Principle of locality

In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of instantaneous, or "non-local" action at a distance. Locality evolved out of the field theories of classical physics. The idea is that for a cause at one point to have an effect at another point, something in the space between those points 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 maximum speed at which causal influence can travel to the speed of light, <math>c</math>. Therefore, the principle of locality implies that an event at one point cannot cause a truly simultaneous result at another point. An event at point <math>A</math> cannot cause a result at point <math>B</math> in a time less than <math>T=D/c</math>, where <math>D</math> is the distance between the points and <math>c</math> is the speed of light in vacuum.

The principle of locality plays a critical role in one of the central results of quantum mechanics. In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen, with their EPR paradox thought experiment, raised the possibility that quantum mechanics might not be a complete theory. They described two systems physically separated after interacting; this pair would be called entangled in modern terminology. They reasoned that without additions, now called hidden variables, quantum mechanics would predict illogical relationships between the physically separated measurements.

In 1964, John Stewart Bell formulated Bell's theorem, an inequality which, if violated in actual experiments, implies that quantum mechanics violates local causality (referred to as local realism in later work), a result now considered equivalent to precluding local hidden variables. Progressive variations on those Bell test experiments have since shown that quantum mechanics broadly violates Bell's inequalities. According to some interpretations of quantum mechanics, this result implies that some quantum effects violate the principle of locality.

Pre-quantum mechanics

During the 17th century, Newton's principle of universal gravitation was formulated in terms of "action at a distance", thereby violating the principle of locality. Newton himself considered this violation to be absurd:

Coulomb's law of electric forces was initially also formulated as instantaneous action at a distance, but in 1880, James Clerk Maxwell showed that field equations – which obey locality – predict all of the phenomena of electromagnetism. These equations show that electromagnetic forces propagate at the speed of light.

In 1905, Albert Einstein's special theory of relativity postulated that no matter or energy can travel faster than the speed of light, and Einstein thereby sought to reformulate physics in a way that 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 developed subsequently from the theory of quantum mechanics, which Einstein himself had helped to create.

Models for locality

thumb|Diagram for locality in quantum mechanics

Simple spacetime diagrams can help clarify the issues related to locality. A way to describe the issues of locality suitable for discussion of quantum mechanics is illustrated in the diagram. A particle is created in one location, then split and measured in two other, spatially separated, locations. The two measurements are named for Alice and Bob. Alice performs measurements (A) and gets a result <math>\mathbf{a}</math>. Bob performs (<math>B</math>) and gets result <math>\mathbf{b}</math>. The experiment is repeated many times and the results are compared.

Alice and Bob in spacetime

upright=1.3|thumb|Alice and Bob in spacetime diagram

A spacetime diagram has a time coordinate going vertical and a space coordinate going horizontal. Alice, in a local region on the left, can affect events only in a cone extending in the future as shown; the finite speed of light prevents her from affecting other areas including Bob's location in this case. Similarly, we can use the diagram to reason that Bob's local circumstances cannot be altered by Alice at the same time: all events that cause an effect on Bob are in the cone below his location on the diagram. Dashed lines around Alice show her valid future locations; dashed lines around Bob show events that could have caused his present circumstance. When Alice measures quantum states in her location she gets the results labeled <math>\mathbf{a}</math>; similarly Bob gets <math>\mathbf{b}</math>. Models of locality attempt to explain the statistical relationship between these measured values.

Action at a distance

thumb|Action at a distance

The simplest locality model is no locality: instantaneous action at a distance with no limits for relativity. The locality model for action at a distance is called continuous action. The concepts of locality are more complex and they are described in the language of probability and correlation.

In the 1935 Einstein–Podolsky–Rosen paradox paper (EPR paper), Albert Einstein, Boris Podolsky and Nathan Rosen imagined such an experiment. They observed that quantum mechanics predicts what is now known as quantum entanglement and examined its consequences. In their view, the classical principle of locality implied that "no real change can take place" at Bob's site as a result of whatever measurements Alice was doing. Since quantum mechanics does predict a wavefunction collapse that depends on Alice's choice of measurement, they concluded that this was a form of action-at-distance and that the wavefunction could not be a complete description of reality. Other physicists did not agree: they accepted the quantum wavefunction as complete and questioned the nature of locality and reality assumed in the EPR paper.

In 1964 John Stewart Bell investigated whether it might be possible to fulfill Einstein's goal—to "complete" quantum theory—with local hidden variables to explain the correlations between spatially separated particles as predicted by quantum theory. Bell established a criterion to distinguish between local hidden-variables theory and quantum theory by measuring specific values of correlations between entangled particles. Subsequent experimental tests have shown that some quantum effects do violate Bell's inequalities and cannot be reproduced by a local hidden-variables theory.

Numerous experiments specifically designed to probe the issues of locality confirm the predictions of quantum mechanics; these include experiments where the two measurement locations are more than a kilometer apart.

The 2022 Nobel Prize in Physics was awarded to Alain Aspect, John Clauser and Anton Zeilinger, in part "for experiments with entangled photons, establishing the violation of Bell inequalities". The specific aspect of quantum theory that leads to these correlations is termed quantum entanglement, and versions of Bell's scenario are now used to verify entanglement experimentally. These different names do not alter the mathematical assumptions.

A review of papers using this phrase suggests that a common (classical) physics definition of realism is

This definition includes classical concepts like "well-defined", which conflicts with quantum superposition, and "prior to ... measurements", which implies (metaphysical) preexistence of properties. Specifically, the term local realism in the context of Bell's theorem cannot be viewed as a kind of "realism" involving locality other than the kind implied by the Bell screening assumption. This conflict between common ideas of realism and quantum mechanics requires careful analysis whenever local realism is discussed. Asher Peres distinguishes between weak and strong nonlocality, the latter referring to the theories that allow faster-than-light communication. Under these terms, quantum mechanics would allow weakly nonlocal correlations but not strong nonlocality.

Relativistic quantum mechanics

One of the main principles of quantum field theory is the principle of locality. The field operators and the Lagrangian density describing the dynamics of the fields are local, in the sense that interactions are not described by action-at-a-distance. This condition can be achieved by avoiding terms in the Lagrangian that are products of two fields that depend on distant coordinates. Specifically, in relativistic quantum field theory, to enforce the principles of locality and causality the following condition is required: if there are two observables, each localized within two distinct spacetime regions which happen to be at a spacelike separation from each other, the operators of observables must commute. This condition is sometimes imposed as one of the axioms of relativistic quantum field theory. Operators that commute are not correlated and so cannot influence each other (they are simultaneously observable), so imposing a requirement that the operators of observables with a spacelike separation should commute guarantees that the quantum field theory will respect causality.

References

External links

Quantum nonlocality vs. Einstein locality by H. Dieter Zeh