Editing De Broglie–Bohm theory (section)

=== 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>