Editing De Broglie–Bohm theory (section)

== Extensions ==

=== Relativity ===
Pilot-wave theory is explicitly nonlocal, which is in ostensible conflict with [[special relativity]]. Various extensions of "Bohm-like" mechanics exist that attempt to resolve this problem. Bohm himself in 1953 presented an extension of the theory satisfying the [[Dirac equation]] for a single particle. However, this was not extensible to the many-particle case because it used an absolute time.<ref>{{Cite journal|arxiv=quant-ph/0611032|last1=Passon|first1=Oliver|title=What you always wanted to know about Bohmian mechanics but were afraid to ask|journal=Physics and Philosophy|volume=3|issue=2006|year=2006|bibcode=2006quant.ph.11032P|doi=10.17877/DE290R-14213|hdl=2003/23108|s2cid=45526627}}</ref>

A renewed interest in constructing [[Lorentz scalar|Lorentz-invariant]] extensions of Bohmian theory arose in the 1990s; see ''Bohm and Hiley: The Undivided Universe''<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> and references therein. Another approach is given by Dürr et al.,<ref>{{cite journal | last1 = Dürr | first1 = D. | last2 = Goldstein | first2 = S. | 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> who use Bohm–Dirac models and a Lorentz-invariant foliation of space-time.

Thus, Dürr et al. (1999) showed that it is possible to formally restore Lorentz invariance for the Bohm–Dirac theory by introducing additional structure. This approach still requires a [[foliation]] of space-time. While this is in conflict with the standard interpretation of relativity, the preferred foliation, if unobservable, does not lead to any empirical conflicts with relativity. In 2013, Dürr et al. suggested that the required foliation could be covariantly determined by the wavefunction.<ref>{{cite journal | last1 = Dürr | first1 = Detlef | last2 = Goldstein | first2 = Sheldon | last3 = Norsen | first3 = Travis | last4 = Struyve | first4 = Ward | last5 = Zanghì | first5 = Nino | year = 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>

The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohm theory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on the instantaneous positions of all other particles. On the other hand, in the theory of relativity the concept of instantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous. The simplest way to achieve this is to introduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines a hypersurface of equal time.

Initially, it had been considered impossible to set out a description of photon trajectories in the de Broglie–Bohm theory in view of the difficulties of describing bosons relativistically.<ref name="ghose-1996">{{cite journal | last1 = Ghose | first1 = Partha | year = 1996 | title = Relativistic quantum mechanics of spin-0 and spin-1 bosons | journal = Foundations of Physics | volume = 26 | issue = 11| pages = 1441–1455 | doi = 10.1007/BF02272366 |bibcode = 1996FoPh...26.1441G | s2cid = 121129680 }}</ref> In 1996, [[Partha Ghose]] presented a relativistic quantum-mechanical description of spin-0 and spin-1 bosons starting from the [[Duffin–Kemmer–Petiau equation]], setting out Bohmian trajectories for massive bosons and for massless bosons (and therefore [[photon]]s).<ref name="ghose-1996" /> In 2001, [[Jean-Pierre Vigier]] emphasized the importance of deriving a well-defined description of light in terms of particle trajectories in the framework of either the Bohmian mechanics or the Nelson stochastic mechanics.<ref>{{cite journal | last1 = Cufaro Petroni | first1 = Nicola | last2 = Vigier | first2 = Jean-Pierre | year = 2001| title = Remarks on Observed Superluminal Light Propagation | journal = Foundations of Physics Letters | volume = 14 | issue = 4| pages = 395–400 | doi = 10.1023/A:1012321402475 | s2cid = 120131595 }}, therein: section ''3. Conclusions'', page 399.</ref> The same year, Ghose worked out Bohmian photon trajectories for specific cases.<ref>{{cite journal | last1 = Ghose | first1 = Partha | last2 = Majumdar | first2 = A. S. | last3 = Guhab | first3 = S. | last4 = Sau | first4 = J. | year = 2001 | title = Bohmian trajectories for photons | url = http://web.mit.edu/saikat/www/research/files/Bohmian-traj_PLA2001.pdf | journal = Physics Letters A | volume = 290 | issue = 5–6| pages = 205–213 | doi=10.1016/s0375-9601(01)00677-6|arxiv = quant-ph/0102071 |bibcode = 2001PhLA..290..205G | s2cid = 54650214 }}</ref> Subsequent [[weak measurement|weak-measurement]] experiments yielded trajectories that coincide with the predicted trajectories.<ref>Sacha Kocsis, Sylvain Ravets, Boris Braverman, Krister Shalm, Aephraim M. Steinberg: [http://www.aip.org.au/Congress2010/Abstracts/Monday%206%20Dec%20-%20Orals/Session_3E/Kocsis_Observing_the_Trajectories.pdf "Observing the trajectories of a single photon using weak measurement"] {{webarchive|url=https://web.archive.org/web/20110626194505/http://www.aip.org.au/Congress2010/Abstracts/Monday%206%20Dec%20-%20Orals/Session_3E/Kocsis_Observing_the_Trajectories.pdf |date=26 June 2011 }} 19th Australian Institute of Physics (AIP) Congress, 2010.</ref><ref>{{cite journal | last1 = Kocsis | first1 = Sacha | last2 = Braverman | first2 = Boris | last3 = Ravets | first3 = Sylvain | last4 = Stevens | first4 = Martin J. | last5 = Mirin | first5 = Richard P. | last6 = Shalm | first6 = L. Krister | last7 = Steinberg | first7 = Aephraim M. | year = 2011 | title = Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer | journal = Science | volume = 332 | issue = 6034| pages = 1170–1173 | doi = 10.1126/science.1202218 | pmid = 21636767 |bibcode = 2011Sci...332.1170K | s2cid = 27351467 }}</ref> The significance of these experimental findings is controversial.<ref>{{cite journal | author = Fankhauser Johannes, Dürr Patrick | year = 2021 | title = How (not) to understand weak measurements of velocity | journal = Studies in History and Philosophy of Science Part A | volume = 85 | pages = 16–29 | doi = 10.1016/j.shpsa.2020.12.002| pmid = 33966771 | bibcode = 2021SHPSA..85...16F |issn=0039-3681| doi-access = free | arxiv = 2309.10395 }}</ref>

Chris Dewdney and G. Horton have proposed a relativistically covariant, wave-functional formulation of Bohm's quantum field theory<ref>{{cite journal | last1 = Dewdney | first1 = Chris | last2 = Horton | first2 = George | year = 2002 | title = Relativistically invariant extension of the de Broglie Bohm theory of quantum mechanics | journal = Journal of Physics A: Mathematical and General | volume = 35 | issue = 47| pages = 10117–10127 | doi = 10.1088/0305-4470/35/47/311 |arxiv = quant-ph/0202104 |bibcode = 2002JPhA...3510117D | s2cid = 37082933 }}</ref><ref>{{cite journal | last1 = Dewdney | first1 = Chris | last2 = Horton | first2 = George | year = 2004 | title = A relativistically covariant version of Bohm's quantum field theory for the scalar field | journal = Journal of Physics A: Mathematical and General | volume = 37 | issue = 49| pages = 11935–11943 | doi = 10.1088/0305-4470/37/49/011 |arxiv = quant-ph/0407089 |bibcode = 2004JPhA...3711935H | s2cid = 119468313 }}</ref> and have extended it to a form that allows the inclusion of gravity.<ref>{{cite journal | last1 = Dewdney | first1 = Chris | last2 = Horton | first2 = George | year = 2010 | title = A relativistic hidden-variable interpretation for the massive vector field based on energy-momentum flows | journal = Foundations of Physics | volume = 40 | issue = 6| pages = 658–678 | doi = 10.1007/s10701-010-9456-9 |bibcode = 2010FoPh...40..658H | s2cid = 123511987 }}</ref>

Nikolić has proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wavefunctions.<ref>{{cite journal | last1 = Nikolić | first1 = Hrvoje | year = 2005 | title = Relativistic Quantum Mechanics and the Bohmian Interpretation | journal = Foundations of Physics Letters | volume = 18 | issue = 6| pages = 549–561 | doi = 10.1007/s10702-005-1128-1 | bibcode=2005FoPhL..18..549N|arxiv = quant-ph/0406173 | citeseerx = 10.1.1.252.6803 | s2cid = 14006204 }}</ref> He has developed a generalized relativistic-invariant probabilistic interpretation of quantum theory,<ref name="nikolicqft" /><ref>{{Cite journal|arxiv=0811.1905|last1= Nikolic|first1= H.|title= Time in relativistic and nonrelativistic quantum mechanics|journal= International Journal of Quantum Information|volume= 7|issue= 3|pages= 595–602|year= 2009|bibcode= 2008arXiv0811.1905N|doi=10.1142/s021974990900516x|s2cid= 17294178}}</ref><ref>{{Cite journal|arxiv=1002.3226|last1= Nikolic|first1= H.|title= Making nonlocal reality compatible with relativity|journal= Int. J. Quantum Inf. |volume= 9|issue= 2011|pages= 367–377|year= 2011|bibcode= 2010arXiv1002.3226N|doi= 10.1142/S0219749911007344|s2cid= 56513936}}</ref> in which <math>|\psi|^2</math> is no longer a probability density in space, but a probability density in space-time. He uses this generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theory without introducing a preferred foliation of space-time. His work also covers the extension of the Bohmian interpretation to a quantization of fields and strings.<ref>Hrvoje Nikolić: [http://iopscience.iop.org/1742-6596/67/1/012035/pdf/jpconf7_67_012035.pdf "Bohmian mechanics in relativistic quantum mechanics, quantum field theory and string theory"], ''2007 Journal of Physics'': Conf. Ser. 67 012035.</ref>
{{See also|Quantum potential#Relativistic and field-theoretic extensions}}

Roderick I. Sutherland at the University in Sydney has a Lagrangian formalism for the pilot wave and its [[wikt:beable|beables]]. It draws on [[Yakir Aharonov]]'s retrocasual weak measurements to explain many-particle entanglement in a special relativistic way without the need for configuration space. The basic idea was already published by [[Olivier Costa de Beauregard]] in the 1950s and is also used by [[John G. Cramer|John Cramer]] in his transactional interpretation except the beables that exist between the von Neumann strong projection operator measurements. Sutherland's Lagrangian includes two-way action-reaction between pilot wave and beables. Therefore, it is a post-quantum non-statistical theory with final boundary conditions that violate the no-signal theorems of quantum theory. Just as special relativity is a limiting case of general relativity when the spacetime curvature vanishes, so, too is statistical no-entanglement signaling quantum theory with the Born rule a limiting case of the post-quantum action-reaction Lagrangian when the reaction is set to zero and the final boundary condition is integrated out.<ref>{{Cite journal|arxiv=1509.02442|last1=Sutherland|first1=Roderick|title=Lagrangian Description for Particle Interpretations of Quantum Mechanics -- Entangled Many-Particle Case|journal=Foundations of Physics|year=2015|doi=10.1007/s10701-016-0043-6|volume=47|issue=2|pages=174–207|bibcode=2017FoPh...47..174S|s2cid=118366293}}</ref>

=== Spin ===
To incorporate [[Spin (physics)|spin]], the wavefunction becomes complex-vector-valued. The value space is called spin space; for a [[spin-1/2]] particle, spin space can be taken to be <math>\Complex^2</math>. The guiding equation is modified by taking [[inner product]]s in spin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding a [[Pauli equation|Pauli spin term]]:

<math display="block">\begin{align}
\frac{d\mathbf{Q}_k}{dt}(t) &= \frac{\hbar}{m_k} \operatorname{Im}\left(\frac{(\psi,D_k \psi)}{(\psi,\psi)}\right)(\mathbf{Q}_1, \ldots, \mathbf{Q}_N, t), \\
  i\hbar\frac{\partial}{\partial t}\psi &= \left(-\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}D_k^2 + V - \sum_{k=1}^{N} \mu_k \frac{\mathbf{S}_k}{\hbar s_k} \cdot \mathbf{B}(\mathbf{q}_k)\right) \psi,
\end{align}</math>

where
* <math>m_k, e_k,\mu_k</math> — the mass, charge and [[magnetic moment]] of the <math>k</math>–th particle
* <math>\mathbf{S}_k</math> — the appropriate [[Spin (physics)#Operator|spin operator]] acting in the <math>k</math>–th particle's spin space
* <math>s_k</math> — [[spin quantum number]] of the <math>k</math>–th particle (<math>s_k = 1/2</math> for electron)
* <math>\mathbf{A}</math> is [[vector potential]] in <math>\R^{3}</math>
* <math>\mathbf{B}=\nabla\times\mathbf{A}</math> is the [[magnetic field]] in <math>\R^{3}</math>
* <math display="inline">D_k = \nabla_k - \frac{ie_k}{\hbar}\mathbf{A}(\mathbf{q}_k)</math> is the covariant derivative, involving the vector potential, ascribed to the coordinates of <math>k</math>–th particle (in [[International System of Units|SI units]])
* <math>\psi</math> — the wavefunction defined on the multidimensional configuration space; e.g. a system consisting of two spin-1/2 particles and one spin-1 particle has a wavefunction of the form <math display="block">\psi: \R^9 \times \R \to \Complex^2 \otimes \Complex^2 \otimes \Complex^3,</math> where <math>\otimes</math> is a [[tensor product]], so this spin space is 12-dimensional
* <math>(\cdot,\cdot)</math> is the [[inner product]] in spin space <math>\Complex^d</math>: <math display="block">(\phi, \psi) = \sum_{s=1}^d \phi_s^* \psi_s.</math>

=== Stochastic electrodynamics ===

[[Stochastic electrodynamics]] (SED) is an extension of the de Broglie–Bohm interpretation of [[quantum mechanics]], with the electromagnetic [[Zero-point energy|zero-point field]] (ZPF) playing a central role as the guiding [[pilot-wave]]. Modern approaches to SED, like those proposed by the group around late Gerhard Grössing, among others, consider wave and particle-like quantum effects as well-coordinated emergent systems. These emergent systems are the result of speculated and calculated sub-quantum interactions with the zero-point field.<ref>{{Cite book|last1=Pena|first1=Luis de la|last2=Cetto|first2=Ana Maria|last3=Valdes-Hernandez|first3=Andrea|title=The Emerging Quantum: The Physics Behind Quantum Mechanics|date=2014|page=95|doi=10.1007/978-3-319-07893-9|url=https://books.google.com/books?id=v0MqBAAAQBAJ|isbn=978-3-319-07893-9}}</ref><ref>{{cite journal|last1=Grössing|first1=G.|last2=Fussy|first2=S.|last3=Mesa Pascasio|first3=J.|last4=Schwabl|first4=H.|title=An explanation of interference effects in the double slit experiment: Classical trajectories plus ballistic diffusion caused by zero-point fluctuations|journal=Annals of Physics|date=2012|volume=327|issue=2|pages=421–437|doi=10.1016/j.aop.2011.11.010|arxiv=1106.5994|bibcode=2012AnPhy.327..421G|s2cid=117642446}}</ref><ref>{{cite journal|last1=Grössing|first1=G.|last2=Fussy|first2=S.|last3=Mesa Pascasio|first3=J.|last4=Schwabl|first4=H.|title=The Quantum as an Emergent System|journal=Journal of Physics: Conference Series|date=2012|volume=361|issue=1|page=012008|doi=10.1088/1742-6596/361/1/012008|arxiv=1205.3393|bibcode=2012JPhCS.361a2008G|s2cid=119307454}}</ref>

=== Quantum field theory ===
In Dürr et al.,<ref name="dgtz04">{{Cite journal|arxiv=quant-ph/0303156|last1=Duerr|first1=Detlef|title=Bohmian Mechanics and Quantum Field Theory|journal=Physical Review Letters|volume=93|issue=9|last2=Goldstein|first2=Sheldon|last3=Tumulka|first3=Roderich|last4=Zanghi|first4=Nino|year=2004|doi=10.1103/PhysRevLett.93.090402|pmid=15447078|page=090402|bibcode=2004PhRvL..93i0402D|citeseerx=10.1.1.8.8444|s2cid=8720296}}</ref><ref>{{Cite journal|arxiv=quant-ph/0407116|last1=Duerr|first1=Detlef|title=Bell-Type Quantum Field Theories|journal=Journal of Physics A: Mathematical and General|volume=38|issue=4|pages=R1|last2=Goldstein|first2=Sheldon|last3=Tumulka|first3=Roderich|last4=Zanghi|first4=Nino|year=2005|doi=10.1088/0305-4470/38/4/R01|bibcode=2005JPhA...38R...1D|s2cid=15547226}}</ref> the authors describe an extension of de Broglie–Bohm theory for handling [[creation and annihilation operators]], which they refer to as "Bell-type quantum field theories". The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under the guiding equation with a fixed number of particles. But under a [[stochastic process]], particles may be created and annihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolving at all times over the full multi-particle configuration space.

Hrvoje Nikolić<ref name="nikolicqft">{{cite journal | last1 = Nikolic | first1 = H | year = 2010 | title = QFT as pilot-wave theory of particle creation and destruction | journal = International Journal of Modern Physics | volume = 25 | issue = 7| pages = 1477–1505 | doi=10.1142/s0217751x10047889|arxiv = 0904.2287 |bibcode = 2010IJMPA..25.1477N | s2cid = 18468330 }}</ref> introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, according to which particle trajectories are continuous, but particle detectors behave as if particles have been created or destroyed even when a true creation or destruction of particles does not take place.

=== Curved space ===
To extend de Broglie–Bohm theory to curved space ([[Riemannian manifolds]] in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as [[gradient]]s and [[Laplacian]]s. Thus, we use equations that have the same form as above. Topological and [[boundary conditions]] may apply in supplementing the evolution of Schrödinger's equation.

For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a [[vector bundle]] over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.<ref>{{cite journal | last1 = Dürr | first1 = D. | last2 = Goldstein | first2 = S. | last3 = Taylor | first3 = J. | last4 = Tumulka | first4 = R. | last5 = Zanghì | first5 = N. | year = 2007 | title = Quantum Mechanics in Multiply-Connected Spaces | journal = J. Phys. A | volume = 40 | issue = 12| pages = 2997–3031 | doi=10.1088/1751-8113/40/12/s08|arxiv = quant-ph/0506173 |bibcode = 2007JPhA...40.2997D | s2cid = 119410880 }}</ref>
The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for curved space-times with torsion.<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>

In a general spacetime with curvature and torsion, the guiding equation for the [[four-velocity]] <math>u^i</math> of an elementary [[fermion]] particle is<math display="block">u^i=\frac{e^i_\mu \bar{\psi}\gamma^\mu \psi}{\bar{\psi}\psi}, </math>where the wave function <math>\psi</math> is a [[Dirac spinor|spinor]], <math>\bar{\psi}</math> is the corresponding [[Dirac adjoint|adjoint]], <math>\gamma^\mu</math> are the [[gamma matrices|Dirac matrices]], and <math>e^i_\mu</math> is a [[tetrad formalism|tetrad]].<ref name="FG">{{cite journal | author=F. R. Benard Guedes, N. J. Popławski | date=2024 | title=General-relativistic wave-particle duality with torsion | journal=Classical and Quantum Gravity | volume=41 | issue=6 | pages=065011 | doi=10.1088/1361-6382/ad1fcb | arxiv=2211.03234 }}</ref> If the wave function propagates according to the [[Dirac equation in curved spacetime|curved]] Dirac equation, then the particle moves according to the [[Mathisson-Papapetrou-Dixon equations|Mathisson-Papapetrou equations]] of motion, which are an extension of the [[geodesics in general relativity|geodesic equation]]. This relativistic wave-particle duality follows from the [[conservation law|conservation laws]] for the [[spin tensor]] and [[stress-energy tensor|energy-momentum tensor]],<ref name="FG" /> and also from the covariant [[Heisenberg picture]] equation of motion.<ref>{{cite journal | author=S. K. Wong | date=1972 | title=Heisenberg equations of motion for spin-1/2 wave equation in general relativity | journal=International Journal of Theoretical Physics | volume=5 | issue=4 | pages=221–230 | doi=10.1007/BF00670477 }}</ref> 

=== Exploiting nonlocality ===
{{Main|Quantum non-equilibrium}}
[[File:Quantum Theory is a special case of a wider physics.svg|thumb|286x286px|right|Diagram made by [[Antony Valentini]] in a lecture about the De Broglie–Bohm theory. Valentini argues quantum theory is a special equilibrium case of a wider physics and that it may be possible to observe and exploit [[quantum non-equilibrium]]<ref>{{cite web|last1=Valentini|first1=Antony|title=Hidden Variables in Modern Cosmology|url=https://www.youtube.com/watch?v=XYZV9crCZM8| archive-url=https://ghostarchive.org/varchive/youtube/20211211/XYZV9crCZM8| archive-date=2021-12-11 | url-status=live|via=YouTube|publisher=Philosophy of Cosmology|access-date=23 December 2016|date=2013}}{{cbignore}}</ref>]]
[[Louis de Broglie|De Broglie]] and Bohm's causal interpretation of quantum mechanics was later extended by Bohm, Vigier, Hiley, Valentini and others to include stochastic properties. Bohm and other physicists, including Valentini, view the Born rule linking <math>R</math> to the [[probability density function]] <math>\rho = R^2</math> as representing not a basic law, but a result of a system having reached ''quantum equilibrium'' during the course of the time development under the [[Schrödinger equation]]. It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the [[Continuity equation#Quantum mechanics|continuity equation]] associated with the Schrödinger evolution of <math>\psi</math>.<ref>See for ex. Detlef Dürr, Sheldon Goldstein, Nino Zanghí: ''Bohmian mechanics and quantum equilibrium'', Stochastic Processes, Physics and Geometry II. World Scientific, 1995 [http://www.ge.infn.it/~zanghi/BMQE.pdf#page=5 page 5]</ref> It is less straightforward to demonstrate whether and how such an equilibrium is reached in the first place.

[[Antony Valentini]]<ref>{{cite journal | last1 = Valentini | first1 = A | year = 1991 | title = Signal-Locality, Uncertainty and the Subquantum H-Theorem. II | journal = Physics Letters A | volume = 158 | issue = 1–2| pages = 1–8 | doi=10.1016/0375-9601(91)90330-b|bibcode = 1991PhLA..158....1V }}</ref> has extended de Broglie–Bohm theory to include signal nonlocality that would allow entanglement to be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" the message encoded in the entanglement. This violates orthodox quantum theory but has the virtue of making the parallel universes of the [[chaotic inflation theory]] observable in principle.

Unlike de Broglie–Bohm theory, Valentini's theory the wavefunction evolution also depends on the ontological variables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heat death". The resulting theory becomes nonlinear and non-unitary. Valentini argues that the laws of quantum mechanics are [[Emergence|emergent]] and form a "quantum equilibrium" that is analogous to thermal equilibrium in classical dynamics, such that other "[[quantum non-equilibrium]]" distributions may in principle be observed and exploited, for which the statistical predictions of quantum theory are violated. It is controversially argued that quantum theory is merely a special case of a much wider nonlinear physics, a physics in which non-local ([[Faster-than-light|superluminal]]) signalling is possible, and in which the uncertainty principle can be violated.<ref name="Valentini2009">{{cite journal|last1=Valentini|first1=Antony|title=Beyond the quantum|journal=Physics World|volume=22|issue=11|year=2009|pages=32–37|issn=0953-8585|doi=10.1088/2058-7058/22/11/36|arxiv=1001.2758|bibcode=2009PhyW...22k..32V|s2cid=86861670}}</ref><ref>{{cite web|last1=Musser|first1=George|title=Cosmological Data Hint at a Level of Physics Underlying Quantum Mechanics|url=https://blogs.scientificamerican.com/critical-opalescence/cosmological-data-hint-at-a-level-of-physics-underlying-quantum-mechanics-guest-post/|website=blogs.scientificamerican.com|publisher=Scientific American|access-date=5 December 2016|date=18 November 2013}}</ref>