During the treatment of the gauge transformations for showing the A(r,t) in the context of Goeppert-Mayer gauge; it is expressed as the difference between the two transverse gauges vector potentials for the light-atom interaction picture, and when we go to derive the Hamiltonian for longer wavelength of the radiation field compared to the average size of the atom, we invoke the long-wavelength approximation which kills the GM gauge vector potential, i.e. the two potentials are equal. Why is that? and how can i derive explicitly this equality for long wavelengths?
Maxwell defined the vector potential as electromagnetic momentum per unit charge. Momentum, inexorably and unequivocally, relates to inertia, which is an electromagnetic phenomenon (so, dependent of the vector potential). Therefore, the vector potential has a close connection with momentum, which from the de Broglie relation, p=h/λ, vanishes in the long-wavelength limit.
The definition of the vector potential as an electromagnetic momentum is in the reference:
[J.C. Maxwell. A dynamical Theory of the Electromagnetic Field. Philosophical Transactions of the Royal Society of London, 1865, 155, 459-512.]
Equation 29
Maxwell did not use vector notation. This notation only appeared after Heaviside, which treated the vector potential as a mere mathematical entity (ignoring its momentum content). The real meaning only resumed in the context of quantum mechanics (as in the reference you send me). See also the Aharonov-Bohm effect.
Fernando, as Remi wrote, we are here more interested in the mechanical-canonical representation of the momentum in Lorentz regime. I didn't get how you really connected the idea of vanishing ΔA which is a difference between the classical em-field potential and the charged atom potential by using the De Broglie relation.
In fact, I don’t have a specific answer to your question. Even so, I suggest read the reference
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