= Vacuum source functional of a quantum oscillator
{title2=$Z[J]/Z[0]=\exp(-iJGJ/2)$}
<Completing the square> in $S[q]=\tfrac12qKq+Jq$ with $K=-\partial_t^2-m^2+i\epsilon$ gives the displayed source factor, where $G=K^{-1}$. In the frequency convention $e^{i\omega t}$, $\widetilde G=(\omega^2-m^2+i\epsilon)^{-1}$ and $G(t)=-ie^{-im|t|}/(2m)$ 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.
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