Helicity ($\lambda$)

Helicity is the projection of angular momentum onto the momentum direction. For a finite-helicity massless particle, the Pauli-Lubanski vector satisfies $W^{\mu }=\lambda P^{\mu }$.

A physical photon has two transverse helicity states. Circular polarization distinguishes their populations; a real photon has no longitudinal helicity-zero polarization.

For a relativistic neutrino, the normalized helicity is approximately $-1$ in ordinary weak production. For a massive particle, helicity and fermion chirality differ; suppressed opposite-helicity amplitudes and frame dependence prevent promoting the massless selection rule to an exact statement about arbitrary boosts.

For $\mathbf{k}=k\hat{z}$, these complex vectors form an orthonormal basis of the transverse plane using the Hermitian inner product. They have opposite phases under rotations about $\hat{z}$, so they describe the two helicity components. Since $\mathbf{e}\cdot m^{(\pm )}=\sin \theta e^{\pm i\phi }/\sqrt{2}$, transverse-vector sources carry only the angular sectors $m=\pm 1$ of spherical harmonics. Streaming by $ik\cos \theta$ preserves $m$ and couples neighboring angular degrees.