Comptes Rendus
Bouncing drops, memory
Hydrodynamic quantum field theory: the free particle
Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 555-571.

We revisit de Broglie’s double-solution pilot-wave theory in light of insights gained from the hydrodynamic pilot-wave system discovered by Couder and Fort []. de Broglie proposed that quantum particles are characterized by an internal oscillation at the Compton frequency, at which rest mass energy is exchanged with field energy. He further proposed that the resulting pilot-wave field satisfies the Klein–Gordon equation. While he developed a guidance equation for the particle, he did not specify how the particle generates the wave. Informed by the hydrodynamic pilot-wave system, we explore a variant of de Broglie’s mechanics in which the form of the Compton-scale dynamic interaction between particle and pilot wave is specified. The particle is modeled as a localized periodic disturbance of the Klein–Gordon field at twice the Compton frequency. We simulate the evolution of the particle position by assuming that the particle is propelled by the local gradient of its pilot wave field. Resonance is achieved between the particle and its pilot wave, leading to self-excited motion of the particle. The particle locks into quasi-steady motion characterized by a mean momentum p ¯=k, where k is the wavenumber of the surrounding matter waves. Speed modulations along the particle path arise with the de Broglie wavelength and frequency ck. The emergent dynamics is strongly reminiscent of that arising in the hydrodynamic pilot-wave system, on the basis of which we anticipate the emergence of quantum statistics in various settings. Our results suggest the potential value of a new hydrodynamically-inspired pilot-wave theory for the motion of quantum particles.

Publié le :
DOI : 10.5802/crmeca.34
Mots clés : Klein–Gordon equation, Matter waves, Hydrodynamic quantum analogs, Pilot-wave theory, Free particle
Yuval Dagan 1 ; John W. M. Bush 2

1 Faculty of Aerospace Engineering, Technion - Israel Institute of Technology, Haifa, Israel
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMECA_2020__348_6-7_555_0,
     author = {Yuval Dagan and John W. M. Bush},
     title = {Hydrodynamic quantum field theory: the free particle},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {555--571},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {6-7},
     year = {2020},
     doi = {10.5802/crmeca.34},
     language = {en},
}
TY  - JOUR
AU  - Yuval Dagan
AU  - John W. M. Bush
TI  - Hydrodynamic quantum field theory: the free particle
JO  - Comptes Rendus. Mécanique
PY  - 2020
SP  - 555
EP  - 571
VL  - 348
IS  - 6-7
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.34
LA  - en
ID  - CRMECA_2020__348_6-7_555_0
ER  - 
%0 Journal Article
%A Yuval Dagan
%A John W. M. Bush
%T Hydrodynamic quantum field theory: the free particle
%J Comptes Rendus. Mécanique
%D 2020
%P 555-571
%V 348
%N 6-7
%I Académie des sciences, Paris
%R 10.5802/crmeca.34
%G en
%F CRMECA_2020__348_6-7_555_0
Yuval Dagan; John W. M. Bush. Hydrodynamic quantum field theory: the free particle. Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 555-571. doi : 10.5802/crmeca.34. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.34/

[1] Y. Couder; S. Protier; E. Fort; A. Boudaoud Dynamical phenomena: walking and orbiting droplets, Nature, Volume 437 (2005) no. 7056, p. 208 | DOI

[2] D. J. Griffiths; D. F. Schroeter Introduction to Quantum Mechanics, Cambridge University Press, 2018 | DOI | Zbl

[3] L. De Broglie (“Recherches sur la théorie des quanta”, PhD thesis, Migration-université en cours d’affectation, 1924)

[4] L. De Broglie Rapport au Veme Congres de Physique Solvay, Gauthier-Villars, 1930

[5] L. De Broglie The reinterpretation of wave mechanics, Found. Phys., Volume 1 (1970) no. 1, pp. 5-15 | DOI

[6] S. Protière; A. Boudaoud; Y. Couder Particle–wave association on a fluid interface, J. Fluid Mech., Volume 554 (2006), pp. 85-108 | DOI | MR | Zbl

[7] J. W. M. Bush Pilot-wave hydrodynamics, Annu. Rev. Fluid Mech., Volume 47 (2015), pp. 269-292 | DOI | MR

[8] J. W. M. Bush The new wave of pilot-wave theory, Phys. Today, Volume 68 (2015), pp. 47-53 | DOI

[9] J. W. M. Bush; Y. Couder; T. Gilet; P. A. Milewski; A. Nachbin Introduction to focus issue on hydrodynamic quantum analogs, Chaos: An Interdiscip. J. Nonlinear Sci., Volume 28 (2018) no. 9 (096001)

[10] A. Eddi; E. Fort; F. Moisy; Y. Couder Unpredictable tunneling of a classical wave-particle association, Phys. Rev. Lett., Volume 102 (2009) (240401) | DOI

[11] A. Nachbin; P. A. Milewski; J. W. M. Bush Tunneling with a hydrodynamic pilot-wave model, Phys. Rev. Fluids, Volume 2 (2017) no. 3 (034801) | DOI

[12] M. Hubert; M. Labousse; S. Perrard Self-propulsion and crossing statistics under random initial conditions, Phys. Rev. E, Volume 95 (2017) no. 6 (062607)

[13] E. Fort; A. Eddi; A. Boudaoud; J. Moukhtar; Y. Couder Path-memory induced quantization of classical orbits, Proc. Natl Acad. Sci. USA, Volume 107 (2010) no. 41, pp. 17515-17520 | DOI

[14] D. M. Harris; J. W. M. Bush Droplets walking in a rotating frame: from quantized orbits to multimodal statistics, J. Fluid Mech., Volume 739 (2014), pp. 444-464 | DOI

[15] A. Eddi; J. Moukhtar; S. Perrard; E. Fort; Y. Couder Level splitting at macroscopic scale, Phys. Rev. Lett., Volume 108 (2012) no. 26 (264503) | DOI

[16] T. Cristea-Platon (“Hydrodynamic analogues of quantum corrals and Friedel oscillations”, PhD thesis, Massachusetts Institute of Technology, 2019)

[17] P. J. Sáenz; T. Cristea-Platon; J. W. Bush A hydrodynamic analog of Friedel oscillations, Sci. Adv., Volume 6 (2020) no. 20 (eaay9234) | DOI

[18] A. U. Oza; D. M. Harris; R. R. Rosales; J. W. M. Bush Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization, J. Fluid Mech., Volume 744 (2014), pp. 404-429 | DOI

[19] S. Perrard; M. Labousse; E. Fort; Y. Couder Chaos driven by interfering memory, Phys. Rev. Lett., Volume 113 (2014) no. 10 (104101) | DOI

[20] M. Labousse; S. Perrard; Y. Couder; E. Fort Build-up of macroscopic eigenstates in a memory-based constrained system, New J. Phys., Volume 16 (2014) no. 11 (113027) | DOI

[21] M. Durey; P. A. Milewski Faraday wave-droplet dynamics: discrete-time analysis, J. Fluid Mech., Volume 821 (2017), pp. 296-329 | DOI | MR | Zbl

[22] D. M. Harris; J. Moukhtar; E. Fort; Y. Couder; J. W. M. Bush Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E, Volume 88 (2013) no. 1 (011001)

[23] P. J. Sáenz; T. Cristea-Platon; J. W. M. Bush Statistical projection effects in a hydrodynamic pilot-wave system, Nat. Phys., Volume 14 (2018) no. 3, p. 315 | DOI

[24] Y. Couder; E. Fort Single-particle diffraction and interference at a macroscopic scale, Phys. Rev. Lett., Volume 97 (2006) no. 15 (154101) | DOI

[25] A. Andersen; J. Madsen; C. Reichelt; S. R. Ahl; B. Lautrup; C. Ellegaard; M. T. Levinsen; T. Bohr Double-slit experiment with single wave-driven particles and its relation to quantum mechanics, Phys. Rev. E, Volume 92 (2015) no. 1 (013006)

[26] M. Rode; J. Madsen; A. Andersen Wave fields in double-slit experiments with wave-driven droplets, Phys. Rev. Fluids, Volume 4 (2019) no. 10 (104801) | DOI

[27] G. Pucci; D. M. Harris; L. M. Faria; J. W. M. Bush Walking droplets interacting with single and double slits, J. Fluid Mech., Volume 835 (2018), pp. 1136-1156 | DOI

[28] M. Labousse; S. Perrard; Y. Couder; E. Fort Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction, Phys. Rev. E, Volume 94 (2016) (042224)

[29] A. U. Oza; R. R. Rosales; J. W. M. Bush Hydrodynamic spin states, Chaos: An Interdiscip. J. Nonlinear Sci., Volume 28 (2018) no. 9 (096106)

[30] R. Valani; A. C. Slim Pilot-wave dynamics of two identical, in-phase bouncing droplets, Chaos, Volume 28 (2018) (096114) | DOI | MR

[31] M. Durey; S. E. Turton; J. W. M. Bush Speed oscillations in classical pilot-wave dynamics, Proc. R. Soc. Lond. A, Volume 476 (2020) no. 2239 (20190884) | MR

[32] E. Fort; Y. Couder Trajectory eigenmodes of an orbiting wave source, Europhys. Lett., Volume 102 (2013) no. 1 (16005) | DOI

[33] C. Borghesi Equivalent quantum equations in a system inspired by bouncing droplets experiments, Found. Phys., Volume 47 (2017), pp. 933-958 | DOI | MR | Zbl

[34] T. Shinbrot Dynamic pilot wave bound states, Chaos: An Interdiscip. J. Nonlinear Sci., Volume 29 (2019) no. 11 (113124) | DOI | MR | Zbl

[35] G. Grössing; S. Fussy; J. M. Pascasio; H. Schwabl Implications of a deeper level explanation of the de Broglie–Bohm version of quantum mechanics, Quantum Stud. Math. Found., Volume 2 (2015), pp. 133-140 | DOI | Zbl

[36] A. M. Cetto; L. de la Peña; A. Valdés-Hernández Specificity of the Schrödinger equation, Quantum Stud.: Math. Found., Volume 2 (2015), pp. 275-287 | DOI | Zbl

[37] M. Hatifi; R. Willox; S. Colin; T. Durt Bouncing oil droplets, de Broglie’s quantum thermostat, and convergence to equilibrium, Entropy, Volume 20 (2018) no. 10, p. 780 | DOI

[38] J. Walleczek; G. Grössing; P. Pylkkänen; B. Hiley Emergent quantum mechanics: David Bohm Centennial Perspectives, Entropy, Volume 21 (2019), p. 113 | DOI | MR

[39] D. Bohm A suggested interpretation of the quantum theory in terms of hidden variables, I, Phys. Rev., Volume 85 (1952a), pp. 66-179 | Zbl

[40] D. Bohm A suggested interpretation of the quantum theory in terms of hidden variables, II, Phys. Rev., Volume 85 (1952b), pp. 180-193 | DOI | MR | Zbl

[41] P. R. Holland The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press, 1995 | Zbl

[42] J. Miles; D. Henderson Parametrically forced surface waves, Annu. Rev. Fluid Mech., Volume 22 (1990) no. 1, pp. 143-165 | DOI | Zbl

[43] P. A. Milewski; C. A. Galeano-Rios; A. Nachbin; J. W. M. Bush Faraday pilot-wave dynamics: modelling and computation, J. Fluid Mech., Volume 778 (2015), pp. 361-388 | DOI | MR | Zbl

[44] S. Turton; M. Couchman; J. W. Bush A review of the theoretical modeling of walking droplets: toward a generalized pilot-wave framework, Chaos: An Interdiscip. J. Nonlinear Sci., Volume 28 (2018) no. 9 (096111) | DOI

[45] J. Moláček; J. W. M. Bush Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory, J. Fluid Mech., Volume 727 (2013a), pp. 612-647 | DOI

[46] J. Moláček; J. W. M. Bush Drops bouncing on a vibrating bath, J. Fluid Mech., Volume 727 (2013b), pp. 582-611 | DOI | Zbl

[47] D. M. Harris; J. W. M. Bush The pilot-wave dynamics of walking droplets, Phys. Fluids, Volume 25 (2013) no. 9 (091112) | DOI

[48] P.-T. Brun; D. M. Harris; V. Prost; J. Quintela; J. W. M. Bush Shedding light on pilot-wave phenomena, Phys. Rev. Fluids, Volume 1 (2016) no. 5 (050510)

[49] A. U. Oza; R. R. Rosales; J. W. M. Bush A trajectory equation for walking droplets: hydrodynamic pilot-wave theory, J. Fluid Mech., Volume 737 (2013), pp. 552-570 | DOI | MR | Zbl

[50] A. Eddi; E. Sultan; J. Moukhtar; E. Fort; M. Rossi; Y. Couder Information stored in Faraday waves: the origin of a path memory, J. Fluid Mech., Volume 674 (2011), pp. 433-463 | DOI | MR | Zbl

[51] M. Durey; P. A. Milewski; J. W. M. Bush Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets, Chaos, Volume 28 (2018) no. 9 (096108) | DOI | MR

[52] Ø. Wind-Willassen; J. Moláček; D. M. Harris; J. W. M. Bush Exotic states of bouncing and walking droplets, Phys. Fluids, Volume 25 (2013) no. 8 (082002) | DOI | Zbl

[53] V. Bacot; S. Perrard; M. Labousse; Y. Couder; E. Fort Multistable free states of an active particle from a coherent memory dynamics, Phys. Rev. Lett., Volume 122 (2019) no. 10 (104303) | DOI

[54] J. Friedel Electronic structure of primary solid solutions in metals, Adv. Phys., Volume 3 (1954) no. 12, pp. 446-507 | DOI | Zbl

[55] K. Kanisawa; M. Butcher; H. Yamaguchi; Y. Hirayama Imaging of Friedel oscillation patterns of two-dimensionally accumulated electrons at epitaxially grown InAs (111) A surfaces, Phys. Rev. Lett., Volume 86 (2001) no. 15, p. 3384 | DOI

[56] T. Gilet Dynamics and statistics of wave-particle interactions in a confined geometry, Phys. Rev. E, Volume 90 (2014) no. 5 (052917)

[57] T. Gilet Quantumlike statistics of deterministic wave-particle interactions in a circular cavity, Phys. Rev. E, Volume 93 (2016) no. 4 (042202)

[58] M. Hubert; M. Labousse; S. Perrard; M. Labousse; N. Vandewalle; Y. Couder Tunable bimodal explorations of space from memory-driven deterministic dynamics, Phys. Rev. E, Volume 100 (2019) no. 032201

[59] E. Schrödinger About the force-free motion in relativistic quantum mechanics, Session Phys. Math., Volume 31 (1930), p. 418

[60] D. Hestenes The zitterbewegung interpretation of quantum mechanics, Found. Phys., Volume 20 (1990) no. 10, pp. 1213-1232 | DOI | MR

[61] L. De la Peña; A. M. Cetto; A. Valdés-Hernández The emerging quantum, The Physics behind Quantum Mechanics, Springer International Publishing, Cham, 2015

[62] C. Qu; C. Hamner; M. Gong; C. Zhang; P. Engels Observation of Zitterbewegung in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A, Volume 88 (2013) no. 2 (021604)

[63] R. Gerritsma; G. Kirchmair; F. Zähringer; E. Solano; R. Blatt; C. Roos Quantum simulation of the Dirac equation, Nature, Volume 463 (2010) no. 7277, p. 68 | DOI

[64] L. De Broglie La mécanique ondulatoire et la structure atomique de la matière et du rayonnement, J. Phys. Radium, Volume 8 (1927a) no. 5, pp. 225-241 | DOI | Zbl

[65] L. De Broglie L’univers à cinq dimensions et la mécanique ondulatoire, J. Phys. Radium, Volume 8 (1927b) no. 2, pp. 65-73 | DOI | Zbl

[66] P. W. Higgs Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett., Volume 13 (1964) no. 16, p. 508 | DOI | MR

[67] D. Greenspan Particle Modeling, Springer Science & Business Media, 2013

[68] W. Greiner et al. Relativistic Quantum Mechanics, Vol. 3, Springer, 1990 | MR

[69] M. Durey; J. W. Bush Hydrodynamic quantum field theory: the onset of particle motion and the form of the pilot wave, Front. Phys., Volume 8 (2020), p. 300 | DOI

[70] J. J. Sakurai; E. D. Commins Modern Quantum Mechanics, 1995

[71] L. E. Ballentine Ensembles in quantum mechanics, Compendium of Quantum Physics (D. Greenberger; K. Hentschel; F. Weinert, eds.), Springer, 2009, pp. 199-201 | DOI

Cité par Sources :

Commentaires - Politique


Ces articles pourraient vous intéresser

Walkers in a wave field with memory

Olivier Devauchelle; Éric Lajeunesse; François James; ...

C. R. Méca (2020)


Resonant interactions in bouncing droplet chains

Lauren Barnes; Giuseppe Pucci; Anand U. Oza

C. R. Méca (2020)


The birth of wave mechanics (1923–1926)

Alain Aspect; Jacques Villain

C. R. Phys (2017)