Editing De Broglie–Bohm theory (section)

== Theory ==

=== Pilot wave ===

The de Broglie–Bohm theory describes a pilot wave <math>\psi(q,t) \in \mathbb{C}</math> in a [[Configuration space (physics) | configuration space]] <math>Q</math> and trajectories <math>q(t) \in Q</math> of particles as in classical mechanics but defined by non-Newtonian mechanics.<ref>{{Cite journal |last=Passon |first=Oliver |date=2004-11-01 |title=How to teach quantum mechanics |url=https://iopscience.iop.org/article/10.1088/0143-0807/25/6/008 |journal=European Journal of Physics |volume=25 |issue=6 |pages=765–769 |doi=10.1088/0143-0807/25/6/008 |issn=0143-0807|arxiv=quant-ph/0404128 }}</ref> At every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrödinger equation). 

The de Broglie–Bohm theory works on particle positions and trajectories like [[classical mechanics]] but the dynamics are different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory, the quantum "field exerts a new kind of "quantum-mechanical" force".<ref>{{cite book |author=Bohm |first=David |title=Causality and Chance in Modern Physics |publisher=Routledge & Kegan Paul and D. Van Nostrand |year=1957 |isbn=978-0-8122-1002-6}}</ref>{{rp|76}} Bohm hypothesized that each particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction by the quantum potential.<ref>D. Bohm and B. Hiley: ''The undivided universe: An ontological interpretation of quantum theory'', p. 37.</ref> Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie–Bohm theory, not localized at the position of the particle.<ref>H. R. Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", ''Foundations of Physics'', 1995, Volume 25, Number 2, pp. 329–347.</ref><ref>J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", ''Foundations of Physics'', March 1999, Volume 29, Issue 3, pp. 333–348.</ref>

The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrödinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...] the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles".<ref>{{Cite book |last1=Bohm |first1=David |url=https://books.google.com/books?id=vt9XKjc4WAQC&pg=PA24 |title=The undivided universe: an ontological interpretation of quantum theory |last2=Hiley |first2=Basil J. |date=1995 |publisher=Routledge |isbn=978-0-415-12185-9 |pages=24 |language=en}}</ref> P. Holland considers this lack of reciprocal action of particles and wave function to be one "[a]mong the many nonclassical properties exhibited by this theory".<ref>{{Cite book |last=Holland |first=Peter R. |url=https://books.google.com/books?id=BsEfVBzToRMC&pg=PA26 |title=The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics |date=1995-01-26 |publisher=Cambridge University Press |isbn=978-0-521-48543-2 |pages=26 |language=en}}</ref> Holland later called this a merely ''apparent'' lack of back reaction, due to the incompleteness of the description.<ref>{{cite journal | last1 = Holland | first1 = P. | year = 2001 | title = Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction | url = http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf#page=31 | journal = Nuovo Cimento B | volume = 116 | issue = 10 | pages = 1143–1172 | bibcode = 2001NCimB.116.1143H | access-date = 1 August 2011 | archive-date = 10 November 2011 | archive-url = https://web.archive.org/web/20111110140052/http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf#page=31 | url-status = dead }}</ref>

In what follows below, the setup for one particle moving in <math>\mathbb{R}^3</math> is given followed by the setup for ''N'' particles moving in 3 dimensions. In the first instance, configuration space and real space are the same, while in the second, real space is still <math>\mathbb{R}^3</math>, but configuration space becomes <math>\mathbb{R}^{3N}</math>. While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space, which is how particles are entangled with each other in this theory.

[[#Extensions|Extensions]] to this theory include spin and more complicated configuration spaces.

We use variations of <math>\mathbf{Q}</math> for particle positions, while <math>\psi</math> represents the complex-valued wavefunction on configuration space.

=== Guiding equation ===
For a spinless single particle moving in <math>\mathbb{R}^3</math>, the particle's velocity is

:<math>\frac{d\mathbf{Q}}{dt}(t) = \frac{\hbar}{m} \operatorname{Im}\left(\frac{\nabla \psi}{\psi}\right)(\mathbf{Q}, t).</math>

For many particles labeled <math>\mathbf{Q}_k</math> for the <math>k</math>-th particle their velocities are

:<math>\frac{d\mathbf{Q}_k}{dt}(t) = \frac{\hbar}{m_k} \operatorname{Im}\left(\frac{\nabla_k \psi}{\psi}\right)(\mathbf{Q}_1, \mathbf{Q}_2, \ldots, \mathbf{Q}_N, t).</math>

The main fact to notice is that this velocity field depends on the actual positions of all of the <math>N</math> particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe.

=== Schrödinger's equation ===
The one-particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on <math>\mathbb{R}^3</math>. The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function <math>V</math> on <math>\mathbb{R}^3</math>:

:<math>i\hbar\frac{\partial}{\partial t}\psi = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi.</math>

For many particles, the equation is the same except that <math>\psi</math> and <math>V</math> are now on configuration space, <math>\mathbb{R}^{3N}</math>:

:<math>i\hbar\frac{\partial}{\partial t}\psi = -\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}\nabla_k^2\psi + V\psi.</math>

This is the same wavefunction as in conventional quantum mechanics.

=== Relation to the Born rule ===
{{Main|Born rule}}
In Bohm's original papers,<ref name=":0">{{Cite journal |last=Bohm |first=David |date=1952-01-15 |title=A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I |url=https://link.aps.org/doi/10.1103/PhysRev.85.166 |journal=Physical Review |language=en |volume=85 |issue=2 |pages=166–179 |doi=10.1103/PhysRev.85.166 |issn=0031-899X}}</ref> he discusses how de Broglie–Bohm theory results in the usual measurement results of quantum mechanics. The main idea is that this is true if the positions of the particles satisfy the statistical distribution given by <math>|\psi|^2</math>. And that distribution is guaranteed to be true for all time by the guiding equation if the initial distribution of the particles satisfies <math>|\psi|^2</math>.

For a given experiment, one can postulate this as being true and verify it experimentally. But, as argued by Dürr et al.,<ref name="dgz92">{{cite journal | last1 = Dürr | first1 = D. | last2 = Goldstein | first2 = S. | last3 = Zanghì | first3 = N. | year = 1992 | title = Quantum Equilibrium and the Origin of Absolute Uncertainty | journal = Journal of Statistical Physics | volume = 67 | issue = 5–6| pages = 843–907 |arxiv = quant-ph/0308039 |bibcode = 1992JSP....67..843D |doi = 10.1007/BF01049004 | s2cid = 15749334 }}</ref> one needs to argue that this distribution for subsystems is typical. The authors argue that <math>|\psi|^2</math>, by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure of typicality for [[initial condition]]s of the positions of the particles. The authors then prove that the vast majority of possible initial configurations will give rise to statistics obeying the Born rule (i.e., <math>|\psi|^2</math>) for measurement outcomes. In summary, in a universe governed by the de Broglie–Bohm dynamics, Born rule behavior is typical.

The situation is thus analogous to the situation in classical statistical physics. A low-[[entropy]] initial condition will, with overwhelmingly high probability, evolve into a higher-entropy state: behavior consistent with the [[second law of thermodynamics]] is typical. There are anomalous initial conditions that would give rise to violations of the second law; however in the absence of some very detailed evidence supporting the realization of one of those conditions, it would be quite unreasonable to expect anything but the actually observed uniform increase of entropy. Similarly in the de Broglie–Bohm theory, there are anomalous initial conditions that would produce measurement statistics in violation of the Born rule (conflicting the predictions of standard quantum theory), but the typicality theorem shows that absent some specific reason to believe one of those special initial conditions was in fact realized, the Born rule behavior is what one should expect.

It is in this qualified sense that the Born rule is, for the de Broglie–Bohm theory, a [[theorem]] rather than (as in ordinary quantum theory) an additional [[postulate]].

It can also be shown that a distribution of particles which is ''not'' distributed according to the Born rule (that is, a distribution "out of quantum equilibrium") and evolving under the de Broglie–Bohm dynamics is overwhelmingly likely to evolve dynamically into a state distributed as <math>|\psi|^2</math>.<ref>{{Cite journal|arxiv=1103.1589|last1= Towler|first1= M. D.|title= Timescales for dynamical relaxation to the Born rule|journal= Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume= 468|issue= 2140|pages= 990|last2= Russell|first2= N. J.|last3= Valentini|first3= A.|year= 2012|doi= 10.1098/rspa.2011.0598|bibcode= 2012RSPSA.468..990T|s2cid= 119178440}}. A video of the electron density in a 2D box evolving under this process is available [http://www.tcm.phy.cam.ac.uk/~mdt26/raw_movie.gif here] {{Webarchive|url=https://web.archive.org/web/20160303230023/http://www.tcm.phy.cam.ac.uk/~mdt26/raw_movie.gif |date=3 March 2016 }}.</ref>

=== The conditional wavefunction of a subsystem ===
In the formulation of the de Broglie–Bohm theory, there is only a wavefunction for the entire universe (which always evolves by the Schrödinger equation). Here, the "universe" is simply the system limited by the same boundary conditions used to solve the Schrödinger equation. However, once the theory is formulated, it is convenient to introduce a notion of wavefunction also for subsystems of the universe. Let us write the wavefunction of the universe as <math>\psi(t, q^\text{I}, q^\text{II})</math>, where <math>q^\text{I}</math> denotes the configuration variables associated to some subsystem (I) of the universe, and <math>q^\text{II}</math> denotes the remaining configuration variables. Denote respectively by <math>Q^\text{I}(t)</math> and <math>Q^\text{II}(t)</math> the actual configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case. The ''conditional wavefunction'' of subsystem (I) is defined by

:<math>\psi^\text{I}(t, q^\text{I}) = \psi(t, q^\text{I}, Q^\text{II}(t)).</math>

It follows immediately from the fact that <math>Q(t) = (Q^\text{I}(t), Q^\text{II}(t))</math> satisfies the guiding equation that also the configuration <math>Q^\text{I}(t)</math> satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wavefunction <math>\psi</math> replaced with the conditional wavefunction <math>\psi^\text{I}</math>. Also, the fact that <math>Q(t)</math> is random with [[Probability density function|probability density]] given by the [[square modulus]] of <math>\psi(t,\cdot)</math> implies that the [[Conditional probability density function|conditional probability density]] of <math>Q^\text{I}(t)</math> given <math>Q^\text{II}(t)</math> is given by the square modulus of the (normalized) conditional wavefunction <math>\psi^\text{I}(t,\cdot)</math> (in the terminology of Dürr et al.<ref>{{Cite journal|arxiv=quant-ph/0308039 |last1=Dürr |first1=Detlef |title=Quantum Equilibrium and the Origin of Absolute Uncertainty |journal=Journal of Statistical Physics |volume=67 |issue=5–6 |pages=843–907 |last2=Goldstein |first2=Sheldon |last3=Zanghí |first3=Nino |year=2003 |doi=10.1007/BF01049004 |bibcode=1992JSP....67..843D|s2cid=15749334 }}</ref> this fact is called the ''fundamental conditional probability formula'').

Unlike the universal wavefunction, the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation, but in many situations it does. For instance, if the universal wavefunction factors as

:<math>\psi(t, q^\text{I} ,q^\text{II}) = \psi^\text{I}(t, q^\text{I}) \psi^\text{II}(t, q^\text{II}),</math>

then the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to <math>\psi^\text{I}</math> (this is what standard quantum theory would regard as the wavefunction of subsystem (I)). If, in addition, the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then <math>\psi^\text{I}</math> does satisfy a Schrödinger equation. More generally, assume that the universal wave function <math>\psi</math> can be written in the form

:<math>\psi(t, q^\text{I}, q^\text{II}) = \psi^\text{I}(t, q^\text{I}) \psi^\text{II}(t, q^\text{II}) + \phi(t, q^\text{I}, q^\text{II}),</math>

where <math>\phi</math> solves Schrödinger equation and, <math>\phi(t, q^\text{I}, Q^\text{II}(t)) = 0</math> for all <math>t</math> and <math>q^\text{I}</math>. Then, again, the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to <math>\psi^\text{I}</math>, and if the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then <math>\psi^\text{I}</math> satisfies a Schrödinger equation.

The fact that the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of standard quantum theory emerges from the Bohmian formalism when one considers conditional wavefunctions of subsystems.