Notes on Quantum Effective Actions

TL;DR

This paper discusses the definition and properties of quantum effective actions in Lorentzian and Euclidean frameworks, emphasizing gauge invariance and the Wilsonian approach, with implications for quantum gravity.

Contribution

It clarifies the Lorentzian definition of the quantum effective action, examines gauge invariance of Vilkovisky-DeWitt effective action, and connects to Wilsonian effective action.

Findings

Lorentzian effective action well-defined in perturbation theory

Gauge invariance of Vilkovisky-DeWitt effective action confirmed

Relation to Wilsonian effective action established

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.