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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-301/1/d/solution