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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( …
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