Editing De Broglie–Bohm theory (section)

== History ==

The theory was historically developed in the 1920s by de Broglie, who, in 1927, was persuaded to abandon it in favour of the then-mainstream Copenhagen interpretation. David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot-wave theory in 1952. Bohm's suggestions were not then widely received, partly due to reasons unrelated to their content, such as Bohm's youthful [[communist]] affiliations.<ref>F. David Peat, ''Infinite Potential: The Life and Times of David Bohm'' (1997), p. 133. James T. Cushing, ''Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony'' (1994) discusses "the hegemony of the Copenhagen interpretation of quantum mechanics" over theories like Bohmian mechanics as an example of how the acceptance of scientific theories may be guided by social aspects.</ref> The de Broglie–Bohm theory was widely deemed unacceptable by mainstream theorists, mostly because of its explicit non-locality. On the theory, [[John Stewart Bell]], author of the 1964 [[Bell's theorem]] wrote in 1982: {{blockquote|Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the "observer", could be eliminated. ...{{pb}}But why then had Born not told me of this "pilot wave"? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing "impossibility" proofs, after 1952, and as recently as 1978?... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?<ref>{{cite journal |last1=Bell |first1=J. S. |title=On the impossible pilot wave |journal=Foundations of Physics |date=1 October 1982 |volume=12 |issue=10 |pages=989–999 |doi=10.1007/BF01889272 |bibcode=1982FoPh...12..989B |s2cid=120592799 |url=https://link.springer.com/article/10.1007/BF01889272 |language=en |issn=1572-9516}}</ref>}}

Since the 1990s, there has been renewed interest in formulating extensions to de Broglie–Bohm theory, attempting to reconcile it with [[special relativity]] and [[quantum field theory]], besides other features such as [[Spin (physics)|spin]] or curved spatial geometries.<ref>David Bohm and Basil J. Hiley, ''The Undivided Universe – An Ontological Interpretation of Quantum Theory'' appeared after Bohm's death, in 1993; [http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/bhr.pdf reviewed] by Sheldon Goldstein in ''Physics Today'' (1994). J. Cushing, A. Fine, S. Goldstein (eds.), ''Bohmian Mechanics and Quantum Theory – An Appraisal'' (1996).</ref>

De Broglie–Bohm theory has a history of different formulations and names. In this section, each stage is given a name and a main reference.

=== Pilot-wave theory ===

[[Louis de Broglie]] presented his [[pilot wave theory]] at the 1927 Solvay Conference,<ref>Solvay Conference, 1928, Electrons et Photons: Rapports et Descussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 October 1927 sous les auspices de l'Institut International Physique Solvay</ref> after close collaboration with Schrödinger,{{cn|date=August 2024}} who developed his wave equation for de Broglie's theory.{{clarify|Which "
his equation"?|date=August 2024}} At the end of the presentation, [[Wolfgang Pauli]] pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the case of inelastic scattering. Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particular technique could not be generalized for Pauli's purpose, although the audience might have been lost in the technical details and de Broglie's mild manner left the impression that Pauli's objection was valid. He was eventually persuaded to abandon this theory nonetheless because he was "discouraged by criticisms which [it] roused".<ref>Louis be Broglie, in the foreword to David Bohm's ''Causality and Chance in Modern Physics'' (1957). p. x.</ref> De Broglie's theory already applies to multiple spin-less particles, but lacks an adequate theory of measurement as no one understood [[quantum decoherence]] at the time. An analysis of de Broglie's presentation is given in Bacciagaluppi et al.{{clarify|And what pray tell did they say?|date=August 2024}}<ref>Bacciagaluppi, G., and Valentini, A., [https://arxiv.org/pdf/quant-ph/0609184.pdf "Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference"]</ref><ref>See the brief summary by Towler, M., [http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/bohm7.pdf "Pilot wave theory, Bohmian metaphysics, and the foundations of quantum mechanics"] {{Webarchive|url=https://web.archive.org/web/20160322141228/http://www.tcm.phy.cam.ac.uk/%7Emdt26/PWT/lectures/bohm7.pdf |date=22 March 2016 }}</ref> Also, in 1932 [[John von Neumann]] published a [[Von Neumann's no hidden variables proof|no hidden variables proof]] in his book ''[[Mathematical Foundations of Quantum Mechanics]]'',<ref>von Neumann, J. 1932 ''Mathematische Grundlagen der Quantenmechanik''</ref> that was widely believed to prove that all hidden-variable theories are impossible.<ref>{{cite journal |title=Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal|year=2010 |last1=Bub |first1=Jeffrey |author-link1=Jeffrey Bub |journal=Foundations of Physics |volume=40|issue=9–10|pages=1333–1340|bibcode = 2010FoPh...40.1333B |doi = 10.1007/s10701-010-9480-9 |arxiv = 1006.0499 |s2cid=118595119 }}</ref> This sealed the fate of de Broglie's theory for the next two decades.

In 1926, [[Erwin Madelung]] had developed a hydrodynamic version of [[Schrödinger's equation]], which is incorrectly{{cn|date=August 2024}} considered as a basis for the density current derivation of the de Broglie–Bohm theory.<ref>{{cite journal |last=Madelung |first=E. |title=Quantentheorie in hydrodynamischer Form |journal=[[Zeitschrift für Physik|Z. Phys.]] |volume=40 |year=1927 |issue=3–4 |pages=322–326 |doi=10.1007/BF01400372 |bibcode = 1927ZPhy...40..322M |s2cid=121537534 }}</ref> The [[Madelung equations]], being quantum analog of [[Euler equations (fluid dynamics)|Euler equations of fluid dynamics]], differ philosophically from the de Broglie–Bohm mechanics<ref>{{cite journal | first = Roumen | last = Tsekov | title = Bohmian Mechanics versus Madelung Quantum Hydrodynamics | journal = Annuaire de l'Université de Sofia | pages = 112–119 | year = 2012 | doi = 10.13140/RG.2.1.3663.8245 | arxiv=0904.0723| bibcode = 2012AUSFP..SE..112T | s2cid = 59399059 }}</ref> and are the basis of the [[stochastic interpretation]] of quantum mechanics.

[[Peter R. Holland]] has pointed out that, earlier in 1927, [[Albert Einstein|Einstein]] had actually submitted a preprint with a similar proposal but, not convinced, had withdrawn it before publication.<ref>{{cite journal | last1 = Holland | first1 = Peter | year = 2005 | title = What's wrong with Einstein's 1927 hidden-variable interpretation of quantum mechanics? | journal = Foundations of Physics | volume = 35 | issue = 2| pages = 177–196 | doi = 10.1007/s10701-004-1940-7 |arxiv= quant-ph/0401017 |bibcode = 2005FoPh...35..177H | s2cid = 119426936 }}</ref> According to Holland, failure to appreciate key points of the de Broglie–Bohm theory has led to confusion, the key point being "that the trajectories of a many-body quantum system are correlated not because the particles exert a direct force on one another (''à la'' Coulomb) but because all are acted upon by an entity – mathematically described by the wavefunction or functions of it – that lies beyond them".<ref>{{cite journal | last1 = Holland | first1 = Peter | year = 2005 | title = What's wrong with Einstein's 1927 hidden-variable interpretation of quantum mechanics? | journal = Foundations of Physics | volume = 35 | issue = 2| pages = 177–196 | doi = 10.1007/s10701-004-1940-7 |  arxiv= quant-ph/0401017 |bibcode = 2005FoPh...35..177H | s2cid = 119426936 }}</ref> This entity is the [[quantum potential]].

After publishing his popular textbook ''Quantum Theory'' that adhered entirely to the Copenhagen orthodoxy, Bohm was persuaded by Einstein to take a critical look at von Neumann's no hidden variables proof. The result was 'A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' [Bohm 1952]. It was an independent origination of the pilot wave theory, and extended it to incorporate a consistent theory of measurement, and to address a criticism of Pauli that de Broglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers that there should be disturbances to this, in the way [[Brownian motion]] disturbs Newtonian mechanics). This stage is known as the ''de Broglie–Bohm Theory'' in Bell's work [Bell 1987] and is the basis for 'The Quantum Theory of Motion' [Holland 1993].

This stage applies to multiple particles, and is deterministic.

The de Broglie–Bohm theory is an example of a [[hidden-variables theory]]. Bohm originally hoped that hidden variables could provide a [[principle of locality|local]], [[causal]], [[objectivity (philosophy)|objective]] description that would resolve or eliminate many of the paradoxes of quantum mechanics, such as [[Schrödinger's cat]], the [[measurement problem]] and the collapse of the wavefunction. However, [[Bell's theorem]] complicates this hope, as it demonstrates that there can be no local hidden-variable theory that is compatible with the predictions of quantum mechanics. The Bohmian interpretation is [[causal]] but not [[principle of locality|local]].

Bohm's paper was largely ignored or panned by other physicists. [[Albert Einstein]], who had suggested that Bohm search for a realist alternative to the prevailing [[Copenhagen interpretation|Copenhagen approach]], did not consider Bohm's interpretation to be a satisfactory answer to the quantum nonlocality question, calling it "too cheap",<ref>(Letter of 12 May 1952 from Einstein to Max Born, in ''The Born–Einstein Letters'', Macmillan, 1971, p. 192.</ref> while [[Werner Heisenberg]] considered it a "superfluous 'ideological superstructure' ".<ref>Werner Heisenberg, ''Physics and Philosophy'' (1958), p. 133.</ref> [[Wolfgang Pauli]], who had been unconvinced by de Broglie in 1927, conceded to Bohm as follows:

<blockquote>I just received your long letter of 20th November, and I also have studied more thoroughly the details of your paper. I do not see any longer the possibility of any logical contradiction as long as your results agree completely with those of the usual wave mechanics and as long as no means is given to measure the values of your hidden parameters both in the measuring apparatus and in the observe [sic] system. As far as the whole matter stands now, your 'extra wave-mechanical predictions' are still a check, which cannot be cashed.<ref>Pauli to Bohm, 3 December 1951, in Wolfgang Pauli, ''Scientific Correspondence'', Vol IV – Part I, [ed. by Karl von Meyenn], (Berlin, 1996), pp. 436–441.</ref></blockquote>

He subsequently described Bohm's theory as "artificial metaphysics".<ref>Pauli, W. (1953). "Remarques sur le probleme des parametres caches dans la mecanique quantique et sur la theorie de l'onde pilote". In A. George (Ed.), ''Louis de Broglie—physicien et penseur'' (pp. 33–42). Paris: Editions Albin Michel.</ref>

According to physicist [[Max Dresden]], when Bohm's theory was presented at the [[Institute for Advanced Study]] in Princeton, many of the objections were [[ad hominem]], focusing on Bohm's sympathy with communists as exemplified by his refusal to give testimony to the [[House Un-American Activities Committee]].<ref>F. David Peat, ''Infinite Potential: The Life and Times of David Bohm'' (1997), p. 133.</ref>

In 1979, Chris Philippidis, Chris Dewdney and [[Basil Hiley]] were the first to perform numeric computations on the basis of the quantum potential to deduce ensembles of particle trajectories.<ref>Statement on that they were in fact the first in: B. J. Hiley: ''Nonlocality in microsystems'', in: Joseph S. King, Karl H. Pribram (eds.): ''Scale in Conscious Experience: Is the Brain Too Important to be Left to Specialists to Study?'', Psychology Press, 1995, pp.&nbsp;318 ff., [https://books.google.com/books?id=jXRub48gzuIC&pg=PA p. 319], which takes reference to: {{Cite journal|doi=10.1007/BF02743566|title=Quantum interference and the quantum potential|journal=Il Nuovo Cimento B|volume=52|issue=1|pages=15|year=2007|last1=Philippidis|first1=C.|last2=Dewdney|first2=C.|last3=Hiley|first3=B. J.|bibcode=1979NCimB..52...15P|s2cid=53575967}}</ref><ref>[[Olival Freire Jr.]]: ''Continuity and change: charting David Bohm's evolving ideas on quantum mechanics'', In: Décio Krause, Antonio Videira (eds.): ''Brazilian Studies in the Philosophy and History of Science'', Boston Studies in the Philosophy of Science, Springer, {{ISBN|978-90-481-9421-6}}, pp.291–300, therein [https://books.google.com/books?id=EAswiGbShCAC&pg=PA296 p. 296–297]</ref> Their work renewed the interests of physicists in the Bohm interpretation of quantum physics.<ref>Olival Freire jr.: ''A story without an ending: the quantum physics controversy 1950–1970'', Science & Education, vol.&nbsp;12, pp.&nbsp;573–586, 2003, [http://www.controversia.fis.ufba.br/index_arquivos/freire-se.pdf#page=4 p. 576] {{Webarchive|url=https://web.archive.org/web/20140310040550/http://www.controversia.fis.ufba.br/index_arquivos/freire-se.pdf#page=4 |date=10 March 2014 }}</ref>

Eventually [[John Stewart Bell|John Bell]] began to defend the theory. In "Speakable and Unspeakable in Quantum Mechanics" [Bell 1987], several of the papers refer to hidden-variables theories (which include Bohm's).

The trajectories of the Bohm model that would result for particular experimental arrangements were termed "surreal" by some.<ref name=ESSW_1992/><ref>{{cite arXiv|eprint=quant-ph/0010020|last1= Hiley|first1= B. J.|title= Quantum trajectories, real, surreal or an approximation to a deeper process?|last2=  E Callaghan|first2= R.|last3=  Maroney|first3= O.|year= 2000}}</ref> Still in 2016, mathematical physicist [[Sheldon Goldstein]] said of Bohm's theory: "There was a time when you couldn't even talk about it because it was heretical. It probably still is the kiss of death for a physics career to be actually working on Bohm, but maybe that's changing."<ref name="newscientist.com" />

=== Bohmian mechanics ===
Bohmian mechanics is the same theory, but with an emphasis on the notion of current flow, which is determined on the basis of the [[quantum equilibrium hypothesis]] that the probability follows the Born rule.{{cn|date=August 2024}} The term "Bohmian mechanics" is also often used to include most of the further extensions past the spin-less version of Bohm.{{cn|date=August 2024}} While de Broglie–Bohm theory has [[Lagrangian mechanics|Lagrangians]] and [[Hamilton-Jacobi equations]] as a primary focus and backdrop, with the icon of the [[quantum potential]], Bohmian mechanics considers the [[continuity equation]] as primary and has the guiding equation as its icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-less particles.

All of non-relativistic quantum mechanics can be fully accounted for in this theory. Recent studies have used this formalism to compute the evolution of many-body quantum systems, with a considerable increase in speed as compared to other quantum-based methods.<ref>Larder et al. (2019) Fast nonadiabatic dynamics of many-body quantum systems  https://doi.org/10.1126/sciadv.aaw1634</ref>

=== Causal interpretation and ontological interpretation ===
Bohm developed his original ideas, calling them the ''Causal Interpretation''. Later he felt that ''causal'' sounded too much like ''deterministic'' and preferred to call his theory the ''Ontological Interpretation''. The main reference is "The Undivided Universe" (Bohm, Hiley 1993).

This stage covers work by Bohm and in collaboration with [[Jean-Pierre Vigier]] and Basil Hiley. Bohm is clear that this theory is non-deterministic (the work with Hiley includes a stochastic theory). As such, this theory is not strictly speaking a formulation of de Broglie–Bohm theory, but it deserves mention here because the term "Bohm Interpretation" is ambiguous between this theory and de Broglie–Bohm theory.

In 1996 [[philosopher of science]] [[Arthur Fine]] gave an in-depth analysis of possible interpretations of Bohm's model of 1952.<ref>A. Fine: "On the interpretation of Bohmian mechanics", in: J. T. Cushing, A. Fine, S. Goldstein (Eds.): ''Bohmian mechanics and quantum theory: an appraisal'', Springer, 1996, pp.&nbsp;231−250.</ref>

William Simpson has suggested a [[Hylomorphism|hylomorphic]] interpretation of Bohmian mechanics, in which the cosmos is an Aristotelian substance composed of material particles and a substantial form. The wave function is assigned a dispositional role in choreographing the trajectories of the particles.<ref>{{cite journal|last=Simpson|first=W.M.R|title=Cosmic Hylomorphism: a powerist ontology of quantum mechanics|date=2021|journal=European Journal for Philosophy of Science|volume=11|issue=28|page=28|doi=10.1007/s13194-020-00342-5 |issn=1879-4912 |pmid=33520035|pmc=7831748}}</ref>

=== Hydrodynamic quantum analogs ===
{{Main|Hydrodynamic quantum analogs}}

Experiments on hydrodynamical analogs of quantum mechanics beginning with the work of Couder and Fort (2006)<ref>{{cite journal|last1=Couder|first1=Yves|last2=Fort|first2=Emmanuel|title=Single-Particle Diffraction and Interference at a Macroscopic Scale|journal=Phys. Rev. Lett.|date=2006|volume=97|issue=15|page=154101|doi=10.1103/PhysRevLett.97.154101|pmid=17155330|url=http://users.isy.liu.se/en/jalar/kurser/QF/assignments/Couder2006.pdf|bibcode=2006PhRvL..97o4101C}}</ref><ref>{{cite web|last1=Hardesty|first1=Larry|title=Fluid mechanics suggests alternative to quantum orthodoxy|url=https://news.mit.edu/2014/fluid-systems-quantum-mechanics-0912|website=news.mit.edu|access-date=7 December 2016|date=12 September 2014}}</ref> have purported to show that macroscopic classical pilot-waves can exhibit characteristics previously thought to be restricted to the quantum realm. Hydrodynamic pilot-wave analogs have been claimed to duplicate the double slit experiment, tunneling, quantized orbits, and numerous other quantum phenomena which have led to a resurgence in interest in pilot wave theories.<ref>{{cite journal|last1=Bush|first1=John W. M.|title=The new wave of pilot-wave theory|journal=Physics Today|date=2015|volume=68|issue=8|page=47|doi=10.1063/PT.3.2882|url=http://newfos.arizona.edu/sites/default/files/uploads/documents/Pilot_Waves_Phys_Today_Aug_2015.pdf|bibcode=2015PhT....68h..47B|hdl=1721.1/110524|s2cid=17882118 |access-date=7 December 2016|archive-url=https://web.archive.org/web/20161125110215/http://newfos.arizona.edu/sites/default/files/uploads/documents/Pilot_Waves_Phys_Today_Aug_2015.pdf|archive-date=25 November 2016|url-status=dead|hdl-access=free}}</ref><ref>{{cite journal|last1=Bush|first1=John W. M.|title=Pilot-Wave Hydrodynamics|journal=Annual Review of Fluid Mechanics|date=2015|volume=47|issue=1|pages=269–292|doi=10.1146/annurev-fluid-010814-014506|bibcode=2015AnRFM..47..269B|hdl=1721.1/89790|hdl-access=free}}</ref><ref>{{cite journal|last1=Wolchover|first1=Natalie|title=Fluid Tests Hint at Concrete Quantum Reality|journal=Quanta Magazine|date=24 June 2014|url=https://www.quantamagazine.org/20140624-fluid-tests-hint-at-concrete-quantum-reality/|access-date=28 November 2016}}</ref> 
The analogs have been compared to the ''[[Faraday wave]]''.<ref>John W. M. Bush: [http://www.tcm.phy.cam.ac.uk/~mdt26/tti_talks/deBB_10/bush_tti2010.pdf "Quantum mechanics writ large"] {{Webarchive|url=https://web.archive.org/web/20171215051636/http://www.tcm.phy.cam.ac.uk/~mdt26/tti_talks/deBB_10/bush_tti2010.pdf |date=15 December 2017 }}.</ref>
These results have been disputed: experiments fail to reproduce aspects of the double-slit experiments.<ref>{{Cite journal |last1=Andersen |first1=Anders |last2=Madsen |first2=Jacob |last3=Reichelt |first3=Christian |last4=Rosenlund Ahl |first4=Sonja |last5=Lautrup |first5=Benny |last6=Ellegaard |first6=Clive |last7=Levinsen |first7=Mogens T. |last8=Bohr |first8=Tomas |date=2015-07-06 |title=Double-slit experiment with single wave-driven particles and its relation to quantum mechanics |url=https://link.aps.org/doi/10.1103/PhysRevE.92.013006 |journal=Physical Review E |language=en |volume=92 |issue=1 |page=013006 |doi=10.1103/PhysRevE.92.013006 |pmid=26274269 |issn=1539-3755}}</ref><ref>{{cite web |url=https://www.quantamagazine.org/famous-experiment-dooms-pilot-wave-alternative-to-quantum-weirdness-20181011/ |title=Famous Experiment Dooms Alternative to Quantum Weirdness |last=Wolchover |first=Natalie |date=11 October 2018 |publisher=Quanta Magazine |access-date=17 October 2018 |quote=Oil droplets guided by "pilot waves" have failed to reproduce the results of the quantum double-slit experiment}}</ref> High precision measurements in the tunneling case point to a different origin of the unpredictable crossing: rather than initial position uncertainty or environmental noise, interactions at the barrier seem to be involved.<ref>{{Cite journal |last1=Tadrist |first1=Loïc |last2=Gilet |first2=Tristan |last3=Schlagheck |first3=Peter |last4=Bush |first4=John W. M. |date=2020-07-09 |title=Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling |url=https://link.aps.org/doi/10.1103/PhysRevE.102.013104 |journal=Physical Review E |language=en |volume=102 |issue=1 |page=013104 |doi=10.1103/PhysRevE.102.013104 |pmid=32795022 |issn=2470-0045}}</ref>  

Another classical analog has been reported in surface gravity waves.<ref>{{cite journal |last1=Rozenman |first1=Georgi Gary |last2=Bondar |first2=Denys I |last3=Schleich |first3=Wolfgang P |last4=Shemer |first4=Lev |last5=Arie |first5=Ady A |title=Observation of Bohm trajectories and quantum potentials of classical waves |journal=Physica Scripta |volume=98 |issue=4 |pages=044004 |date=10 March 2023 |doi=10.1088/1402-4896/acb408 |publisher=IOP Publishing Ltd |issn=|pmc= |pmid=|doi-access=free }}</ref>

{| class="wikitable" style="text-align: center;"
|+ A comparison by Bush (2015)<ref>{{cite journal|last1=Bush|first1=John W.M.|title=Pilot-Wave Hydrodynamics|journal=Annual Review of Fluid Mechanics|date=2015|volume=47|issue=1|pages=269–292|doi=10.1146/annurev-fluid-010814-014506|url=http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Bush-AnnRev2015.pdf|bibcode=2015AnRFM..47..269B|hdl=1721.1/89790|hdl-access=free}}</ref> among the walking droplet system, de Broglie's double-solution pilot-wave theory<ref>{{cite journal|last1=De Broglie|first1=Louis|title=Une tentative d'interprétation causale et non linéaire de la mécanique ondulatoire: (la théorie de la double solution)|journal=Gauthier-Villars|date=1956}}</ref><ref>{{cite journal|last1=de Broglie|first1=Louis|title=Interpretation of quantum mechanics by the double solution theory|journal=Annales de la Fondation|date=1987|volume=12|issue=4|pages=399–421|url=http://aflb.ensmp.fr/AFLB-classiques/aflb124p001.pdf|issn=0182-4295}}</ref> and its extension to SED<ref>{{cite book|last1=de la Peña|first1=Luis|last2=Cetto|first2=A.M.|title=The Quantum Dice: An Introduction to Stochastic Electrodynamics|date=1996|publisher=Springer|isbn=978-90-481-4646-8|url=https://books.google.com/books?id=XPHqCAAAQBAJ|doi=10.1007/978-94-015-8723-5}}</ref><ref>{{cite journal|last1=Haisch|first1=Bernard|last2=Rueda|first2=Alfonso|title=On the relation between a zero-point-field-induced inertial effect and the Einstein-de Broglie formula|journal=Physics Letters A|date=2000|volume= 268|issue=4–6|pages=224–227|doi=10.1016/S0375-9601(00)00186-9|arxiv=gr-qc/9906084|bibcode=2000PhLA..268..224H|citeseerx=10.1.1.339.2104|s2cid=2030449}}</ref>
!
! Hydrodynamic walkers
! de Broglie
! SED pilot wave
|-
! Driving
|| bath vibration || internal clock || vacuum fluctuations
|-
! Spectrum
|| monochromatic || monochromatic || broad
|-
! Trigger
|| bouncing || [[zitterbewegung]] || zitterbewegung
|-
! Trigger frequency
|| <math>\omega_F</math> || <math>\omega_c=mc^2/\hbar</math> || <math>\omega_c=mc^2/\hbar</math>
|-
! Energetics
|| GPE <math>\leftrightarrow</math> wave ||<math>mc^2\leftrightarrow\hbar\omega</math> || <math>mc^2\leftrightarrow</math> EM
|-
! Resonance
|| droplet-wave|| harmony of phases || unspecified
|-
! Dispersion <math>\omega(k)</math>
|| <math>\omega_F^2\approx\sigma k^3/\rho</math> || <math>\omega^2\approx\omega_c^2+c^2k^2</math> || <math>\omega=ck</math>
|-
! Carrier <math>\lambda</math>
|| <math>\lambda_F</math> || <math>\lambda_{dB}</math> || <math>\lambda_{c}</math>
|-
! Statistical <math>\lambda</math>
|| <math>\lambda_F</math> || <math>\lambda_{dB}</math> || <math>\lambda_{dB}</math>
|}