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In the Dirac equation for a massless fermion, for example, in the Weyl representation, we can split the equation into two separate equations for left-handed and right-handed electrons. In the Weyl represenation, the spinor is split as $\psi=(\psi_L,\psi_R)$

For the Maxwell equation for a massless photon, can we split this into two equations, for each of the two polarizations of light? I'm not sure how one would split the 4-vector potential $A_\mu$ into the two polarization states e.g. $\phi_L$ and $\phi_R$.

Also, is there some operator analagous to the $\frac{1}{2}(1+\gamma^5)$ operator for the Dirac equation that gives us one of the two polarization states?

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Yes. The Riemann-Silberstein form of Maxwell's equations is precisely the analogue of the decomposition of Dirac into its left and right handed parts. See my answer here.

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