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Restored revision 1297076964 by OAbot (talk): Many worlds is completely outside the issue being discussed—i.e. which hidden variable theories are possible given bell's theorem. many worlds is the plainest schrödinger fundamentalist theory possible |
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{{Short description|Type of quantum mechanics theory}}
{{about|a class of quantum mechanics theories|other uses|Hidden variable (disambiguation){{!}}Hidden variable}}
{{Quantum mechanics |interpretations}}
In [[physics]], a '''hidden-variable theory''' is a [[Determinism|deterministic]]
The [[mathematical formulation of quantum
In their 1935 [[Einstein–Podolsky–Rosen paradox|EPR paper]], [[Albert Einstein]], [[Boris Podolsky]], and [[Nathan Rosen
== Motivation ==
Macroscopic physics requires classical mechanics which allows accurate predictions of mechanical motion with reproducible, high precision. Quantum phenomena require quantum mechanics, which allows accurate predictions of statistical averages only. If quantum states had hidden-variables awaiting ingenious new measurement technologies, then the latter (statistical results) might be convertible to a form of the former (classical-mechanical motion).<ref>{{Cite journal |last=Bell |first=John S. |date=1966-07-01 |title=On the Problem of Hidden Variables in Quantum Mechanics |url=https://link.aps.org/doi/10.1103/RevModPhys.38.447 |journal=Reviews of Modern Physics |language=en |volume=38 |issue=3 |pages=447–452 |doi=10.1103/RevModPhys.38.447 |bibcode=1966RvMP...38..447B |osti=1444158 |issn=0034-6861}}</ref>
In other words, whereas a hidden-variable theory would imply intrinsic particle properties, in quantum mechanics an electron has no definite position and velocity to even ''be'' revealed.
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=== "God does not play dice" ===
In June 1926, [[Max Born]] published a paper,<ref>{{Cite journal |last=Born |first=Max |date=1926 |title=Zur Quantenmechanik der Stoßvorgänge |url=http://link.springer.com/10.1007/BF01397477 |journal=Zeitschrift für Physik |language=de |volume=37 |issue=12 |pages=863–867 |doi=10.1007/BF01397477 |bibcode=1926ZPhy...37..863B |s2cid=119896026 |issn=1434-6001|url-access=subscription }}</ref> in which he was the first to clearly enunciate the probabilistic interpretation of the quantum [[wavefunction|wave function]], which had been introduced by [[Erwin Schrödinger]] earlier in the year. Born concluded the paper as follows:{{blockquote|Here the whole problem of determinism comes up. From the standpoint of our quantum mechanics there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which conditions a definite outcome for the collision. Ought we to hope later to discover such properties ... and determine them in individual cases? Or ought we to believe that the agreement of theory and experiment—as to the impossibility of prescribing conditions for a causal evolution—is a pre-established harmony founded on the nonexistence of such conditions? I myself am inclined to give up determinism in the world of atoms. But that is a philosophical question for which physical arguments alone are not decisive.}}Born's interpretation of the wave function was criticized by Schrödinger, who had previously attempted to interpret it in real physical terms, but [[Albert Einstein]]'s response became one of the earliest and most famous assertions that quantum mechanics is incomplete:{{blockquote|Quantum mechanics is very worthy of respect. But an inner voice tells me this is not the genuine article after all. The theory delivers much but it hardly brings us closer to the Old One's secret. In any event, I am convinced that ''He'' is not playing dice.<ref name="Einstein letter, 4 Dec 1926">[https://einsteinpapers.press.princeton.edu/vol15-trans/437 The Collected Papers of Albert Einstein, Volume 15: The Berlin Years: Writings & Correspondence, June 1925-May 1927 (English Translation Supplement), p. 403]</ref><ref>{{cite book|title=The Born–Einstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 1916–1955, with commentaries by Max Born|year=1971|publisher=Macmillan|page=91}}</ref>}}[[Niels Bohr]] reportedly replied to Einstein's later expression of this sentiment by advising him to "stop telling God what to do."<ref>This is a common paraphrasing. Bohr recollected his reply to Einstein at the 1927 [[Solvay Congress]] in his essay "Discussion with Einstein on Epistemological Problems in Atomic Physics", in ''Albert Einstein, Philosopher–Scientist'', ed. Paul Arthur Shilpp, Harper, 1949, p. 211: "...in spite of all divergencies of approach and opinion, a most humorous spirit animated the discussions. On his side, Einstein mockingly asked us whether we could really believe that the providential authorities took recourse to dice-playing ("''ob der liebe Gott würfelt''"), to which I replied by pointing at the great caution, already called for by ancient thinkers, in ascribing attributes to Providence in everyday language." Werner Heisenberg, who also attended the congress, recalled the exchange in ''Encounters with Einstein'', Princeton University Press, 1983, p. 117,: "But he [Einstein] still stood by his watchword, which he clothed in the words: 'God does not play at dice.' To which Bohr could only answer: 'But still, it cannot be for us to tell God, how he is to run the world.'"</ref>
=== Early attempts at hidden-variable theories ===
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=== Declaration of completeness of quantum mechanics, and the Bohr–Einstein debates ===
{{Main|Bohr–Einstein debates}}
Also at the Fifth Solvay Congress, Max Born and [[Werner Heisenberg]] made a presentation summarizing the recent tremendous theoretical development of quantum mechanics. At the conclusion of the presentation, they declared:{{blockquote|[W]hile we consider ... a quantum mechanical treatment of the electromagnetic field ... as not yet finished, we consider quantum mechanics to be a closed theory, whose fundamental physical and mathematical assumptions are no longer susceptible of any modification...
On the question of the 'validity of the law of causality' we have this opinion: as long as one takes into account only experiments that lie in the ___domain of our currently acquired physical and quantum mechanical experience, the assumption of indeterminism in principle, here taken as fundamental, agrees with experience.<ref>Max Born and Werner Heisenberg, "Quantum mechanics", proceedings of the Fifth Solvay Congress.</ref>}}Although there is no record of Einstein responding to Born and Heisenberg during the technical sessions of the Fifth Solvay Congress, he did challenge the completeness of quantum mechanics at various times. In his tribute article for Born's retirement he discussed the quantum representation of a macroscopic ball bouncing elastically between rigid barriers. He argues that such a quantum representation does not represent a specific ball, but "time ensemble of systems". As such the representation is correct, but incomplete because it does not represent the real individual macroscopic case.<ref>{{Cite arXiv |last=Einstein |first=Albert |title=Elementary Considerations on the Interpretation of the Foundations of Quantum Mechanics |date=2011 |class=physics.hist-ph |eprint=1107.3701 |quote=This paper, whose original title was “Elementare Uberlegungen zur Interpretation ¨ der Grundlagen der Quanten-Mechanik”, has been translated from the German by Dileep Karanth, Department of Physics, University of Wisconsin-Parkside, Kenosha, USA}}</ref> Einstein considered quantum mechanics incomplete "because the state function, in general, does not even describe the individual event/system".<ref>{{Cite journal |last=Ballentine |first=L. E. |date=1972-12-01 |title=Einstein's Interpretation of Quantum
▲Although there is no record of Einstein responding to Born and Heisenberg during the technical sessions of the Fifth Solvay Congress, he did challenge the completeness of quantum mechanics at various times. In his tribute article for Born's retirement he discussed the quantum representation of a macroscopic ball bouncing elastically between rigid barriers. He argues that such a quantum representation does not represent a specific ball, but "time ensemble of systems". As such the representation is correct, but incomplete because it does not represent the real individual macroscopic case.<ref>{{Cite arXiv |last=Einstein |first=Albert |title=Elementary Considerations on the Interpretation of the Foundations of Quantum Mechanics |date=2011 |class=physics.hist-ph |eprint=1107.3701 |quote=This paper, whose original title was “Elementare Uberlegungen zur Interpretation ¨ der Grundlagen der Quanten-Mechanik”, has been translated from the German by Dileep Karanth, Department of Physics, University of Wisconsin-Parkside, Kenosha, USA}}</ref> Einstein considered quantum mechanics incomplete "because the state function, in general, does not even describe the individual event/system".<ref>{{Cite journal |last=Ballentine |first=L. E. |date=1972-12-01 |title=Einstein's Interpretation of Quantum Mechanics |url=https://pubs.aip.org/ajp/article/40/12/1763/527506/Einstein-s-Interpretation-of-Quantum-Mechanics |journal=American Journal of Physics |language=en |volume=40 |issue=12 |pages=1763–1771 |doi=10.1119/1.1987060 |bibcode=1972AmJPh..40.1763B |issn=0002-9505|doi-access=free }}</ref>
=== Von Neumann's proof ===
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Einstein argued that quantum mechanics could not be a complete theory of physical reality. He wrote,
▲<blockquote>Consider a mechanical system consisting of two partial systems ''A'' and ''B'' which interact with each other only during a limited time. Let the ''ψ'' function [i.e., [[wavefunction]]] before their interaction be given. Then the Schrödinger equation will furnish the ''ψ'' function after the interaction has taken place. Let us now determine the physical state of the partial system ''A'' as completely as possible by measurements. Then quantum mechanics allows us to determine the ''ψ'' function of the partial system ''B'' from the measurements made, and from the ''ψ'' function of the total system. This determination, however, gives a result which depends upon which of the physical quantities (observables) of ''A'' have been measured (for instance, coordinates or momenta). Since there can be only one physical state of ''B'' after the interaction which cannot reasonably be considered to depend on the particular measurement we perform on the system ''A'' separated from ''B'' it may be concluded that the ''ψ'' function is not unambiguously coordinated to the physical state. This coordination of several ''ψ'' functions to the same physical state of system ''B'' shows again that the ''ψ'' function cannot be interpreted as a (complete) description of a physical state of a single system.<ref>{{Cite journal |author=Einstein A |year=1936 |title=Physics and Reality |journal=Journal of the Franklin Institute |volume=221}}</ref></blockquote>
Together with [[Boris Podolsky]] and [[Nathan Rosen]], Einstein published a paper that gave a related but distinct argument against the completeness of quantum mechanics.<ref>{{Cite journal |first1=A. |last1=Einstein |first2=B. |last2=Podolsky |first3=N. |last3=Rosen |year=1935 |title=Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? |journal=Physical Review |volume=47 |issue= 10|pages=777–780 |doi=10.1103/physrev.47.777 |bibcode=1935PhRv...47..777E |doi-access=free }}</ref> They proposed a [[thought experiment]] involving a pair of particles prepared in what would later become known as an [[Quantum entanglement|entangled]] [[quantum state|state]]. Einstein, Podolsky, and Rosen pointed out that, in this state, if the position of the first particle were measured, the result of measuring the position of the second particle could be predicted. If instead the momentum of the first particle were measured, then the result of measuring the momentum of the second particle could be predicted. They argued that no action taken on the first particle could instantaneously affect the other, since this would involve information being transmitted faster than light, which is impossible according to the [[theory of relativity]]. They invoked a principle, later known as the "EPR criterion of reality", positing that: "If, without in any way disturbing a system, we can predict with certainty (i.e., with [[probability]] equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity." From this, they inferred that the second particle must have a definite value of both position and of momentum prior to either quantity being measured. But quantum mechanics considers these two observables [[Observable#Incompatibility of observables in quantum mechanics|incompatible]] and thus does not associate simultaneous values for both to any system. Einstein, Podolsky, and Rosen therefore concluded that quantum theory does not provide a complete description of reality.<ref>{{cite book |last=Peres |first=Asher |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |pages=149 |publisher=Kluwer |year=2002}}</ref>
Bohr answered the Einstein–Podolsky–Rosen challenge as follows:
Bohr is here choosing to define a "physical reality" as limited to a phenomenon that is immediately observable by an arbitrarily chosen and explicitly specified technique, using his own special definition of the term 'phenomenon'. He wrote in 1948:
This was, of course, in conflict with the EPR criterion of reality.
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{{Main|de Broglie–Bohm theory}}
In 1952, [[David Bohm]] proposed a hidden variable theory. Bohm unknowingly rediscovered (and extended) the idea that Louis de Broglie's [[pilot wave theory]] had proposed in 1927 (and abandoned) – hence this theory is commonly called "de Broglie-Bohm theory".
Bohm posited ''both'' the quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles. When a [[double-slit experiment]] is performed, the electron goes through either one of the slits. Also, the slit passed through is not random but is governed by the (hidden) pilot wave, resulting in the wave pattern that is observed.
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