Editing De Broglie–Bohm theory (section)

=== Pilot wave ===

The de Broglie–Bohm theory describes a pilot wave <math>\psi(q,t) \in \mathbb{C}</math> in a [[Configuration space (physics) | configuration space]] <math>Q</math> and trajectories <math>q(t) \in Q</math> of particles as in classical mechanics but defined by non-Newtonian mechanics.<ref>{{Cite journal |last=Passon |first=Oliver |date=2004-11-01 |title=How to teach quantum mechanics |url=https://iopscience.iop.org/article/10.1088/0143-0807/25/6/008 |journal=European Journal of Physics |volume=25 |issue=6 |pages=765–769 |doi=10.1088/0143-0807/25/6/008 |issn=0143-0807|arxiv=quant-ph/0404128 }}</ref> At every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrödinger equation). 

The de Broglie–Bohm theory works on particle positions and trajectories like [[classical mechanics]] but the dynamics are different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory, the quantum "field exerts a new kind of "quantum-mechanical" force".<ref>{{cite book |author=Bohm |first=David |title=Causality and Chance in Modern Physics |publisher=Routledge & Kegan Paul and D. Van Nostrand |year=1957 |isbn=978-0-8122-1002-6}}</ref>{{rp|76}} Bohm hypothesized that each particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction by the quantum potential.<ref>D. Bohm and B. Hiley: ''The undivided universe: An ontological interpretation of quantum theory'', p. 37.</ref> Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie–Bohm theory, not localized at the position of the particle.<ref>H. R. Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", ''Foundations of Physics'', 1995, Volume 25, Number 2, pp. 329–347.</ref><ref>J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", ''Foundations of Physics'', March 1999, Volume 29, Issue 3, pp. 333–348.</ref>

The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrödinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...] the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles".<ref>{{Cite book |last1=Bohm |first1=David |url=https://books.google.com/books?id=vt9XKjc4WAQC&pg=PA24 |title=The undivided universe: an ontological interpretation of quantum theory |last2=Hiley |first2=Basil J. |date=1995 |publisher=Routledge |isbn=978-0-415-12185-9 |pages=24 |language=en}}</ref> P. Holland considers this lack of reciprocal action of particles and wave function to be one "[a]mong the many nonclassical properties exhibited by this theory".<ref>{{Cite book |last=Holland |first=Peter R. |url=https://books.google.com/books?id=BsEfVBzToRMC&pg=PA26 |title=The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics |date=1995-01-26 |publisher=Cambridge University Press |isbn=978-0-521-48543-2 |pages=26 |language=en}}</ref> Holland later called this a merely ''apparent'' lack of back reaction, due to the incompleteness of the description.<ref>{{cite journal | last1 = Holland | first1 = P. | year = 2001 | title = Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction | url = http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf#page=31 | journal = Nuovo Cimento B | volume = 116 | issue = 10 | pages = 1143–1172 | bibcode = 2001NCimB.116.1143H | access-date = 1 August 2011 | archive-date = 10 November 2011 | archive-url = https://web.archive.org/web/20111110140052/http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf#page=31 | url-status = dead }}</ref>

In what follows below, the setup for one particle moving in <math>\mathbb{R}^3</math> is given followed by the setup for ''N'' particles moving in 3 dimensions. In the first instance, configuration space and real space are the same, while in the second, real space is still <math>\mathbb{R}^3</math>, but configuration space becomes <math>\mathbb{R}^{3N}</math>. While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space, which is how particles are entangled with each other in this theory.

[[#Extensions|Extensions]] to this theory include spin and more complicated configuration spaces.

We use variations of <math>\mathbf{Q}</math> for particle positions, while <math>\psi</math> represents the complex-valued wavefunction on configuration space.