RESUMO
The information loss paradox associated with black hole Hawking evaporation is an unresolved problem in modern theoretical physics. In a recent brief essay, we revisited the evolution of the black hole entanglement entropy via the Euclidean path integral (EPI) of the quantum state and allow for the branching of semi-classical histories along the Lorentzian evolution. We posited that there exist at least two histories that contribute to EPI, where one is an information-losing history, while the other is an information-preserving one. At early times, the former dominates EPI, while at the late times, the latter becomes dominant. By doing so, we recovered the essence of the Page curve, and thus, the unitarity, albeit with the turning point, i.e., the Page time, much shifted toward the late time. In this full-length paper, we fill in the details of our arguments and calculations to strengthen our notion. One implication of this modified Page curve is that the entropy bound may thus be violated. We comment on the similarity and difference between our approach and that of the replica wormholes and the islands' conjectures.
RESUMO
We study the two-dimensional free symplectic fermion theory with antiperiodic boundary condition. This model has negative norm states with a naive inner product. This negative norm problem can be cured by introducing a new inner product. We demonstrate that this new inner product follows from the connection between the path integral formalism and the operator formalism. This model has a negative central charge, c=-2, and we clarify how two-dimensional conformal field theory with negative central charge can have a non-negative norm. Furthermore, we introduce α vacua in which the Hamiltonian is seemingly non-Hermitian. In spite of non-Hermiticity, we find that the energy spectrum is real. We also compare a correlation function with respect to the α vacua with that of the de Sitter space.
