is usually viewed as associating to every state vector $\phi(t)$ a state $\phi^\prime(-t)\equiv A_T^{-1}\phi(-t)$ at the negative time $-t$ (relative to a distinguished point of time $t=0$). Irreversibility and time-reversibility thus appear to be in conflict with each other. Here we want to discuss the resolution of this conflict, based on the empirical fact, that (due to the boundary and initial conditions) for a given state one can usually not (experimentally) prepare its time-reversed state, and that the experimentally tested time-reversal invariance (like, e.g., reciprocity relations) refers to the relations of $A_T$ with the observables and not the action of $A_T$ on the states.

Irreversibility, initial-boundary conditions, the time-evolution semigroup and Gamow vectors 
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Standard (Hilbert Space) quantum mechanics admits only reversible time evolution because time evolution is represented by a group (generated by a self-adjoint Hamiltonian). In contrast to this mathematical theory, there is ample empirical evidence of intrinsically irreversible time evolution of microphysical systems, e.g., the decay of quasistationary states or resonances. Truly stationary states of such quantum physical systems like stable elementary particles are rare. Most relativistic or non-relativistic elementary particles are decaying states (weakly or electromagnetically) or hadron resonances. Empirically, stability or the values of the 