# Lorentzian Quantum Cosmology

## TL;DR

This paper advocates for using Lorentzian path integrals in quantum cosmology, employing complex contour deformation and Picard-Lefschetz theory to identify relevant saddle points, leading to a new semiclassical result contrasting the Hartle-Hawking proposal.

## Contribution

It introduces a Lorentzian path integral approach with complex contour deformation for quantum gravity, providing a clearer method to select relevant saddle points in quantum cosmology.

## Key findings

- Lorentzian path integral is more suitable than Euclidean for quantum cosmology.
- Revealed the dominant saddle point yields a semiclassical factor inverse to Hartle-Hawking.
- Applied Picard-Lefschetz theory to determine relevant saddle points unambiguously.

## Abstract

We argue that the Lorentzian path integral is a better starting point for quantum cosmology than the Euclidean version. In particular, we revisit the mini-superspace calculation of the Feynman path integral for quantum gravity with a positive cosmological constant. Instead of rotating to Euclidean time, we deform the contour of integration over metrics into the complex plane, exploiting Picard-Lefschetz theory to transform the path integral from a conditionally convergent integral into an absolutely convergent one. We show that this procedure unambiguously determines which semiclassical saddle point solutions are relevant to the quantum mechanical amplitude. Imposing "no-boundary" initial conditions, i.e., restricting attention to regular, complex metrics with no initial boundary, we find that the dominant saddle contributes a semiclassical exponential factor which is precisely the {\it inverse} of the famous Hartle-Hawking result.