The quadratic momentum-space operator is inverted by the <Proca propagator>. With the <Feynman propagator> boundary prescription,
$$
\Delta_{\mu\nu}(x-y)
=\int\frac{d^4p}{(2\pi)^4}
\frac{-i}{p^2-m^2+i\epsilon}
\left(\eta_{\mu\nu}-\frac{p_\mu p_\nu}{m^2}\right)
e^{-ip\cdot(x-y)}.
$$
Multiplication by the Fourier-space Proca operator gives $i\delta_\mu{}^\nu$, with the overall sign determined by the convention for the Green function.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / d / Solution

{c}

For metric signature $(+---)$, the momentum-space Feynman propagator of a Proca field is
$$
\widetilde\Delta_{\mu\nu}(p)=\frac{-i}{p^2-m^2+i\epsilon}\left(\eta_{\mu\nu}-\frac{p_\mu p_\nu}{m^2}\right).
$$
The longitudinal numerator term inverts the massive vector kinetic operator without gauge fixing.


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 Incoming links: Proca propagator