Principle of locality
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In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must either violate the principle of locality or engage in superluminal communication.
Pre-quantum mechanics 
In the 17th Century Newton's law of universal gravitation was formulated in terms of "action at a distance", thereby violating the principle of locality.
It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.—Isaac Newton, Letters to Bentley, 1692/3
In 1905 Albert Einstein's Special Theory of Relativity postulated that no material or energy can travel faster than the speed of light, and Einstein thereby sought to reformulate physical laws in a way which obeyed the principle of locality. He later succeeded in producing an alternative theory of gravitation, General Relativity, which obeys the principle of locality.
However, a different challenge to the principle of locality subsequently emerged from the theory of Quantum Mechanics, which Einstein himself had helped to create.
Quantum mechanics 
EPR Paradox 
Albert Einstein argued that quantum mechanics was an incomplete physical theory. Using the principle of locality, he, Podolsky, and Rosen articulated the Einstein-Podolsky-Rosen Paradox which showed that quantum mechanics predicts non-locality unless position and momentum were simultaneous "real" properties of a particle. The locality question remained unverifiable for several decades. Then in 1964, John Stewart Bell derived his eponymous theorem, which describes quantum mechanical predictions that no theory of local hidden variables, no local realism, could ever reproduce.
Einstein assumed that the principle of locality was necessary, and that there could be no violations of it. He said:
"(...) The following idea characterises the relative independence of objects far apart in space, A and B: external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible. (...)"
Local realism 
Local realism is the combination of the principle of locality with the "realistic" assumption that all objects must objectively have a pre-existing value for any possible measurement before the measurement is made.
Local realism is a significant feature of classical mechanics, of general relativity, and of electrodynamics; but quantum mechanics largely rejects this principle due to the theory of distant quantum entanglements, an interpretation rejected by Einstein in the EPR paradox but subsequently proven by Bell's inequalities. Any theory, such as quantum mechanics, that violates Bell's inequalities must abandon either local realism or counterfactual definiteness; but some physicists dispute that experiments have demonstrated Bell's violations, on the grounds that the sub-class of inhomogeneous Bell inequalities has not been tested or due to experimental limitations in the tests. Different interpretations of quantum mechanics violate different parts of local realism and/or counterfactual definiteness.
Realism in the sense used by physicists does not equate to realism in metaphysics. The latter is the claim that the world is in some sense mind-independent: that even if the results of a possible measurement do not pre-exist the act of measurement, that does not require that they are the creation of the observer (contrary to the "consciousness causes collapse" interpretation of quantum mechanics). Furthermore, a mind-independent property does not have to be the value of some physical variable such as position or momentum. A property can be dispositional (or potential), i.e. it can be a tendency: in the way that glass objects tend to break, or are disposed to break, even if they do not actually break. Likewise, the mind-independent properties of quantum systems could consist of a tendency to respond to particular measurements with particular values with ascertainable probability. Such an ontology would be metaphysically realistic, without being realistic in the physicist's sense of "local realism" (which would require that a single value be produced with certainty).
A closely related term is counterfactual definiteness (CFD), used to refer to the claim that one can meaningfully speak of the definiteness of results of measurements that have not been performed (i.e. the ability to assume the existence of objects, and properties of objects, even when they have not been measured).
Copenhagen interpretation 
In most of the conventional interpretations, such as the Copenhagen interpretation and the interpretation based on Consistent Histories, where the wavefunction is not assumed to physically exist in real spacetime, it is local realism that is rejected. These interpretations propose that actual definite properties of a physical system "do not exist" prior to the measurement; and the wavefunction has a restricted interpretation, as nothing more than a mathematical tool used to calculate the probabilities of experimental outcomes, hence in agreement with positivism in philosophy as the only topic that science should discuss.
If the wavefunction is assumed to physically exist in real spacetime, the principle of locality is violated during the measurement process via wavefunction collapse. This is a non-local process because Born's Rule, when applied to the system's wavefunction, yields a probability density for all regions of space and time. Upon actual measurement of the physical system, the probability density vanishes everywhere instantaneously, except where (and when) the measured entity is found to exist. This "vanishing" is postulated to be a real physical process, and clearly non-local (i.e. faster than light) if the wavefunction is considered physically real and the probability density has converged to zero at arbitrarily far distances during the finite time required for the measurement process.
Bohm interpretation 
The Bohm interpretation preserves realism, hence it needs to violate the principle of locality in order to achieve the required correlations.
Many-worlds interpretation 
In the many-worlds interpretation both realism and locality are retained, but counterfactual definiteness is rejected by the extension of the notion of reality to allow the existence of parallel universes.
Because the differences between the different interpretations are mostly philosophical ones (except for the Bohm and many-worlds interpretations), physicists usually employ language in which the important statements are neutral with regard to all of the interpretations. In this framework, only the measurable action at a distance - a superluminal propagation of real, physical information - would usually be considered in violation of the principle of locality by physicists. Such phenomena have never been seen, and they are not predicted by the current theories.
Locality is one of the axioms of relativistic quantum field theory, as required for causality. The formalization of locality in this case is as follows: if we have two observables, each localized within two distinct spacetime regions which happen to be at a spacelike separation from each other, the observables must commute. Alternatively, a solution to the field equations is local if the underlying equations are either Lorentz invariant or, more generally, generally covariant or locally Lorentz invariant.
See also 
- J-D. Bancal, S. Pironio, A. Acín, Y-C. Liang, V. Scarani & N. Gisin (Nature Physics, 2012). Quantum non-locality based on finite-speed causal influences leads to superluminal signalling
- Berkovitz, Joseph (2008). "Action at a Distance in Quantum Mechanics". In Edward N. Zalta. The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).
- "Quantum Mechanics and Reality" ("Quanten-Mechanik und Wirklichkeit", Dialectica 2:320-324, 1948)
- Ben Dov, Y. Local Realism and the Crucial experiment.
- "Quantum crypto still not proven, claim Cambridge experts."
- Norsen, T. - Against "Realism"
- Ian Thomson's dispositional quantum mechanics