Automatic hermiticity for mixed states

Abstract

Abstract We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product IQ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator Q. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time TA but also a future state at the final time TB is given, we study a couple of candidates for it, and 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 ${\cal O}$ matches a normalized matrix element of ${\cal O}$. We argue that the skew density matrix defined with IQ at the present time t for large TB − t and large t − TA 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 − TA.