ABSTRACT
Unitary group representations were used for symmetry transformations of spacetime. The Hamiltonian and other observables are derived from them. Vice versa, if the dynamical equations (Schrödinger, Heisenberg, etc.) are solved under Hilbert space boundary conditions the solutions lead to an unitary group. This is in conflict with: 1. time asymmetry and Einstein causality, 2. the iε prescription for Lippmann-Schwinger states and for propagators, 3. exponential evolution for decaying Gamow states, 4. Breit-Wigner lineshape (of constant width) for decaying states. A unified theory of resonances and decaying states, requires new boundary conditions replacing the Hilbert space axiom by the Hardy space axiom. This leads to semigroup representations of space-time symmetries. We will argue that experiments require semigroup, too. The semigroup property can be seen in experiments with single ions.
