Proca field

A Proca field is a massive spin-one vector field. Its field equation enforces transversality and leaves three physical polarizations.

For metric signature $(+---)$, the momentum-space Feynman propagator of a Proca field isThe longitudinal numerator term inverts the massive vector kinetic operator without gauge fixing.

A vector field is transverse when ${\partial }_{\mu }{V}^{\mu }=0$ in spacetime, or $\mathbf{k}\cdot \mathbf{V}=0$ for a spatial Fourier mode.