Entanglement in field theory and gravity

Abstract

In the study of von Neumann algebra, we often work with modu lar operator and associated modular Hamiltonian In fact, they play an important role in the studies of the sub-regions of a quantum field theory. We find it interesting because in the studies of entanglement and related ideas, the expectation value of the modular Hamiltonian becomes equal to the entanglement entropy which is regarded as a measure of entan glement in a system. By definition, the modular Hamiltonian is associated with the definition of states. Therefore, modular Hamiltonian is a quantity that changes with the definition of states. In the case of quantum field theory, there must exist a vacuum modular Hamiltonian and one may ask how this modular Hamiltonian di↵ers from an excited state. To answer this question, one can construct an excited state by perturbing the vacuum and then define a modular Hamiltonian for the excited state. At this point, we would like to study the change between the vacuum modular Hamiltonian and the excited state modular Hamiltonian to figure out any surprising property if exists at all. It can be shown that for a particular type of per turbation the excited state modular Hamiltonian can be written in terms of a unitary transformation of the vacuum state modular Hamiltonian. We compute the change in the modular Hamiltonian in a 2-D con formal field theory (CFT2) and we expect that the result will be similar to as obtained from the unitary transformation. But, to our surprise, we find that the change contains an additional term apart from whatever we got from calculating the commutator. We call this term ‘endpoint contri bution’. In this project, we extend the endpoint contribution to several orders of perturbation and try to formulate a scheme to generalize the perturbation for any arbitrary order. We also try to construct an elegant approach to express the endpoint contribution as well as the commutator contribution in a compact form for any arbitrary order of perturbation. We further develop an operator which can express the endpoint contribution for a stress-tensor perturbation along the null horizon x+ in a commutator form similar to the commutator contribution.