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propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum.
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Bessel function representation of spacelike KG propagator
This can be seen by partial integration
$$\frac{\partial}{\partial \rho}\sqrt{\rho^2-m^2}=\frac{\rho}{\sqrt{\rho^2-m^2}}$$
OP edit: More explicitly, we use this to write $(3)$ as
\begin{align}
D( …