= Euclidean scalar source functional and Laplacian sign
{c}
{title2=$Z[J]=\mathcal N\exp[-\tfrac12JGJ],\quad\Delta G=\delta$}
For $S[\phi]=\tfrac12\int(\partial\phi)^2$ and a source appearing as $-\int J\phi$ in the exponent, the positive quadratic operator is $A=-\Delta$. Completing the square gives $Z[J]=\mathcal N\exp[\tfrac12JA^{-1}J]$. If one writes it with a negative quadratic source exponent, its kernel must be $G=-A^{-1}=\Delta^{-1}$. In the plane $G(z,w)=(2\pi)^{-1}\log(|z-w|/L)$, up to harmonic additions and infrared conventions. The physical covariance is its negative. Zero modes are regulated or separated; oscillatory sources in a string path integral produce <momentum conservation> from the constant mode.
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