  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
bra-ket 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 that fulfills the time-asymmetry condition which
could not be realized in the Hilbert space.
