Notes on Quantum Effective Actions

TL;DR

The paper develops a rigorous, Lorentzian-focused framework for the quantum effective action in perturbative QFT, including curved spacetime and gravity, highlighting heat-kernel regularization and the backreaction on kinetic terms. It also clarifies Euclidean convexity issues by examining dependence on initial and final states, and introduces the Vilkovisky–DeWitt construction to ensure gauge- and field-redefinition invariance of the effective action. A covariant perturbative expansion is presented, together with a heat-kernel regularization scheme that extends naturally to curved space via world-function techniques. Finally, the Wilsonian viewpoint is integrated into the DWV framework, showing how integrating out high-energy modes yields a scale-dependent action $S_{\Lambda'}[\phi_c]$ with explicit regulator structure, while remaining consistent in the $\Lambda' \to \Lambda$ limit. Altogether, the work provides a cohesive, covariant treatment linking 1PI and Wilsonian pictures in gauge theories and gravity with a robust gauge-fixing independent effective action.

Abstract

We first note that, at least in perturbation theory, there is a well-defined (subject to regularization) Lorentzian definition of the quantum effective action in both flat and curved space including (perturbative) gravity. The advantage of the latter is that we do not need to deal with the conformal factor problems of Euclidean quantum gravity. We then make some remarks on the Euclidean version (in flat space) and convexity and resolve a puzzle that highlights the importance of keeping the initial and final states in the functional integral. Next we discuss the gauge invariant effective action of Vilkovisky and DeWitt and show its gauge fixing independence. We conclude with the expression for the Wilsonian effective action in this framework.