 Normalization of a complex scalar propagator

For positive kinetic coefficient $\zeta$ in ${\mathcal{L}}_0=\zeta ({\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi )$, the canonical momentum is ${\pi }_{\varphi }=\zeta {\dot{\varphi }}^{\ast }$. Positive-energy oscillator modes with unit oscillator commutators have amplitude $1/\sqrt{2\zeta E_{\mathbf{p}}}$. The mixed time-ordered propagator is $C={\zeta }^{-1}{D}_F$, with $D_F$ the ordinary scalar propagator. This also follows by inverting the quadratic form in the path integral. Rescaling $\varphi$ changes interaction coefficients as well as external correlation-function factors.