (2024) Symmetry 16:149)—allow us to investigate the limits of classical systems and offer a touchstone between quantum and classical dynamics. In this work, we present a general result that bridges Bohmian mechanics with this classical pilot-wave theory. Namely, Darrow and Bush ((2024) Symmetry 16:149) recently introduced a Lagrangian pilot-wave framework to study quantum-like behaviours in classical systems; with a particular choice of particle-wave coupling, they recover key dynamics hypothesised in de Broglie’s early double-solution theory (de Broglie (1970) Foundations Phys. 1:5–15). We here show that, with a different choice of coupling, their de Broglie-like system reduces exactly to single-particle Bohmian mechanics in the non-relativistic limit. Our result clarifies that, while multi-particle entanglement is impossible to replicate in general with local, classical theories, no such restriction exists for single-particle quantum mechanics. Moreover, connecting with the previous work of Darrow and Bush, our work demonstrates that de Broglie’s and Bohm’s theories can be connected naturally within a single Lagrangian framework. Finally, we present an application of the present work in developing a single-particle analogue for position measurement in a de Broglie-like setting.