Editing Pilot wave theory (section)

==Mathematical foundations==
To derive the de Broglie–Bohm pilot-wave for an electron, the quantum [[Lagrangian mechanics|Lagrangian]]

:<math>L(t)={\frac{1}{2}}mv^2-(V+Q),</math>

where <math>V</math> is the potential energy, <math>v</math> is the velocity and <math>Q</math> 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}}:

:<math>K^Q(X_1, t_1; X_0, t_0) = \frac{1}{J(t)^ {\frac{1}{2}} } \exp\left[\frac{i}{\hbar}\int_{t_0}^{t_1}L(t)\,dt\right].</math>

This [[propagator]] allows one to precisely track the electron over time under the influence of the quantum potential <math>Q</math>.

===Derivation of the Schrödinger equation===
Pilot wave theory is based on [[Hamilton–Jacobi equation|Hamilton–Jacobi dynamics]],<ref>{{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 }}</ref> rather than [[Lagrangian mechanics|Lagrangian]] or [[Hamiltonian dynamics]]. Using the Hamilton–Jacobi equation

:<math> H\left(\,\vec{x}\,, \;\vec{\nabla}_{\!x}\, S\,, \;t \,\right) + {\partial S \over \partial t}\left(\,\vec{x},\, t\,\right) = 0</math>

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 <math>\rho (\vec{x},t)</math> is known. Probability must be conserved, i.e. <math>\int\rho\,\mathrm{d}^3\vec{x} = 1</math> for each <math>t</math>. Therefore, it must satisfy the continuity equation

:<math>\frac{\, \partial \rho \,}{ \partial t } = - \vec{\nabla} \cdot (\rho \,\vec{v} ) \qquad\qquad (1)</math>

where <math>\,\vec{v}(\vec{x},t)\,</math> is the velocity of the particle.

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

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

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

:<math>i\, \hbar\, \frac{\,\partial \psi\,}{\partial t} = \left( - \frac{\hbar^2}{2m} \nabla^2 +\tilde{V} - Q \right)\psi \quad</math>
with
:<math> \; Q = - \frac{\;\hbar^2\,}{\,2m\,} \frac{\nabla^2 \sqrt{\rho\,}}{\sqrt{\rho\,}}~.</math>

The time-dependent Schrödinger equation is obtained if we start with <math>\;\tilde{V} = V + Q \;,</math> the usual potential with an extra [[quantum potential]] <math>Q</math>. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the [[Curvature#Graph of a function|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.

===Mathematical formulation for a single particle===
The matter wave of de Broglie is described by the time-dependent Schrödinger equation:

:<math> i\, \hbar\, \frac{\,\partial \psi\,}{\partial t} = \left( - \frac{\hbar^2}{\,2m\,} \nabla^2 + V \right)\psi \quad</math>

The complex wave function can be represented as:

<math>\psi = \sqrt{\rho\,} \; \exp \left( \frac{i \, S}{\hbar} \right) ~</math>

By plugging this into the Schrödinger equation, one can derive two new equations for the real variables. The first is the [[Probability current#Continuity equation for quantum mechanics|continuity equation for the probability density]] <math>\,\rho\,:</math><ref name=Bohm1952a/>

:<math>\frac{\, \partial \rho \,}{\, \partial t \,} + \vec{\nabla} \cdot \left( \rho\, \vec{v} \right) = 0 ~ ,</math>

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

:<math>\vec{v}\left(\,\vec{r},\,t\,\right) = \frac{1}{\,m\,} \, \vec{\nabla} S\left(\, \vec{r},\, t \,\right) ~ .</math>

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 <math>\; \rho = |\psi|^2 ~ .</math> 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}}:

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

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

:<math> Q = - \frac{\hbar^2}{\,2m\,} \frac{\nabla^2 \sqrt{\rho \,} }{\sqrt{ \rho \,} } ~.</math>

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

:<math>m \, \frac{d}{dt} \, \vec{v} = - \vec{\nabla}( V + Q ) ~ ,</math>

where the hydrodynamic time derivative is defined as

:<math>\frac{d}{dt} = \frac{ \partial }{\, \partial t \,} + \vec{v} \cdot \vec{\nabla} ~ .</math>

===Mathematical formulation for multiple particles===
The Schrödinger equation for the many-body wave function <math> \psi(\vec{r}_1, \vec{r}_2, \cdots, t) </math> is given by

:<math> i \hbar \frac{\partial \psi}{\partial t} =\left( -\frac{\hbar^2}{2} \sum_{i=1}^{N} \frac{\nabla_i^2}{m_i} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \right) \psi </math>

The complex wave function can be represented as:

:<math>\psi = \sqrt{\rho\,} \; \exp \left( \frac{i \, S}{\hbar} \right) </math>

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

:<math> \vec{v}_j = \frac{\nabla_j S}{m_j}\; .</math>

The velocity of the jth particle explicitly depends on the positions of the other particles.
This means that the theory is nonlocal.

===Relativity===

An extension to the [[De Broglie–Bohm theory#Relativity|relativistic case]] with spin has been developed since the 1990s.<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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 }}</ref><ref>{{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 }}</ref><ref>{{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 }}</ref><ref>{{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 }}</ref>

===Empty wave function===
[[Lucien Hardy]]<ref>{{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}}</ref> and [[John Stewart Bell]]<ref name="bell-1992"/> 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,<ref name="Selleri">{{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}}</ref> and not associated with a particle. The same concept was called ''ghost waves'' (or "Gespensterfelder", ''ghost fields'') by [[Albert Einstein]].<ref name="Selleri"/> The empty wave function notion has been discussed controversially.<ref>{{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
}}</ref><ref>{{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 }}</ref><ref>{{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 }}</ref> In contrast, the [[many-worlds interpretation]] of quantum mechanics does not call for empty wave functions.<ref name="bell-1992"/>