  Many useful concepts for a quantum theory of scattering and decay (like
Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially
decaying Gamow vectors, causality) are not well defined in the mathematical
frame set by the conventional (Hilbert space) axioms of quantum mechanics.
Using the Lippmann-Schwinger equations as the takeoff point and aiming for a
theory that unites resonances and decay, we conjecture a new axiom for quantum
mechanics that distinguishes mathematically between prepared states and
detected observables. Suggested by the two signs $\pm i\epsilon$ of the
Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of
conventional quantum mechanics by two Hardy spaces. The new Hardy space theory
automatically provides Gamow kets with exponential time evolution derived from
the complex poles of the $S$-matrix. It solves the causality problem since it
results in a semigroup evolution. But this semigroup brings into quantum
physics a new concept of the semigroup time $t=0$, a beginning of time. Its
interpretation and observations are discussed in the last section.
