This paper discusses geometric and renormalized entropy in conformal field theory (CFT), focusing on the challenges of defining entropy in relativistic quantum field theories. The authors consider the problem of divergent entropy in sharply localized states and propose a renormalized entropy as the entropy relative to the ground state. They analyze this in the context of a conformal quantum field theory excited by a moving mirror, showing how geometric entropy arises naturally in a dynamical context and how ultraviolet divergences occur. They also define a useful renormalized entropy that is finite and meaningful. The paper begins by discussing the general concept of entropy in statistical mechanics and quantum physics, emphasizing the role of coarse-graining and the limitations of finite-volume measurements. It then introduces the concept of geometric entropy in quantum field theory, which measures the correlations between a subsystem and the rest of the universe. The authors show that geometric entropy can be calculated using conformal field theory techniques, and they apply this to the moving mirror model, demonstrating how the entropy arises from the boundary conditions imposed by the mirror. The paper then discusses the application of the moving mirror model to black hole physics, noting that the entropy calculated in this model has interesting implications for black hole entropy. The authors show that the renormalized entropy calculated for the moving mirror model can be related to the entropy of a black hole, and they argue that the divergence of geometric entropy in the absence of an ultraviolet cutoff is a fundamental feature of quantum field theory. The paper concludes by discussing the causal structure of the mirror problem and how it relates to black hole physics. The authors show that the moving mirror model can be used to simulate the behavior of a black hole, with the mirror representing the black hole's event horizon. They argue that the correlations between the thermal state on one side of the mirror and the apparently empty region on the other side are essential for maintaining the purity of the overall state. The paper also highlights the importance of renormalized entropy in understanding the information content of quantum states in relativistic quantum field theories.This paper discusses geometric and renormalized entropy in conformal field theory (CFT), focusing on the challenges of defining entropy in relativistic quantum field theories. The authors consider the problem of divergent entropy in sharply localized states and propose a renormalized entropy as the entropy relative to the ground state. They analyze this in the context of a conformal quantum field theory excited by a moving mirror, showing how geometric entropy arises naturally in a dynamical context and how ultraviolet divergences occur. They also define a useful renormalized entropy that is finite and meaningful. The paper begins by discussing the general concept of entropy in statistical mechanics and quantum physics, emphasizing the role of coarse-graining and the limitations of finite-volume measurements. It then introduces the concept of geometric entropy in quantum field theory, which measures the correlations between a subsystem and the rest of the universe. The authors show that geometric entropy can be calculated using conformal field theory techniques, and they apply this to the moving mirror model, demonstrating how the entropy arises from the boundary conditions imposed by the mirror. The paper then discusses the application of the moving mirror model to black hole physics, noting that the entropy calculated in this model has interesting implications for black hole entropy. The authors show that the renormalized entropy calculated for the moving mirror model can be related to the entropy of a black hole, and they argue that the divergence of geometric entropy in the absence of an ultraviolet cutoff is a fundamental feature of quantum field theory. The paper concludes by discussing the causal structure of the mirror problem and how it relates to black hole physics. The authors show that the moving mirror model can be used to simulate the behavior of a black hole, with the mirror representing the black hole's event horizon. They argue that the correlations between the thermal state on one side of the mirror and the apparently empty region on the other side are essential for maintaining the purity of the overall state. The paper also highlights the importance of renormalized entropy in understanding the information content of quantum states in relativistic quantum field theories.