Quantum theory in the rigged hilbert space — Irreversibility from causality

After a review of the arrows of time, we describe the possibilities of a time-asymmetry in quantum theory. Whereas Hilbert space quantum mechanics is time-symmetric, the rigged Hilbert space formulation, which arose from Dirac’s braket formalism, allows the choice of asymmetric boundary conditions analogous to the retarded solutions of the Maxwell equations for the radiation arrow of time. This led to irreversibility on the microphysical level as exemplified by decaying states or resonances. Resonances are mathematically represented by Gamow kets, functionals over a space of very well-behaved (Hardy class) vectors, which have been chosen by a boundary condition (outgoing for decaying states). Gamow states have all the properties that one heuristically needs for quasistable states. For them a Golden Rule can be derived from the fundamental probabilities P(t)=Tr(Λ(t)WGamow (t0)) that fulfills the time-asymmetry condition t≥t 0 which could not be realized in the Hilbert space.