Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 43 / 3 / 1 / Solution

Measure the statistical Hamiltonian in units of $k_{B}T$ so that the weight is $e^{-H_0}$, and take $\alpha >0$, $m^2>0$ for convergence. Use the Fourier transform conventionThe last identity is the reality condition, so modes at $p$ and $-p$ are not independent complex fields. Orthogonality of the Fourier exponentials gives $\int d^{D}x{\varphi }^2={\int }_p|\tilde{\varphi }(p){|}^2$, while a derivative multiplies a Fourier mode by $ip$. HenceThe kernel is positive for the stated parameters. Keeping an explicit inverse temperature instead would replace this kernel in the statistical weight by ${\beta }_{\mathrm{th}}\tilde{\Delta }$; the following source convention absorbs that factor.