Let us denote by \(\Phi_+\) the set of detected observables and by \(\Phi_-\) the set of prepared states in scattering experiments. The authors explain that the standard quantum mechanics based on the Hilbert space axiom (\(\Phi_+= \Phi_-= H\) (Hilbert space)) cannot describe exponential decay and resonance scattering. They also explain that time asymmetry for quantum systems is a manifestation of causality, which standard quantum mechanics cannot possess. Then they change the Hilbert space axiom by the Hardy space axion, where \(\Phi_{\pm}\subset H\) are equivalent to the spaces of energy wave functions \((H^2_{\pm}\cap{\mathcal S})|_{\mathbb{R}_+}\), \(H^2_{\pm}\) are the Hardy function spaces and \({\mathcal S}\) is the Schwartz space. In this way one obtains a mathematically consistent theory that provides a refinement of quantum theory by distinguishing between states and observables, unifies resonance and decay phenomena and that has a causal, asymmetric time evolution.