Editing De Broglie–Bohm theory (section)

== Results ==
Below are some highlights of the results that arise out of an analysis of de Broglie–Bohm theory. Experimental results agree with all of quantum mechanics' standard predictions insofar as it has them. But while standard quantum mechanics is limited to discussing the results of "measurements", de Broglie–Bohm theory governs the dynamics of a system without the intervention of outside observers (p.&nbsp;117 in Bell<ref name="bell" />).

The basis for agreement with standard quantum mechanics is that the particles are distributed according to <math>|\psi|^2</math>. This is a statement of observer ignorance: the initial positions are represented by a statistical distribution so deterministic trajectories will result in a statistical distribution.<ref name="dgz92" />

=== Measuring spin and polarization ===
According to ordinary quantum theory, it is not possible to measure the [[Spin (physics)|spin]] or [[Polarization (waves)|polarization]] of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or −1, meaning that it is aligned the opposite way. An ensemble of particles prepared by a polarizer to be in state 1 will all measure polarized in state 1 in a subsequent apparatus. A polarized ensemble sent through a polarizer set at angle to the first pass will result in some values of 1 and some of −1 with a probability that depends on the relative alignment. For a full explanation of this, see the [[Stern–Gerlach experiment]].

In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup. It is possible<ref>Albert, D. Z., 1992, Quantum Mechanics and Experience, Cambridge, MA: Harvard University Press.</ref> to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin-up, while in the other setup it registers as spin-down. Thus, for the de Broglie–Bohm theory, the particle's spin is not an intrinsic property of the particle; instead spin is, so to speak, in the wavefunction of the particle in relation to the particular device being used to measure the spin. This is an illustration of what is sometimes referred to as contextuality and is related to naive realism about operators.<ref>{{cite journal | last1 = Daumer | first1 = M. | last2 = Dürr | first2 = D. | last3 = Goldstein | first3 = S. | last4 = Zanghì | first4 = N. | year = 1997 | title = Naive Realism About Operators | arxiv = quant-ph/9601013| journal = Erkenntnis | volume = 45 | issue = 2–3 | pages = 379–397 | doi = 10.1007/BF00276801 | bibcode = 1996quant.ph..1013D }}</ref> Interpretationally, measurement results are a deterministic property of the system and its environment, which includes information about the experimental setup including the context of co-measured observables; in no sense does the system itself possess the property being measured, as would have been the case in classical physics.

=== Measurements, the quantum formalism, and observer independence ===
De Broglie–Bohm theory gives almost the same results as (non-relativisitic) quantum mechanics. It treats the wavefunction as a fundamental object in the theory, as the wavefunction describes how the particles move. This means that no experiment can distinguish between the two theories. This section outlines the ideas as to how the standard quantum formalism arises out of quantum mechanics.<ref name=":0" /><ref name="dgz92" />

==== Collapse of the wavefunction ====
{{Unreferenced section|date=September 2024}}
De Broglie–Bohm theory is a theory that applies primarily to the whole universe. That is, there is a single wavefunction governing the motion of all of the particles in the universe according to the guiding equation. Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe. In some situations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theory in which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial <math>|\psi|^2</math> distribution for the particles in the system (see the section on [[#The conditional wavefunction of a subsystem|the conditional wavefunction of a subsystem]] for details).

It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a system interacts with its environment, such as through a measurement, the conditional wavefunction of the system evolves in a different way. The evolution of the universal wavefunction can become such that the wavefunction of the system appears to be in a superposition of distinct states. But if the environment has recorded the results of the experiment, then using the actual Bohmian configuration of the environment to condition on, the conditional wavefunction collapses to just one alternative, the one corresponding with the measurement results.

[[Wavefunction collapse|Collapse]] of the universal wavefunction never occurs in de Broglie–Bohm theory. Its entire evolution is governed by Schrödinger's equation, and the particles' evolutions are governed by the guiding equation. Collapse only occurs in a [[Phenomenology (physics)|phenomenological]] way for systems that seem to follow their own Schrödinger's equation. As this is an effective description of the system, it is a matter of choice as to what to define the experimental system to include, and this will affect when "collapse" occurs.

==== Operators as observables ====
In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory). In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.<ref>{{Cite journal|arxiv=quant-ph/0308038|last1=Dürr|first1=Detlef|title=Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory|journal=Journal of Statistical Physics |volume=116|issue=1–4|pages=959|last2=Goldstein|first2=Sheldon|last3=Zanghì|first3=Nino|year=2003|doi=10.1023/B:JOSS.0000037234.80916.d0|bibcode=2004JSP...116..959D|citeseerx=10.1.1.252.1653|s2cid=123303}}</ref> A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction.

In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory is impossible. Such arguments are generally based on inappropriate analysis of operators as observables. If one believes that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature of the particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimental apparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant.
There are also objections to this theory based on what it says about particular situations usually involving eigenstates of an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation, this means that the electron is at rest when in this state. Nevertheless, it is distributed according to <math>|\psi|^2</math>, and no contradiction to experimental results is possible to detect.

Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, from this perspective, chooses the position observable as a favored observable rather than, say, the momentum observable. Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie–Bohm theory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of those particles at all times. Other observables do not have this compelling ontological status. Having definite positions explains having definite results such as flashes on a detector screen. Other observables would not lead to that conclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman et al.<ref>{{Cite journal|doi=10.1088/0305-4470/37/44/L02|title=Bohmian mechanics with discrete operators|journal=Journal of Physics A: Mathematical and General|volume=37|issue=44|pages=L547|year=2004|last1=Hyman|first1=Ross|last2=Caldwell|first2=Shane A|last3=Dalton|first3=Edward|bibcode=2004JPhA...37L.547H|arxiv=quant-ph/0401008|s2cid=6073288}}</ref> for an exploration of the fact that a probability density and probability current can be defined for any set of commuting operators.

==== Hidden variables ====
De Broglie–Bohm theory is often referred to as a "hidden-variable" theory. Bohm used this description in his original papers on the subject, writing: "From the point of view of the [[Copenhagen interpretation|usual interpretation]], these additional elements or parameters [permitting a detailed causal and continuous description of all processes] could be called 'hidden' variables." Bohm and Hiley later stated that they found Bohm's choice of the term "hidden variables" to be too restrictive. In particular, they argued that a particle is not actually hidden but rather "is what is most directly manifested in an observation [though] its properties cannot be observed with arbitrary precision (within the limits set by [[uncertainty principle]])".<ref>David Bohm, Basil Hiley: ''The Undivided Universe: An Ontological Interpretation of Quantum Theory'', edition published in the Taylor & Francis e-library 2009 (first edition Routledge, 1993), {{ISBN|0-203-98038-7}}, [https://books.google.com/books?id=vt9XKjc4WAQC&pg=PA2 p. 2].</ref> However, others nevertheless treat the term "hidden variable" as a suitable description.<ref>"While the testable predictions of Bohmian mechanics are isomorphic to standard Copenhagen quantum mechanics, its underlying hidden variables have to be, in principle, unobservable. If one could observe them, one would be able to take advantage of that and signal faster than light, which – according to the special theory of relativity – leads to physical temporal paradoxes." J. Kofler and A. Zeiliinger, "Quantum Information and Randomness", ''European Review'' (2010), Vol. 18, No. 4, 469–480.</ref>

Generalized particle trajectories can be extrapolated from numerous weak measurements on an ensemble of equally prepared systems, and such trajectories coincide with the de Broglie–Bohm trajectories. In particular, an experiment with two entangled photons, in which a set of Bohmian trajectories for one of the photons was determined using weak measurements and postselection, can be understood in terms of a nonlocal connection between that photon's trajectory and the other photon's polarization.<ref>{{cite journal | doi = 10.1126/science.1501466 | pmid=26989784 | pmc=4788483 | volume=2 | title=Experimental nonlocal and surreal Bohmian trajectories | year=2016 | journal=Sci Adv | page=e1501466 | last1 = Mahler | first1 = DH | last2 = Rozema | first2 = L | last3 = Fisher | first3 = K | last4 = Vermeyden | first4 = L | last5 = Resch | first5 = KJ | last6 = Wiseman | first6 = HM | last7 = Steinberg | first7 = A| issue=2 }}</ref><ref name="newscientist.com">Anil Ananthaswamy: [https://www.newscientist.com/article/2078251-quantum-weirdness-may-hide-an-orderly-reality-after-all/ Quantum weirdness may hide an orderly reality after all], newscientist.com, 19 February 2016.</ref> However, not only the De Broglie–Bohm interpretation, but also many other interpretations of quantum mechanics that do not include such trajectories are consistent with such experimental evidence.

=== Different predictions ===
A specialized version of the [[double slit experiment]] has been devised to test characteristics of the trajectory predictions.<ref>Golshani, M., and O. Akhavan. "Bohmian prediction about a two double-slit experiment and its disagreement with standard quantum mechanics." Journal of Physics A: Mathematical and General 34.25 (2001): 5259.</ref>
Results from one such experiment agreed with the predictions of standard quantum mechanics and disagreed with the Bohm predictions when they conflicted.<ref>{{Cite journal|arxiv=quant-ph/0206196|last1= Brida|first1= G.|title= A first experimental test of de Broglie-Bohm theory against standard quantum mechanics|journal= Journal of Physics B: Atomic, Molecular and Optical Physics|volume= 35|issue= 22|pages= 4751|last2=  Cagliero|first2= E.|last3=  Falzetta|first3= G.|last4=  Genovese|first4= M.|last5=  Gramegna|first5= M.|last6=  Novero|first6= C.|year= 2002|doi= 10.1088/0953-4075/35/22/316|bibcode=2002JPhB...35.4751B|s2cid= 250773374}}</ref> These conclusions have been the subject of debate.<ref>{{Cite book|arxiv=quant-ph/0108038|last1= Struyve|first1= W.|chapter= Comments on some recently proposed experiments that should distinguish Bohmian mechanics from quantum mechanics|title=Quantum Theory: Reconsideration of Foundations |publisher=Vaxjo University Press |location=Vaxjo|page= 355|year= 2001 |last2=  De Baere|first2= W.|bibcode= 2001quant.ph..8038S}}</ref><ref>{{Cite journal |last1=Brida |first1=G |last2=Cagliero |first2=E |last3=Genovese |first3=M |last4=Gramegna |first4=M |date=2004-09-28 |title=Reply to Comment on Experimental realization of a first test of de Broglie–Bohm theory |url=https://iopscience.iop.org/article/10.1088/0953-4075/37/18/N02 |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=37 |issue=18 |pages=3781–3783 |doi=10.1088/0953-4075/37/18/N02 |issn=0953-4075}}</ref>

=== Heisenberg's uncertainty principle ===
{{Unreferenced section|date=September 2024}}
The Heisenberg's uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy. As an example, if one measures the position with an accuracy of <math>\Delta x</math> and the momentum with an accuracy of <math>\Delta p</math>, then <math>\Delta x \Delta p \gtrsim h.</math>

In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle. Each particle has a well-defined trajectory, as well as a wavefunction. Observers have limited knowledge as to what this trajectory is (and thus of the position and momentum). It is the lack of knowledge of the particle's trajectory that accounts for the uncertainty relation. What one can know about a particle at any given time is described by the wavefunction. Since the uncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can be likewise derived (in the [[Epistemology|epistemic]] sense mentioned above) on the de Broglie–Bohm theory.

To put the statement differently, the particles' positions are only known statistically. As in [[classical mechanics]], successive observations of the particles' positions refine the experimenter's knowledge of the particles' [[initial conditions]]. Thus, with succeeding observations, the initial conditions become more and more restricted. This formalism is consistent with the normal use of the Schrödinger equation.

For the derivation of the uncertainty relation, see [[Heisenberg uncertainty principle]], noting that this article describes the principle from the viewpoint of the [[Copenhagen interpretation]].

=== Quantum entanglement, Einstein–Podolsky–Rosen paradox, Bell's theorem, and nonlocality ===
De Broglie–Bohm theory highlighted the issue of [[Quantum nonlocality|nonlocality]]: it inspired [[John Stewart Bell]] to prove his now-famous [[Bell's theorem|theorem]],<ref>{{cite journal | author = Bell J. S. | year = 1964 | title = On the Einstein Podolsky Rosen Paradox | url = http://www.drchinese.com/David/Bell_Compact.pdf | journal = Physics Physique Fizika | volume = 1 | issue = 3| page = 195 | doi = 10.1103/PhysicsPhysiqueFizika.1.195 | doi-access = free }}</ref> which in turn led to the [[Bell test experiments]].

In the [[EPR paradox|Einstein–Podolsky–Rosen paradox]], the authors describe a thought experiment that one could perform on a pair of particles that have interacted, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory.<ref>{{cite journal |last1=Einstein |last2=Podolsky |last3=Rosen |title=Can Quantum Mechanical Description of Physical Reality Be Considered Complete? |journal=[[Physical Review|Phys. Rev.]] |volume=47 |issue=10 |pages=777–780 |year=1935 |doi=10.1103/PhysRev.47.777 |bibcode = 1935PhRv...47..777E |url=https://cds.cern.ch/record/405662 |doi-access=free }}</ref>

Decades later [[John Stewart Bell|John Bell]] proved [[Bell's theorem]] (see p.&nbsp;14 in Bell<ref name="bell">{{cite book |last=Bell |first=John S. |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |year=1987 |isbn=978-0-521-33495-2 }}</ref>), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such "hidden-variable" completions of quantum mechanics must either be nonlocal (as the Bohm interpretation is) or give up the assumption that experiments produce unique results (see [[counterfactual definiteness]] and [[many-worlds interpretation]]). In particular, Bell proved that any local theory with unique results must make empirical predictions satisfying a statistical constraint called "Bell's inequality".

[[Alain Aspect]] performed a series of [[Bell test experiments]] that test Bell's inequality using an EPR-type setup. Aspect's results show experimentally that Bell's inequality is in fact violated, meaning that the relevant quantum-mechanical predictions are correct. In these Bell test experiments, entangled pairs of particles are created; the particles are separated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the apparent nonlocality of the effect.

The de Broglie–Bohm theory makes the same (empirically correct) predictions for the Bell test experiments as ordinary quantum mechanics. It is able to do this because it is manifestly nonlocal. It is often criticized or rejected based on this; Bell's attitude was: "It is a merit of the de Broglie–Bohm version to bring this [nonlocality] out so explicitly that it cannot be ignored."<ref>Bell, page 115.</ref>

The de Broglie–Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction. The particles in the experiment follow the guidance of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus.  [[Tim Maudlin#Philosophical work|Maudlin]] provides an analysis of exactly what kind of nonlocality is present and how it is compatible with relativity.<ref>{{cite book |last=Maudlin |first=T. |year=1994 |title=Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics |location=Cambridge, Mass. |publisher=Blackwell |isbn=978-0-631-18609-0 }}</ref> Bell has shown that the nonlocality does not allow [[superluminal communication]]. Maudlin has shown this in greater detail.

=== Classical limit ===
Bohm's formulation of de Broglie–Bohm theory in a classical-looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in 1952. Modern methods of [[decoherence]] are relevant to an analysis of this limit. See Allori et al.<ref>{{cite journal | last1 = Allori | first1 = V. | last2 = Dürr | first2 = D. | last3 = Goldstein | first3 = S. | last4 = Zanghì | first4 = N. | year = 2002 | title = Seven Steps Towards the Classical World | journal = Journal of Optics B | volume = 4 | issue = 4| pages = 482–488 | doi=10.1088/1464-4266/4/4/344|arxiv = quant-ph/0112005 |bibcode = 2002JOptB...4S.482A | s2cid = 45059773 }}</ref> for steps towards a rigorous analysis.

=== Quantum trajectory method ===
Work by [[Robert E. Wyatt]] in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to the Bohm equations of motion. At each time step, one then re-synthesizes the wavefunction from the points, recomputes the quantum forces, and continues the calculation. (QuickTime movies of this for H + H<sub>2</sub> reactive scattering can be found on the [http://research.cm.utexas.edu/rwyatt/movies/qtm/index.html Wyatt group web-site] at UT Austin.)
This approach has been adapted, extended, and used by a number of researchers in the chemical physics community as a way to compute semi-classical and quasi-classical molecular dynamics. A 2007 issue of [[The Journal of Physical Chemistry A]] was dedicated to Prof. Wyatt and his work on "computational Bohmian dynamics".<ref>{{cite journal |last1= Wyatt |first1= Robert |date= 11 Oct 2007 |title= The Short Story of My Life and My Career in Quantum Propagation |url= https://pubs.acs.org/doi/full/10.1021/jp079540%2B  |journal= The Journal of Physical Chemistry A |volume= 111 |issue= 41 |pages= 10171–10185 |doi= 10.1021/jp079540+ |pmid= 17927265 |bibcode= 2007JPCA..11110171. |access-date= 2023-03-18}}</ref>

[[Eric R. Bittner]]'s group<ref name="h523">{{cite web | title=Bittner Group Webpage | website=k2.chem.uh.edu | date=2021-03-10 | url=http://k2.chem.uh.edu/ | archive-url=http://web.archive.org/web/20210805161220/http://k2.chem.uh.edu/ | archive-date=2021-08-05 | url-status=dead | access-date=2024-07-10}}</ref> at the [[University of Houston]] has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points. This technique was recently used to estimate quantum effects in the heat capacity of small clusters Ne<sub>n</sub> for ''n'' ≈ 100.

There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effects lead to the case where <math>R^{-1}\nabla^2R \to \infty.</math> This results in an infinite force on the sample particles forcing them to move away from the node and often crossing the path of other sample points (which violates single-valuedness). Various schemes have been developed to overcome this; however, no general solution has yet emerged.

These methods, as does Bohm's Hamilton–Jacobi formulation, do not apply to situations in which the full dynamics of spin need to be taken into account.

The properties of trajectories in the de Broglie–Bohm theory differ significantly from the [[method of quantum characteristics|Moyal quantum trajectories]] as well as the [[Quantum stochastic calculus#Quantum trajectories|quantum trajectories]] from the unraveling of an open quantum system.