Hidden variable theory
Quantum mechanics 

Introduction Glossary · History 
Equations

Interpretations

Scientists

Historically, in physics, hidden variable theories were espoused by some physicists who argued that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system; i.e., that quantum mechanics is ultimately incomplete, and that a complete theory would provide descriptive categories to account for all observable behavior and thus avoid any indeterminism. The existence of indeterminacy for some measurements is a characteristic of prevalent interpretations of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle.
Albert Einstein, the most famous proponent of hidden variables, objected to the fundamentally probabilistic nature of quantum mechanics,^{1} and famously declared "I am convinced God does not play dice".^{2} Einstein, Podolsky, and Rosen argued that "elements of reality" (hidden variables) must be added to quantum mechanics to explain entanglement without action at a distance.^{3}^{4} Later, Bell's theorem would suggest (in the opinion of most physicists and contrary to Einstein's assertion) that local hidden variables of certain types are impossible. The most famous nonlocal theory is de BroglieBohm theory.
Contents
Motivation
Under the orthodox Copenhagen interpretation, quantum mechanics is nondeterministic, meaning that it generally does not predict the outcome of any measurement with certainty. Instead, it indicates what the probabilities of the outcomes are, with the indeterminism of observable quantities constrained by the uncertainty principle. The question arises whether there might be some deeper reality hidden beneath quantum mechanics, to be described by a more fundamental theory that can always predict the outcome of each measurement with certainty: if the exact properties of every subatomic particle were known the entire system could be modeled exactly using deterministic physics similar to classical physics, a la Laplace's demon.
In other words, it is conceivable that the Copenhagen interpretation of quantum mechanics is an incomplete description of nature. The designation of variables as underlying "hidden" variables depends on the level of physical description (so, for example, "if a gas is described in terms of temperature, pressure, and volume, then the velocities of the individual atoms in the gas would be hidden variables".^{5}). Physicists supporting de BroglieBohm theory maintain that underlying the observed probabilistic nature of the universe is a deterministic objective foundation/property—the hidden variable. Others, however, believe that there is no deeper deterministic reality in quantum mechanics—experiments have shown a vast class of hidden variable theories to be incompatible with observations. Kirchmair and colleagues have shown that, in a system of trapped ions, quantum mechanics conflicts with hidden variable theories regardless of the quantum state of the system.^{6}
Although determinism was initially a major motivation for physicists looking for hidden variable theories, nondeterministic theories trying to explain what the supposed reality underlying the quantum mechanics formalism looks like are also considered hidden variable theories; for example Edward Nelson's stochastic mechanics.
"God does not play dice"
In June 1926, Max Born published a paper, "Zur Quantenmechanik der Stoßvorgänge" ("Quantum Mechanics of Collision Phenomena") in the scientific journal Zeitschrift für Physik, in which he was the first to clearly enunciate the probabilistic interpretation of the quantum wavefunction, which had been introduced by Erwin Schrödinger earlier in the year. Born concluded the paper as follows:
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 preestablished 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 wavefunction 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:
Quantum mechanics is very worthy of regard. But an inner voice tells me that this is not yet the right track. The theory yields much, but it hardly brings us closer to the Old One's secrets. I, in any case, am convinced that He does not play dice.^{1}^{7}
Shortly after making his famous "God does not play dice" comment, Einstein attempted to formulate a deterministic counterproposal to quantum mechanics, presenting a paper at a meeting of the Academy of Sciences in Berlin, on 5 May 1927, titled "Bestimmt Schrödinger's Wellenmechanik die Bewegung eines Systems vollständig oder nur im Sinne der Statistik?" (“Does Schrödinger’s wave mechanics determine the motion of a system completely or only in the statistical sense?”).^{8} However, as the paper was being prepared for publication in the academy's journal, Einstein decided to withdraw it, possibly because he discovered that implied nonseparability of entangled systems could not be eliminated, as he had hoped.^{9}
At the Fifth Solvay Congress, held in Belgium in October 1927 and attended by all the major theoretical physicists of the era, Louis de Broglie presented his own version of a deterministic hiddenvariable theory, apparently unaware of Einstein's aborted attempt earlier in the year. In his theory, every particle had an associated, hidden "pilot wave" which served to guide its trajectory through space. The theory was subject to criticism at the Congress, particularly by Wolfgang Pauli, which de Broglie did not adequately answer. de Broglie abandoned the theory shortly thereafter.
Declaration of completeness of quantum mechanics
Also at the Fifth Solvay Congress, Max Born and Werner Heisenberg made a presentation summarizing the recent tremendous theoretical development of the subject. At the conclusion of the presentation, they declared:
[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.^{10}
BohrEinstein debates
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 during informal discussions over meals, presenting a thought experiment intended to demonstrate that quantum mechanics could not be entirely correct. He did likewise during the Sixth Solvay Congress held in 1930. Both times, Niels Bohr is generally considered to have successfully defended quantum mechanics by discovering errors in Einstein's arguments.
EPR paradox
The debates between Bohr and Einstein essentially concluded in 1935, when Einstein finally expressed what is widely considered his best argument against the completeness of quantum mechanics. Einstein, Podolsky, and Rosen had proposed their definition of a "complete" description as one which uniquely determines the values of all its measurable properties. Einstein later summarized their argument as follows:
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.^{11}
Bohr answered Einstein's challenge by pointing out that, in accordance with the principle of complementarity, one of Einstein's assumptions, regarding the act of measurement, is not true in quantum mechanics: a quantum mechanical measurement not only states, but also prepares the physics of a system:
[The argument of] Einstein, Podolsky and Rosen contains an ambiguity as regards the meaning of the expression "without in any way disturbing a system." ... [E]ven at this stage [i.e., the measurement of, for example, a particle that is part of an entangled pair], there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. Since these conditions constitute an inherent element of the description of any phenomenon to which the term "physical reality" can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that quantummechanical description is essentially incomplete."^{12}
Quantum mechanics is also not "local", essentially because the state of a system is described by the Hilbert vector , which includes the value at every site, . So in this instance Einstein was simply wrong, although he did sharply identify implications of the formalism of quantum mechanics that had been previously overlooked.
Bell's theorem
In 1964, John Bell showed through his famous theorem that if local hidden variables exist, certain experiments could be performed involving quantum entanglement where the result would satisfy a Bell inequality. If, on the other hand, statistical correlations resulting from quantum entanglement could not be explained by local hidden variables, the Bell inequality would be violated. Another nogo theorem concerning hidden variable theories is the KochenSpecker theorem.
Physicists such as Alain Aspect and Paul Kwiat have performed experiments that have found violations of these inequalities up to 242 standard deviations^{13} (excellent scientific certainty). This rules out local hidden variable theories, but does not rule out nonlocal ones. Theoretically, there could be experimental problems that affect the validity of the experimental findings.
Gerard 't Hooft has disputed the validity of Bell's theorem on the basis of the superdeterminism loophole and proposed some ideas to construct local deterministic models.^{14}
Assuming the validity of Bell's theorem, any deterministic hiddenvariable theory which is consistent with quantum mechanics would have to be nonlocal, maintaining the existence of instantaneous or fasterthanlight relations (correlations) between physically separated entities. The currently bestknown hiddenvariable theory, the "causal" interpretation of the physicist and philosopher David Bohm, originally published in 1952, is a nonlocal hidden variable theory. Bohm unknowingly rediscovered (and extended) the idea that Louis de Broglie had proposed in 1927 (and abandoned)  hence this theory is commonly called "de BroglieBohm 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 doubleslit experiment is performed, its trajectory goes through one slit rather than the other. Also, the slit passed through is not random but is governed by the (hidden) guiding wave, resulting in the wave pattern that is observed.
Such a view does not contradict the idea of local events that is used in both classical atomism and relativity theory as Bohm's theory (and quantum mechanics) are still locally causal (that is, information travel is still restricted to the speed of light) but allow nonlocal correlations. It points to a view of a more holistic, mutually interpenetrating and interacting world. Indeed Bohm himself stressed the holistic aspect of quantum theory in his later years, when he became interested in the ideas of Jiddu Krishnamurti.
In Bohm's interpretation, the (nonlocal) quantum potential constitutes an implicate (hidden) order which organizes a particle, and which may itself be the result of yet a further implicate order: a superimplicate order which organizes a field.^{15} Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics which give a realist interpretation, and not merely a positivistic one, to quantummechanical calculations. Some consider it the simplest theory to explain quantum phenomena.^{16} Nevertheless it is a hidden variable theory, and necessarily so.^{17} The major reference for Bohm's theory today is his posthumous book with Basil Hiley.^{18}
A possible weakness of Bohm's theory is that some (including Einstein, Pauli, and Heisenberg) feel that it looks contrived.^{19} (Indeed, Bohm thought this of his original formulation of the theory.^{20}) It was deliberately designed to give predictions that are in all details identical to conventional quantum mechanics.^{20} Bohm's original aim was not to make a serious counterproposal but simply to demonstrate that hiddenvariable theories are indeed possible.^{20} (It thus provided a supposed counterexample to the famous proof by John von Neumann that was generally believed to demonstrate that no deterministic theory reproducing the statistical predictions of quantum mechanics is possible.) Bohm said he considered his theory to be unacceptable as a physical theory due to the guiding wave's existence in an abstract multidimensional configuration space, rather than threedimensional space.^{20} His hope was that the theory would lead to new insights and experiments that would lead ultimately to an acceptable one;^{20} his aim was not to set out a deterministic, mechanical viewpoint, but rather to show that it was possible to attribute properties to an underlying reality, in contrast to the conventional approach to quantum mechanics.^{21}
Recent developments
In August 2011, Roger Colbeck and Renato Renner published a proof that any extension of quantum mechanical theory, whether using hidden variables or otherwise, cannot provide a more accurate prediction of outcomes, assuming that observers can freely choose the measurement settings.^{22} Colbeck and Renner write: "In the present work, we have ... excluded the possibility that any extension of quantum theory (not necessarily in the form of local hidden variables) can help predict the outcomes of any measurement on any quantum state. In this sense, we show the following: under the assumption that measurement settings can be chosen freely, quantum theory really is complete".
In January 2013, GianCarlo Ghirardi and Raffaele Romano described a model which, "under a different free choice assumption [...] violates [the statement by Colbeck and Renner] for almost all states of a bipartite twolevel system, in a possibly experimentally testable way".^{23}
See also
 Local hidden variable theory
 Bell's theorem
 Bell test experiments
 Quantum mechanics
 Bohm interpretation
 Spekkens Toy Model
References
 ^ ^{a} ^{b} The BornEinstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 19161955, with commentaries by Max Born. Macmillan. 1971. p. 158., (Private letter from Einstein to Max Born, 3 March 1947: "I admit, of course, that there is a considerable amount of validity in the statistical approach which you were the first to recognize clearly as necessary given the framework of the existing formalism. I cannot seriously believe in it because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance. ... I am quite convinced that someone will eventually come up with a theory whose objects, connected by laws, are not probabilities but considered facts, as used to be taken for granted until quite recently".)
 ^ private letter to Max Born, 4 December 1926, Albert Einstein Archives reel 8, item 180
 ^ Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can QuantumMechanical Description of Physical Reality Be Considered Complete?". Physical Review 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
 ^ "The debate whether Quantum Mechanics is a complete theory and probabilities have a nonepistemic character (i.e. nature is intrinsically probabilistic) or whether it is a statistical approximation of a deterministic theory and probabilities are due to our ignorance of some parameters (i.e. they are epistemic) dates to the beginning of the theory itself". See: arXiv:quantph/0701071v1 12 Jan 2007
 ^ Senechal M, Cronin J (2001). "Social influences on quantum mechanics?I". The Mathematical Intelligencer 23 (4): 15–17. doi:10.1007/BF03024596.
 ^ Kirchmair, G., et al. (2009). "Stateindependent experimental test of quantum contextuality". Nature 460 (7254): 494–497. arXiv:0904.1655. Bibcode:2009Natur.460..494K. doi:10.1038/nature08172.
 ^ Cache of the Einstein section of the American Museum of Natural History
 ^ Albert Einstein Archives reel 2, item 100
 ^ Baggott, Jim (2011). The Quantum Story: A History in 40 Moments. New York: Oxford University Press. pp. 116117.
 ^ Max Born and Werner Heisenberg, "Quantum mechanics", proceedings of the Fifth Solvay Congress.
 ^ Einstein A (1936). "Physics and Reality". Journal of the Franklin Institute 221.
 ^ Bohr N (1935). "Can QuantumMechanical Description of Physical Reality be Considered Complete?". Physical Review 48: 700.
 ^ Kwiat, P. G., et al. (1999) Ultrabright source of polarizationentangled photons, Physical Review A 60, R773R776
 ^ G 't Hooft, The FreeWill Postulate in Quantum Mechanics [1]; Entangled quantum states in a local deterministic theory [2]
 ^ David Pratt: David Bohm and the Implicate Order. Appeared in: Sunrise magazine, February/March 1993, Theosophical University Press
 ^ Michael K.H. Kiessling: Misleading Signposts Along the de BroglieBohm Road to Quantum Mechanics, Foundations of Physics, volume 40, number 4, 2010, pp. 418429 (abstract)
 ^ "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.
 ^ D.Bohm and B.J.Hiley, The Undivided Universe, Routledge, 1993, ISBN 0415065887.
 ^ Wayne C. Myrvold (2003). "On some early objections to Bohm's theory". International Studies in the Philosophy of Science 17 (1): 8–24.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} David Bohm (1957). Causality and Chance in Modern Physics. Routledge & Kegan Paul and D. Van Nostrand. p. 110. ISBN 0812210026.
 ^ B. J. Hiley: Some remarks on the evolution of Bohm's proposals for an alternative to quantum mechanics, 30 January 2010
 ^ Roger Colbeck; Renato Renner (2011). "No extension of quantum theory can have improved predictive power". Nature Communications 2 (8). arXiv:1005.5173. Bibcode:2011NatCo...2E.411C. doi:10.1038/ncomms1416.
 ^ Giancarlo Ghirardi; Raffaele Romano (2013). Onthological models predictively inequivalent to quantum theory. arXiv:1301.2695. Bibcode:2013PhRvL.110q0404G. doi:10.1103/PhysRevLett.110.170404.
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