Chapter 2

The State of Play in Hydrodynamic
Quantum Analogs
John W. M. Bush, Konstantinos Papatryfonos, and Valeri Frumkin

Abstract We describe the manner in which the physical picture emerging fromhydrodynamic quantum analogs (HQAs) may serve to resolve some of the longstanding difficulties of quantum mechanics. We enumerate some of the mostsignificant intellectual cul-de-sacs of quantum mechanics, and the manner in which
HQA suggests a route past them. Particular attention is given to enumerating themany guises of quantum nonlocality as it appears in the standard quantum interpretations. We illustrate how one might misinfer such nonlocality from the walkingdroplet system if one had an incomplete description of the system dynamics, if thevariables required for its complete description were hidden rather than in plain sight.
We highlight recent work that illustrates how phenomena typically attributed tononlocality in quantum systems may be rationalized in terms of classical, pilot-wave
dynamics. Finally, we define the frontiers of the field of hydrodynamic quantum
analogs, including attempts to achieve classical entanglement by demonstrating
Bell violations in pilot-wave hydrodynamics, and attempts to develop a model ofquantum dynamics informed by the walking-droplet system.
Keywords Pilot-wave hydrodynamics · Walking droplets · Quantum analogs ·Local realism

J. W. M. Bush (🖂) · V. Frumkin
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USAe-mail: bush@math.mit.edu
K. Papatryfonos
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, Paris, France© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
P. Castro et al. (eds.), Advances in Pilot Wave Theory, Boston Studies
in the Philosophy and History of Science 344,
https://doi.org/10.1007/978-3-031-49861-9_2

78J. W. M. Bush et al.

2.1 Introduction
The walking-droplet system discovered in 2005 by Yves Couder and EmmanuelFort (Couder et al., 2005; Couder and Fort, 2006) has since captured many features
previously thought to be exclusive the the microscopic, quantum realm (Bush, 2015;
Bush and Oza, 2020). The growing grocery list of hydrodynamic quantum analogs
includes single-particle diffraction and interference (Couder and Fort, 2006), quantized orbits (Fort et al., 2010; Perrard et al., 2014), unpredictable tunneling (Eddiet al., 2009a), Friedel oscillations (Sáenz et al., 2020), spin lattices (Sáenz et al.,2021), statistical projection effects (Sáenz et al., 2018) in corrals (Harris et al.,2013), surreal trajectories (Frumkin et al., 2022) and superradiance (Papatryfonos
et al., 2022a; Frumkin et al., 2023). This pilot-wave hydrodynamic system has
extended the range of classical systems and so provided a platform for delineating
between what can and cannot be understood about quantum systems from a classicalperspective.
The hydrodynamic pilot-wave system consists of a millimetric liquid droplet suspended on the surface of a vibrating liquid bath (Couder et al., 2005) (Fig. 2.1a–c). In
certain parameter regimes (specifically, for certain drop sizes, liquid properties, andvibrational accelerations), the bouncing droplet achieves resonance with the bath’s
most unstable Faraday wave mode and destabilizes into a walking state in which it
is guided or ‘piloted’ by its own wave field. The resulting ‘walker’ then consists ofa droplet self-propelling along the bath surface, dressed in a quasi-monochromatic
pilot-wave field (Fig. 2.1c). The key features of the system are two-fold (Bush andOza, 2020). First, the resonance between the droplet and its wave ensures that

Fig. 2.1 (a) A millimetric droplet bounces in place on a vibrating bath. (b) When the droplet’s
bouncing frequency matches that of the vibrating bath’s most unstable Faraday mode, thedrop destabilizes into a ‘walker’, and is self-propelled across the bath by its pilot wave. (c)Comparison between experimental measurements (top) and simulation (bottom) of the wave fieldaccompanying a walker moving from left to right (Damiano et al., 2016). Color bar indicates thewave height in microns. (d) The pilot-wave field computed for a quantum particle propelled bya wave field that evolves according to the Klein-Gordon equation and is excited locally by theparticle vibrating at the Compton frequency, .ωc = mc2 /h̄ (Durey and Bush, 2020)

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the pilot-wave field is quasi-monochromatc, a feature that is responsible for the
emergence of quantized states in many settings. Second, the persistence of this pilotwave field renders the droplet dynamics non-Markovian: computing its trajectory
requires that one consider its history. The droplet is thus endowed with ‘pathmemory’ (Eddi et al., 2011), and navigates a quasi-monochromatic potential of its
own making, as is ultimately responsible for all of the system’s emergent quantum
features (Bush and Oza, 2020).
The most valuable aspect of the walking-droplet system is that it furnishes amacroscopic example of wave-particle interaction, and so a physical picture for
how quantum dynamics might conceivably look. Importantly, this physical picture,of a vibrating particle moving in resonance with its own wave field, is not newand has features of several extant realist theories of quantum dynamics, including
de Broglies double-solution pilot-wave theory (de Broglie, 1926, 1970, 1987),
stochastic electrodynamics (de la Peña et al., 2015) and the Zitterbewegung theoryof quantum mechanics (Hestenes, 1990). Each of these theories invokes the particles
internal vibration at the Compton frequency as the source of its guiding wave. The
commonality of this physical picture has allowed the HQA community to connect toothers attempting to make sense of quantum mechanics. Moreover, it has motivatedthe development of a generalized classical pilot-wave theory that allows one to
explore parameter regimes inaccessible in the laboratory with the walking-droplet
system (Bush, 2015; Oza et al., 2018; Durey and Bush, 2021), as well as forge links
with and extend extant quantum pilot-wave theories (Dagan and Bush, 2020; Durey
and Bush, 2020).
The formalisms of pilot-wave hydrodynamics and quantum mechanics aremarkedly different. In the former, the HQA community has developed a hierarchyof progressively more sophisticated theories to describe the droplet and wave
dynamics, both of which are required for an adequate desription of the droplet’s
trajectory (Turton et al., 2018). The resulting particle statistics are viewed as
an emergent feature, the rationalization of which is not always straightforward.
Conversely, quantum mechanics provides an explicit theory for the evolution of thesystem’s statistics, as prescribed by the wave function. According to the standard
Copenhagen Interpretation, there is no notion of an underlying dynamics, so no needfor a trajectory equation. Bohmian mechanics furnishes a trajectory equation bypositing that the particle moves in response to the quantum wavefunction (Holland,1995). Throughout this review, we describe recent attempts to reconcile the very
different theoretical formalisms developed to describe pilot-wave hydrodynamics
and quantum mechanics. Specifically, we describe recent attempts to develop astatstical theory for pilot-wave hydrodynamics and a dynamical theory for quantum
mechanics informed by the walker system.
We proceed by enumerating a number of concepts that may seem beguilingfrom the point of view of quantum mechanics, but become less problematic when
considered from the new perspective offered by HQA. We advance in increasingorder of difficulty, commencing with notions that transform from inscrutable totrivial, such as wave-particle duality and wave-function collapse. We move on toshow how quantized states and coherent wave-like statistics naturally emerge from

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classical pilot-wave dynamics. We highlight the manner in which quantum nonlocality in its various guises might be misinfered from the walking-droplet system.
Finally, we discuss current attempts to demonstrate violation of Bell’s inequality
with the pilot-wave hydrodynamic system, which remains the central challenge to
any local realist theory of quantum mechanics. We conclude with a discussion of
the benefits of the perspective and enhanced physical intuition offered by HQAs onquantum mechanics and quantum foundations.

2.2 Wave-Particle Duality and Complementarity
Both matter and radiation possess a remarkable duality of character, as they sometimes
exhibit the properties of waves, at other times those of particles. Now it is obvious that athing cannot be a form of wave motion and composed of particles at the same time—thetwo concepts are too different.
– Heisenberg, ‘The Physical Principles of the Quantum Theory’ (Heisenberg, 1930).

Wave-particle duality is a notion common in both optics and quantum mechanics.
It first arose in the debate over the nature of light, where it became apparent that lightsometimes behaves like a particle, other times like a wave. Huygens and Fresnel
were two of the most prominent proponents of the wave nature of light. In 1678,
Huygens proposed that every point on a wavefront of a light beam may be seen
as a new source, emitting spherical waves in the forward direction (Miller, 1991).
These secondary wave sources interfere with each other to produce the advancing
wavefront. Huygens thus explained linear and spherical wave propagation andderived the laws of reflection and refraction, but failed to rationalize other opticalphenomena, such as the diffraction from an edge or an aperture (Miller, 1991).
More than a century later, Fresnel combined his own theory of interference with
Huygens’s principle, which enabled him to rationalise these diffraction effects (Santos et al., 2018). Newton championed the corpuscular view, that light consistedof a series of discrete, localized corpuscles, now known as photons, ‘skipping on
the ether like stones on a pond’ (Newton, 1704). Subsequently, Thomas Young’s
ripple tank experiments demonstrated that the diffraction of light was consistent
with its having a wave nature. The wave view became dominant when Maxwell(1873) demonstrated that all forms of light (infrared, visible and ultraviolet) could
be described as electromagnetic waves oscillating at different frequencies. When
the wave-particle debate seemed all but settled, it was again revived in 1905 by
Einstein’s explanation of the photoelectric effect in terms of ‘light quanta’, a criticalstep in the development of quantum mechanics (McKagan et al., 2009).
The concept of complementarity was introduced by Niels Bohr as an essential
feature of quantum theory (Folse, 1985). It asserts that a complete knowledgeof quantum phenomena requires a simultaneous description of particle and wave
properties, hence a quantum version of wave-particle duality. Specifically, prior to2 The State of Play in Hydrodynamic Quantum Analogs

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being measured, a single quantum particle is described in terms of its wave functionthat evolves according to Schrodinger’s equation. Measurement forces the wave tocollapse to a particle, for example when particles passing through slits arrive at
the detection screen. Bohr asserted that it is impossible to observe both wave andparticle aspects simultaneously, but both notions need be retained. Finally, quantum
systems have certain pairs of ‘complementary’ properties that cannot be observed
simultaneously. These are also known as non-commuting observables and include,for example, position and momentum, and different components of a particle’s spin.
De Broglie’s theory of matter waves was built upon his premise that the universe
is composed of two elements, light and matter. Since light has both corpuscularand wave aspects, so too must matter. His was an attempt to reconcile quantum
mechanics with Einstein’s theory of relativity. He proposed that a particle of mass
m has an internal frequency that may be deduced by equating its rest mass energy
mc2 to its wave energy h̄ω. The resulting de Broglie-Einstein equation defines thefrequency of particle vibration to be the Compton frequency, ωc = mc2 /h̄. He thus
envisioned microscopic particles generating, then moving in response to, their own
wave fields. He proposed that a free particle moves in response to gradients in thephase of its monochromatic guiding or ‘pilot’ wave, with a wavelength prescribed by
the de Broglie relation, p = h̄k. He imagined, but never proved, that this pilot-wave
dynamics might give rise to statistical behavior consistent with that described by thestandard formulation of quantum mechanics. His theory thus involved two waves,the real pilot-wave responsible for guiding the particle, and the emergent statisticalwave, and his unfinished theoretical program was known as the double-solution
pilot-wave theory (Hatifi et al., 2018; Colin et al., 2017). De Broglie’s answer tothe question of ‘particle or waves?’ was thus simply ‘particle and wave’ (Bell,1987). On the basis of this physical picture, he predicted electron diffraction and
was awarded the Nobel prize in 1929. Nevertheless, this physical picture has been
largely ignored by the physics community (Bell, 1987) until its recent revival by the
HQA community (Bush, 2015; Bush and Oza, 2020).
The walking droplet is inarguably a classical realization of wave-particle duality(Figs. 2.1b and 2.2). The ‘walker’ has both particle and wave aspects. Without the
droplet, there would be no source of waves, and without the accompanying wave
field, the droplet wouldn’t self propel. The walker is, moreover, an embodimentof the physical picture proposed by de Broglie in his double-solution pilot-wave
theory (Bush, 2015). In the walker system, the Faraday frequency plays the role of
the Compton frequency, the Faraday wavelength that of the de Broglie wavelength.
The walker moves in response to gradients in the wave amplitude rather than phase,but the resonance condition respected by the free walker effectively renders this
distinction a moot point. The pilot-wave of the walker is quasi-monochromatic,of a distinctive, horseshoe-like form (Fig. 2.1c), while de Broglie envisaged amonochromatic plane wave of the form illustrated in Fig. 2.1d (Durey and Bush,2020).

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2.3 Quantized States
I wish first to show in the simplest case of the hydrogen atom that the usual rates forquantization can be replaced by another requirement, in which mention of “whole numbers”
no longer occurs. Instead the integers occur in the same natural way as the integers
specifying the number of nodes in a vibrating string.
– Schrödinger, ‘Quantisierung als Eigenwertproblem’ (Schrödinger, 1926).

In the pilot-wave hydrodynamic system, quantized dynamical states are a central
feature, apparent in both the structure of droplet aggregates (Fig. 2.2) and orbital
dynamics (Fig. 2.3). Bouncing droplets may form either static bound states, pairs,
trios, rings or lattices (Eddi et al., 2009b), or dynamic bound states comprised ofratcheting (Eddi et al., 2008; Galeano-Rios et al., 2018) orbiting (Couder et al.,2005; Protière et al., 2008; Oza et al., 2017) or promenading pairs (Borghesi
et al., 2014; Arbelaiz et al., 2018). In all such states, the interdrop distance isquantized, prescribed by the Faraday wave length. In both rings and lattices, thedrop aggregates are most stable when the droplets bounce in local minima of theircollective wave field (Couchman and Bush, 2020). A critical requirement for the
emergence of quantization is the synchronization of the droplets, which ensures acoherent quasi-monochromatic wave form that serves as the trapping self-potentialof the aggregate.
Quantization is also a key feature of orbital pilot-wave dynamics, as arises
when walkers move subject to constraints. Three such systems have been exploredFig. 2.2 Quantized bound states, both static and dynamic, arise owing to the quasimonochromatic wave field generated by the droplets bouncing in resonance with the bath’s most
unstable Faraday mode. (a) Two walkers locked into a circular orbit. (b) A promenading pair: drops
move together in the same direction, with the lateral distance between them varying periodicallywith time (Arbelaiz et al., 2018). (c) A colinear trio of stationary bouncers. (d) Rings of bouncing
droplets may form with quantized radii. We show here the circular motion arising at the onset of
instability of a stable ring induced as the driving acceleration exceeds a critical value (Couchman
and Bush, 2020)

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Fig. 2.3 Quantized orbital states arise for (a) walkers in a rotating frame or (b) walkers confinedby a central spring force. (a) The solution curve for the orbital radius, r0 , as a function of rotation
rate, Ω, has both stable (blue) and unstable (red) branches. The absence of stable solutions atcertain radii leads to orbital quantization (Fort et al., 2010; Harris and Bush, 2014; Oza et al.,2014a). Inset: the walker moves along a circularly corrugated wave field, whose form imposes thequantization (Fort et al., 2010). (b) When a walker is confined by a radial spring force, a variety of
periodic orbits are accessible, all of which are quantized in both mean radius R̄ = R/λF and mean
orbital angular momentum L̄z = Lz /(mu0 λF ) (Perrard et al., 2014; Labousse et al., 2014a)

both experimentally and theoretically. When walkers move in a rotating frame, theCoriolis force plays a role analogous to the Lorentz force acting on a charge moving
in a uniform magnetic field (Fort et al., 2010). The expected continuum of circularinertial orbits arising at low memory are replaced by quantized circular orbits as thememory increases (Fort et al., 2010; Harris and Bush, 2014). The requirement fororbital quantization is that the orbital period of the drop is less than the memory
time; thus, the drop feels its own capillary wake, which serves to quantize its orbital
radius. The drop thus navigates its own potential, a circularly symmetric wave fieldwith the Faraday wavelength centered on the orbital center (Fig. 2.3a, inset). In thesequantized circular orbits, the Faraday wavelength plays a role analogous to the de
Broglie wavelength in Landau levels (Fort et al., 2010).
When walkers move in a central force, in addition to quantized circular orbits,a family of more elaborate orbits arise, including lemniscates and trefoils (Perrard
et al., 2014; Labousse et al., 2014a). As in the 2D quantum harmonic oscillator,
these orbits are quantized in both energy and mean angular momentum. A similar
progression of quantized orbits arise when a walker is confined to a small corral at
relative low memory, when the bounding geometry plays the role of the confining
potential (Cristea-Platon et al., 2018). Once again, the key requirement for the
emergence of quantized states is resonance between drop and bath, as assuresa quasi-monochromatic self-potential. Quantized orbital states emerge when thememory time exceeds the orbital period, so that the drop continuoulsy navigatesa highly structured potential of its own making.

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2.4 Single-Particle Diffraction
A phenomenon which is impossible, absolutely impossible to explain in any classical way,
and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.
– Richard Feynman (Feynman, 1964).

The most prominent example of particle-wave duality is found in single-particle
diffraction. The first experimental investigation of single-photon diffraction wasundertaken by a graduate student, G.I. Taylor, who subsequently went on to becomea prominent and influential fluid and solid mechanician (Batchelor, 2008). The
question posed him by his supervisor, J.J. Thomson, was whether a diffraction
pattern would emerge when light passed through a slit, even when the photons
passed through one at a time. The results of his experiments were conclusive: adiffraction pattern ultimately emerged. So, despite the photons passing throughthe slits one at a time, their superposition corresponded to a continuous wave
pattern (Taylor, 1909), indicating single-particle diffraction.
Feynman’s (Feynman, 1964) insistence on the inscrutability of the electron
double-slit experiments is somewhat puzzling in light of de Broglie’s work onelectron diffraction (de Broglie, 1926; Bell, 1987). The relevant experiments werefirst performed with electrons diffracting around a pair of obstacles, the complementof the double slit (Tonomura et al., 1989). The build-up of the resulting diffraction
pattern, into alternating bright and dark bands, has been numbered among the mostbeautiful experiments in the history of physics, the beauty presumably being rooted
in the common conception that the phenomenon is impossible to understand froma classical perspective. The mystery is not the particular form of the diffraction
pattern: the mystery is that the pattern emerges at all.
The original experiments of walkers passing through slits (Fig. 2.4) and accompanying simulations of Couder and Fort (2006) clearly showed a diffraction pattern.
As the drop approaches the slit, the distortion of its pilot wave by the barrierscauses the drop to be deflected, with certain deflection angles being preferred.
A more exhaustive series experiments performed in different laboratories havemade clear that single-particle diffraction is indeed a robust feature of pilot-wavehydrodynamics (Andersen et al., 2015; Pucci et al., 2018; Ellegaard and Levinsen,2020). Moreover, the presence of a second slit alters the emergent diffraction
pattern, another keystone of its quantum counterpart. The fact that the emerging
diffraction pattern does not generally conform to the Fraunhofer pattern, is neither
troubling nor surprising when one considers that the pilot-wave form is markedly
different in the walker systems than what one would expect to arise in de Broglie’s
mechanics (Fig. 2.1c, d). The diffraction of walking droplets reminds us that thephysical picture of de Broglie’s is sufficient to understand the basic mysteries ofelectron diffraction.

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Fig. 2.4 Single-particle diffraction and interference of a walking drop. (a) A walker passing
through a single slit is deviated from its initial path owing to the influence of the submerged
topography on its pilot wave. Histogram for the final deflection angle α in (b) the single-slit
experiments and (c) the double-slit arrangement. The droplet trajectories emerging from (d) the
single-slit and (f) the double slit experiments of Pucci et al. (2018). (e) The distribution ofdeflection angles in the single-slit arrangement. In the double-slit arrangement, the presence ofthe second slit alters the diffraction pattern of walkers through the first slit because the pilot wave
is influenced by both slits (Pucci et al., 2018). Panels (a)–(c) adapted from Couder and Fort (2006);
panels (d)–(f) from Pucci et al. (2018)

2.5 Wave-Like Statistics
The wave character of light is not vibrating stuff like a wave of water but rather a wavelike
function encoding information about where you’ll find the photon of light once it isdetected.
– Marcus du Sautoy, ‘The Great Unknown: Seven Journeys to the Frontiers of Science’(Sautoy, 2017).

One of the most compelling features of pilot-wave hydrodynamics is that it
naturally leads to the emergence of wave-like particle statistics reminiscent of
those arising in many quantum systems. We have already seen the wave patterns
arising in walker diffraction through slits (Fig. 2.4). Superpositions of dynamical
states have been demonstrated in a number of settings involving chaotic pilot-wavehydrodynamics. In the orbital pilot-wave systems first considered by Couder andFort, specifically walker motion in a rotating frame (Fort et al., 2010) and a central
spring force (Perrard et al., 2014), analogs of a charge moving in a uniform magnetic
field and the 2D quantum harmonic oscillator, respectively, quantized orbital states

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Fig. 2.5 The hydrodynamic corral consists of a single walker moving within a bounded domain,either circular (a)–(b) (Harris et al., 2013) or elliptical (d)–(g) (Sáenz et al., 2018). (a) The build-upof the particle trajectory, which is color-coded according to droplet speed. The resulting correlation
between droplet position and speed gives rise to a statistical signature (b) strongly reminiscent ofthat arising in its quantum counterpart (c) (Crommie et al., 1993b). In the elliptical corral, similar
speed maps (e) and histograms (f) emerge. The instantaneous wave field (d) is complex and timedependent, while the mean pilot-wave (g) closely resembles the droplet histogram (f)emerge as the memory is increased (Fig. 2.3). Eventually, at sufficiently highmemory, these orbital states destabilize, giving rise to chaotic states marked by
the drop switching intermittently between a number of finite accessible quantizedorbital states (Harris and Bush, 2014; Oza et al., 2014b; Labousse et al., 2014a).
The emergent wave-like statistics thus reflect a superposition of dynamical states,
and the precise form of the emergent statistics reflects the relative instability of theaccessible unstable orbits (Harris and Bush, 2014; Oza et al., 2014a; Labousse et al.,2014a).
Robust wave-like statistics also arose in Harris et al. (2013)’s experimental
investigation of a walker in a circular corral (Fig. 2.5a–c). In the high-memory
limit arising just below the Faraday threshold, the drop executes an erratic, chaotic
trajectory. Ultimately, the correlation between drop position and speed give rises
to a statistical signature of comparable form to the most unstable Faraday modeof the cavity. The emergent statistics in this ergodic system is virtually identicalto that arising when electrons are confined to the quantum corral (Crommie et al.,1993b; Fiete and Heller, 2003), with the Faraday wavelength again playing a roleanalogous to the de Broglie wavelength. The walker corral is marked by three
distinct timescales, those of droplet bouncing (∼0.01 s), droplet translation (∼2 s)
and statistical convergence (∼1 h). Given the vast difference in scales between thisexperiment and its quantum counterpart (e.g. the corral diameter is 3 cm rather than
75 Angstrom), the ability to resolve all three timescales in the laboratory is quite
remarkable.
While theoretical models have yet to capture satisfactorily the wave-like statistical behavior in corrals (Durey et al., 2020), its robustness was demonstrated ina subsequent study of an elliptical corral (Sáenz et al., 2018), where additional2 The State of Play in Hydrodynamic Quantum Analogs

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Fig. 2.6 The hydrodynamic analog of Friedel oscillations (Sáenz et al., 2020). (a) When a walker
approaches a deep well, it is drawn inward along a spiral path, then exits the well radially. (b)In-line speed oscillations along its outgoing path are evident in the droplet’s speed map. The
associated correlation between droplet speed and radial position leads to a coherent statisticalsignature similar in form to Friedel oscillations (Crommie et al., 1993)

quantum-like features were revealed, including superposition of statistical statesand statistical projection effects (Fig. 2.5d–g). Moreover, it was noted that the mean
pilot wave was very similar in form to the emergent particle histogram, a result
later rationalized by Durey et al. (2018, 2020), who demonstrated that the mean
pilot-wave field for either periodic or ergodic walker mation may be deduced froma convolution of the particle histogram and the stationary bouncer wave field. This
result provides the means to deduce the particle statistics from the mean pilot-wave
form, so plays a role analogous to that of Born’s Rule in quantum mechanics if one
identifies the mean pilot wave with the wave function (Kutz et al., 2023). Bush andOza (2020) discuss the relation between the mean-pilot-wave potential in the walker
system and the quantum potential in Bohmian mechanics.
Robust wave-like statistics were also revealed in the hydrodynamic analog ofFriedel oscillations (Sáenz et al., 2020, Fig. 2.6), waves in the density of states
surrounding impurities in the electron sea on a metal surface (Crommie et al., 1993).
The analog system consisted of a walker interacting with a deep well. The drop was
drawn inwards along a spiral path until crossing the center of the well then exiting
radially. As the drop exited the well, in-line speed oscillations were excited, and theresulting correlation between radial position and speed along the outgoing trajectory
gave rise to a statistical signature identical to that arising in Friedel oscillations, with
the Faraday wavelength again playing the role of the de Broglie wavelength. With
this concrete physical mechanism for the emergent statistics in the analog quantum
corral and Friedel oscillations, one can imagine that a similar mechanism might also
be at play in their quantum counterparts. At the very least, one can conclude that the
emergent statistics in both systems are not inconsistent with the notion of particle
trajectories.
According to the Copenhagen Interpretation, the act of measurement causes thewave function to collapse to a particular eigenstate of the system, the associatedprobability cloud describing the particle position to collapse instantaneously to apoint. Given the spatially extended nature of the associated wave form, this collapse

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would seem to violate the basic tenets of relativity (Einstein et al., 1935). Adherents
to the Copenhagen Interpretation typically sidestep this issue by stating that thewave function collapse does not represent a physical process, but merely an updateof information. However, if the wave function represents a complete description ofa quantum system, there should be no new information according to which it need
be updated (Hance and Hossenfelder, 2022). This criticism of the completenessof quantum theory was originally launched by Einstein et al. (1935), prompting
an exchange now referred to as the Debate over the Nature of Reality. Despite
the obscurity of Bohr’s response (Bohr, 1935), history has judged him to be thewinner (Bricmont, 2017, 2016).

2.6 The Measurement Problem
The electron, as it leaves the atom, crystallises out of Schrödinger’s mist like a genie
emerging from his bottle.
– Sir Arthur Stanley Eddington, Gifford Lectures (Eddington, 1927).

The state of a quantum system is described by the wave function, a vector
in a Hilbert space that evolves deterministically in time according to the linear
Schrodinger equation:
i h̄

.

∂ψ
= Ĥ ψ
∂t(2.1)

where Ĥ is the system’s Hamiltonian. Due to the linearity of the Schrodinger
equation, any superposition (linear combination) of solutions will also be a solution
and evolve in time according to Eq. (1). During measurement, the wave function“collapses” onto an eigenstate of the Hamiltonian that corresponds to a particularmeasurement outcome, the value measured in the laboratory (Griffiths, 2014).
Unlike the time evolution described by Eq. (2.1), the collapse process is non-linear,non-deterministic, and typically non-local (Bassi et al., 2013). Importantly, thetheory does not specify what precisely constitutes a measurement, which raises
the following fundamental questions. What exactly happens during measurement?
Is the “collapse” a physical process or merely an update of information? These
puzzles in the foundations of quantum mechanics are collectively referred to as‘the measurement problem’, and embodied in widely known paradoxes such as
Schrodinger’s cat (Schrödinger, 1935) and Wigner’s friend (Wigner, 1995).
Attempts to solve the measurement problem have led to the various interpretations of quantum mechanics. In some, such as the de Broglie–Bohm theory (Bohm,
1952a, 1925b; Holland, 1995) (a.k.a. Bohmian mechanics, not to be confused withde Broglie’s original double solution pilot-wave theory) and the Many Worlds
Interpretation (Everett, 1957), there is no measurement problem since the wave
function never undergoes a collapse process. In others, like the objective-collapse
models, a stochastic term is added to the Schrodinger equation in order to induce2 The State of Play in Hydrodynamic Quantum Analogs

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a spontaneous collapse of the wavefunction at some characteristic time scale.
The result is an emergent “classical” behavior for many particle systems, andan approximate quantum behavior for microscopic isolated systems (Bassi and
Ghirardi, 2003). Thus, while collapse is continuously occurring, it is not the resultof interaction with a measurement device.
An attempt to reconcile the Copenhagen interpretation with the measurement
problem has given rise to the notion of decoherence. Roughly speaking, decoherence
posits that, through interaction with the environment, a quantum system loses its
coherent properties, resulting in a classical superposition of probabilities. In termsof the density matrix formalism, the off-diagonal terms that represent quantum
interference disappear due to interaction with the environment, yielding a diagonal
matrix with the Born probabilities as its entries (Zurek, 2003). Decoherence thus
attempts to explain how quantum states evolve into classical probabilities, andso rationalize why we do not observe quantum superpositions in the laboratory.
However, it does not explain why we do not observe classical superpositions instead.
Schrodinger’s cat evolves from being simultaneously dead and alive, to being half
dead and half alive. However, when we observe the cat, it is either 100% deador 100% alive, corresponding to a density matrix consisting of a single non-zero
entry on its diagonal. Thus, while decoherence describes how quantum probabilities
become classical, it does not provide an entirely satisfactory resolution of themeasurement problem (Adler, 2003).
If we adopt the physical picture suggested by de Broglie and the walking droplets,
the measurement problem vanishes from consideration. In particular, wave function
collapse becomes a nonproblem (Bush and Oza, 2020). Consider the robust wavelike statistics emerging in the hydrodynamic corrals (Harris et al., 2013; Sáenz et al.,2018) (Fig. 2.5) or the analog Friedel oscillations (Fig. 2.6). If one were to assert that
this statistical waveform were a complete description of the system, then its collapse
into a discrete droplet in response to the act of observation might be troubling.
With the knowledge of the underlying pilot-wave dynamics, it is obvious that wave
function collapse is a feature of any statistical theory. Bush and Oza (2020) referto this as ‘statistical nonlocality’, the misinference of nonlocality owing to one’s
insistence on the completeness of a statistical theory. Wave-function collapse may
thus be seen as a dilemma only for those insistent on the completeness of a statisticaltheory, be it quantum or classical. Ditto for the measurement problem.

2.7 Quantum Superposition
One cannot in the classical sense picture a system being partly in each of two states andsee the equivalence of this to the system being completely in some other state. There isan entirely new idea involved, to which one must get accustomed and in terms of which
one must proceed to build up an exact mathematical theory, without having any detailed
classical picture.
– P.A.M. Dirac, Principles of Quantum Mechanics (Dirac, 1958).

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J. W. M. Bush et al.

Due to the linearity of the Schrodinger equation (Eq. (2.1)), any linear combination of its solutions also constitutes a solution of the equation, and thus represent avalid quantum state. These linearly combined states are called superpositions, andthey play a key role in the formalism and phenomenology of quantum mechanics.
Formally, if ψ1 and ψ2 represent two different eigenstates of the Hamiltonian, thenthe superposed state, ψ = aψ1 +bψ2 , is also a solution of the Schrodinger equation,with a, b being complex numbers such that |a|2 + |b|2 = 1. According to the
Copenhagen interpretation, during measurement the wave function collapses onto
one of the eigenstates ψ1 , ψ2 with probabilities |a|2 , |b|2 respectively. Notably, thecoefficients a, b are generally complex numbers, and can assume negative values.
As a result, the two eigenstates constituting the superposed state can destructively
interfere with one another, yielding phenomena such as single particle interference
as seen in the double-slit experiment (Sect. 2.4).
In their study of walker motion in an elliptical corral (Fig. 2.5d–g), Sáenz
et al. (2018) demonstrated that in the high-memory, chaotic regime, the emergent
statistics may be simply expressed in terms of the superposition of two cavity
modes, one being the corrals most unstable Faraday mode at the systems driving
frequency, the other being the most unstable mode at a nearby frequency. This
then represents a superposition of statistical rather than dynamical states. By using
bottom topography with high symmetry, they demonstrated that the relative weightsof the two modes could be tuned. Specifically, by placing a submerged well at thefocus of the ellipse, it favored one mode over the other. The analogous procedurein the quantum corral (Fiete and Heller, 2003), undertaken by manipulating
magnetic impurities on the metal surface, leads to so-called ‘statistical projection
effects’ (Moon et al., 2008; Manoharan et al., 2000). When an impurity is located at
the focus of an ellipse, the preferred mode is that with extrema at the foci, leading
to a projection effect referred to as the ‘quantum mirage’ (Manoharan et al., 2000).
The walker system thus provides a rational means of interpreting both statisticalprojection and mirage effects.
We have seen that pilot-wave hydrodynamics can account for wave-particle
duality and related phenomena such as single-particle interference. An important
open question is thus whether one can derive a description of the emergent statistics
equivalent to the Schrodinger formalism in quantum mechanics, from the dynamical
description of the walker in HQA. The fact that the walker system exhibits both
single-particle interference phenomena and the superposition of states suggests thatsuch may be the case. In the HQA community, efforts are currently being made to
develop a theory of walker statistics through consideration only of the pilot-wave
field (Kutz et al., 2023). Specifically, for a walker confined to a one-dimensionalwell, the evolution of the pilot-wave field is characterized as the system memory(vibration forcing) is increased progressively. A discrete set of wave modes emerge
sequentially, analogous to the new eigenstates of the wavefunciton arising as theparticle energy is increased. Finally, the mean wavefield is inverted to yield theparticle statistics, using Durey’s convolution result (Durey et al., 2018, 2020). This
recent study takes one step closer towards one current goal of the HQA community,
developing a wave theory for the statistics of walking droplets (Kutz et al., 2023).

2 The State of Play in Hydrodynamic Quantum Analogs

21

2.8 Nonlocality
That particular interpretation (Bohmian mechanics) has indeed a grossly non-local
structure. This is characteristic, according to the result to be proved here, of any suchtheory which reproduces exactly the quantum mechanical predictions.
– John S. Bell, On the Einstein Podolsky Rosen paradox (Bell, 1964).

The inference made from the experimental violation of Bell’s Theorem (seeSect. 2.10) has generally been that any hidden variable theory of quantum mechanics
must be non-local; thus, non-locality is seen by some as being a necessary featureof a theory of quantum dynamics (Maudlin, 2014). For example, having seen theexperimental violations of Bell’s Inequality (Aspect et al., 1982a), Bell was an
advocate of Bohmian mechanics on the grounds that it was non-local (Bricmont,2016). Nonlocality has different guises in the various interpretations of quantum
mechanics. For example, we have already seen in §2.5 that proponents of the
Copenhagen Interpretation must contend with the instantaneous collapse of the
wave function, a form of nonlocality simply rationalized from the new perspective
offered from HQAs (Bush and Oza, 2020).
Another manifestation of quantum nonlocality takes the form of an apparent
action at a distance, as arises in Bohmian mechanics. Specifically, if the position of
one particle in an entangled pair is changed, the position of its entangled counterpart
is instantaneously affected. In HQA, a number of systems have been considered
in which one would infer action-at-a-distance if the pilot-wave dynamics were notadequately resolved. For example, in the double-slit walker diffraction experiments,
the change prompted by the influence of the second slit on a droplet passing
through the first would be considered nonlocal (Fig. 2.4d). When walkers approach
submerged pillars (Harris et al., 2018) and wells (Sáenz et al., 2020, Fig. 2.6), theyexperience long-range lift forces that depend in a particular fashion on the distancefrom the obstacle. Were it not known that such forces are wave-mediated, one might
misinfer that they imply action at a distance.
In single-particle Bohmian mechanics, nonlocality enters through the quantum
potential. In classical mechanics, Newtons law of gravitation and Coulombs law
both express non-local force laws, specifically, action at a distance. While themotion of a massive or charged particle in response to either force field is local (inthe sense that it responds only to the local field), the field itself is nonlocal, in thesense that its origins cannot be rationalized without appealing to deeper theoreticaldevelopments, specifically, quantum electrodynamics or general relativity. The same
could be said of Bohmian mechanics, according to which a particle moves in
response to the quantum potential, whose form is prescribed by the quantum wave
function. While the particle’s motion may be considered local, it is moving in
response to a nonlocal field imposed by fiat.
A recent example illustrates how the HQA perspective allows one to achieve
quantum effects and rationalize them without appealing to nonlocality. ‘Surreal
trajectories’ is a term coined by Englert et al. (1992) (ESSW) to describe Bohmian

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Fig. 2.7 Surreal trajectories in the hydrodynamic pilot-wave system (Frumkin et al., 2022). (a)A variant of the interferometer setup considered by ESSW (Englert et al., 1992). (b) A single
particle trajectory, along with the instantaneous pilot wave field. (c) In a symmetric arrangement,
the droplet enters the right or left channel with equal probability, after which it is deflected awayfrom the system centerline, resulting in a ‘surreal’ trajectory. Twenty such trajectories are shown.
(d) When one of the barriers is removed, the symmetry of the system is broken. The walking
droplet is then reflected away from the remaining barrier, resulting in the trajectory that one might
expect. (e) Numerical simulations of an ensemble of initially vertical trajectories with different
values of the impact parameter x. (f) The mean pilot-wave field generated by averaging simulated
droplet trajectories with a Gaussian distribution of initial impact parameters

trajectories predicted to arise in the interferometer arrangement illustrated inFig. 2.7a. The term was intended to point out that the trajectories predicted byBohmian mechanics are counterintuitive, and so cannot be real. Mahler et al.
(2016) measured mean trajectories in the geometry proposed by ESSW via weakmeasurement, and found that they were consistent with those predicted by Bohmianmechanics. The authors concluded that ‘the trajectories seem surreal only if one
ignores their manifest nonlocality’. In Bohmian mechanics, surreal trajectories
arise as a result of the particles being guided by the quantum potential, a nonlocal field. In the standard formulation, there is no notion of trajectories, but theexperimental observations of mean trajectories consistent with surreal trajectories
led (Mahler et al., 2016) to seek a rationale in terms of quantum nonlocality,
specifically entanglement with the measurement device. In the walker system,‘real surreal trajectories’ arise naturally from non-Markovian, classical dynamics inwhich the droplet navigates its pilot-wave field, a local potential of its own making.
Nonlocality need not be invoked. Our study showed that the designation of Bohmian
trajectories as surreal is based on misconceptions concerning the limitations of classical dynamics and a lack of familiarity with pilot-wave hydrodynamics (Frumkin2 The State of Play in Hydrodynamic Quantum Analogs

23

et al., 2022). Moreover, it made clear that the physical picture furnished by thewalker system allows one to see how to side-step the invocation of the non-local
quantum potential required in Bohmian mechanics.
HQA suggests that one can avoid the invocation of a nonlocal field, as is donein Bohmian mechanics, provided one follows the suggestion of Holland (1995):“We can envisage a more active role for the particle, something which is not evenadmitted as conceivable in the conventional view. This may, for instance, enteras a source of the pilot-wave field through an inhomogeneous term in the wave
equation”. This conceptual leap would constitute a conformance to de Broglie’s
double-solution theory rather than the provisional theory now known as de BroglieBohm theory or Bohmian mechanics, and would seem to be a critical step in
restoring locality to quantum pilot-wave theories. Considering the particle as thesource of its own wave renders the resulting dynamics local: the particle generates
its pilot wave and moves in response to it. This approach was recently followedby Dagan and Bush (2020) and Durey and Bush (2020), who considered particles
with an internal oscillation at the Compton frequency generating, then moving in
response to, waves satisfying a forced Klein-Gordon equation. While exploratory,
the results yielded a striking result, a physical picture for the origins of the de
Broglie relation. If a particle vibrating at the Compton frequency is dressed in aquasi-monochromatic wavefield that is a solution of the Klein-Gordon equation,then the group velocity of that wavefield must match the particle speed; thus,
p = h̄k. The emergent pilot-wave form for a free, uniformly translating particle,
illustrated in Fig. 2.1d, roughly conforms to that envisaged by de Broglie (1987).
The potential and limitations of this approach, of developing a model of quantum
dynamics informed by the walker system, are currently being explored more widely.

2.9 Non-separable States
Often a pair of quantum systems may be represented mathematically (by a vector) in a
way each system alone cannot: the mathematical representation of the pair is said to be
non-separable: Schrödinger called this feature of quantum theory entanglement.
– Richard Healey, The Quantum Revolution in Philosophy (Healey, 2017).

Non-separable states arise in multi-partite systems when the state of the wholecannot be simply defined in terms of the state of its subsystems (Horodecki et al.,2009). For example, if a quantum system S is comprised of two subsystems, A
and B, then the system will be considered non-separable if ψS /= ψA ⊗ ψB ,where ψS , ψA and ψB are the wave functions of the respective systems, and ⊗represents a tensor product. A canonical example is that of a singlet state, namely,
two particles with opposite spins: ψ = √1 (|↓↑> − |↑↓>) . Such states cannot
2
be factored into a product of two independent states, specifically, there are nocomplex numbers a, b, c, d such that: ψ = (a |↓> + b |↑>) ⊗ (c |↓> + d |↑>) . The
singlet state is one of the maximally entangled Bell states (Bohm, 1951), and has

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J. W. M. Bush et al.

been used historically to illustrate the difference between classical and quantum
correlations through violations of Bell’s inequality (see Sect. 2.10). The outcomeof a spin measurement on one of these particles will necessarily be correlated withthe outcome of a measurement on the other. When non-separability persists overarbitrarily large distances (Weihs et al., 1998; Yin et al., 2017), it is considered to
be a signature of entanglement. According to the Copenhagen interpretation (Homeand Nair, 1994), particles in a singlet state do not have a well-defined spin prior
to measurement. Once one of the two particles is measured, the wave function
collapses instantaneously for both. One might then ask how the second particle, thatwas not measured, knows that a measurement took place at the location of the distantfirst particle. The Copenhagen Interpretation suggests that there is an instantaneoustransfer of information between the two particles, spcifically the information that
one of the particles was measured. This peculiar “non-local” feature of entangledstates in quantum mechanics, is what Einstein referred to as “spooky action at a
distance” (Einstein et al., 1935).
In recent years, it has been shown that non-separability is not limited solely
to the quantum realm. Indeed, it is possible to achieve non-separable states withdifferent degrees of freedom of a classical electromagnetic field, and to use these
non-separable states to violate Bell’s inequality (Qian et al., 2015; Spreeuw, 1998).
Non-separable acoustical states were also recently demonstrated in coupled elastic
waveguides (Hasan et al., 2019), extending the notion of non-separability to thefield of phononics. Due to the commonalities with the standard formalism ofquantum entanglement, the above demonstrations have been described in terms of‘classical entanglement’ (Spreeuw, 1998; Qian et al., 2015; Hasan et al., 2019).
However, it has been convincingly argued that a different term, specifically ‘nonseparable states’, would be preferable to distinguish these classical states from thoseexhibiting standard bipartite quantum entanglement (Karimi and Boyd, 2015).
Prior work on classical non-separable systems has made clear the distinction
between classical and quantum superpositions of states: while non-separability
may arise in both, entanglement (specifically, the persistence of non-separable
states to arbitrary distances Weihs et al., 1998; Yin et al., 2017) is peculiar toquantum mechanics. While the classical non-separable states discovered to datecannot be used for testing local realism or quantum computing, they may be usefulfor a number of other quantum information related applications (Schmid et al.,2010; Simon et al., 2010; Töppel et al., 2014; Pinheiro et al., 2013). Unlike theaforementioned examples of classical non-separability, the bouncing droplet systemallows for the possibility of non-separable states in a spatially separated bipartite
system (Nachbin, 2022), and so for the application of these states in quantuminspired classical computing.
In quantum mechanics, tunneling is unpredictable, but the probability of its
occurrence is well-defined by the system geometry (Papatryfonos et al., 2015). Eddiet al. (2009a) and later Tadrist et al. (2020) demonstrated that such is also the casefor bouncing droplets, and Nachbin et al. (2017) captured this behavior using atheoretical model for one-dimensional walker motion over a vibrating liquid bathwith complex topography. Subsequently, Nachbin (2018) extended this model to2 The State of Play in Hydrodynamic Quantum Analogs

25

study correlations between two droplets separated by an intervening cavity. The
coupling strength was prescribed by the geometry of the central cavity, which
served as a nearly resonant transmission line. Using this system, the author showedthat the two-particle system is non-separable, by demonstrating that each particle’s
phase space dynamics is given by the system as a whole and cannot be described
separately. He further showed that after the correlations were established, removing
one of the particles from the system resulted in a drastic alteration of the phase spacepicture. The correlations arising in this model persisted even when the droplets wereseparated by large distances. Most recently, this model was extended to combine
the aspects of unpredictable tunneling and bipartite non-separability in order todemonstrate a hydrodynamic analog of superradiance (Papatryfonos et al., 2022a).
Superradiant photon emission is the anomalous emission arising when a pair or
assemblage of ions is in close proximity. Superradiance was originally attributed
to quantum interference arising from entangled atoms (DeVoe and Brewer, 1996;
Makarov and Letokhov, 2004; Karnieli et al., 2021), but has since been rationalisedin terms of electromagnetic wave interference (Tanji-Suzuki et al., 2011). Papatryfonos et al. (2022a) have recently used the walker system to develop a hydrodynamic
analog of superradiance in a numerical model of bipartite walker tunneling. Theirsystem consists of a pair of walkers, each in a subsystem consisting of a pair of cavities across which the walker may tunnel between ground (outer) and excited (inner)
states (see Fig. 2.8). While the bottom topography precludes the drops from escaping
their respective subsystems, the two subsystems communicate through the wave
field spanning the coupling cavity, allowing for strongly correlated states induced by
the wave-mediated forces. By identifying the tunneling transitions between excitedand ground states with photon emission events, they demonstrated that the waveinduced droplet tunneling may lead to an analog of superradiant photon emission.
Moreover, they demonstrated that the emission rate varies sinusoidally with distance
between the two subsystems, as is also the case in superradiant photon emission
from ion pairs (DeVoe and Brewer, 1996).

2.10 Entanglement
What Bell’s theorem, together with the experimental results, proves to be impossible . . . is
not determinism or hidden variables or realism but locality, in a perfectly clear sense. What
Bell proved, and what theoretical physics has not yet properly absorbed, is that the physicalworld itself is non-local.
– Tim Maudlin, “What Bell Did”, J. Phys. A: Mathematical and Theoretical (Maudlin,
2014).

Those that assert that quantum mechanics in its standard form is incomplete
appeal to ‘hidden variable theories’ for its completion. The hidden variables arethose variables needed to provide a complete description of the quantum dynamics
underlying quantum statistics. For example, in pilot-wave theories of the form

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Fig. 2.8 Photonic and hydrodynamic Bell test arrangements. (a) For optical Bell tests, pairs ofentangled photons are produced at the source and sent in opposite directions to two measurement
stations. At each station one of two measurement settings is selected randomly and a measurement
is performed. The measurement outcomes, which can take on two possible values, +1 or −1,
are noted. (b) The hydrodynamic Bell test consists of a pair of drops (red and green) walking
on the surface of a vibrating fluid bath (blue) that spans the solid substrate (grey). Each drop
is confined to its subsystem, a pair of wells separated by barriers across which they may tunnel
unpredictably at a rate prescribed by the barrier depths α and β, as may assume values of a, a 'or b, b' , respectively. The two subsystems are coupled through the intervening wave field. Fora limited range of measurement settings (barrier depths), Bell violations are obtained, with amaximum violation of 2.49. (Figure adapted from Ref. Papatryfonos et al., 2022b)

suggested by pilot-wave hydrodynamics, the hidden variables would be the particle
position and momentum and the form of the real pilot-wave field, as distinct from thewave function. In 1964, John Bell derived an inequality, the experimental violationof which is widely taken to indicate that no local hidden variable theory can accountfor the correlations apparent in quantum spin measurements (Bell, 1964). We referhere to a variant of Bell’s theorem that was introduced by Clauser, Horne, Shimony,and Holt (CHSH) (Clauser et al., 1969; Clauser and Horne, 1974). Consider two

2 The State of Play in Hydrodynamic Quantum Analogs

27

entangled particles in a singlet state that are measured at two spatially separated
detectors, A and B. Each detector can have two possible measurement settings,
α ∈ {a, a ' } for A, and β ∈ {b, b' } for B. If E(a, b) denotes the expectation valueof the product of outcomes of measurements (+1 for spin-up and -1 for spin-down)with settings a and b, then the CHSH-Bell inequality states that any local hidden
variable theory must satisfy the condition:|||S| = |E(a, b) + E(a, b' ) + E(a ' , b) − E(a ' , b' )| ≤ 2

.

(2.2)

where S quantifies the correlations between measurement outcomes at the two
detectors. One can demonstrate theoretically that for certain choices of measurement
settings, the singlet state √can violate the CHSH-Bell inequality, with a maximal
correlation value of S = 2 2.
The predicted violation of the CHSH-Bell inequality by quantum mechanics
has been verified experimentally in a wide range of physical systems (Brunner
et al., 2014), thus leading to the conclusion that quantum mechanics does notsatisfy at least one of the assumptions made in the derivation of Bell’s theorem.
Given that these assumptions include the basic tenets of reality (a physical world
exists independent of human observation) and locality (nothing travels faster than
the speed of light), some have even proclaimed Bell’s Theorem to be ‘the mostprofound in science’ (Stapp, 1975). The experimental violation thereof by Aspect
et al. (1982a,b) was rewarded with the most recent Nobel prize in physics, andwas interpreted by the Nobel Prize Committee as proving that ‘quantum mechanics
cannot be replaced by a theory that uses hidden variables’.
While the experimental violation of Bell’s inequality is widely taken to imply
that quantum mechanics is inescapably non-local, less drastic conclusions are
currently being explored through careful scrutiny of the assumptions made, either
explicitly or implicitly, in the derivation of Bell’s inequality. In particular, Vervoort(2018) has questioned whether the assumption of ‘measurement-independence’,
according to which the hidden variables that prescribe the measurement outcomes
are independent of α and β, is valid in systems with a background field. Vervoort
argues that such may not be the case in pilot-wave systems, wherein the hidden
variables characterizing the pilot-wave field might in principle be influenced by the
analyzer settings. A similar line of questioning was originally put forth by workers
in stochastic electrodynamics (de la Peña et al., 1972).
To inform this debate, we have recently adopted the model geometry used in thehydrodynamic analog of superradiance (Papatryfonos et al., 2022a) to administer
the first static Bell test in the hydrodynamic pilot-wave system (see Fig. 2.8). To
that end, we identified the location of the drop (inner or outer cavity) with thedichotomic property, corresponding to spin (+1 or −1) in traditional Bell tests. We
identified the barrier depths in the two subsystems with the measurement settings(α, β). Judicious choice of combinations of measurement settings, including one
for which strongly synchronized tunneling arises, allowed for violation of Bell’s
Inequality, with a maximum violation of 2.49±0.04. This violation is made possibleby the fact that in the walker system, the assumption of measurement-independence

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J. W. M. Bush et al.

is violated. Specifically, owing to the wave-mediated coupling between the two
subsystems, the probability of a droplet occupying a given cavity in either of thesubsystems depends on both measurement settings (Papatryfonos et al., 2022b).
Pilot-wave hydrodynamics thus illustrates how static Bell violations may arise ina classical setting. This particular system also shows how non-separable states with
non-classical correlations can be established in classical bipartite systems.
The question remains open as to whether the violations deduced in the static Belltest will survive isolation of the two subsystems, as would effectively eliminate
communication between them. A promising result in this direction has recently
been demonstrated by Nachbin (2022), who considered correlations of two droplets
confined to individual cavities linked through a coupling cavity. He demonstrated
that the wave-mediated forces between the droplets allowed them to achievestrongly correlated energetic states inaccessible to either droplet in the other’sabsence. By decreasing the depth of the coupling cavity during the course of thesimulations, he demonstrated that these correlated excited states could survive
even when the droplets are no longer communicating. We are currently taking
a similar approach to the two-cavity geometry considered in our static Bell test
that will determine whether violations of Bell’s inequality can persist even when
the two subsystems are isolated. Such tests should settle the matter as to whether
memory might provide a viable mechanism for establishing classical entanglement.
Such violations might in principle be rationalized through the measurementindependence loophole (Vervoort, 2018): the hidden variables (specifically, thedroplet trajectories and pilot-wave field) depend on both measurement settings,
whose influence is imposed prior to, but persists after, isolation of the subsystems.
Concurrently, numerous efforts are being made in the HQA community towards
achieving Bell violations experimentally with the walker system.

2.11 Conclusion
I think it very probable that the solution to our problem will come through the back door:some person who is not addressing himself to these difficulties with which I am concerned
will probably see the light. An analogy that I like is that of a fly buzzing against a window
when the door is open.
– John. S. Bell (Davies and Brown, 1993).

Several pronouncements concerning the inscrutability of quantum mechanics,
made by some of its most distinguished practitioners, have been revisited in lightof insights gained from pilot-wave hydrodynamics. Many now fall flat. If one isunwilling to make one’s peace with quantum nonlocality, there would appear to
be two options. First, one may ascribe to one of the more extravagant quantum
interpretations that side-step the nonlocality problem by offering up an even lessappealing prospect, for example, the Multiverse (Everett, 1957). Second, one may
follow the progressive approach offered by HQA, and explore the possibility of2 The State of Play in Hydrodynamic Quantum Analogs

29

a hidden variable theory that violates Bell’s inequality without being explicitly
nonlocal, specifically without having to appeal to superluminal signaling or anonlocal quantum field. The authors are currently following this approach, and so
persist in the stubborn view that while nonlocality is undoubtedly a feature of thecurrent quantum theory, it need not be a feature of an adequately resolved, complete
theory of quantum physics (Kupczynski, 2020).
The proliferation of quantum interpretations has arisen because there is presently
no experimental means by which to distinguish between them. Conversely, theories
of quantum mechanics can be falsified or verified via experimental test. The
hope is that HQAs will provide additional insight into the quantum pilot-wave
theories of de Broglie (1987), de la Peña et al. (2015), and Hestenes (1990),
that will ultimately be tested experimentally. Given that all such theories appeal
to an unresolved dynamics on the Compton scale, it is conceivable that future
experimental investigations, achieved for example in electron channeling (Catillon
et al., 2008), might validate or refute the physical picture suggested by walking
droplets.
Finally, it is noteworthy that the field of hydrodynamic analogs of generalrelativity was initiated by Unruh (1981) and Schützhold and Unruh (2002) andhas subsequently been well-established (Euve et al., 2016; Rousseaux and Kellay,2020). The field of HQAs is relatively recent, but burgeoning. A primary difficultyin modern physics is reconciling general relativity and quantum mechanics. Should
the subject of fluid mechanics provide the church at which to marry the two, one
imagines that the subject might become a more prominent component of the modern
undergraduate physics curriculum.

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