Wick rotation in the lapse, admissible complex metrics, and foliation changing diffeomorphisms
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909572970577920 |
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| author | Banerjee, Rudrajit Niedermaier, Max |
| author_facet | Banerjee, Rudrajit Niedermaier, Max |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wick rotation in the lapse, admissible complex metrics, and foliation changing diffeomorphisms Banerjee, Rudrajit Niedermaier, Max Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Analysis of PDEs 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. |
| title | Wick rotation in the lapse, admissible complex metrics, and foliation changing diffeomorphisms |
| topic | Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2406.06047 |