= Euclidean effective action
{c}
{title2=$\Gamma_E[\varphi]$}
For $Z_E[J]=\int\mathcal D\phi\,e^{-S_E[\phi]+\int J\phi}$, define $W_E=\log Z_E$, $\varphi=\delta W_E/\delta J$ and the <Legendre transform> $\Gamma_E[\varphi]=\int J\varphi-W_E[J]$. Its functional derivatives generate Euclidean <one-particle irreducible> vertices. A bosonic <Gaussian integral> about a background gives the one-loop term $\tfrac12\operatorname{Tr}\log S_E^{(2)}[\varphi]$. This is the Euclidean version of the <quantum effective action>, with signs fixed by $e^{-S_E}$ rather than $e^{iS}$.
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