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

Rudrajit Banerjee, Max Niedermaier

TL;DR

The authors introduce a lapse-based Wick rotation on foliated manifolds that keeps the manifold’s real atlas while smoothly interpolating between Lorentzian and Euclidean geometries via a phase rotation of the lapse. By formulating the rotated metric as a rank-one perturbation and analyzing foliation-changing diffeomorphisms, they achieve a covariant description of $g^{\theta}$ across equivalent foliations. They prove three notions of admissibility for the complex metric in a minimally coupled scalar theory: damping of the action, positivity of the linear response, and sectoriality of the complex Hessian, with the Hessian spectrum confined to a wedge in the upper half-plane and explicit behavior near Minkowski and Friedmann-Lemaître backgrounds. The work predicts modified propagators compared to the standard $i\epsilon$ prescription and suggests broad applicability to other fields and to gravity via heat-kernel methods, offering a unified framework for Lorentzian-to-Euclidean transitions on foliated spaces.

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 $iε$ prescription and on Friedmann-Lemaître backgrounds the difference to a Wick rotation in time is illustrated.

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

TL;DR

The authors introduce a lapse-based Wick rotation on foliated manifolds that keeps the manifold’s real atlas while smoothly interpolating between Lorentzian and Euclidean geometries via a phase rotation of the lapse. By formulating the rotated metric as a rank-one perturbation and analyzing foliation-changing diffeomorphisms, they achieve a covariant description of across equivalent foliations. They prove three notions of admissibility for the complex metric in a minimally coupled scalar theory: damping of the action, positivity of the linear response, and sectoriality of the complex Hessian, with the Hessian spectrum confined to a wedge in the upper half-plane and explicit behavior near Minkowski and Friedmann-Lemaître backgrounds. The work predicts modified propagators compared to the standard prescription and suggests broad applicability to other fields and to gravity via heat-kernel methods, offering a unified framework for Lorentzian-to-Euclidean transitions on foliated spaces.

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 prescription and on Friedmann-Lemaître backgrounds the difference to a Wick rotation in time is illustrated.
Paper Structure (10 sections, 5 theorems, 64 equations)

This paper contains 10 sections, 5 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Phase rotated lapse and foliation changing diffeomorphisms
  3. Foliation changing diffeomorphisms
  4. Complexified metric as a rank one perturbation
  5. Admissible metrics for scalar field theories.
  6. Linearised and nonlinear admissibility
  7. The complexified Hessian
  8. Conclusions
  9. Foliation geometry
  10. Lapse-Wick rotation for metrics with preferred foliation

Key Result

Proposition 2.1

The lapse Wick rotated metric $g^{\theta}_{\mu\nu} dy^{\mu} dy^{\nu} = - N_{\theta}^2 dt^2 + {\rm g}_{ab} (dx^a + N^a dt) (dx^b + N^b dt)$ in the fiducial foliation $t \mapsto \Sigma_{t}$ gives in a new (equivalent) foliation $t' \mapsto \Sigma'_{t'}$ rise to ${{g'}^{\theta}}_{\!\!\!\!\mu\nu} d{y'}^ We shall refer to (lwick3) as the complexified metric defined by lapse Wick rotation. It is invaria

Theorems & Definitions (5)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3