Wick rotation in the lapse, admissible complex metrics, and foliation changing diffeomorphisms

TL;DR

This paper introduces a novel Wick rotation in the lapse function for manifolds with foliation, leading to admissible complex metrics with well-behaved spectra, and explores implications for scalar field theories and spacetime backgrounds.

Contribution

It defines a new Wick rotation method in the lapse, ensuring admissible complex metrics and analyzing the spectral properties of the resulting operators.

Findings

The Wick rotation in the lapse produces admissible complex metrics.

The Hessian spectrum lies within a wedge of the upper complex half plane.

Differences in propagators are shown compared to traditional Feynman prescriptions.

Abstract

A Wick rotation in the lapse (not in time) is introduced that interpolates between Riemannian and Lorentzian metrics on real manifolds admitting a codimension-one foliation. The definition refers to a fiducial foliation but covariance under foliation changing diffeomorphisms can be rendered explicit in a reformulation as a rank one perturbation. Applied to scalar field theories a Lorentzian signature action develops a positive imaginary part thereby identifying the underlying complex metric as ``admissible''. This admissibility is ensured in non-fiducial foliations in technically distinct ways also for the variation with respect to the metric and for the Hessian. The Hessian of the Wick rotated action is a complex combination of a generalized Laplacian and a d'Alembertian, which is shown to have spectrum contained in a wedge of the upper complex half plane. Specialized to near Minkowski space the induced propagator differs from the one with the Feynman $i\epsilon$ prescription and on Friedmann-Lema\^{i}tre backgrounds the difference to a Wick rotation in time is illustrated.