= Derivative jump of a free scalar time-ordered two-point function
{title2=$g'_E(0^+)-g'_E(0^-)=-i$}
At fixed spatial <momentum> with mode <energy> $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second <distributional derivative> has a $-i\delta(t)$ term, while away from zero $g''_E+E^2g_E=0$. Therefore $(\partial_t^2+E^2)g_E=-i\delta(t)$ and, after the spatial <Fourier transform>, $(\Box+m^2)\Delta_F=-i\delta^4$. This checks the propagator numerator, contour signs and equal-time <canonical commutation relations> independently of residue calculation.
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