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"The theory of local beables" is a scientific paper authored the physicist John Stewart Bell in June 1975, and reprinted in his book Speakable and Unspeakable in Quantum Mechanics in 2004.^[1]^[2]^[3] It introduced a new way to think of the principle of locality in quantum mechanics. Specifically it introduced the concept of beables (pronounced separately be-ables) to discuss the locality and causality in physics, in the context of Bell's theorem.

Context

Physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) published a paper in 1935 discussing the phenomena of quantum entanglement. According to the EPR paper, quantum mechanics was incomplete and speculated that it should be possible to construct a local hidden-variable theory that would explain the phenomena of entanglement. Other physicists like Niels Bohr, argued against the conclusion of EPR.

In 1964, John Stewart Bell came with a response to the EPR paper, in which he presented a inequality that every local hidden-variable theory should comply. Quantum mechanics predicted a violation of the inequalities, meaning that it was not possible to explain entanglement using hidden classical physics. According to Bell this implied a violation of the principle of locality. "The theory of local beables" was written by Bell while working at CERN in 1975, in order to deep further into the implications of his theorem.

The Aspect's experiment of 1982 demonstrated experimentally confirmed these violations. Further Bell tests have been carried out reducing most of the possible loopholes.

Definitions

Observables and beables

The concept of beable is derived from the concept of observable in quantum mechanics. An observable is a property of a system that is revealed after a measurement (e.g the position of a particle).^[1] A beable is defined as something that correspond to physically real elements, independently of observation.^[4]

According to Bell, the electric and magnetic fields are beables, while the electric potential and the magnetic vector potential are not due to gauge fixing, which means that these potentials are only defined up to a choice of gauge.^[1] These potentials could propagate faster than the speed of light but not the electromagnetic fields.^[1]^[4]

Bell's paper concerns local beables. According to Bell, these are beable that can be localized in a bounded region of spacetime.^[1]

Local determinism

Local determinism is defined as theories where the values of the beables in a localized space region are determined by the backward light cone to single point.

Local causality

Let $P(A|\Lambda )$ be the probability of getting A for a given beable $\Lambda$ .

References

^ ^a ^b ^c ^d ^e Goldstein, Sheldon; Norsen, Travis; Tausk, Daniel Victor; Zanghi, Nino (2011). "Bell's theorem". Scholarpedia. 6 (10): 8378. doi:10.4249/scholarpedia.8378. ISSN 1941-6016.{{cite journal}}: CS1 maint: unflagged free DOI (link)
^ Bell, J. S. (1975), The theory of local beables, Geneva: CERN, CERN-TH-2053, retrieved 2024-04-24
^ Bell, J. S., ed. (2004), "The theory of local beables", Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy (2 ed.), Cambridge: Cambridge University Press, pp. 52–62, ISBN 978-0-521-81862-9, retrieved 2024-04-24
^ ^a ^b Norsen, Travis (2011-12-01). "John S. Bell's concept of local causality". American Journal of Physics. 79 (12): 1261–1275. doi:10.1119/1.3630940. ISSN 0002-9505.

[:0-1] Goldstein, Sheldon; Norsen, Travis; Tausk, Daniel Victor; Zanghi, Nino (2011). "Bell's theorem". Scholarpedia. 6 (10): 8378. doi:10.4249/scholarpedia.8378. ISSN 1941-6016.{{cite journal}}: CS1 maint: unflagged free DOI (link)

[2] Bell, J. S. (1975), The theory of local beables, Geneva: CERN, CERN-TH-2053, retrieved 2024-04-24

[3] Bell, J. S., ed. (2004), "The theory of local beables", Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy (2 ed.), Cambridge: Cambridge University Press, pp. 52–62, ISBN 978-0-521-81862-9, retrieved 2024-04-24

[:1-4] Norsen, Travis (2011-12-01). "John S. Bell's concept of local causality". American Journal of Physics. 79 (12): 1261–1275. doi:10.1119/1.3630940. ISSN 0002-9505.

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