Maxwell's equations govern the dynamics of electromagnetic field produced by moving charges (particle). What will be these equations for antiparticle? for antimagnetic charge?
One needs to apply the charge conjugation operator to Maxwell equations.
Charge conjugation is the operation that replaces all particles by their antiparticles in
the same state, so that momenta and positions are unchanged. Since
the charge and magnetic moment of every particle are reversed,
the electromagnetic interaction remains invariant under this operation.
Under charge conjugation the electromagnetic 4-vector transform as:
A^{\mu} \to -A^{\mu}
and therefore the electromagnetic field tensor transform as:
F^{\mu\nu} to -F^{\mu\nu}
In conclusion, the QED remains invariant under this operation
Maxwell's equations describe electromagnetism as a classical phenomenon. This theory is based on the principle of action at a distance. In other words if you have a positive charge and I have a negative charge, they will start interacting at no time at all. However, according to the relativity theory no signal can run faster than the speed of light and hence there has to be a finite (however small) time lag between cause and effect. Therefore in a theory one has to include a mediator which can connect two charges at most at the speed of light. In the theory of Quantum Electrodynamics (QED) photon is precisely a mediator of electromagnetic interaction or in other words a gauge boson. We can restate your question as, is QED also a theory of positron ?
The QED lagrangian density is:
L=FμνFμν+ ψ̅ (iγμ∂μ-m) ψ - eψ̅ γμAμψ,
using properties of charge conjugation operator (see any textbook)
L--> L' = FμνFμν+ ψ̅c (iγμ∂μ -m) ψc + eψ̅c γμAμψc
clearly, L' is not equal to L. That means the Lagrangian density becomes different under the action of charge conjugation operator. But notice that if you also change e --> -e then the two Lagrangian densities, namely L and L' agree. Now reinterpret that you are working with a particle of same mass and opposite charge. Then this form of Lagrangian density (QED) also provides a theory of positron.
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