Source differentiation inserts time-ordered field operators ($G_r={i^{-r}{\delta }^{r}Z/\delta J(x_1)\cdots \delta J(x_r)|}_{J=0}$)

A functional derivative of the source term in a Minkowski scalar field path integral inserts $i\varphi (x)$. Time-slicing identifies the resulting product with the time-ordered product of field operators. Dividing by the zero-source vacuum functional removes disconnected vacuum diagrams. The normalized formula has one factor $1/i$ for every insertion.