Euclidean scalar source functional and Laplacian sign ($Z[J]=\mathcal{N}\exp [-\frac{1}{2}JGJ],\Delta G=\delta$)

For $S[\varphi ]=\frac{1}{2}\int (\partial \varphi {)}^2$ and a source appearing as $-\int J\varphi$ in the exponent, the positive quadratic operator is $A=-\Delta$. Completing the square gives $Z[J]=\mathcal{N}\exp [\frac{1}{2}J{A}^{-1}J]$. 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.