Particle polarization ($ϵ$)

A particle polarization specifies a physical spin state at fixed four-momentum. Massive spin states transform under the rotation little group $SO(D-1)$; finite-spin massless states transform under its transverse rotation part $SO(D-2)$. A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ such states, while a two-form gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$. Polarization components must be counted after field equations and any gauge equivalence are imposed.

For an on-shell massive vector with $k^{\mu }=(E,|\mathbf{k}|\hat{\mathbf{k}})$ in Minkowski metric signature $+---$, the displayed vector satisfies $k\cdot {ϵ}_L=0$ and ${ϵ}_L^2=-1$. It specifies the longitudinal one of the three physical polarization states. It is not a replacement for a final polarization sum when other states are unobserved.