World-Line Path Integral for the Propagator Expressed as an Ordinary Integral: Concept and Applications

The (Feynman) propagator $G(x_2,x_1)$ encodes the entire dynamics of a massive, free scalar field propagating in an arbitrary curved spacetime. usual procedures for computing -- either as time ordered correlator or from partition function defined through path integral requires introduction $\phi(x)$ and its action functional $A[\phi(x)]$. An alternative, more geometrical, procedure is to define terms world-line which only uses curves, $x^i(s)$, on manifold. I show how can be reinterpreted ordinary by introducing concept effective number quantum paths given length. Several manipulations become algebraically tractable this approach. In particular, derive explicit expression $G_{\rm QG}(x_2,x_1)$, incorporates structure spacetime zero-point-length, standard std}(x_2,x_1)$, This approach also helps clarify interplay between amplitude measure determining form propagator. illustrated with several examples.