# Principle of locality

(Redirected from Locality principle)

In physics, the principle of locality states that an object is directly influenced only by its immediate surroundings. A theory which includes the principle of locality is said to be a "local theory". This is an alternative to the older concept of instantaneous "action at a distance". Locality evolved out of the field theories of classical physics. The concept is that for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.

The special theory of relativity limits the speed at which all such influences can travel to the speed of light, ${\displaystyle c}$. Therefore, the principle of locality implies that an event at one point cannot cause a simultaneous result at another point. An event at point ${\displaystyle A}$ cannot cause a result at point ${\displaystyle B}$ in a time less than ${\displaystyle T=D/c}$, where ${\displaystyle D}$ is the distance between the points.

In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen in their EPR paradox theorised that quantum mechanics might not be a local theory, because a measurement made on one of a pair of separated but entangled particles causes a simultaneous effect, the collapse of the wave function, in the remote particle (i.e. an effect exceeding the speed of light). But because of the probabilistic nature of wave function collapse, this violation of locality cannot be used to transmit information faster than light. In 1964 John Stewart Bell formulated the "Bell inequality", which, if violated in actual experiments, implies that quantum mechanics violates either locality or statistical independence ("free will").

Experimental tests of the Bell inequality, beginning with Alain Aspect's 1972 experiments, show that quantum mechanics seems to violate the inequality, so it must violate either locality or statistical independence. However, critics have noted these experiments contained "loopholes", which prevented a definitive answer to this question. This might now be resolved: in 2015 Dr Ronald Hanson at Delft University performed what has been called the first loophole-free experiment.[1] On the other hand, some loopholes might persist, and may continue to persist to the point of being fundamentally untestable.[2]

## 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.[3]

— Isaac Newton, Letters to Bentley, 1692/3

Coulomb's law of electric forces was initially also formulated as instantaneous action at a distance, but was later superseded by Maxwell's Equations of electromagnetism which obey locality.

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.

## Relativity

Locality is a key axiom of Einstein's relativistic classical field theory, where it is essential to causality that effects do not propagate faster than the speed of light.

In Einstein's theory, two observable objects are localised, each within its own distinct spacetime region (frame), which regions are separated from each other in space, and effects pass from one object to the other at the speed of light or slower. This is a key property of spacetime flowing from the special theory of relativity.

A solution of Einstein's field equations is local if the underlying equations are invariant (a condition where the laws of physics are invariant – that is, the same – in all frames which are moving with uniform velocity with respect to one another).

Alternatively, a solution of Einstein's field equations is still local if the underlying equations are co-variant: i.e. if all (non-gravitational) laws make the same predictions for identical experiments taking place at the same time in two different inertial (that is, non-accelerating) frames; such that the variations from the resting state are the same (i.e. vary equally) for each frame.

## Quantum mechanics

Albert Einstein argues that quantum mechanics is an incomplete theory, because he has shown that, as originally formulated, it leads to a violation of locality.

Einstein, Podolsky and Rosen (dubbed the "EPR" group) outlined a logical argument: quantum mechanics predicts non-locality (which they considered a contradiction with special relativity), unless the hypothesis that the wave function is a complete description of the system is rejected.

Einstein based his conclusion on two assumptions, which he termed axioms: the "free will" hypothesis, namely the fact that Alice and Bob can choose the kind of measurement and the instant to do it independently from each other and on anything else in lab, and the principle of locality. He termed this "the principle of Local Action":

The ... idea characterises the relative independence of objects far apart in space, A and B: external influence on A has no direct influence on B.[4]

He said that without this principle, the idea of the existence of quasi-enclosed systems, and thereby the formulation of laws which can be checked experimentally, would be impossible.

Einstein's conclusion was unverifiable experimentally until, in 1964, John Stewart Bell derived a theorem that states that every physical model able to reproduce quantum mechanical predictions has to reject locality or statistical independence. This assumption require that ontic variables have to be independent from Bob and Alice's space-like separated choices.

### Realism

It is usually thought that in the EPR argument there is a hidden hypothesis of "realism"[5]. However such an hypothesis is actually a very weak hypothesis that is necessary to make sense to the operational notion of "locality". Indeed it is sufficient to assume that, once Alice has become aware of the outcomes of her distant pre-positioned and the programmed robot, they can be considered as real events that have taken place in her space-time. So EPR does not require neither that events are real regardless of whether anyone has observed them, nor that the choices and outcomes of Alice's robot are real for her even when she is unaware of them. Every observer able to set-up an EPR experiment and to calculate the correlations must conclude that at least one of the hypotheses of the argument ("free will", "locality" and quantum mechanical correctness) has to be rejected in his space-time. Of course rejecting the assumption that space-time and therein observed events (by Alice) exist, makes it difficult to give sense to the notion of “speed of influences in space” or to the notion of “No space-like influence”. So this choice is definitely not an option to save locality.

A concept related to realism is "counterfactual definiteness", the idea that it is possible to meaningfully describe as definite the result of a measurement which, in fact, has not been performed (i.e. the ability to assume the existence of objects, and assign values to their properties, even when they have not been measured). However in EPR there is actually no such assumption, but only the counterfactual reasoning necessary to deduce the existence of the information to calculate the distant outcomes, assuming "free will" (a kind of non-contextuality assumption), "locality" and "correctness of quantum mechanical predictions in every experimental context". Such a counterfactual reasoning is nothing but the usual way the predictions of a physical model are calculated.

### Interpretation of the wave function

In some interpretations, such as Relational quantum mechanics and the interpretation based on Consistent histories, the wavefunction is not assumed to physically exist in real spacetime. These interpretations propose that actual definite properties of a physical system "do not exist" prior to the measurement, and the wavefunction is nothing more than a mathematical tool used to calculate the probabilities of experimental outcomes.

However such assumption doesn't change the implications of the EPR argument, since no hypothesis about the ontological interpretation of the wave function is taken here.

### Bohm interpretation

The Bohm interpretation preserves statistical independence and the Born's rule, hence it needs to violate the principle of locality in order to achieve the required correlations. It does so by maintaining that both the position and momentum of a particle are determinate, in that they correspond to the definite trajectory of the particle, but that trajectory depends on all other particle positions of the entire universe.

### Many-worlds interpretation

In the many-worlds interpretation there isn't a well defined notion of "locality in space-time", so the interpretation is agnostic about that.

### Loophole-free violation

In 2015, Ronald Hanson reported observing a loophole-free violation in an experimental test of Bell's theorem: in other words, a result which – for the first time – is free of any additional assumptions (previous experiments, going all the way back to 1972, had required that various assumptions be made in order to obtain an unambiguous contradiction of local realism).[6]

Hanson was reporting on entanglement regarding distant electron spins, in 245 trials, which found that S = 2.42 (± 0.20), with a p-value of 0.039. This result rules out large classes of local realism theories.

### Non-locality

A 2016 experiment demonstrated a single photon, in certain conditions, can have non-local observable physical effects in two places at once.[7][8]