The scalar kinetic operator is proportional in momentum space to $k^2(\widetilde c k^2-m^2)$. Its inverse has the partial-fraction decomposition
$$
\frac{i}{k^2(\widetilde c k^2-m^2)}
=-\frac{i}{m^2}\left(
\frac1{k^2}-\frac{\widetilde c}{\widetilde c k^2-m^2}
\right),
$$
up to the overall convention-dependent factor allowed in the question. Thus $\widetilde c=\widetilde a+\widetilde b$. Under the physical relation $\widetilde b=-\widetilde a$, one has $\widetilde c=0$, so the extra massive pole and its ghost residue disappear.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 2 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-301/2/b/ii/solution