Editing Aharonov–Bohm effect (section)

=== Locality of electromagnetic effects ===
The Aharonov–Bohm effect shows that the local '''E''' and '''B''' fields do not contain full information about the electromagnetic field, and the [[electromagnetic four-potential]], (''Φ'', '''A'''), must be used instead. By [[Stokes' theorem]], the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, ''or'' using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded. In contrast, when using just the four-potential, the effect only depends on the potential in the region where the test particle is allowed.  Therefore, one must either abandon the [[principle of locality]], which most physicists are reluctant to do, or accept that the electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields can. On the other hand, the Aharonov–Bohm effect is crucially quantum mechanical; quantum mechanics is well known to feature [[Quantum nonlocality|non-local effects]] (albeit still disallowing superluminal communication), and Vaidman has argued that this is just a non-local quantum effect in a different form.<ref name="Vaidman2012" />

In [[classical electromagnetism]] the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials ''Φ'' and '''A''' are seen as being more fundamental.<ref>{{cite book|author=Feynman, R|title=The Feynman Lectures on Physics|volume=2 |pages=15–25|quote=knowledge of the classical electromagnetic field acting locally on a particle is not sufficient to predict its quantum-mechanical behavior.'' and ''...is the vector potential a "real" field? ... a real field is a mathematical device for avoiding the idea of action at a distance. .... for a long time it was believed that '''A''' was not a "real" field. .... there are phenomena involving quantum mechanics which show that in fact '''A''' is a "real" field in the sense that we have defined it..... '''E''' and '''B''' are slowly disappearing from the modern expression of physical laws; they are being replaced by '''A''' [the vector potential] and <math>\varphi</math>[the scalar potential]|title-link=The Feynman Lectures on Physics}}</ref> Despite this, all observable effects end up being expressible in terms of the electromagnetic fields, '''E''' and '''B'''.  This is interesting because, while you can calculate the electromagnetic field from the four-potential, due to [[gauge freedom]] the reverse is not true.