Abstract
We explore the properties of non-local effective actions which include gravitational couplings. Non-local functions originally defined in flat space can not be easily generalized to curved space. The problem is made worse by the calculational impossibility of providing closed form expressions in a general metric. The technique of covariant perturbation theory (CPT) has been pioneered by Vilkovisky, Barvinsky and collaborators whereby the effective action is displayed as an expansion in the generalized curvatures similar to the Schwinger-De Witt local expansion. We present an alternative procedure to construct the non-local action which we call non-linear completion. Our approach is in one-to-one correspondence with the more familiar diagrammatic expansion of the effective action. This technique moreover enables us to decide on the appropriate non-local action that generates the QED trace anomaly in 4D. In particular we discuss carefully the curved space generalization of ln □, and show that the anomaly requires both the anomalous logarithm as well as 1/□ term where the latter is related to the Riegert anomaly action.
Highlights
Fields [2,3,4,5,6,7,8,9,10,11,12]
Non-local functions originally defined in flat space can not be generalized to curved space
The technique of covariant perturbation theory (CPT) has been pioneered by Vilkovisky, Barvinsky and collaborators whereby the effective action is displayed as an expansion in the generalized curvatures similar to the Schwinger-De Witt local expansion
Summary
In flat-space the one loop effective action for a photon, obtained by integrating out a massless charged scalar or fermion, has the form. The two related transformations, rescaling the coordinates and rescaling the metric, act differently in the effective action yet both yielding the same anomaly relation We see that both types of non-locality, i.e. the logarithm and the massless pole in eq (2.7) are required by direct calculation as well as by anomaly considerations. Because one is starting out with the F ln ∇2F expression as the covariant form which is second order in the curvature, one needs to add and subtract correction terms in order to reproduce the actual calculated result. In order to match the full result found in the direct one-loop calculation [23], one must add a nonanomalous term that has no relation to the beta function This is different for fermions and scalars and is invariant under both scaling and conformal transformations.
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