The vector potential revisited

Abstract The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts: 1) the so‐called transverse part represented by a divergenceless vector; and 2) the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when the Helmholtz theorem is used are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, the Aharonov‐Bohm effect is chosen here. As for the Aharonov‐Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z‐axis. It is shown that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is determined uniquely by the magnetic field and does not alter by a gauge transformation. On the other hand, the longitudinal part depends on the choice of special gauge. It is shown that the Aharonov‐Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.