Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / e / Solution

Use ${\partial }_x\mapsto iq$ under the Fourier transform and the reality conditions $\varphi (-q)=\varphi (q{)}^{\ast }$ and $p(-q)=p(q{)}^{\ast }$. With $\Psi =(\varphi ,p)$, the quadratic free energy iswhereThe opposite imaginary off-diagonal entries make $G(q)$ a Hermitian matrix. Reversing the Fourier-sign convention reverses both of those signs without changing any correlator.