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Maxwell Electrodynamics in Terms of Physical Potentials

Indian Association for the Cultivation of Science, Kolkata 700032, India
Department of Physics, Presidency University, Kolkata 700032, India
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 915;
Received: 11 June 2019 / Revised: 2 July 2019 / Accepted: 12 July 2019 / Published: 14 July 2019
(This article belongs to the Special Issue Symmetry in Electromagnetism)
PDF [267 KB, uploaded 19 July 2019]


A fully relativistically covariant and manifestly gauge-invariant formulation of classical Maxwell electrodynamics is presented, purely in terms of gauge-invariant potentials without entailing any gauge fixing. We show that the inhomogeneous equations satisfied by the physical scalar and vector potentials (originally discovered by Maxwell) have the same symmetry as the isometry of Minkowski spacetime, thereby reproducing Einstein’s incipient approach leading to his discovery of special relativity as a spacetime symmetry. To arrive at this conclusion, we show how the Maxwell equations for the potentials follow from stationary electromagnetism by replacing the Laplacian operator with the d’Alembertian operator, while making all variables dependent on space and time. We also establish consistency of these equations by deriving them from the standard Maxwell equations for the field strengths, showing that there is a unique projection operator which projects onto the physical potentials. Properties of the physical potentials are elaborated through their iterative Nöther coupling to a charged scalar field leading to the Abelian Higgs model, and through a sketch of the Aharonov–Bohm effect, where dependence of the Aharonov–Bohm phase on the physical vector potential is highlighted. View Full-Text
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Majumdar, P.; Ray, A. Maxwell Electrodynamics in Terms of Physical Potentials. Symmetry 2019, 11, 915.

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