A conformally invariant derivation of average electromagnetic helicity

The average helicity of a given electromagnetic field measures the difference between the number of left- and right-handed photons contained in the field. In here, the average helicity is derived using the conformally-invariant inner-product for Maxwell fields. Several equivalent integral expressions in momentum space, in $(\mathbf{r},t)$ space, and in the time-harmonic $(\mathbf{r},\omega)$ space are obtained, featuring Riemann-Silberstein-like fields and potentials. The time-harmonic expressions can be directly evaluated using the outputs of common numerical solvers of Maxwell equations. The results are shown to be equivalent to the well-known volume integral for the average helicity, featuring the electric and magnetic fields and potentials.