Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: Dependence on the DeWitt Parameter
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| Format: | Preprint |
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2023
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| _version_ | 1866929515410751488 |
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| author | Liu, Xiaoyi Marolf, Donald Santos, Jorge E. |
| author_facet | Liu, Xiaoyi Marolf, Donald Santos, Jorge E. |
| contents | Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a non-trivial choice of contour. The present work examines a generalization of a recently proposed rule-of-thumb \cite{Marolf:2022ntb} for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinite-signature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter $α=-1$. This choice was made to match previous results, but was otherwise admittedly {\it ad hoc}. To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions for $α\neq -1$. We study such contours for Euclidean gravity linearized about AdS-Schwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare path-integral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameter $α$, the precise agreement between these two notions of stability found at $α=-1$ continues to hold over the finite interval $(-2,-2/d)$, where $d$ is the dimension of the bulk spacetime. This agreement manifestly fails for $α> -2/d$ when the DeWitt metric becomes positive definite. However, we also find dramatic failures for $α< -2$ that correlate with breakdowns of the de Donder-like gauge condition defined by $α$, and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physically-useful contours in more general settings. Along the way, we also identify an interesting error in \cite{Marolf:2022ntb}, though we show this error to be harmless. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_08555 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: Dependence on the DeWitt Parameter Liu, Xiaoyi Marolf, Donald Santos, Jorge E. High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Physics Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a non-trivial choice of contour. The present work examines a generalization of a recently proposed rule-of-thumb \cite{Marolf:2022ntb} for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinite-signature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter $α=-1$. This choice was made to match previous results, but was otherwise admittedly {\it ad hoc}. To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions for $α\neq -1$. We study such contours for Euclidean gravity linearized about AdS-Schwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare path-integral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameter $α$, the precise agreement between these two notions of stability found at $α=-1$ continues to hold over the finite interval $(-2,-2/d)$, where $d$ is the dimension of the bulk spacetime. This agreement manifestly fails for $α> -2/d$ when the DeWitt metric becomes positive definite. However, we also find dramatic failures for $α< -2$ that correlate with breakdowns of the de Donder-like gauge condition defined by $α$, and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physically-useful contours in more general settings. Along the way, we also identify an interesting error in \cite{Marolf:2022ntb}, though we show this error to be harmless. |
| title | Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: Dependence on the DeWitt Parameter |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Physics |
| url | https://arxiv.org/abs/2310.08555 |