# Quantum fields without Wick rotation

## TL;DR

This paper presents a method for calculating one-loop effective actions in Lorentzian spacetimes that aligns with Euclidean methods, using steepest descent and renormalization group techniques, applicable to quantum gravity.

## Contribution

It introduces a straightforward approach to compute effective actions in Lorentzian spacetimes that matches Euclidean results and is independent of coarse-graining choices.

## Key findings

- Lorentzian and Euclidean calculations agree for static spacetimes.
- The method is consistent with analytic continuation techniques.
- Effective actions are independent of coarse-graining definitions.

## Abstract

We discuss the calculation of one-loop effective actions in Lorentzian spacetimes, based on a very simple application of the method of steepest descent to the integral over the field. We show that for static spacetimes this procedure agrees with the analytic continuation of Euclidean calculations. We also discuss how to calculate the effective action by integrating a renormalization group equation. We show that the result is independent of arbitrary choices in the definition of the coarse-graining and we see again that the Lorentzian and Euclidean calculations agree. When applied to quantum gravity on static backgrounds, our procedure is equivalent to analytically continuing time and the integral over the conformal factor.