What would Maxwell's equations look like if photons had only a single helicity?

Question

There are two types of photons, positive and negative helicity photons. What would Maxwell's equations look like say if there were only negative helicity photons? It would be interesting to see this in any of the forms of Maxwell's equations; e.g. in the form of the field strength tensor $F^{\mu\nu}$, or in the form of the gauge field $A^{\mu}$, or in the form of the electric and magnetic fields $\vec{E},\vec{B}$.

Answer 1

One possible interpretation of this question is that we add a new law, like $\nabla B = 0$, prohibiting electromagnetic waves of one polarization. Any field can be decomposed into a sum of circularly polarized fields at various wavevectors. In Fourier space, I think the left circularly polarized field satisfies $$i\alpha\vec{E} = \vec{B}$$ while the right circularly polarized field satisfies $$-i\alpha\vec{E} = \vec{B}$$ where $ \alpha = \mathrm{sgn}(\vec{k} \cdot (\vec{E} \times \vec{B})) $ (signs might be flipped.) It's not obvious to me what real-space equation corresponds to this - if you can figure it out, let me know. But to eliminate right-circularly-polarized photons, we could then add $\vec{B} + i\alpha\vec{E} = 0$ (or the real-space equivalent) as our fifth Maxwell's equation.

Notice that this set of equations will in general only have consistent solutions in vacuum. In the presence of charges, the usual four Maxwell's equations specify a unique solution (up to vacuum fields). If that unique solution doesn't already satisfy our added equation, then no solution of all five exists.

In vacuum, this new law only changes which initial conditions are allowed, not time evolution. So the physics here is a strict subset of the physics in the real world.

Edit: Actually, I think in the presence of charges it would make sense to take a solution of the usual four equations and decompose into a sum of parts, one with $\vec{B} + i\alpha\vec{E} = 0 $ and the other with $\vec{B} - i\alpha\vec{E} = 0 $. By taking only one part or the other, we have a solution to "right-handed Maxwell's equations." But is there a way to specify this solution in terms of a set of differential equations instead?

Answer 2

We have a way to study the helicity of photons. For this purpose it is helpful that we obtain polarized electromagnetic radiation when accelerating electrons in an antenna rod. Means, all emitted photons during the acceleration of the electrons in one direction on the surface of the rod are aligned with their electric field component (directed parallel to the rod and after a half period of the wave generator directed “anti-parallel” to the rod).

Now there are two types of receiving antennas. In an antenna rod, the electric field of the radio wave couples with the radiation. In a ring antenna, the magnetic field is coupled in.

The conclusion is that electrons emit photons, all with the same helicity. But what about photons from the other charged particles? What about radiation from anti-proton? What about protons and positrons?

The thing is that there is no research. My guess due to symmetry is that electrons and and anti-protons emit photons of the same helicity; protons and positrons of the other. To proof it a possible way would be to measure the Lorentz force for these particles, moving in an external magnetic field.