Vector Potential Revisited

The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts; the one is the so-called transverse part represented by a divergenceless vector and the other 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 we use the Helmholtz theorem are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, we choose here the Aharonov-Bohm effect. 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. We show 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 uniquely determined 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. We show 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.