If we plug the "elementary" propagator (which also known as the Wightman function) \begin{align}D(t',\vec{x}', t,\vec{x}) \equiv \langle{0| \phi(t, \vec x) \phi(t', \vec x') |0} \rangle&= \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{i k^\mu (x_\mu' -x_\mu) } \end{align} into the Klein-Gordon equation we find zero. In other words, it's a kernel.
In contrast, the Feynman propagator $$ D_F (t',\vec{x}', t,\vec{x}) \equiv \langle{0| T \phi(t, \vec x) \phi(t', \vec x') |0}\rangle$$ is a Green's function. This means we find $\delta(x_\mu-y_\mu)$ if we plug it into the Klein-Gordon equation.
Is there any way to interpret this fact in physical terms?
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