Abstract:
In recent years, significant advancements have emerged at the intersection of quantum
information theory and gravitational physics. These developments encompass a variety
of topics, including but not limited to the AdS/CFT correspondence and the Hawking
information paradox. Within this domain, this thesis presents work that focuses on re-
search that has been inspired by recent insights related to the island formula, as well as
applications stemming from algebraic quantum field theory. The work comprises of two papers, the first of which addresses the notion of theory de- pendence in the econstruction of black hole interiors utilizing Hawking radiation. This work is set in the context of a two-sided black hole that is coupled to a thermal bath, specifically in the framework of Jackiw-Teitelboim gravity. The question asked is of how theory dependence influences reconstruction efforts, framing it through the lens of the
island formula. The results tell us that at extremely late times, reconstructing the interior of a black hole using Hawking radiation necessitates an understanding of the microscopic details of the theory, which can be interpreted as a dependence on the ultraviolet details of a quantum theory of gravity.
In the next work, I introduce an operator decomposition formula derived from the Zassenhaus decomposition applicable to unbounded operators within two-dimensional conformal field theory in Minkowski spacetime. This work uses half-sided translations a construction that has been important in recent theoretical developments as the prototypical example. The derivation of this result requires formulation of a regularization procedure aimed at transforming ill-defined operators, such as density matrices in quantum field theory, into well-defined entities. Furthermore, a ’centred’ version of the Zassenhaus formula is derived to achieve the result. It is demonstrated how this decomposition can be applied to an infinite class of operators, and the governing differential equations
