Shifting the Classical-quantum Boundary: Insights From Pilot-wave Hydrodynamics

In the legal proceedings of a criminal case, the prosecution must prove their point “beyond a shadow of a doubt.” In civil litigation, the burden of proof is much less stringent: the prosecution need only prove that a stated claim is “more likely than not.” Let us proceed by litigating the completeness of quantum theory at the level of a civil rather than criminal case; thus, we seek to show that it is more likely than not that quantum theory is incomplete.

To do so, we emphasize three points:

(i) A newly discovered hydrodynamic pilot-wave system has provided classical analogs for some of the most beguiling quantum systems.
(ii) This hydrodynamic system is a macroscopic realization of the double-solution pilot-wave theory proposed by French physicist Louis de Broglie in the 1920s.
(iii) New classical pilot-wave models—inspired by pilot-wave hydrodynamics—are providing fresh insights into longstanding conceptual difficulties with the standard quantum formalism.

For context, geophysicist Alfred Wegener’s 1912 theory of continental drift won over some of his contemporaries but was not widely accepted by the scientific community until 1962. Wegener could thus be said to have won the civil case but not the criminal case. Let us try to do likewise.

The field of pilot-wave hydrodynamics was pioneered by French physicists Yves Couder and Emmanuel Fort in 2005 [5], advanced by my group at the Massachusetts Institute of Technology, and is currently under investigation in several research groups worldwide [2]. The list of established hydrodynamic quantum analogs (HQAs) is steadily growing and now includes single-particle diffraction and interference, quantized orbits, probabilistic tunneling, statistical projection effects in corrals, interaction-free measurement, spin lattices, Kapitza-Dirac diffraction, super-radiance, Anderson localization, and static Bell violations [3, 4]. The evolution of their reception has been noteworthy. The first HQAs were generally met with derision: “Of course one cannot analogize quantum systems with classical mechanics; quantum mechanics is nonlocal!” As we continue to expand the list of HQAs, new challenges are presented: “Okay, you’ve done A-Q, but you’ll never do R.” And so it goes as we march through the alphabet.

The analogs can be either quantitative or qualitative, but they collectively suggest that there need be no conceptual distinction between microscopic and macroscopic physics; when a particle serves as a resonant excitation of a field, it may exhibit quantum features at either micro- or macroscopic scales. Of course, it is a logical possibility that the similarities between the two systems are mere coincidence. But as the compendium of HQAs grows, this possibility becomes progressively less likely, and it becomes more likely that the similarities arise because the two systems have comparable underlying dynamics.

In pilot-wave hydrodynamics, a millimetric droplet self-propels along the surface of a vibrating liquid bath (see Figures 1a-1c) [5]. The droplet interacts with a wave field of its own making, its self-potential. A key feature of the system is that the droplet bounces at or near the Faraday frequency, in resonance with its pilot wave. At each impact, it thus generates a quasi-monochromatic standing waveform with the Faraday wavelength whose longevity is prescribed by the bath’s vibrational acceleration. The resulting physical picture is of a vibrating particle acting as a moving source of monochromatic waves in a medium that serves as the system’s memory. The droplet dynamics are non-Markovian because the self-potential that pilots the droplet depends on the droplet’s history. In an ever-growing number of settings, these non-Markovian dynamics give rise to features previously thought to be peculiar to the quantum realm, where they are often regarded as evidence of quantum nonlocality [3, 4].

The walking-droplet system has three distinct timescales — those of particle vibration, particle self-propulsion, and statistical convergence [2]. Strobing the system at the Faraday frequency reveals a droplet that surfs on a monochromatic pilot-wave envelope with the Faraday wavelength, but eliminates from consideration the timescale of wave generation (see Figure 1b). Resolving all three timescales has proven to be critical when rationalizing the emergent quantum-like features. For example, variability in the bouncing dynamics—associated with the disruption of resonance between droplet and wave—directly affects the emergent statistical behavior in the analogs of both Kapitza-Dirac diffraction and the quantum corral (see Figures 1d and 1e). While standard quantum theory correctly predicts the statistical behavior of quantum systems, it offers no description of the underlying dynamics. HQAs thus suggest that quantum theory is incomplete in the sense that it only resolves one of the three timescales that would be required for a rational, local quantum theory.

An important signpost along the HQA path was the realization that the physical picture suggested by pilot-wave hydrodynamics has a historical precedent. Specifically, it corresponds to that proposed by de Broglie in the 1920s in his double-solution pilot-wave theory [7]. As in pilot-wave hydrodynamics (see Figures 1d and 1e), de Broglie’s theory has two waves: the instantaneous pilot wave and the emergent statistical wave. Also as in pilot-wave hydrodynamics, there are three characteristic timescales — enabling a mapping between the two systems [2].

In de Broglie’s theory, particle vibration at the Compton frequency \((\omega_c=mc^2 \hbar)\) generates a pilot wave that conforms to the Klein-Gordon equation. If the particle is dressed in a monochromatic pilot wave, then the particle speed and group velocity of its wave must be equal, yielding the de Broglie relation \(p=\hbar k\). In all HQAs, the Faraday wavelength plays the role of the de Broglie wavelength \(\lambda=2\pi/k\) in their quantum counterparts. Strobing the quantum system at the Compton frequency would reveal a pilot-wave envelope that satisfies the linear Schrödinger equation, but eliminate from consideration the timescale of wave generation. De Broglie proposed—but never proved—that this type of pilot-wave dynamics could yield emergent statistics of the form predicted by standard quantum theory. A century later, pilot-wave hydrodynamics has provided evidence in support of his proposal.

The notion of nonlocality, or action at a distance, should be anathema to any rational scientist. Nevertheless, most physicists have made peace with it; they either remain agnostic on the subject or believe it to be an essential, inescapable feature of quantum physics. Because standard quantum theory describes probabilities but not particle dynamics, nonlocality is perceived to be everywhere — in wavefunction collapse, single-particle interference, the quantum mirage, and interaction-free measurement. Correlation at a distance is taken as evidence of action at a distance. HQAs have demonstrated that if we adopt de Broglie’s physical picture of quantum dynamics, we need not invoke nonlocality for any such effects. In short, HQAs suggest that quantum nonlocality is a misinference that is rooted in the incompleteness of quantum theory. While nonlocality is a feature of quantum theory, it need not be a feature of quantum physics.

What about Bell’s theorem? The experimental violation of Bell’s theorem is widely accepted as proof that any hidden variable theory for quantum mechanics must be nonlocal. Of course, one is more inclined to believe a proof that something exists if one perceives it to be everywhere. I have been puzzling over Bell’s theorem (its lacunas, loopholes, and implications) for more than 40 years and have come to the conclusion that the matter is not yet settled. For example, it is not entirely clear that the assumptions made in its derivation apply to non-Markovian, pilot-wave systems. Moreover, Bell’s theorem was first derived a century earlier by George Boole and numbered among his “conditions of possible experience,” from which one might reasonably infer that Bell violations hint at a deep misunderstanding [11]. While mathematical proofs are nonnegotiable, one should be deeply suspicious of impossibility proofs as they pertain to the physical world. Indeed, the history of impossibility proofs in quantum mechanics should fuel one’s skepticism [1]. In my view, such impossibility proofs smack of the best laid plans. “Our holiday will be perfect and here’s the proof: I have booked our flights, the finest hotels and restaurants. What could possibly go wrong?”

De Broglie did not specify the nature of the quantum pilot wave. Stochastic electrodynamics is a modern extension of de Broglie’s theory, in which the pilot wave is sought in the electromagnetic quantum vacuum [8]. Others have suggested the appealing prospect of pilot waves being gravitational in origin, in which case de Broglie’s pilot waves would exist in the fabric of spacetime. By redefining the boundary between classical and quantum, HQAs are prompting a revisitation of quantum foundations [11]. Most importantly, they have furnished a physical picture that allows one to develop physical insight into quantum systems. When we adopt this physical picture, many conceptual problems in the standard formalism simply vanish. For instance, wave function collapse and the superposition of states are simply attributable to the incompleteness of quantum theory. Beyond the many quantum features that HQAs capture directly, recent theoretical explorations of generalized pilot-wave dynamics have suggested viable, classical rationales for Bohr-Sommerfeld orbital quantization, spin (see Figures 1f and 1g), and single-particle Fraunhofer diffraction (see Figures 1h and 1i). Recent static Bell violations—achieved with classical pilot-wave models—raise the following question: Might wave-mediated chaotic synchronization account for entanglement?

Bohmian mechanics have provided another valuable touchstone for HQAs. In his monograph on the subject, theoretical physicist Peter Holland enumerates the central conceptual difficulties of quantum physics [10], which I paraphrase here. First, individual events are unpredictable, but coherent statistics emerge when one considers a large number of such events. Central puzzling quantum phenomena include self-interference of particles and the tunneling of particles across a barrier that would be forbidden to classical particles. Atoms and molecules exist only in certain discrete energy states and do not collapse, as would their classical counterparts. Quantum particles may possess spin — a form of intrinsic, nonclassical angular momentum. Finally, quantum particles might be entangled: the properties of one particle can depend on those of an arbitrarily distant particle with which it has interacted in the past. Pilot-wave hydrodynamics suggests a physical picture that would provide classical rationale for all such features while averting the need to invoke nonlocality.

I rest my case in the civil prosecution of the completeness of quantum theory and invite the applied mathematics, dynamical systems, physics, philosophy, and engineering communities to join in its criminal prosecution.

A senior colleague of mine liked to cast applied mathematicians as the sherpas of physics. He evoked an image of sherpas waiting patiently at the summit, smoking cheroots, their cargo strewn willy-nilly, wondering when their glorious expedition leader might be joining them. In the case of HQAs, a more appropriate metaphor would be that of explorers surveying new terrain with a view to pushing back the border between the classical and quantum realms, transforming terra incognita into terra firma. While the landscape is vast—and Mount Bell casts a long shadow—we now have a well-established base camp from which to launch our expeditions and ascents. Come join the adventure, enjoy the spectacular views — and pass the cheroot.

Acknowledgments: I gratefully acknowledge the financial support of the U.S. National Science Foundation through grant CMMI-2154151 and the Office of Naval Research through grant N000014-24-1-2232.

References 
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