Unraveling Quantum Gravity Through the Gravitational Path Integral: Geometries, Entropies, and Algebras [electronic resource]

다국어 초록 (Multilingual Abstract)

The gravitational path integral has long served as a crucial tool in deciphering mysteries within quantum gravity. In recent years, studies of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence have offered many valuable insights into comprehending those mysteries, and many fruitful results have been yielded from utilizing the gravitational path integral within the framework of AdS/CFT.This dissertation is devoted to studying certain aspects of the gravitational path integral, discussing its relation with gravitational entropies, spacetime geometries, and its algebraic aspects. We explore contexts from Euclidean to Lorentz signature, from holographic theories to general theories, with a goal of understanding quantum gravity in the real world.In Part I, we discuss the fixed-(HRT)-area states in the gravitational path integral. The fixed-area states are holographic states where the area of the Hubeny-Rangamani-Takayanagi (HRT) surface, the holographic dual of entanglement entropy for a region in the boundary CFT, is constrained to a small window when prepared by the gravitational path integral. The study of those fixed-area states helps understand quantum gravity beyond the leading semiclassical order. We first show that by decomposing a general holographic state into fixed-area states, an important subleading correction appears to the entanglement entropy near phase transitions. Then we explore the intrinsic spacetime geometries of fixed-area states under Lorentz-signature time evolution.In Part II, we study saddle-point geometries of the real-time gravitational path integral, in the context of computing holographic R´enyi entropies. Unlike their Euclidean counterparts, these real-time saddles necessarily have complex metrics, giving an example where the saddle point is off the original contour of integration. We first present the formalism of this setup, illustrating the relevant variational problem, and features of the complex saddles. Then we demonstrate explicitly the structure of those saddles by showing examples in low dimensions by direct calculation. We also find that it is possible to deform the original integration contour to pass through saddles of this kind constructed in two-dimensional Jackiw-Teitelboim gravity. Finally, we show that the existence of these saddles results in a consequence which is necessary for unitarity to hold in quantum gravity.In Part III, we take a step towards explaining the origin of gravitational entropies, by utilizing the mathematical tool of von Neumann algebras. In particular, we give an explanation of the HRT formula purely from the bulk perspective, without making any reference to holography. This is done by constructing Hilbert spaces and von Neumann algebras from boundary conditions of the gravitational path integral with several natural axioms. The von Neumann algebras we find from this construction allows us to define a notion of entropy, which matches the HRT formula in the semiclassical limit. One of the axioms we assume which is crucial for the construction of von Neumann algebras - the trace inequality, is proven in the semiclassical limit, and it leads to novel positivity conjectures for the gravitational action. .