Local momentum space: scalar field and gravity

We use the local momentum space technique to obtain an expansion of the Feynman propagators for scalar field and graviton up to first order in the background curvature. The expressions for the propagators are cross-checked with the past literature as well as with the expressions for the traced heat kernel coefficients. The propagators so obtained are used to compute one-loop divergences in the Vilkovisky-Dewitt's (VD's) effective action for a scalar field non-minimally coupled with gravity for an arbitrary spacetime metric background. The VD effective action is then compared with the standard effective action in the limit kappa = 0, where kappa=2/MP in terms of the Planck mass. The comparison yields the important result that taking the limit kappa = 0 after computing the VD effective action is not equivalent to computing the VD effective action for the same theory in the absence of gravity.

Automated summary

This study employs the local momentum space technique to derive the first-order expansion of Feynman propagators for scalar field and graviton in the presence of background curvature. The obtained propagators are validated by comparing them with previous literature and traced heat kernel coefficients. These propagators are then utilized to calculate the one-loop divergences in the Vilkovisky-Dewitt's effective action for a scalar field non-minimally coupled with gravity in an arbitrary spacetime metric. The comparison between the VD effective action and the standard effective action reveals an important result regarding the limit kappa = 0 and the presence of gravity.