On defect CFT and path integral methods for entanglement in quantum field theories

In the first few chapters of the thesis, we will study defect CFT methods based on the replica trick for characterizing quantum information in quantum field theories. We calculate a coefficient that characterizes the strength of the two point function of the displacement operator in the replica twist defect placed in a holographic CFT, which controls the second order shape dependence of Renyi entropy. We introduce defect CFT methods for calculating correlation functions involving the modular Hamiltonian together with probe operators inserted at lightcone separation. We use these methods to further calculate correlation functions involving modular flows of these probe operators. Tomita-Takesaki theory constrains these correlation functions, which when combined with our defect CFT calculations, provides a proof of the Quantum Null Energy Condition. In the last few chapters of this thesis, we will calculate entanglement measures for states that are defined by a Euclidean path integral together with a source for an operator inserted in the path integral. We provide a purely Lorentzian formula for the modular Hamiltonians for these states for flat entangling cuts which systematizes the task of writing time-ordered expressions for relative entropy of these states with respect to the vacuum to all orders in the source. We further apply this method to calculate a formula for shape deformed modular Hamiltonian for the vacuum state to all orders in the shape deformation. In the case of null shape deformation, we recover the formula for the vacuum modular Hamiltonian for null cuts. We then calculate the shape deformation of relative entropy and provide evidence for the presence of a shock in the stress tensor expectation value when one performs the Connes cocycle flow of the state.U of I OnlyAuthor requested U of Illinois access only (OA after 2yrs) in Vireo ETD syste