= Planck-constant loop counting with source normalization
{c}
{title2=$\hbar^{L-1}$}
In a <scalar field path integral> with weight $\exp[i(S+\int J\phi)/\hbar]$, each internal <propagator> contributes $\hbar$, each interaction vertex contributes $\hbar^{-1}$, and each source insertion contributes $\hbar^{-1}$. A connected graph with $I$ internal lines and $V$ interaction vertices has <loop order> $L=I-V+1$. Its amputated amplitude, or its coefficient in $\log Z[J]$ with the source insertions included, is proportional to $\hbar^{I-V}=\hbar^{L-1}$. An ordinary unamputated connected $n$-field <correlation function> instead has the additional $n$ external propagator factors and scales as $\hbar^{L+n-1}$. Multiplication by $-i\hbar$ to form the <connected generating functional> shifts the source-kernel counting to $\hbar^L$. These are compatible normalization conventions.
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