Pseudo Entropy in $U(1)$ gauge theory

Abstract: We study the properties of pseudo entropy, a new generalization of entanglement entropy, in free Maxwell field theory in $d = 4$ dimension. We prepare excited states by the different components of the field strengths located at different Euclidean times acting on the vacuum. We compute the difference between the pseudo R\'{e}nyi entropy and the R\'{e}nyi entropy of the ground state and observe that the difference changes significantly near the boundary of the subsystems and vanishes far away from the boundary. Near the boundary of the subsystems, the difference between pseudo R\'{e}nyi entropy and R\'{e}nyi entropy of the ground state depends on the ratio of the two Euclidean times where the operators are kept. To begin with, we develop the method to evaluate pseudo entropy of conformal scalar field in $d=4$ dimension. We prepare two states by two operators with fixed conformal weight acting on the vacuum and observe that the difference between pseudo R\'{e}nyi entropy and ground state R\'{e}nyi entropy changes only near the boundary of the subsystems. We also show that a suitable analytical continuation of pseudo R\'{e}nyi entropy leads to the evaluation of real-time evolution of R\'{e}nyi entropy during quenches.