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

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-Lemaître backgrounds the difference to a Wick rotation in time is illustrated.

arXiv comment28 pages. The original paper has been split into two parts. This part is to be published in Classical and Quantum Gravity. The other part, entitled `Analytic semigroups approaching a Schrödinger group on real foliated metric manifolds', has been published in the Journal of Functional Analysis 289 (2025) 110898; it will also be posted as a separate arXiv entry