The quadratic <Proca action> is
$$
\mathcal L=-\frac14X_{\mu\nu}X^{\mu\nu}+\frac12M_X^2X_\mu X^\mu,
\qquad
X_{\mu\nu}=\partial_\mu X_\nu-\partial_\nu X_\mu.
$$
Its momentum-space kinetic operator is
$$
K_{\mu\nu}=-(p^2-M_X^2)\eta_{\mu\nu}+p_\mu p_\nu.
$$
Inverting it on transverse and longitudinal projectors gives the <Proca propagator>
$$
\boxed{D_{\mu\nu}(p)=\frac{-i}{p^2-M_X^2+i\epsilon}
\left(\eta_{\mu\nu}-\frac{p_\mu p_\nu}{M_X^2}\right)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-305/1/b/solution