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arXiv · 2406.06047

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

Rudrajit Banerjee·Max Niedermaier
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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.

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Rudrajit Banerjee, Max Niedermaier. 2024-06-10. Wick rotation in the lapse, admissible complex metrics, and foliation changing diffeomorphisms. https://arxiv.org/abs/2406.06047

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