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, a result now considered equivalent to no local hidden variables. Later work refers to local causality as local realism.
Bell test experiments show that quantum mechanics broadly violates the inequalities established in Bell's theorem. 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:
Isaac Newton
— Letters to Bentley, 1692/3
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.[citation needed] 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
Simple spacetime diagrams can help clarify the issues related to locality.<ref name="WhartonArgaman">Wharton, K. B.; Argaman, N. (May 18, 2020). "Colloquium : Bell's theorem and locally mediated reformulations of quantum mechanics". Reviews of Modern Physics. 92 (2): 021002. arXiv:1906.04313. doi:10.1103/RevModPhys.92.021002. ISSN 0034-6861.</ref> 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
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 prevent 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
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.<ref name="WhartonArgaman" /> The gray area (a circle here) is a mathematical concept called a "screen". Any path from a location through the screen becomes part of the physical model at that location. The gray ring indicates events from all parts of space and time can affect the probability measured by Alice or Bob. So in the case of continuous action, events at all times and places affect Alice's and Bob's model. This simple model is highly successful for solar planetary dynamics with Newtonian gravity and in electrostatics, cases where relativistic effects are insignificant.
No future-input dependence
Many locality models explicitly or implicitly ignore the possible effect of future events. The spacetime diagram at the right shows the effect of such a restriction when combined with continuous action. Inputs from the future (above the dashed line) are no longer considered part of Alice's or Bob's model. Comparing this diagram with the one for continuous action makes it clear that these are not the same locality model.<ref name="WhartonArgaman" /> Common sense arguments about the future not affecting the present are reasonable criteria but such assumptions alter the mathematical character of the models.
Bell's local causality
John Stewart Bell when discussing his Bell's theorem uses the screening model shown at the right. Events in the common past of Alice and Bob are part of the model used in calculating probabilities for Alice and for Bob as indicated the way the screen absorbs those events. However events at Bob's location during Alice measurement and events in the future are excluded. Bell called this assumption local causality, but with the diagram we can reason about the meaning of the assumption without getting tripped up by other meanings of local combined with other meanings of causal.<ref name="WhartonArgaman" /> Dash lines show relativistically valid regions in the past of Alice or Bob. The gray arc is the assumed Bell "screen".
Quantum mechanics
The relative positions of our few, easily distinguishable planets (for example) can be seen directly: understanding and measuring their relative location poses only technical issues. The submicroscopic world on the other hand is known only by measurements that average over many seemingly random ('statistical' or 'probabilistic') events and measurements can show either particle-like or wave-like results depending on their design. This world is governed by quantum mechanics.<ref name="FeynmanIII">Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew L. (2007). Quantum Mechanics. The Feynman Lectures on Physics. Vol. 3. Reading/Mass.: Addison-Wesley. ISBN 978-0-201-02118-9.</ref> 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),<ref>Einstein, A.; Podolsky, B.; Rosen, N. (May 15, 1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?". Physical Review. 47 (10): 777–780. doi:10.1103/PhysRev.47.777. ISSN 0031-899X.</ref> 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.<ref name=ReidDrummond >Reid, M. D.; Drummond, P. D.; Bowen, W. P.; Cavalcanti, E. G.; Lam, P. K.; Bachor, H. A.; Andersen, U. L.; Leuchs, G. (December 10, 2009). "Colloquium : The Einstein-Podolsky-Rosen paradox: From concepts to applications". Reviews of Modern Physics. 81 (4): 1727–1751. doi:10.1103/RevModPhys.81.1727. hdl:10072/37941. ISSN 0034-6861. S2CID 53407634.</ref> 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 Bob'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.<ref name=ClauserShimony1978>Clauser, John F., and Abner Shimony. "Bell's theorem. Experimental tests and implications." Reports on Progress in Physics 41.12 (1978): 1881.</ref>
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 criteria 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.<ref name=ReidDrummond/> Bell's theorem depends on careful defined models of locality.
Bell described local causality in terms of probability needed for analysis of quantum mechanics. Using the notation that <math>P(\mathbf{r}|g)</math> for the probability of a result <math>\mathbf{r}</math> with given state <math>g</math>, Bell investigated the probability distribution <math display=block>P(\mathbf{ab}|AB,\lambda)</math> where <math>\lambda</math> represents hidden state variables set (locally) when the two particles are initially co-located. If local causality holds, then the probabilities observed by Alice and by Bob should be only coupled by the hidden variables, and we can show that <math display=block>P(\mathbf{ab}|AB,\lambda) = P(\mathbf{a}|A,\lambda)P(\mathbf{b}|B,\lambda),</math> Bell proved that a consequence of this factorization are limits on the correlations observed by Alice and Bob known as Bell inequalities. Since quantum mechanics predicts correlations stronger than this limit, locally set hidden variables cannot be added to "complete" quantum theory as desired by the EPR paper.<ref name=Brunner>Brunner, Nicolas; Cavalcanti, Daniel; Pironio, Stefano; Scarani, Valerio; Wehner, Stephanie (April 18, 2014). "Bell nonlocality". Reviews of Modern Physics. 86 (2): 419–478. arXiv:1303.2849. doi:10.1103/RevModPhys.86.419. ISSN 0034-6861. S2CID 119194006.</ref>
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.<ref name=Brunner/><ref>Holmes, Rebecca (2017). "Local realism is dead, long live local realism?". Physics World. 30 (6): 21–25. Bibcode:2017PhyW...30f..21H. doi:10.1088/2058-7058/30/6/41.</ref> 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".<ref>"The Nobel Prize in Physics 2022". Nobel Foundation. October 4, 2022. Archived from the original on October 4, 2022. Retrieved October 6, 2022.</ref> 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.<ref name=Brunner/>
Terminology
Bell's mathematical results, when compared to experimental data, eliminate local hidden variable mathematical quantum theories. But the interpretation of the math with respect to the physical world remains under debate. Bell described the assumptions behind his work as "local causality", shortened to "locality"; later authors referred to the assumptions as local realism.<ref>Laudisa, Federico (February 2023). "How and when did locality become 'local realism'? A historical and critical analysis (1963–1978)". Studies in History and Philosophy of Science. 97: 44–57. arXiv:2205.05452. doi:10.1016/j.shpsa.2022.11.008. PMID 36549108. S2CID 248693366.</ref> These different names do not alter the mathematical assumptions.
A review of papers<ref name=Lambare>Lambare, Justo Pastor (October 2022). "On the Meaning of Local Realism". Foundations of Physics. 52 (5). doi:10.1007/s10701-022-00618-1. ISSN 0015-9018. S2CID 252107202.</ref> using this phrase suggests that a common (classical) physics definition of realism is
the assumption that measurement outcomes are well defined prior to and independent of the measurements.<ref>Paterek, Tomasz; Fedrizzi, Alessandro; Gröblacher, Simon; Jennewein, Thomas; Żukowski, Marek; Aspelmeyer, Markus; Zeilinger, Anton (November 21, 2007). "Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption". Physical Review Letters. 99 (21): 210406. arXiv:0708.0813. doi:10.1103/PhysRevLett.99.210406. ISSN 0031-9007. PMID 18233201. S2CID 21746600.</ref>
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.<ref name=Lambare/>: 98 Adding a "locality" modifier, that the results of two spatially well-separated measurements cannot causally affect each other,<ref name=ReidDrummond/> does not make the combination relate to Bell's proof; the only interpretation that Bell assumed was the one he called local causality.<ref name=Lambare/>: 98 Consequently, Bell's theorem does not restrict the possibility of nonlocal variables as well as theories based on retrocausality or superdeterminism.<ref name="WhartonArgaman" />
Because of the probabilistic nature of wave function collapse, this apparent violation of locality in quantum mechanics cannot be used to transmit information faster than light, in accordance to the no communication theorem.<ref>Susskind, Leonard; Friedman, Art (February 25, 2014). Quantum Mechanics: The Theoretical Minimum. Penguin Books Limited. ISBN 978-0-14-197782-9.</ref> Asher Peres distinguishes between weak and strong nonlocality, the latter referring to the theories that allow to faster-than-light communication. Under these terms, quantum mechanics would allow weakly nonlocal correlations but not strong nonlocality.<ref>Peres, A. (June 1, 2006). Quantum Theory: Concepts and Methods. Springer Science & Business Media. ISBN 978-0-306-47120-9.</ref>
Relativistic quantum mechanics
One of the main principles of quantum field theory is the principle of locality.<ref name=":0">Tong, David (2006). "Quantum Field Theory" (PDF). University of Cambdridge.</ref> 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 can be achieved by avoiding terms in the Lagrangian that are products of two fields that depend on distant coordinates.<ref name=":0" /><ref>Bonneau, Guy (2009). "Local operator". Scholarpedia. 4 (9): 9669. doi:10.4249/scholarpedia.9669. ISSN 1941-6016.</ref> 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 observables must commute. This condition is sometimes imposed as one of the axioms of relativistic quantum field theory.<ref name=":0" /><ref>Streater, Raymond Frederick (May 20, 2009). "Wightman quantum field theory". Scholarpedia. 4 (5): 7123. doi:10.4249/scholarpedia.7123. ISSN 1941-6016.</ref>
See also
References
External links
