  Non-unitary quantum mechanics has been used in the past to study
irreversibility, dissipation and decay in a variety of physical systems. In
this letter, we propose a general scheme to deal with systems governed by
non-Hermitian Hamiltonians. We argue that the Schwinger-Keldysh formalism gives
a natural description for those problems. To elucidate the method, we study a
simple model inspired by mesoscopic physics --an asymmetric junction. The
system is governed by a non-Hermitian Hamiltonian which captures essential
aspects of irreversibility.
