On Contours and Boundary Conditions for the Gravitational Path Integral.

다국어 초록 (Multilingual Abstract)

The gravitational path integral has been a useful tool in studying quantum gravity. This thesis is devoted to studying certain aspects of the gravitational path integral, discussing the choices of contours of integration and boundary conditions for the gravitational path integral, in both Euclidean and Lorentzian signatures.In Part I, we discuss the choice of contours of integration for the gravitational path integral. The Euclidean gravitational action is unbounded from below, which means we cannot take the contour of the Euclidean path integral to all real Euclidean geometries. We propose a Wick-rotation contour prescription for the Euclidean gravitational path integral, and test this prescription by computing the stability of black hole saddles and comparing it with standard thermodynamic stability. We also study contours for Lorentzian gravitational path integral, where the original contour of integration is taken to be over all real Lorentzian geometries. We use Jackiw-Teitelboim (JT) gravity as an example, and explicitly demonstrate that it is possible to deform the original contour to pass through complex saddles that reproduce the correct Renyi entropy of JT gravity. In addition, we consider more complicated wormhole geometries in Euclidean anti-de Sitter spacetimes, which resemble the wormholes that are important in ensuring the black hole Hilbert space dimension is finite. We discuss their genericity with large Euclidean sources at the conformal infinity and compute the stability of these Euclidean wormholes.In Part II, we discuss the choice of boundary conditions for the gravitational path integral. We propose a one-parameter family of boundary conditions for Euclidean gravity that yields a well-posed elliptic system, and study Euclidean stability of various saddles with such boundary conditions. These boundary conditions can be used to define a generalized thermal canonical ensemble. They are then applied in Lorentzian signature to study real-time evolution in a spherical cavity. By analyzing the quasi-normal modes of empty flat space, we demonstrate that the same boundary conditions fail to define a well-posed initial-boundary value problem in Lorentzian signature.