We cannot reduce complex electric phenomena to the local field

Suppose that we have an electric point charge. We move an infinitesimal test charge in its field.

The inertia of the test charge depends on the amount of energy that the test charge displaces in the combined field of the two charges.

Since the field of a single point charge is so simple, a spherically symmetric field, it may appear that we can determine the inertia based on just the value of the local electric field vector at the test charge. However, that does not work in the general case.

Consider a system of two charges. If we move a test charge, its inertia depends on the energy that is displaced in the individual fields of the two charges. Clearly, we cannot know the displacement from the local field vector at the test charge. We must know the field from a large volume.

Our Minkowski & newtonian model explains the metric of spacetime from the inertia which a test mass has in the newtonian gravity field. It seems to work ok for the Schwarzschild metric around a spherical mass.

Just like in the case of the electric field, we cannot determine the inertia from the local gravitational field at the test mass. We must know the field in a large volume.

Now we face a question both for electromagnetism and for gravity: how do we determine the inertia of a test charge or mass inside a wave? In the case of gravity, the inertia determines the metric. In the case of electromagnetism, the inertia is a phenomenon which has not been measured yet.

The inertia may be additive if we have several charges around the test charge.

Gravitational waves in the linearized Einstein equations are additive.

However, if the local field does not determine the inertia, how do we determine the inertia inside a wave? It looks like that we have to assume an individual field for each charge. That is, the field of each charge extends to infinity, and we cannot reduce a set of fields to the sum of the fields. Then there would not exist such a thing as "the electromagnetic field" in a spatial volume. It would always be a collection of separate fields for each charge in the universe.

The notion of a separate field for each charge reminds us again of the Wheeler-Feynman absorber theory. A photon is always associated with the field of a single elementary charge.

The Aharonov-Bohm effect and the classical limit

Question. Is the Aharonov-Bohm effect a consequence of the "electromagnetic fields cannot be summed" principle? Even though the magnetic field outside a solenoid is negligible, the zero field still affects the phase of an electron.

If the answer to the Question is yes, then the Aharonov-Bohm effect is classical - not a quantum effect.

         e- ● -------->  O  solenoid

When an electron approaches a solenoid, it acquires inertia relative to the electrons on the near side of the solenoid, as well as inertia relative to electrons on the far side. The combined result is that the electron acquires angular momentum and tends to start rotating along the electrons in the solenoid. It is like frame dragging in general relativity.

However, the Aharonov-Bohm effect depends on the sign of the charge carriers in the solenoid. Frame dragging probably is independent of the sign.

The Aharonov-Bohm effect is a consequence of the "minimal coupling" in the Dirac equation. Frame dragging seems to be a multipole effect, and is probably different from Aharonov-Bohm.

The Aharonov-Bohm effect may be due to the following very simple classical effect.

                       B

                 ×    ×    ×                 ^

                 ×    ×    ×             ×  |  •

                 ×    ×    ×                 e-

The crosses at B denote a magnetic field whose vector points inside the screen. The cross and the dot close to the electron denote its magnetic field.

If the electron flies past the field on the right, then the summed magnetic field of the electron and B has more energy than if the electron flies past on the left. Thus, there is a higher potential for the different paths. We have to calculate if the potential difference explains the Aharonov-Bohm effect.

The diagram above is yet another example where one cannot determine the effect of an electromagnetic field from its local value at the electron. One must consider a large volume.

People sometimes claim that classically, the electron in the diagram feels no force and moves along a straight path, while in quantum mechanics there is a phase shift which is due to the nonzero vector potential.

Let us think about the classical limit. If we increase the charge and the mass by some factor C, then in quantum mechanics, the path of the electron stays qualitatively the same. In the case of a tunneling perticle, the path is drastically changed by increasing the required energy, but for a particle under a moderate force field there is little change.

We conclude that at the classical limit, the charge cannot move along a straight path. A classical particle must feel the vector potential just like a quantum particle does.