Proca equation

A massive vector field coupled to a conserved current obeysThe mass term breaks electromagnetic gauge invariance.

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.

For a free massive Proca field in the mostly-plus Minkowski metric, the covariant inverse of the quadratic action and the canonical time-ordered product of vector potentials differ locally. With Fourier transform $e^{iq\cdot (x-y)}$, the covariant Proca propagator is $-i({\eta }_{ab}+q_a{q}_b/m^2)/(q^2+m^2-i0)$, while the literal canonical two-point correlation function subtracts $i{\delta }_a^0{\delta }_b^0/m^2$ in momentum space. The time component is nondynamical; its physical-mode numerator is $|\mathbf{q}{|}^2/m^2$, rather than $-1+(q^0{)}^2/m^2$. The difference is an instantaneous Dirac delta function, not another propagating state. Both expressions agree away from coincidence. This distinction matters when identifying a covariant Green function with a canonical correlator in a constrained field theory.

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.

Taking a divergence and using antisymmetry of $F^{\mu \nu }$ and current conservation gives $m^2{\partial }_{\mu }{A}^{\mu }=0$. For nonzero mass, the Lorenz condition follows from the field equation rather than from a gauge choice.

The static Green function of ${\nabla }^2-m^2$ in three dimensions is proportional to $e^{-mr}/r$. It describes an interaction screened beyond range $m^{-1}$ and tends to the Coulomb potential as $m\to 0$.