The classical potential associated with the magnetic vector potential A: Aharonov-Bohm effect is classical

In the previous blog post we had the configuration where an electron is passing a tube of magnetic flux from the outside. Let us calculate if we can explain the vector potential A by the effect that the magnetic field of the electron has on the energy of the magnetic field B in the tube.

                   B

             ×    ×    ×               ^

             ×    ×    ×               |   v

             ×    ×    ×               e-

                             r

https://en.wikipedia.org/wiki/Magnetic_vector_potential

Suppose that B is constant over an area S. The magnetic flux is

       Φ = S B,

and the vector potential at a distance r is

       A = Φ / (2 π r).

https://en.wikipedia.org/wiki/Hamiltonian_mechanics

The hamiltonian H is:

Suppose that r is large. Then q A² is negligible. The contribution of A to the hamiltonian is

       W = -1 / m * m v * q A

            = -q v A

            = -1 / (2 π) * v q S B / r.

Is the value equal to the energy which the magnetic field of the electron adds to the field B?

The energy density of the field B is

       1/2 * 1 / μ₀ * B².

The magnetic field of the electron at the distance r is

       B' = 1 / c² * v × E

           = 1 / c² * v * 1 / (4 π ε₀) * q / r²

           = μ₀ / (4 π) * v q /  r².

If |B'| is much smaller than |B|, then B' changes the energy density by

       -1 / μ₀ * B B'.

In how big a volume does the change of energy density take place? The volume might be 2 r S.

The energy change in the magnetic field in the tube is

      W' = -1 / (2 π) * v q S B / r

We see that W = W'.

We get an intuitive definition for the vector potential (at least in this case):

The "potential"

       -q v A

is the energy which the magnetic field of a moving small charge q adds to the magnetic field B which generates the vector potential A.

That is, if we are moving the charge at a velocity v, we must do that much work against the magnetic field B to bring q to a location where the vector potential is A.

The Aharonov-Bohm effect

The Aharonov-Bohm effect is due to the classical "potential" -q v A. The effect is not quantum mechanical. Of course, the interference pattern on the screen is quantum mechanical but that is not the essence of the effect. The essence is that a magnetic field B, which the electron does not touch, still affects the path of the electron.

https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect

The Wikipedia article claims that the effect proves that the vector potential A is "physical" in quantum mechanics.

But we showed that quantum mechanics is not relevant here. The hamiltonian of a classical charge does depend on A, and consequently, A does affect the path of a classical charge just like a quantum particle.

The excerpt above is from Wikipedia. Let us comment on them.

1. We could claim that the magnetic field B is "physical", as well as the field of the electron. There is no need to claim that the vector potential A is "physical". 

2. The same holds for action principles. There is no need to claim that they are "fundamental".

3. The field of the electron spans the whole universe. To find out how the electron moves, we cannot restrict ourselves to the immediate vicinity of the electron.