Vacuum source functional of a quantum oscillator ($Z[J]/Z[0]=\exp (-iJGJ/2)$)

Completing the square in $S[q]=\frac{1}{2}qKq+Jq$ with $K=-{\partial }_t^2-m^2+iϵ$ gives the displayed source factor, where $G=K^{-1}$. In the frequency convention $e^{i\omega t}$, $\tilde{G}=({\omega }^2-m^2+iϵ{)}^{-1}$ and $G(t)=-i{e}^{-im|t|}/(2m)$ for $m>0$. The physical time-ordered oscillator quantum field theory propagator is $iG$. Sources are first taken as test functions so the pairings are defined.