Physics Today The new wave of pilot-wave theoryJohn W. M. Bush Citation: Physics Today 68(8), 47 (2015); doi: 10.1063/PT.3.2882 View online: http://dx.doi.org/10.1063/PT.3.2882 View Table of Contents: http://scitation.aip.org/content/aip/magazine/physicstoday/68/8?ver=pdfcov Published by the AIP Publishing

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www.physicstoday.org August 2015 Physics Today 47

If particle physics is thedazzling crown prince ofscience, fluid mechanicsis the cantankerous queenmother: While her loyal

subjects flatter her as beingrich, mature, and insightful,many consider her to be dé-modé, uninteresting, and difficult. In her youth, shewas more attractive. Her inconsistencies were takenas paradoxes that bestowed on her an air of depthand mystery. The resolution of her paradoxes lefther less beguiling but more powerful, and markedher coming of age. She has since seen it all and hasweighed in on topics ranging from cosmology to astronautics. Scientists are currently exploringwhether she has any wisdom to offer on the contro-versial subject of quantum foundations.

Metaphor provides a means of using one sub-ject to gain insight into another. Even at that impre-

cise level of comparison,fluid mechanics is a rich re-source—for example, IsaacNewton spoke of corpusclesof light skipping throughether like stones on the sur-face of a pond. The mathe-matical description of phys-

ical systems allows for more exacting comparisons.Dynamic similarity, the cornerstone of laboratorymodeling in fluid dynamics, arises between twofluid systems when a strict mathematical equiva-lence is achieved: The systems are governed by pre-cisely the same equations. Thus, with meter-scaleexperiments, one can explore everything from as-trophysical flows to the swimming of bacteria.

John W. M. Bush

Small drops bouncing across a vibrating liquid bathdisplay many features reminiscent of quantum systems.

John W. M. Bush (http://math.mit.edu/~bush) is a professorof applied mathematics at the Massachusetts Institute ofTechnology in Cambridge.

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While the founding fathers agonizedover the question “particle” or “wave”de Broglie in 1925 proposed the obvi-ous answer “particle” and “wave.” . . .This idea seems so natural and simple,to resolve the wave–particle dilemmain such a clear and ordinary way, that itis a great mystery to me that it was sogenerally ignored.

—John S. Bell, Speakable and Unspeakable in Quantum Mechanics

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48 August 2015 Physics Today www.physicstoday.org

Pilot-wave theory

Stronger than metaphor but weaker than math-ematical equivalence is physical analogy, whichmay be drawn between two systems that are com-parable in significant respects owing to similaritiesin their essential physics and underlying mathemat-ical structure. At that level, fluid mechanics pro-vides a framework for describing a broader class ofnonfluidic systems. For example, ripple-tank exper-iments were used by Thomas Young to illustrate thewave nature of light.

A decade ago Yves Couder and Emmanuel Fortdiscovered that a millimeter-sized droplet may pro-pel itself along the surface of a vibrating fluid bathby virtue of a resonant interaction with its ownwave field; figure 1 shows several examples. Thishydrodynamic system has since been shown to exhibit several features previously thought to be peculiar to the microscopic, quantum realm: single-particle diffraction, tunneling, wave-like statistics inconfined geometries, quantized orbits, spin states,orbital-level splitting, and more. In this article I de-scribe the walking-droplet system and, where pos-sible, provide rationale for its quantum-like features.Further, I discuss the physical analogy between thishydrodynamic system and its closest relations inquantum theory, Louis de Broglie’s pilot-wave the-ory and its modern extensions.

Louis de Broglie’s pilot wavesThe Copenhagen interpretation of quantum me-chanics asserts that the well-established statisticaldescription of quantum particles provided by stan-dard quantum theory is the full story. Conversely,realist interpretations assert that a concrete dynam-ics underlies the statistical description; thus micro-scopic quantum particles follow trajectories just asdo their macroscopic counterparts. The descriptionof that hitherto unresolved dynamics would consti-tute a hidden-variable theory. The hydrodynamicsystem of Couder and Fort harkens back to a theory

that predates the Copenhagen interpretation: thepilot-wave theory of de Broglie.

Given only the speed of light c and Planck’sconstant ħ, dimensional analysis dictates that thenatural frequency of a particle of mass m be propor-tional to the Compton frequency, ωC = mc2/ħ. The result is consistent with the de Broglie–Einstein relation, mc2 = ħωC, which de Broglie took as an ex-pression of the wave nature of matter, the link be-tween relativity and quantum mechanics.

In 1923 de Broglie proposed the first pilot-wavetheory, according to which a quantum object suchas an electron is a localized, vibrating particle mov-ing in concert with a spatially extended, particle-centered pilot wave.1 The particle vibration, or zitterbewegung, is characterized by an exchange, atthe Compton frequency, between rest-mass energyand wave energy. De Broglie suggested that theguiding pilot wave is monochromatic, characterizedby the de Broglie wavelength λB.

De Broglie did not detail the wave–particle in-teraction. He did, however, envisage the particle asa singularity following the rays of the guiding waveand moving at the wave’s phase speed, λBω/2π, fromwhich follows the de Broglie relation for the particlemomentum, p = 2πħ/λB. He stressed the importanceof the “harmony of phases,” by which a particle’s vi-bration stays in phase with its guiding wave; thewave and vibrating particle thus maintain a state ofresonance.1 He asserted that the pilot-wave dynam-ics could give rise to a statistical behavior consistentwith standard quantum theory. The result was deBroglie’s double-solution theory, which involvedtwo distinct waves—the pilot wave and the statisti-cal wave of standard quantum mechanics.

Based on his physical picture, de Broglie pre-dicted single-particle diffraction and interference,the experimental confirmation of which led to hisbeing awarded the Nobel Prize in Physics in 1929.As an electron passes through a narrow gap, he

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Figure 1. Riding the wave. Faraday waves (a) appear when a vibrating bath is driven by an acceleration γ thatexceeds the Faraday threshold, γF. Red and blue reflected light allows for easier visualization of surface ripples.The phenomena illustrated in the following panels occur at accelerations slightly below γF. (b) A millimeter-sizeddroplet bounces in place on a vibrating bath. (c) A droplet walks along the bath surface, propelled by its ownwave field. (d) The walking drop, whose reflection is also visible, bounces along the liquid surface, as shown by the oscillatory black line. (e) This strobed image of the walker illustrates the drop surfing on its own wave field. (f) This strobed image shows two walkers locked in orbit. (Images courtesy of Daniel Harris.)

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imagined, its guiding wave is diffracted, whichleads to a divergence of particle paths. The pilot-wave picture provided de Broglie with a frameworkfor rationalizing other quantum phenomena, in-cluding the uncertainty relations and the Bohr–Sommerfeld rule for quantized orbits. It garnerednotable supporters, including Albert Einstein, wholikewise sought to reconcile quantum mechanicsand relativity through consideration of the wave nature of matter and declared that de Broglie had“lifted a corner of the great veil.” Nevertheless, theCopenhagen interpretation gained ascendancy, itsposition bolstered by the proofs of John von Neu-mann and others that erroneously suggested the im-possibility of hidden-variable theories and so dis-couraged their development. (See, for example,reference 2 and the article by Reinhold Bertlmann,PHYSICS TODAY, July 2015, page 40.)

David Bohm’s variantOne year after the 1925 formulation of theSchrödinger equation, Erwin Madelung demon-strated that a particular transformation of the wave-function provides a means of recasting the equationinto hydrodynamic form. The corresponding sys-tem is a shallow, inviscid fluid layer evolving underthe action of surface tension, if one associates theprobability density with the fluid depth and thequantum velocity of probability (proportional to thegradient of the wavefunction phase) with the depth-averaged fluid velocity. Planck’s constant ħ thenplays a role analogous to surface tension in shallow-water hydrodynamics.

In 1952 David Bohm demonstrated that if oneinterprets the quantum velocity of probability as aparticle velocity, one obtains statistical predictionsconsistent with those of standard quantum mechan-ics.3 Bohmian mechanics has just a single wave: Theguiding wave and the statistical wave are one andthe same. It is thus significantly less rich dynami-cally than de Broglie’s double-solution theory.Bohm’s formulation played an important historicalrole in providing a counterexample to the impossi-bility proofs that held sway at the time. It also re-minds us that any hidden-variable theory consistentwith the statistical predictions of quantum mechan-ics has the feature of quantum nonlocality.

Jean-Pierre Vigier and others extended the orig-inal Bohmian mechanics by including in the dynam-ics an additional stochastic element that arisesthrough the particle’s interaction with a subquan-tum realm.4 According to these theories, the quan-tum velocity of probability represents a mean ve -locity about which the real particles jostle, just as,because of their Brownian motion, gas moleculesjostle about streamlines in a gas flow. In his lateryears, de Broglie also sought the origins of his pilotwave in a stochastic subquantum realm.1

Vacuum-based pilot-wave theoriesNowadays de Broglie’s realm is called the quantumvacuum; it is a turbulent sea, roiling with waves as-sociated with a panoply of force-mediating fieldssuch as the photon and Higgs fields. Insofar as theyinteract with quantum particles, all such fields arecandidates for de Broglie’s pilot wave. Stochastic

electrodynamics posits electromagnetic energy inthe quantum vacuum at 0 K, in the form of a sto-chastic, fluctuating “zero-point field” (ZPF), whosespectrum has an energy U(ω) = ħω/2 per normalmode. The ZPF gives an alternative rationale for nu-merous quantum mechanical phenomena, includ-ing the Casimir effect, van der Waals forces, and theblackbody radiation spectrum.5

According to Luis de la Peña and colleagues,the ZPF might also excite a quantum particle’s zitterbewegung.6 As the particle translates, the vibra-tion interacts selectively with waves in the vacuumfield, a resonant interaction that amplifies an elec-tromagnetic pilot wave. In this picture, the deBroglie pilot wave consists of a carrier wave with theCompton wavelength λC = 2πħ/mc, modulated overthe speed-dependent de Broglie wavelength. Thecoupling between the particle and the electromag-netic field results in nonlocal dynamics. For multi-particle systems exhibiting quantum nonlocality,entanglement would be an expression of wave-induced correlations.6 The concept of a particle as aZPF-driven oscillating charge with a resonance atωC was further explored by Bernard Haisch and colleagues, who suggested that ZPF–particle inter-actions might offer insight into the origins of inertialmass; in that way, they sought to link the ZPF, thequantum wave nature of matter, and relativistic mechanics.7

Bouncing and walking on a liquidSurface tension σ specifies the energy per area of aliquid–gas surface. Owing to surface tension, non-planar distortions of an initially flat liquid surfaceare energetically costly, as are aspherical distortionsof small drops. According to dimensional analysis,an inviscid drop with mass m and surface tension σwill have a natural frequency proportional to √―σ/m.The drop is an oscillator: When perturbed, it will os-cillate at its natural frequency, exchanging surfaceenergy and kinetic energy.

When a fluid bath is vibrated vertically, its sur-face becomes unstable to a field of so-called Faradaywaves, such as shown in figure 1a, when the drivingacceleration γ exceeds the Faraday threshold γF. Asthe threshold is crossed, the first waves to appearare subharmonic (having a frequency equal to halfthe driving frequency) with a wavelength λF deter-mined by the dispersion relation for water waves.The phenomena of interest arise when a millimeter-sized drop is placed on a bath of silicone oil vibrat-ing below the Faraday threshold, so the bath’s sur-face would remain flat if not for the drop. The dropmay avoid coalescing with the bath owing to thepersistence of a thin air layer between drop and bathduring impact (see figure 1b). As the bouncing dropstrikes the bath, the dominant force resisting its in-trusion is that generated by the surface tension: Thebath surface behaves roughly like a linear springthat reverses its direction.

When the bouncing frequency becomes com-mensurate with the bath’s most unstable Faradaymode (the first to appear as γ is progressively in-creased), the drop generates at each impact a radi-ally expanding wavefront behind which is triggereda decaying field of Faraday waves. Couder and Fort

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discovered that the bouncing state can then desta-bilize into a dynamic walking state; figures 1c–1e il-lustrate the walking drop. A critical feature of thewalker dynamics is the “path memory” that arisesdue to the longevity of the pilot wave.8 At impact,the walker receives a lateral propulsive force pro-portional to the local slope of the interface. For lowmemory—that is, when γ is well below γF—the wavesare quickly damped. As γ approaches γF (high mem-ory), the waves are relatively persistent, and theforce acting on the walker depends on the drop’sdistant past and its environment, both of which areencoded in the pilot wave. The walker’s quantum-like features emerge at high memory, when this dy-namical nonlocality is most pronounced.

The bouncing and walking behavior of thedroplets has been well characterized both experi-mentally and theoretically.9 Periodic or chaoticbouncing states can arise, depending on the vi -brational frequency and amplitude, drop size, andfluid properties. A drop will bounce most readilywhen the driving frequency is commensurate withits natural frequency√―σ/m, and it will walk mostreadily when resonance is achieved between thebouncing drop and its pilot wave. Theorists havedeveloped increasingly sophisticated models to de-scribe both the bouncing dynamics and the wavegeneration. The most recent models have predictive

power: Given the control parameters, they specify ifa droplet will coalesce, bounce, or walk. When thedroplet walks, the theory determines both its gaitand its speed. When a walker is in a resonant state,for which the drop and wave are synchronized, theso-called stroboscopic approximation describes itshorizontal motion in terms of a droplet surfing onits pilot wave and provides rationale for the stabilityof various dynamical states, including the rectilinear-walking and circular-orbiting states shown in figures 1d–1f.

Orbital dynamicsFort and colleagues demonstrated in 2010 that whena vibrating bath is caused to rotate at a constant an-gular speed Ω, a walker with speed v travels in a cir-cular orbit with radius r ≈ v/2Ω, provided the mem-ory is sufficiently low.10 As the memory increases,however, the orbiting walker begins to interact withits own wake. As figure 2 shows, the result is a kindof orbital quantization. Guided by the identicalforms of the Coriolis force acting on a mass in a rotating frame and the Lorentz force acting on acharge in a uniform magnetic field, Fort and com-pany drew the physical analogy between the angu-lar quantization arising in their system and that ofthe Landau levels that arise when a charged particleorbits in a magnetic field. The Faraday wavelengthλF in the fluid system plays the role of the de Brogliewavelength λB for the orbiting charged particle.

A few years later, my colleagues and I examinedthe high-memory regime of the rotating system bothexperimentally11 and theoretically.12 At high memory,virtually all orbital states become unstable, typicallyvia a period-doubling transition to chaos. Figure 2cshows a typical chaotic trajectory, which tends tomove along arcs whose radii of curvature correspondto those of the unstable orbits. Thus the trajectory’sradius of curvature is characterized by multimodalstatistics, with peaks arising at integer multiples ofλF/2, as shown in figure 2d. One can thus rationalizethe quantum-like statistics in terms of chaotic pilot-wave dynamics: In its chaotic state, the walkerswitches between accessible unstable orbital states.

Antonin Eddi and coworkers studied walkerpairs locked in circular orbits by their wave field;figure 1f shows an example. When the bath was ro-tated, they found that the orbital radius increased ordecreased according to the relative sense of the or-bital motion of the walkers and the bath; the phe-nomenon is reminiscent of Zeeman splitting.13 Thestroboscopic model indicates the possibility of hy-drodynamic spin states and level splitting for singlewalkers.12 As memory increases, even in the absenceof rotation, orbital solutions arise in which the waveforce balances the radial inertial force. The stabilityof such hydrodynamic spin states is a subject of cur-rent interest.

By encapsulating a magnetic suspension in adroplet and applying a spatially varying magneticfield, Couder’s group investigated a walker in a cen-tral force field—that of a two-dimensional simpleharmonic oscillator.14 Their study revealed a varietyof complex orbital forms that, as shown in figure 3,are quantized in both mean radius and angular mo-mentum. As for the case of the rotating bath, in the

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Figure 2. A walker in a rotating frame. The orbital radius r of a walker(normalized by the Faraday wavelength λF) as a function of bath rotationrate Ω and walking speed v demonstrates qualitatively different behaviorat (a) low memory and (b) intermediate memory. Data are filled circles.Theoretical curves are solid lines; blue segments correspond to stable regions, red to unstable regions. The data and theoretical curve in panel a are offset from the dashed curve, which represents r = v/2Ω. The displacement is due to an increased effective mass for the drop,18 inducedby its pilot wave. (c) At high memory the walker trajectory is chaotic, butmost of the trajectory arcs have a radius of curvature R roughly of theform R = (n + 1) λF/2, resulting in the multimodal statistics shown in (d). (For additional detail see refs. 11 and 12.)

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chaotic regime prevalent at high memory, the walkerdrifts between accessible unstable orbital states andmanifests multimodal statistics.

Interactions with boundariesA couple of years ago, Couder’s group and minejoined forces to examine a walker in a circularcorral.15 We found that at low memory, the walkerorbits along the bounding walls. As memory in-creases, the walker executes progressively morecomplex trajectories that include wobbling circularorbits, drifting elliptical orbits, and epicycles. At thehighest memory we examined, the trajectories be-come irregular and chaotic, as shown in figure 4. De-spite the complexity of the trajectories, a coherentstatistical behavior emerges. Specifically, the his-togram (figure 4b) of the walker’s position is roughlyprescribed by the amplitude of the Faraday modethat arises when the bath is driven just above theFaraday threshold. If one views the concentric cir-cular peaks evident in the histogram as orbital states of the system between which the walkerswitches when it is in a chaotic state, the multimodalstatistics may again be rationalized in terms ofchaotic pilot-wave dynamics.16 That the walker sys-tem is closer to de Broglie’s mechanics than toBohm’s is particularly evident in the corral, wherethe distinction between the complex pilot wave thatguides the walker and the axially symmetric statis-tical wave is clear.

Couder’s group examined the dynamics of awalker impinging on a region with a submergedbarrier and found that the walker occasionallydemonstrates a kind of tunneling (see figure 5a). Asthe simulation in figure 5b reveals, the incidentpilot-wave field is partially reflected at the bound-

ary but also has an evanescent tail. The reflectedwave typically causes the approaching walker to bereflected; however, the droplet occasionally crossesthe barrier, with a probability that decreases expo-nentially with barrier width. This statistical behavior,reminiscent of quantum tunneling, is presumablyrooted in the chaotic pilot-wave dynamics but hasyet to be rationalized theoretically. Characterizingthe interaction between walkers and boundaries isa central goal of ongoing theoretical work.

Couder and Fort demonstrated that when awalker passes through a slit—more precisely, over

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a Figure 4. A walker in a circular corral. The high-memory trajectories of increasing length depicted in (a) are color coded accord-ing to droplet speed in millimeters per second, as indicated on the sidebar. The corral has a radius of 14.3 mm. (b) The histogramof the walker’s position corresponds roughly to theamplitude of the corral’smost unstable Faradaymode. (Panels a and badapted from ref. 15.) Theemerging statistics of theconfined walker are reminis-cent of (c) the statistics ofelectrons in a quantum corral. (Panel c courtesy ofDonald Eigler.)

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Figure 3. A walker in a harmonic potential. The observed orbits of awalker subject to a two-dimensional harmonic-oscillator force assumemany shapes, including circles (black), lemniscates (red), dumbbells(green), and trefoils (blue). A double quantization emerges when the orbits of a walker of mass m and speed v are classified by R‾, the root-mean-square radius (normalized by the Faraday wavelength λF) and L‾,the mean angular momentum (normalized by mvλF). (Adapted from S. Perrard et al., Nat. Commun. 5, 3219, 2014.)

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a gap in a submerged barrier—its rectilinear motionis disturbed, as illustrated in figure 5c, owing to thedisruption of its pilot-wave field.17 The behavior isanalogous to that envisioned by de Broglie in hisprediction of electron diffraction and in his ration-alization of the position–momentum uncertainty relation. In the double-slit geometry, the pilot wavepasses through both slits even though the dropletpasses through only one. The walker thus interactswith both slits because of the spatial delocalizationof its pilot wave.

Bridging the micro–macro chasmThe walker represents an example of an oscillatingparticle moving in resonance with its own wavefield. The droplet moves in a state of energetic equi-librium with the vibrating bath, navigating a wavefield sculpted by its motion. The walker system con-tinues to extend the range of classical systems to in-clude features previously thought to be exclusive tothe quantum realm. What might one infer if unawarethat it is a driven, dissipative pilot-wave system?

One would be puzzled by the prevalence ofquantization and multimodal statistics. Inferring aconsistent trajectory equation would be possibleonly in certain limits. Doing so in the limit of weakwalker acceleration would suggest that the droplet’seffective mass depends on its speed.18 Multiple-particle interactions would be characterized by in-explicable scattering events and bound states, andbaffling correlations. If one could detect a walkeronly by interacting with the fluid bath, the measure-ment process would become intrusive. If a detectorconfined the walker spatially, one would infer a po-sition–momentum uncertainty relation. If detectionrequired collisions with other droplets, disruptionof the pilot wave would destroy any coherent statis-tical behavior that might otherwise arise.

The walker dynamics are markedly different

from those described by Bohmian mechanics butsimilar to those imagined in de Broglie’s double-solution theory. An oscillating particle is piloted byits self-generated wave field, and its wave energy isrelated to its frequency by ħ in de Broglie’s mechan-ics and by surface tension σ for the walker. In both,the pilot wave and statistical wave have the samewavelength but different geometric form. In thewalker system, the wavelength is fixed by the bath’sdriving frequency, whereas in its quantum prede-cessor, it is speed dependent, prescribed by the deBroglie relation, p = 2πħ/λB. One common criticismof de Broglie’s pilot-wave theory—that it is too complicated to ever work—would seem to be put torest by its hydrodynamic variant. The walker sys-tem demonstrates that when an oscillating particlemoves in resonance with a monochromatic pilotwave, its nonlocal dynamics may give rise to quan-tization and multimodal statistics.

De Broglie did not specify the physical originsof the pilot wave, but modern extensions of his me-chanics have sought an electromagnetic pilot waveoriginating in the quantum vacuum. A marked dif-ference between the walker system and the vacuum-based models is that in the latter the length scales ofthe pilot wave and statistical wave are different.Nevertheless, the physical analogy between the twosystems is intriguing: The vibrating bath plays therole of the zero-point field in driving the system; thedrop’s bouncing, that of the particle’s zitterbewegungin triggering the pilot wave.9 Like the electromag-netic pilot-wave system envisaged by de la Peña andcompany,6 the walker system is a driven, dissipativesystem in which an oscillator is excited by back-ground vibration and moves in a field structured byits motion, giving rise to a pilot-wave dynamics thatis nonlocal in both space and time.

The resolution of the fluid mechanical para-doxes invariably arose through the elucidation of

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Figure 5. A walker encountering obstacles. When a walker in a confined geometry encounters a region with asubmerged barrier, it can occasionally “tunnel” across the barrier. (a) The experimental trajectories shown hererepresent 110 trials in which a walker encountered a submerged barrier; it crossed the obstacle 14 times. Therhombus-shaped frame is 45 mm across at its waist. (Adapted from A. Eddi et al., Phys. Rev. Lett. 102, 240401,2009; courtesy of Yves Couder.) (b) In this simulation, the walker encountering the barrier is confined to a line.Space and time coordinates x and t trace the evolution of both the walker (red) and the pilot wave (blue). Notethe tunneling event midway through the simulation. (Courtesy of André Nachbin and Paul Milewski.) (c) Observedwalker trajectories are splayed by the spatial confinement associated with a single 14.7-mm-wide slit. (Courtesy of Daniel Harris and Giuseppe Pucci.)

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unimagined dynamics at an unanticipated scale.The hydrodynamic pilot-wave system, when con-sidered in light of vacuum-based pilot-wave theo-ries, would seem to suggest the possibility of an un-resolved dynamics on the Compton scale, that asuccessful nonlocal hidden-variable theory mightbe based on the physical picture of particles inter-acting with the vacuum and propagating in an equi-librium state of energy exchange within it. Since theCompton frequency—which is approximately 1021 Hzfor an electron, for example—sets the time and lengthscales for particle-pair production from the vacuum,the resolution of such Compton-scale dynamicsposes serious experimental challenges.

One is invited to believe that the macroscopicand microscopic worlds are separated by a philo-sophical chasm so deep that there is no hope of evercrossing it. In the depths of the abyss, profound andobscure, lurk the impossibility proofs. To those whocannot see beyond strict mathematical equivalence,one must concede that bouncing droplets are notquantum particles. To those with the relativelymodest goal of establishing philosophical equiva-lence through physical analogy, the walking-dropletsystem has extended a speculative bridge from themacroscopic toward the microscopic and linkedwith a structure emerging from the other side: themodern extensions of de Broglie’s pilot-wave theory.Rickety though this bridge may be, under construc-tion, it offers striking new views to workers fromboth sides.

I gratefully acknowledge the financial support of NSF andthank Mason Biamonte, Yves Couder, Emmanuel Fort,Daniel Harris, Anand Oza, Giuseppe Pucci, and RodolfoRuben Rosales for valuable input.

References1. L. de Broglie, Ann. Fond. Louis Broglie 12, 1 (1987).2. J. S. Bell, Speakable and Unspeakable in Quantum Me-

chanics, 2nd ed., Cambridge U. Press (2004).3. D. Dürr, S. Goldstein, N. Zanghì, Quantum Physics

Without Quantum Philosophy, Springer (2013).4. L. V. Chebotarev, in Jean-Pierre Vigier and the Stochastic

Interpretation of Quantum Mechanics, S. Jeffers et al.,eds., Apeiron (2000), p. 1.

5. T. H. Boyer, Am. J. Phys. 79, 1163 (2011).6. L. de la Peña, A. M. Cetto, A. Valdés Hernández, The

Emerging Quantum: The Physics Behind Quantum Me-chanics, Springer (2015).

7. B. Haisch, A. Rueda, Y. Dobyns, Ann. Phys. 10, 393(2001).

8. A. Eddi et al., J. Fluid Mech. 674, 433 (2011).9. J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269 (2015).

10. E. Fort et al., Proc. Natl. Acad. Sci. USA 107, 17515(2010).

11. D. M. Harris, J. W. M. Bush, J. Fluid Mech. 739, 444(2014).

12. A. Oza et al., J. Fluid Mech. 744, 404 (2014). 13. A. Eddi et al., Phys. Rev. Lett. 108, 264503 (2012).14. S. Perrard et al., Phys. Rev. Lett. 113, 104101 (2014).15. D. M. Harris et al., Phys. Rev. E 88, 011001 (2013). 16. T. Gilet, Phys. Rev. E 90, 052917 (2014). 17. Y. Couder, E. Fort, Phys. Rev. Lett. 97, 154101 (2006).18. J. W. M. Bush, A. Oza, J. Moláček, J. Fluid Mech. 755, R7

(2014). ■

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Cremat's low noise charge sensitive preamplifiers (CSPs) can be used to read out pulse signals from p-i-n photodiodes, avalanche photodiodes (APDs), SiPM photodiodes, semiconductor radiation detectors (e.g. Si, CdTe, CZT), ionization chambers, proportional counters, surface barrier/PIPS detectors and PMTs. Our CSPs and shaping amplifiers are small epoxy-sealed plug-in modules less than 1 in2 in area. We also provide evaluation boards for these modules, letting you easily and quickly integrate these parts into your instrumenta-tion.

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perfect for pulse detection!

andshaping

amplifiers

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18.111.86.206 On: Mon, 03 Aug 2015 15:28:52