Editing De Broglie–Bohm theory (section)

=== 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.