James Clerk Maxwell discovered classical electromagnetism (http://en.wikipedia.org/wiki/James_Clerk_Maxwell). In 1864 he established the physical fact that light is simply a form of electromagnetic waves. Then in 1905 Einstein established the physical fact that light is a quantum of energy called the photon. Although the rest mass of a photon is zero (meaning that a photon is always in motion) its linear momentum is given as its energy E divided by its speed c: p = E/c and E = hnwhere h is Planck’s constant of action and n is the spatiotemporal frequency. However, the product of wavelength l and n is simply c. Substituting and combining these facts led Louis de Broglie in 1924 to associate wave properties to matter. It is the works related to the quantum theory of the electromagnetic field that both Einstein and de Broglie were each awarded the Nobel Prize in Physics, respectively in 1921 and in 1929. Nonetheless, no one described the physical connection of the magnetic vector potential to the nature of matter and energy.
Although the magnetic vector potential was much studied by Maxwell, later investigators completely swept it under the rug until several physical effects which cannot be described completely by Maxwell’s equations emerged.

According to Terence W Barrett, There are 6 effects that go beyond Maxwell theory. They are listed as the following: (1) The Aharonov-Bohm and Altshuler-Aronov-Spivak effects. (2) The topological phase effects of Berry, Aharonov, Anandan, Pancharatnam, Chiao and Wu. (3) The Josephson Effect, (4) the quantum Hall Effect, (5) the De Haas-Van alphen effect, and (6) the Sagnac effect.

Reference: Terence W Barrett, Topological Foundations of Electromagnetism, Volume 26, Page 3-4, World Scientific Series in Contemporary Chemical Physics, 2008.