Is it possible to describe Electromagnetism in a way such that the electric field is defined as the curl of a vector potential and the magnetic field is the grad of a scalar potential? Will such a theory be consistent?
Dear Carringtone,
What is new is that you introduce a scalar potential for magnetic and this will give you a Laplacian of such potential equal zero, which has solution zero without taking into account constants. Thus is is not a new solution for electrodynamics.
Since there are electric monopoles and there is no magnetic monopoles (as yet), we call components of the electromagnetic with the scalar field the electric field and components with no potential the magnetic field. Otherwise the electromagnetic field is quite symmetrical. An magnetic monopole would relieve the magnetic field from being relativistic part.
Dear Christian,
There are many different kind of magnetic monopoles: Dirac (half a string), Schwinger (full string) or t'Hooft non-Abelian, Wu-Yang, etc...
All of them assume the non-existance of free mangetic poles is not possible as Maxwell equations says, the only thing that they add is that the manifold associated to electrodynamics is not simply connected and it has a non-trivial topology.
Thus even if the magnetic monopole could find experimentally (as Cabrera claimed a long time ago), the scalar potential would be not necessary and meaningless in classical or quantum electrodynamics.
Dear Christian,
Many years ago I worked on the symmetry electricity-magnetism within Maxwell equations. Mathematically is known as duallity rotations which depend only on the motion equations but not for the action. Thus you have that they are symmetries of the Hamiltonian but not to the action, i.e. there is not a Legendre transformation which could associated Noether currents with motion. This is one issue that in simple form and within an exterior algebra of differential forms we were developing in my book "Geometry of Electromagnetic Systems" of Clarendon Press.
Thus I think that I know this subject although I have left it many years ago. Curiosly Heaviside found this inner symmetry of electrodynamics and he tried to introduce magnetic sources, although he failed but sometimes this issue is confused with the magnetic monopole when both things are absolutly different. Magnetic monopole was always associated with quantum mechanics and no with classical electrodynamics, and they were introduced (1931) by Dirac for explaining the quantization of the electric charge so exactly in such a form that matter is neutral.
PS I like your paper that I have read it some time ago. It is very clear and very well written.
@Baldomir Thank you very much for your answer I however didn't require that it be new, only that it be consistent.
@Baldomir @Christian @Baumgarten If we are considering the classical electromagnetic field far from any sources, can we sidestep the monopole problem?
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