Microphysical irreversibility in quantum mechanics

Irreversibility on the microphysical level is described by a semigroup which is generated by the Hamiltonian and is distinguished from the irreversibility of open systems. The mathematical tool of this quantum mechanical irreversibility is the rigged Hilbert space. The physical basis is found in the distinct initial and boundary conditions for the states and the observables, which can be formulated in terms of an arrow of time. The principal objects of this theory are resonances which are defined by resonance poles of the S-matrix and described by generalized eigenvectors which are Jordan vectors of higher degree. Though there is no physical evidence for these, they have an intriguing semigroup time evolution.