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The vector potential revisited |
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| Authors: | Toshimi Adachi Shigeru Sasabe Toshio Inagaki Masao Ozaki |
| Affiliation: | 1. Tokyo Metropolitan Institute of Technology Toshimi Adachi graduated in 1962 from the Department of Science Research, Tokyo Metropolitan University. He also has a Ph.D. In 1968, he became a Professor at Tokyo Metropolitan College of Technology, and he is currently a Professor at Tokyo Metropolitan Institute of Technology. His main area is the theory of particle physics. He is a member of the Physical Society of Japan.;2. Tokyo Metropolitan Institute of Technology Shigeru Sasabe obtained a PhD. in 1976 from the Department of Science Research, Kyoto University. He is currently an Associate Professor at Tokyo Metropolitan Institute of Technology. He is engaged in research on quantum electrodynamics and physical properties of electrons. He is a member of the Physical Society of Japan.;3. Nippon Veterinary and Animal Science University Toshio Inagaki obtained a Ph.D. in 1973 from the Department of Science Research, Tokyo Metropolitan University. In 1989 he became an Associate Professor at Nippon Veterinary and Animal Science University. His main area is theoretical physics. He is a member of the Physical Society of Japan.;4. Institute of Industrial Science, University of Tokyo Masao Ozaki graduated in 1976 from the Department of Pure and Applied Sciences, Fac. Education, University of Tokyo. In 1982, he left the Ph.D. program in Geophysics, Fac. Science, University of Tokyo. He has been with the Institute of Industrial Science, University of Tokyo, since 1984. He is engaged in research on optical heterodyne microscope and nonlinear wave propagation in optical fibers. He is a member of the Physical Society of Japan and the Inst. Electr. Inform. Comm. Eng., Japan. |
| 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. |
| Keywords: | Vector potential Aharonov-Bohm effect Helmholz theorem gauge transformations |