Pilot wave theory

Louis de Broglie's early results on the pilot wave theory were presented in his thesis (1924) in the context of atomic orbitals where the waves are stationary. Early attempts to develop a general formulation for the dynamics of these guiding waves in terms of a relativistic wave equation were unsuccessful until in 1926 Schrödinger developed his non-relativistic wave equation. He further suggested that since the equation described waves in configuration space, the particle model should be abandoned.{{Cite arXiv|last1=Valentini|first1=Antony|last2=Bacciagaluppi|first2=Guido|date=2006-09-24|title=Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference|eprint=quant-ph/0609184|language=en}} Shortly thereafter,{{cite journal |last=Born |first=M. |year=1926 |title=Quantenmechanik der Stoßvorgänge |journal=Zeitschrift für Physik |volume=38 |issue=11–12 |pages=803–827 |bibcode=1926ZPhy...38..803B |doi=10.1007/BF01397184|s2cid=126244962 }} Max Born suggested that the wave function of Schrödinger's wave equation represents the probability density of finding a particle. Following these results, de Broglie developed the dynamical equations for his pilot wave theory.{{cite journal |last=de Broglie |first=L. |year=1927 |title=La mécanique ondulatoire et la structure atomique de la matière et du rayonnement |journal=Journal de Physique et le Radium |volume=8 |issue=5 |pages=225–241 |bibcode= 1927JPhRa...8..225D

|doi=10.1051/jphysrad:0192700805022500|url=https://hal.archives-ouvertes.fr/jpa-00205292/document}} Initially, de Broglie proposed a double solution approach, in which the quantum object consists of a physical wave (u-wave) in real space which has a spherical singular region that gives rise to particle-like behaviour; in this initial form of his theory he did not have to postulate the existence of a quantum particle.{{cite journal |last1=Dewdney |first1=C. |last2=Horton |first2=G. |last3=Lam |first3=M. M. |last4=Malik |first4=Z. |last5=Schmidt |first5=M. |year=1992 |title=Wave–particle dualism and the interpretation of quantum mechanics |journal=Foundations of Physics |volume=22 |issue=10 |pages=1217–1265 |bibcode=1992FoPh...22.1217D |doi=10.1007/BF01889712|s2cid=122894371 }} He later formulated it as a theory in which a particle is accompanied by a pilot wave.

De Broglie presented the pilot wave theory at the 1927 Solvay Conference.{{cite book |author=Institut International de Physique Solvay |year=1928 |title=Electrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique tenu à Bruxelles du 24 au 29 Octobre 1927 |publisher=Gauthier-Villars

}} However, Wolfgang Pauli raised an objection to it at the conference, saying that it did not deal properly with the case of inelastic scattering. De Broglie was not able to find a response to this objection, and he abandoned the pilot-wave approach. Unlike David Bohm years later, de Broglie did not complete his theory to encompass the many-particle case. The many-particle case shows mathematically that the energy dissipation in inelastic scattering could be distributed to the surrounding field structure by a yet-unknown mechanism of the theory of hidden variables.{{clarify|date=May 2017}}

In 1932, John von Neumann published a book,{{cite book |last1=von Neumann |first1=J. |year=1932 |title=Mathematische Grundlagen der Quantenmechanik |publisher=Springer}} part of which claimed to prove that all hidden variable theories were impossible. This result was found to be flawed by Grete Hermann{{Cite book |last=Seevinck |first=Michiel |url=https://doi.org/10.1007/978-94-024-0970-3_7 |title=Grete Hermann - Between Physics and Philosophy |date=2016 |publisher=Springer Netherlands |isbn=978-94-024-0970-3 |editor-last=Crull |editor-first=Elise |location=Dordrecht |pages=107–117 |language=en |doi=10.1007/978-94-024-0970-3_7 |editor-last2=Bacciagaluppi |editor-first2=Guido}}Hermann, G.: Die naturphilosophischen Grundlagen der Quantenmechanik (Auszug). Abhandlungen

der Fries’schen Schule 6, 75–152 (1935). English translation: Chapter 15 of “Grete Hermann —

Between physics and philosophy”, Elise Crull and Guido Bacciagaluppi, eds., Springer, 2016, 239-

278. [Volume 42 of Studies in History and Philosophy of Science] three years later, though for a variety of reasons this went unnoticed by the physics community for over fifty years.

In 1952, David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot wave theory. Bohm developed pilot wave theory into what is now called the de Broglie–Bohm theory.{{cite journal |last1=Bohm |first1=D. |year=1952 |title=A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, I |journal=Physical Review |volume=85 |issue=2 |pages=166–179 |bibcode=1952PhRv...85..166B |doi=10.1103/PhysRev.85.166

}}{{cite journal |last1=Bohm |first1=D. |year=1952 |title=A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, II |journal=Physical Review |volume=85 |issue=2 |pages=180–193 |bibcode=1952PhRv...85..180B |doi=10.1103/PhysRev.85.180}} The de Broglie–Bohm theory itself might have gone unnoticed by most physicists, if it had not been championed by John Bell, who also countered the objections to it. In 1987, John Bell rediscovered Grete Hermann's work,{{cite book |last1=Bell |first1=J. S. |year=1987 |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |isbn=978-0521334952}} and thus showed the physics community that Pauli's and von Neumann's objections only showed that the pilot wave theory did not have locality.

File:A pilot-wave walker in a circular corral.png in a circular corral. Trajectories of increasing length are colour-coded according to the droplet's local speed (b) The probability distribution of the walker's position corresponds roughly to the amplitude of the corral's Faraday wave mode.{{cite journal|last1=Harris|first1=Daniel M.|last2=Bush|first2=John W. M.|title=The pilot-wave dynamics of walking droplets|journal=Physics of Fluids|date=2013|volume=25|issue=9|pages=091112–091112–2|doi=10.1063/1.4820128|url=https://pdfs.semanticscholar.org/5659/d7cee01a8e55930b6895b11702705bb013fc.pdf|archive-url=https://web.archive.org/web/20161127215926/https://pdfs.semanticscholar.org/5659/d7cee01a8e55930b6895b11702705bb013fc.pdf|url-status=dead|archive-date=2016-11-27|access-date=27 November 2016|bibcode=2013PhFl...25i1112H|hdl=1721.1/92913|s2cid=120607553|hdl-access=free}}]]

The pilot wave theory is a hidden-variable theory. Consequently:

• the theory has realism (meaning that its concepts exist independently of the observer);
• the theory has determinism.

The positions of the particles are considered to be the hidden variables. The observer doesn't know the precise values of these variables; they cannot know them precisely because any measurement disturbs them. On the other hand, the observer is defined not by the wave function of their own atoms but by the atoms' positions. So what one sees around oneself are also the positions of nearby things, not their wave functions.

A collection of particles has an associated matter wave which evolves according to the Schrödinger equation. Each particle follows a deterministic trajectory, which is guided by the wave function; collectively, the density of the particles conforms to the magnitude of the wave function. The wave function is not influenced by the particle and can exist also as an empty wave function.{{cite journal |last=Bell |first=J. S. |year=1992 |title=Six possible worlds of quantum mechanics |journal=Foundations of Physics |volume=22 |issue=10 |pages=1201–1215 |bibcode=1992FoPh...22.1201B |doi=10.1007/BF01889711|s2cid=119542806 }}

The theory brings to light nonlocality that is implicit in the non-relativistic formulation of quantum mechanics and uses it to satisfy Bell's theorem. These nonlocal effects can be shown to be compatible with the no-communication theorem, which prevents use of them for faster-than-light communication, and so is empirically compatible with relativity.{{cite thesis |type=PhD |last=Westman |first=Hans |date=2004-10-29 |publisher=University of Gothenburg |title=Topics in the Foundations of Quantum Theory and Relativity |hdl=2077/16325 }}

Couder, Fort, et al. claimed{{Citation |title=Yves Couder . Explains Wave/Particle Duality via Silicon Droplets [Through the Wormhole] | date=2 August 2011 |url=https://www.youtube.com/watch?v=W9yWv5dqSKk |access-date=2023-08-26 |language=en}} that macroscopic oil droplets on a vibrating fluid bath can be used as an analogue model of pilot waves; a localized droplet creates a periodical wave field around itself. They proposed that resonant interaction between the droplet and its own wave field exhibits behaviour analogous to quantum particles: interference in double-slit experiment,{{Cite journal |last1=Couder |first1=Yves |last2=Fort |first2=Emmanuel |year=2006 |title=Single-Particle Diffraction and Interference at a Macroscopic Scale |journal=Physical Review Letters |volume=97 |issue=15 |pages=154101 |bibcode=2006PhRvL..97o4101C |doi=10.1103/PhysRevLett.97.154101 |pmid=17155330|url=https://www.repository.cam.ac.uk/handle/1810/350127 }} unpredictable tunneling{{Cite journal |last1=Eddi |first1=A. |last2=Fort |first2=E. |last3=Moisy |first3=F. |last4=Couder |first4=Y. |year=2009 |title=Unpredictable Tunneling of a Classical Wave-Particle Association |journal=Physical Review Letters |volume=102 |issue=24 |pages=240401 |bibcode=2009PhRvL.102x0401E |doi=10.1103/PhysRevLett.102.240401 |pmid=19658983}} (depending in a complicated way on a practically hidden state of field), orbit quantization{{cite journal |last1=Fort |first1=E. |last2=Eddi |first2=A. |last3=Boudaoud |first3=A. |last4=Moukhtar |first4=J. |last5=Couder |first5=Y. |year=2010 |title=Path-memory induced quantization of classical orbits |journal=PNAS |volume=107 |issue=41 |pages=17515–17520 |arxiv=1307.6051 |bibcode=2010PNAS..10717515F |doi=10.1073/pnas.1007386107 |pmc=2955113 |s2cid=53462533 |doi-access=free}} (that a particle has to 'find a resonance' with field perturbations it creates—after one orbit, its internal phase has to return to the initial state) and Zeeman effect.{{Cite journal |last1=Eddi |first1=A. |last2=Moukhtar |first2=J. |last3=Perrard |first3=S. |last4=Fort |first4=E. |last5=Couder |first5=Y. |year=2012 |title=Level Splitting at Macroscopic Scale |journal=Physical Review Letters |volume=108 |issue=26 |pages=264503 |bibcode=2012PhRvL.108z4503E |doi=10.1103/PhysRevLett.108.264503 |pmid=23004988}} While attempts to reproduce these experiments have shown some aspects to be questionable{{cite journal |last1=Pucci |first1=G. |date=2018 |title=Walking droplets interacting with single and double slits |url=http://math.mit.edu/~bush/wordpress/wp-content/uploads/2017/12/Pucci-Slits-2017.pdf |journal=Journal of Fluid Mechanics |volume=835 |issue=835 |pages=1136–1156 |bibcode=2018JFM...835.1136P |doi=10.1017/jfm.2017.790 |s2cid=37760205}}

and the interpretation with respect to quantum mechanics has been challenged,{{cite journal |last1=Andersen |first1=Anders |date=2016 |title=Double-slit experiment with single wave-driven particles and its relation to quantum mechanics |url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.013006 |journal=Phys. Rev. E |volume=92 |issue=1 |pages=013006 |doi=10.1103/PhysRevE.92.013006 |pmid=26274269}}

work on the concept has continued with some success.{{cite journal |last1=Bush |first1=John |date=2024 |title=Perspectives on pilot-wave hydrodynamics|url=http://thales.mit.edu/bush/wp-content/uploads/2024/08/Bush-AppliedPhysRev2024.pdf |journal=Applied Physics Letters|volume=125 |issue=1 |pages=030503 |doi=10.1063/5.0210055}}

To derive the de Broglie–Bohm pilot-wave for an electron, the quantum Lagrangian

:$L(t)=\{\backslash frac\{1\}\{2\}\}mv^2-(V+Q),$

where $V$ is the potential energy, $v$ is the velocity and $Q$ is the potential associated with the quantum force (the particle being pushed by the wave function), is integrated along precisely one path (the one the electron actually follows). This leads to the following formula for the Bohm propagator{{Citation needed|date=July 2016}}:

:$K^Q(X\_1,\; t\_1;\; X\_0,\; t\_0)\; =\; \backslash frac\{1\}\{J(t)^\; \{\backslash frac\{1\}\{2\}\}\; \}\; \backslash exp\backslash left[\backslash frac\{i\}\{\backslash hbar\}\backslash int\_\{t\_0\}^\{t\_1\}L(t)\backslash ,dt\backslash right].$

This propagator allows one to precisely track the electron over time under the influence of the quantum potential $Q$.

Pilot wave theory is based on Hamilton–Jacobi dynamics,{{cite web |last=Towler |first=M. |date=10 February 2009 |title=De Broglie-Bohm pilot-wave theory and the foundations of quantum mechanics |url=http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html |publisher=University of Cambridge |access-date=2014-07-03 |archive-date=10 April 2016 |archive-url=https://web.archive.org/web/20160410173517/http://www.tcm.phy.cam.ac.uk/%7Emdt26/pilot_waves.html |url-status=dead }} rather than Lagrangian or Hamiltonian dynamics. Using the Hamilton–Jacobi equation

:$H\backslash left(\backslash ,\backslash vec\{x\}\backslash ,,\; \backslash ;\backslash vec\{\backslash nabla\}\_\{\backslash !x\}\backslash ,\; S\backslash ,,\; \backslash ;t\; \backslash ,\backslash right)\; +\; \{\backslash partial\; S\; \backslash over\; \backslash partial\; t\}\backslash left(\backslash ,\backslash vec\{x\},\backslash ,\; t\backslash ,\backslash right)\; =\; 0$

it is possible to derive the Schrödinger equation:

Consider a classical particle – the position of which is not known with certainty. We must deal with it statistically, so only the probability density $\backslash rho\; (\backslash vec\{x\},t)$ is known. Probability must be conserved, i.e. $\backslash int\backslash rho\backslash ,\backslash mathrm\{d\}^3\backslash vec\{x\}\; =\; 1$ for each $t$. Therefore, it must satisfy the continuity equation

:$\backslash frac\{\backslash ,\; \backslash partial\; \backslash rho\; \backslash ,\}\{\; \backslash partial\; t\; \}\; =\; -\; \backslash vec\{\backslash nabla\}\; \backslash cdot\; (\backslash rho\; \backslash ,\backslash vec\{v\}\; )\; \backslash qquad\backslash qquad\; (1)$

where $\backslash ,\backslash vec\{v\}(\backslash vec\{x\},t)\backslash ,$ is the velocity of the particle.

In the Hamilton–Jacobi formulation of classical mechanics, velocity is given by $\backslash ;\; \backslash vec\{v\}(\backslash vec\{x\},t)\; =\; \backslash frac\{1\}\{\backslash ,m\backslash ,\}\; \backslash ,\; \backslash vec\{\backslash nabla\}\_\{\backslash !x\}\; S(\backslash vec\{x\},\backslash ,t)\; \backslash ;$ where $\backslash ,\; S(\backslash vec\{x\},t)\; \backslash ,$ is a solution of the Hamilton-Jacobi equation

:$-\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; t\}\; =\; \backslash frac\{\backslash ;\backslash left|\backslash ,\backslash nabla\; S\backslash ,\backslash right|^2\backslash ,\}\{2m\}\; +\; \backslash tilde\{V\}\; \backslash qquad\backslash qquad\; (2)$

$\backslash ,(1)\backslash ,$ and $\backslash ,(2)\backslash ,$ can be combined into a single complex equation by introducing the complex function $\backslash ;\; \backslash psi\; =\; \backslash sqrt\{\backslash rho\backslash ,\}\; \backslash ,\; e^\backslash frac\{\backslash ,i\backslash ,S\backslash ,\}\{\backslash hbar\}\; \backslash ;,$ then the two equations are equivalent to

:$i\backslash ,\; \backslash hbar\backslash ,\; \backslash frac\{\backslash ,\backslash partial\; \backslash psi\backslash ,\}\{\backslash partial\; t\}\; =\; \backslash left(\; -\; \backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash nabla^2\; +\backslash tilde\{V\}\; -\; Q\; \backslash right)\backslash psi\; \backslash quad$

with

:$\backslash ;\; Q\; =\; -\; \backslash frac\{\backslash ;\backslash hbar^2\backslash ,\}\{\backslash ,2m\backslash ,\}\; \backslash frac\{\backslash nabla^2\; \backslash sqrt\{\backslash rho\backslash ,\}\}\{\backslash sqrt\{\backslash rho\backslash ,\}\}~.$

The time-dependent Schrödinger equation is obtained if we start with $\backslash ;\backslash tilde\{V\}\; =\; V\; +\; Q\; \backslash ;,$ the usual potential with an extra quantum potential $Q$. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the curvature of the amplitude of the wave function.

Note this potential is the same one that appears in the Madelung equations, a classical analog of the Schrödinger equation.

The matter wave of de Broglie is described by the time-dependent Schrödinger equation:

:$i\backslash ,\; \backslash hbar\backslash ,\; \backslash frac\{\backslash ,\backslash partial\; \backslash psi\backslash ,\}\{\backslash partial\; t\}\; =\; \backslash left(\; -\; \backslash frac\{\backslash hbar^2\}\{\backslash ,2m\backslash ,\}\; \backslash nabla^2\; +\; V\; \backslash right)\backslash psi\; \backslash quad$

The complex wave function can be represented as:

$\backslash psi\; =\; \backslash sqrt\{\backslash rho\backslash ,\}\; \backslash ;\; \backslash exp\; \backslash left(\; \backslash frac\{i\; \backslash ,\; S\}\{\backslash hbar\}\; \backslash right)\; ~$

By plugging this into the Schrödinger equation, one can derive two new equations for the real variables. The first is the continuity equation for the probability density $\backslash ,\backslash rho\backslash ,:$

:$\backslash frac\{\backslash ,\; \backslash partial\; \backslash rho\; \backslash ,\}\{\backslash ,\; \backslash partial\; t\; \backslash ,\}\; +\; \backslash vec\{\backslash nabla\}\; \backslash cdot\; \backslash left(\; \backslash rho\backslash ,\; \backslash vec\{v\}\; \backslash right)\; =\; 0\; ~\; ,$

where the velocity field is determined by the “guidance equation”

:$\backslash vec\{v\}\backslash left(\backslash ,\backslash vec\{r\},\backslash ,t\backslash ,\backslash right)\; =\; \backslash frac\{1\}\{\backslash ,m\backslash ,\}\; \backslash ,\; \backslash vec\{\backslash nabla\}\; S\backslash left(\backslash ,\; \backslash vec\{r\},\backslash ,\; t\; \backslash ,\backslash right)\; ~\; .$

According to pilot wave theory, the point particle and the matter wave are both real and distinct physical entities (unlike standard quantum mechanics, which postulates no physical particle or wave entities, only observed wave-particle duality).

The pilot wave guides the motion of the point particles as described by the guidance equation.

Ordinary quantum mechanics and pilot wave theory are based on the same partial differential equation. The main difference is that in ordinary quantum mechanics, the Schrödinger equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by $\backslash ;\; \backslash rho\; =\; |\backslash psi|^2\; ~\; .$ Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept.

The second equation is a modified Hamilton–Jacobi equation for the action {{mvar|S}}:

:$-\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; t\}\; =\; \backslash frac\{\backslash ;\backslash left|\backslash ,\; \backslash vec\{\backslash nabla\}\; S\; \backslash ,\backslash right|^2\backslash ,\}\{\backslash ,2m\backslash ,\}\; +\; V\; +\; Q\; ~\; ,$

where {{mvar|Q}} is the quantum potential defined by

:$Q\; =\; -\; \backslash frac\{\backslash hbar^2\}\{\backslash ,2m\backslash ,\}\; \backslash frac\{\backslash nabla^2\; \backslash sqrt\{\backslash rho\; \backslash ,\}\; \}\{\backslash sqrt\{\; \backslash rho\; \backslash ,\}\; \}\; ~.$

If we choose to neglect {{mvar|Q}}, our equation is reduced to the Hamilton–Jacobi equation of a classical point particle.{{efn|Strictly speaking, this is only a semiclassical limit;{{clarify|date=March 2012}} because the superposition principle still holds, one needs a “decoherence mechanism” to get rid of it. Interaction with the environment can provide this mechanism.}} So, the quantum potential is responsible for all the mysterious effects of quantum mechanics.

One can also combine the modified Hamilton–Jacobi equation with the guidance equation to derive a quasi-Newtonian equation of motion

:$m\; \backslash ,\; \backslash frac\{d\}\{dt\}\; \backslash ,\; \backslash vec\{v\}\; =\; -\; \backslash vec\{\backslash nabla\}(\; V\; +\; Q\; )\; ~\; ,$

where the hydrodynamic time derivative is defined as

:$\backslash frac\{d\}\{dt\}\; =\; \backslash frac\{\; \backslash partial\; \}\{\backslash ,\; \backslash partial\; t\; \backslash ,\}\; +\; \backslash vec\{v\}\; \backslash cdot\; \backslash vec\{\backslash nabla\}\; ~\; .$

The Schrödinger equation for the many-body wave function $\backslash psi(\backslash vec\{r\}\_1,\; \backslash vec\{r\}\_2,\; \backslash cdots,\; t)$ is given by

:$i\; \backslash hbar\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; t\}\; =\backslash left(\; -\backslash frac\{\backslash hbar^2\}\{2\}\; \backslash sum\_\{i=1\}^\{N\}\; \backslash frac\{\backslash nabla\_i^2\}\{m\_i\}\; +\; V(\backslash mathbf\{r\}\_1,\backslash mathbf\{r\}\_2,\backslash cdots\backslash mathbf\{r\}\_N)\; \backslash right)\; \backslash psi$

The complex wave function can be represented as:

:$\backslash psi\; =\; \backslash sqrt\{\backslash rho\backslash ,\}\; \backslash ;\; \backslash exp\; \backslash left(\; \backslash frac\{i\; \backslash ,\; S\}\{\backslash hbar\}\; \backslash right)$

The pilot wave guides the motion of the particles. The guidance equation for the jth particle is:

:$\backslash vec\{v\}\_j\; =\; \backslash frac\{\backslash nabla\_j\; S\}\{m\_j\}\backslash ;\; .$

The velocity of the jth particle explicitly depends on the positions of the other particles.

This means that the theory is nonlocal.

An extension to the relativistic case with spin has been developed since the 1990s.{{Cite journal|arxiv=quant-ph/0208185|last1= Nikolic|first1= H.|title= Bohmian particle trajectories in relativistic bosonic quantum field theory|journal= Foundations of Physics Letters|volume= 17|issue= 4|pages= 363–380|year= 2004|doi= 10.1023/B:FOPL.0000035670.31755.0a|bibcode= 2004FoPhL..17..363N|citeseerx= 10.1.1.253.838|s2cid= 1927035}}{{Cite journal|arxiv=quant-ph/0302152|last1= Nikolic|first1= H.|title= Bohmian particle trajectories in relativistic fermionic quantum field theory|journal= Foundations of Physics Letters|volume= 18|issue= 2|pages= 123–138|year= 2005|doi= 10.1007/s10702-005-3957-3|bibcode= 2005FoPhL..18..123N|s2cid= 15304186}}{{cite journal | last1 = Dürr | first1 = D. | last2 = Goldstein | first2 = S. |author-link2=Sheldon Goldstein |last3 = Münch-Berndl | first3 = K. | last4 = Zanghì | first4 = N. | year = 1999 | title = Hypersurface Bohm–Dirac Models | journal = Physical Review A | volume = 60 | issue = 4| pages = 2729–2736 | doi=10.1103/physreva.60.2729|arxiv = quant-ph/9801070 |bibcode = 1999PhRvA..60.2729D | s2cid = 52562586 }}{{cite journal | last1 = Dürr | first1 = Detlef | last2 = Goldstein | first2 = Sheldon | last3 = Norsen | first3 = Travis | last4 = Struyve | first4 = Ward | last5 = Zanghì | first5 = Nino | date= 2014 | title = Can Bohmian mechanics be made relativistic? | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences| volume = 470| issue = 2162| pages = 20130699| doi = 10.1098/rspa.2013.0699 | pmid = 24511259 | pmc = 3896068 | arxiv = 1307.1714 | bibcode = 2013RSPSA.47030699D }}{{cite journal | last1 = Fabbri | first1 = Luca | date= 2022 | title = de Broglie-Bohm formulation of Dirac fields | journal = Foundations of Physics| volume = 52| issue = 6 | pages = 116| doi = 10.1007/s10701-022-00641-2| arxiv = 2207.05755 | bibcode = 2022FoPh...52..116F | s2cid = 250491612 }}{{cite journal | last1 = Fabbri | first1 = Luca | date= 2023 | title = Dirac Theory in Hydrodynamic Form | journal = Foundations of Physics| volume = 53| issue = 3 | pages = 54| doi = 10.1007/s10701-023-00695-w | arxiv = 2303.17461 | bibcode = 2023FoPh...53...54F | s2cid = 257833858 }}

Lucien Hardy{{cite journal |last=Hardy |first=L. |year=1992 |title=On the existence of empty waves in quantum theory |journal=Physics Letters A |volume=167 |issue=1 |pages=11–16 |bibcode=1992PhLA..167...11H |doi=10.1016/0375-9601(92)90618-V}} and John Stewart Bell have emphasized that in the de Broglie–Bohm picture of quantum mechanics there can exist empty waves, represented by wave functions propagating in space and time but not carrying energy or momentum,{{cite book |last1=Selleri |first1=F. |last2=Van der Merwe |first2=A. |year=1990 |title=Quantum paradoxes and physical reality |url=https://books.google.com/books?id=qUgX3B02ofAC&pg=PA85 |pages=85–86 |publisher=Kluwer Academic Publishers |isbn=978-0-7923-0253-7}} and not associated with a particle. The same concept was called ghost waves (or "Gespensterfelder", ghost fields) by Albert Einstein. The empty wave function notion has been discussed controversially.{{cite journal |last=Zukowski |first=M. |year=1993 |title="On the existence of empty waves in quantum theory": a comment |journal=Physics Letters A |volume=175 |issue=3–4 |pages=257–258 |bibcode=1993PhLA..175..257Z |doi=10.1016/0375-9601(93)90837-P

}}{{cite journal |last=Zeh |first=H. D. |year=1999 |title=Why Bohm's Quantum Theory? |journal=Foundations of Physics Letters |volume=12 |issue=2 |pages=197–200 |arxiv=quant-ph/9812059 |bibcode= 1999FoPhL..12..197Z |doi=10.1023/A:1021669308832|s2cid=15405774 }}{{cite journal |last=Vaidman |first=L. |year=2005 |title=The Reality in Bohmian Quantum Mechanics or Can You Kill with an Empty Wave Bullet? |journal=Foundations of Physics |volume=35 |issue=2 |pages=299–312 |arxiv=quant-ph/0312227 |bibcode=2005FoPh...35..299V |doi=10.1007/s10701-004-1945-2|s2cid=18990771 }} In contrast, the many-worlds interpretation of quantum mechanics does not call for empty wave functions.

• Hydrodynamic quantum analogues
• Quantum potential

{{notelist}}

{{reflist|30em}}

• [http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html "Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics"] {{Webarchive|url=https://web.archive.org/web/20160410173517/http://www.tcm.phy.cam.ac.uk/%7Emdt26/pilot_waves.html |date=10 April 2016 }}, lecture course on pilot wave theory by Mike Towler, Cambridge University (2009).
• {{SEP|qm-bohm|Bohmian Mechanics|Sheldon Goldstein|Fall 2021}}
• Klaus von Bloh's [https://demonstrations.wolfram.com/author.html?author=Klaus+von+Bloh Bohmian mechanics demonstrations] in: [https://demonstrations.wolfram.com/index.html Wolfram Demonstrations Project]

{{DEFAULTSORT:Pilot wave}}

Category:Hidden variable theory

Category:Interpretations of quantum mechanics

Category:Quantum measurement