Editing De Broglie–Bohm theory (section)

=== 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>