Editing De Broglie–Bohm theory (section)

=== 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.