Aptly named Aharonov-Bohm effect has classical analogue, long history

I am an author of the 1961 paper that first called the Aharonov-Bohm effect by that name. ¹ The term has since appeared in more than 1000 published papers. However, it has been suggested a few times, most recently by Peter Sturrock and Timothy Groves, that the name does not do justice to Werner Ehrenberg and Raymond Siday, who found in 1949 that the motion of electrons can be influenced by magnetic fields confined to regions that the electrons do not enter. ² I argue that the name Aharonov-Bohm effect is appropriate, although for a reason that I did not fully appreciate in 1961.

The 1959 Aharonov-Bohm paper profoundly changed the way we think about electromagnetic fields in quantum mechanics. ³ The gauge-invariant part of the vector potential was promoted to a real physical field, not just a convenient device for summarizing certain information about the electric and magnetic fields. In the words of C. N. Yang, “The electromagnetic field strength f_µv in quantum mechanics underdescribes electromagnetism, as the Bohm-Aharonov effect demonstrates … information about the phase factor exp $\left[ie\oint \left(A\cdot dx\right)/{\hslash }_c\right]$ for all closed loops correctly describes electromagnetism.” ⁴

A century earlier, James Clerk Maxwell changed the way we think about action at a distance by identifying what we now call the Maxwell fields as real physical things that contain energy and momentum and that enable microscopic conservation laws; they are not just mathematical functions that summarize the necessary information about the past motions of charges.

Later those physical fields had to be quantized. Yakir Aharonov and David Bohm found that in quantum mechanics the Maxwell fields in a multiply connected region do not contain all the physics; the vector potential must also be endowed with reality to make sense of the subtler interactions in quantum mechanics.

Neither Aharonov and Bohm nor Ehrenberg and Siday were the first to observe that magnetic fields in places where a charged particle’s wave function vanishes may influence the motion of that particle. ⁵ Paul Dirac, in his 1931 paper on magnetic monopoles, noted that the electron’s wavefunction must vanish on singular flux lines but said nothing about that raising any question of nonlocality. Fritz London pointed out in the 1930s that the motion of electrons in a superconducting ring depends on an external magnetic flux through the hole in the ring, where the electrons cannot go. He wrote in his 1937 paper, “The most stable state of a ring has no current, unless an external magnetic field is applied.” Like Dirac, he said nothing about a nonlocal action of the magnetic field being surprising or unusual. Of course, London, like Dirac, was focusing on something else.

Ehrenberg and Siday were also focusing on something else—electron optics—when they found in 1949 that the motion of an electron can depend on the magnetic field in a region from which the electron is excluded. They chose not to mention that curious phenomenon in the abstract of their paper, although they did explicitly say elsewhere that it was “curious.”

None of those earlier authors went on to conclude that such a phenomenon implies that the vector potential has to be seen as a real physical field in quantum mechanics. Only Aharonov and Bohm did that.