Principle of locality

Last updated December 13, 2024

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, $c$ . 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 $A$ cannot cause a result at point $B$ in a time less than $T=D/c$ , where $D$ is the distance between the points and $c$ 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:

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.^[1]
— 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.^[2] 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 $\mathbf {a}$ ); Bob performs ( $B$ ) and gets result $\mathbf {b}$ . 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 $\mathbf {a}$ ; similarly Bob gets $\mathbf {b}$ . 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.^[2] 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.^[2] 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.^[2] 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.^[3] 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),^[4] 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.^[5] 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.^[6]

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.^[5] Bell's theorem depends on careful defined models of locality.

Locality and hidden variables

Bell described local causality in terms of probability needed for analysis of quantum mechanics. Using the notation that $P(\mathbf {r} \mid g)$ for the probability of a result $\mathbf {r}$ with given state $g$ , Bell investigated the probability distribution $P(\mathbf {ab} \mid AB,\lambda ),$ where $\lambda$ 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 $P(\mathbf {ab} \mid AB,\lambda )=P(\mathbf {a} \mid A,\lambda )P(\mathbf {b} \mid B,\lambda ).$ 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.^[7]

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.^[7]^[8] 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".^[9] 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.^[7]

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.^[10] These different names do not alter the mathematical assumptions.

A review of papers^[11] 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.^[12]

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.^[11]^: 98 Adding a "locality" modifier, that the results of two spatially well-separated measurements cannot causally affect each other,^[5] does not make the combination relate to Bell's proof; the only interpretation that Bell assumed was the one he called local causality.^[11]^: 98 Consequently, Bell's theorem does not restrict the possibility of nonlocal variables as well as theories based on retrocausality or superdeterminism.^[2]

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.^[13] 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.^[14]

Relativistic quantum mechanics

One of the main principles of quantum field theory is the principle of locality.^[15] 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.^[15]^[16] 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.^[15]^[17]

Related Research Articles

The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen, which argues that the description of physical reality provided by quantum mechanics is incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these hidden variables. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

In quantum mechanics, counterfactual definiteness (CFD) is the ability to speak "meaningfully" of the definiteness of the results of measurements that have not been performed. The term "counterfactual definiteness" is used in discussions of physics calculations, especially those related to the phenomenon called quantum entanglement and those related to the Bell inequalities. In such discussions "meaningfully" means the ability to treat these unmeasured results on an equal footing with measured results in statistical calculations. It is this aspect of counterfactual definiteness that is of direct relevance to physics and mathematical models of physical systems and not philosophical concerns regarding the meaning of unmeasured results.

The de Broglie–Bohm theory is an interpretation of quantum mechanics which postulates that, in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

In philosophy, the philosophy of physics deals with conceptual and interpretational issues in physics, many of which overlap with research done by certain kinds of theoretical physicists. Historically, philosophers of physics have engaged with questions such as the nature of space, time, matter and the laws that govern their interactions, as well as the epistemological and ontological basis of the theories used by practicing physicists. The discipline draws upon insights from various areas of philosophy, including metaphysics, epistemology, and philosophy of science, while also engaging with the latest developments in theoretical and experimental physics.

Causality is the relationship between causes and effects. While causality is also a topic studied from the perspectives of philosophy and physics, it is operationalized so that causes of an event must be in the past light cone of the event and ultimately reducible to fundamental interactions. Similarly, a cause cannot have an effect outside its future light cone.

John Stewart Bell FRS was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum physics regarding hidden-variable theories.

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that

In physics, a hidden-variable theory is a deterministic physical model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional variables.

A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables to explain the behavior of particles like photons and electrons. The test empirically evaluates the implications of Bell's theorem. As of 2015, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.

In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These models attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent.

In physics, the no-communication theorem is a no-go theorem in quantum information theory. It asserts that during the measurement of an entangled quantum state, it is impossible for one observer to transmit information to another observer, regardless of their spatial separation. This conclusion preserves the principle of causality in quantum mechanics and ensures that information transfer does not violate special relativity by exceeding the speed of light.

In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–KS theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden-variable theories, which try to explain the predictions of quantum mechanics in a context-independent way. The version of the theorem proved by Kochen and Specker also gave an explicit example for this constraint in terms of a finite number of state vectors.

In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.

In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not allow an interpretation with local realism. Quantum nonlocality has been experimentally verified under a variety of physical assumptions.

In quantum mechanics, superdeterminism is a loophole in Bell's theorem. By postulating that all systems being measured are correlated with the choices of which measurements to make on them, the assumptions of the theorem are no longer fulfilled. A hidden variables theory which is superdeterministic can thus fulfill Bell's notion of local causality and still violate the inequalities derived from Bell's theorem. This makes it possible to construct a local hidden-variable theory that reproduces the predictions of quantum mechanics, for which a few toy models have been proposed. In addition to being deterministic, superdeterministic models also postulate correlations between the state that is measured and the measurement setting.

This is a glossary for the terminology applied in the foundations of quantum mechanics and quantum metaphysics, collectively called quantum philosophy, a subfield of philosophy of physics.

Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

References

↑ Berkovitz, Joseph (2008). "Action at a Distance in Quantum Mechanics". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter ed.).
1 2 3 4 5 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 . Bibcode:2020RvMP...92b1002W. doi:10.1103/RevModPhys.92.021002. ISSN 0034-6861.
↑ 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.
↑ 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. Bibcode:1935PhRv...47..777E. doi: 10.1103/PhysRev.47.777 . ISSN 0031-899X.
1 2 3 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. arXiv: 0806.0270 . Bibcode:2009RvMP...81.1727R. doi:10.1103/RevModPhys.81.1727. hdl: 10072/37941 . ISSN 0034-6861. S2CID 53407634.
↑ Clauser, John F., and Abner Shimony. "Bell's theorem. Experimental tests and implications". Reports on Progress in Physics 41.12 (1978): 1881.
1 2 3 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 . Bibcode:2014RvMP...86..419B. doi:10.1103/RevModPhys.86.419. ISSN 0034-6861. S2CID 119194006.
↑ 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.
↑ "The Nobel Prize in Physics 2022". Nobel Foundation. October 4, 2022. Archived from the original on October 4, 2022. Retrieved October 6, 2022.
↑ 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.
1 2 3 Lambare, Justo Pastor (October 2022). "On the Meaning of Local Realism". Foundations of Physics. 52 (5): 98. Bibcode:2022FoPh...52...98L. doi:10.1007/s10701-022-00618-1. ISSN 0015-9018. S2CID 252107202.
↑ 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 . Bibcode:2007PhRvL..99u0406P. doi:10.1103/PhysRevLett.99.210406. ISSN 0031-9007. PMID 18233201. S2CID 21746600.
↑ Susskind, Leonard; Friedman, Art (February 25, 2014). Quantum Mechanics: The Theoretical Minimum. Penguin Books Limited. ISBN 978-0-14-197782-9.
↑ Peres, A. (June 1, 2006). Quantum Theory: Concepts and Methods. Springer Science & Business Media. ISBN 978-0-306-47120-9.
1 2 3 Tong, David (2006). "Quantum Field Theory" (PDF). University of Cambdridge.
↑ Bonneau, Guy (2009). "Local operator". Scholarpedia. 4 (9): 9669. Bibcode:2009SchpJ...4.9669B. doi: 10.4249/scholarpedia.9669 . ISSN 1941-6016.
↑ Streater, Raymond Frederick (May 20, 2009). "Wightman quantum field theory". Scholarpedia. 4 (5): 7123. Bibcode:2009SchpJ...4.7123S. doi: 10.4249/scholarpedia.7123 . ISSN 1941-6016.

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[sep-qm-action-distance-1] Berkovitz, Joseph (2008). "Action at a Distance in Quantum Mechanics". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter ed.).

[WhartonArgaman-2] 1 2 3 4 5 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 . Bibcode:2020RvMP...92b1002W. doi:10.1103/RevModPhys.92.021002. ISSN 0034-6861.

[FeynmanIII-3] 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.

[4] 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. Bibcode:1935PhRv...47..777E. doi: 10.1103/PhysRev.47.777 . ISSN 0031-899X.

[ReidDrummond-5] 1 2 3 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. arXiv: 0806.0270 . Bibcode:2009RvMP...81.1727R. doi:10.1103/RevModPhys.81.1727. hdl: 10072/37941 . ISSN 0034-6861. S2CID 53407634.

[ClauserShimony1978-6] Clauser, John F., and Abner Shimony. "Bell's theorem. Experimental tests and implications". Reports on Progress in Physics 41.12 (1978): 1881.

[Brunner-7] 1 2 3 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 . Bibcode:2014RvMP...86..419B. doi:10.1103/RevModPhys.86.419. ISSN 0034-6861. S2CID 119194006.

[8] 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.

[9] "The Nobel Prize in Physics 2022". Nobel Foundation. October 4, 2022. Archived from the original on October 4, 2022. Retrieved October 6, 2022.

[10] 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.

[Lambare-11] 1 2 3 Lambare, Justo Pastor (October 2022). "On the Meaning of Local Realism". Foundations of Physics. 52 (5): 98. Bibcode:2022FoPh...52...98L. doi:10.1007/s10701-022-00618-1. ISSN 0015-9018. S2CID 252107202.

[12] 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 . Bibcode:2007PhRvL..99u0406P. doi:10.1103/PhysRevLett.99.210406. ISSN 0031-9007. PMID 18233201. S2CID 21746600.

[13] Susskind, Leonard; Friedman, Art (February 25, 2014). Quantum Mechanics: The Theoretical Minimum. Penguin Books Limited. ISBN 978-0-14-197782-9.

[14] Peres, A. (June 1, 2006). Quantum Theory: Concepts and Methods. Springer Science & Business Media. ISBN 978-0-306-47120-9.

[:0-15] 1 2 3 Tong, David (2006). "Quantum Field Theory" (PDF). University of Cambdridge.

[16] Bonneau, Guy (2009). "Local operator". Scholarpedia. 4 (9): 9669. Bibcode:2009SchpJ...4.9669B. doi: 10.4249/scholarpedia.9669 . ISSN 1941-6016.

[17] Streater, Raymond Frederick (May 20, 2009). "Wightman quantum field theory". Scholarpedia. 4 (5): 7123. Bibcode:2009SchpJ...4.7123S. doi: 10.4249/scholarpedia.7123 . ISSN 1941-6016.

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