Speaker
Description
We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product $I_Q$ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator $Q$. We studied it for pure states. In this talk we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly. In particular, in the future-included theories, where not only a past state at the initial time $T_A$ but also a future state at the final time $T_B$ is given, we introduce a “skew density matrix” composed of both ensembles of the future and past states such that the trace of the product of it and an operator $O$ matches a normalized matrix element of $O$. We argue that the skew density matrix defined with $I_Q$ at the present time t for large $T_B − t$ and large $t − T_A$ approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product $I_{Q_J}$ for large $t − T_A$. This talk is based on the collaboration with Holger Bech Nielsen [Prog. Theor. Exp. Phys. 2023 (3) 031B01] (arXiv:arXiv:2209.11619 [hep-th]).
