This dissertation discusses a consequence of the limitation on causality originated in smallness of a quantum system. In quantum physics, disturbance due to a measurement is not negligible. Because of this fact, the time parameter t cannot be identified as a time continuum of experimenter's clock T on which observed events are recorded. Indeed, it will be shown that t represents an ensemble of time intervals on T during which a microsystem travels undisturbed. In particular t = 0 represents the ensemble of preparation events that we refer to as the ensemble of beginnings of time. This restricts t to range the positive real line only, but such a time evolution of quantum states cannot be achieved in the Hilbert space. Hence one needs the time asymmetric boundary condition in which only the semigroup time evolution is allowed. This boundary condition is characterized by the energy wave functions of quantum state (and of observables) satisfying the Hilbert transform, called in physics the dispersion relation.

The time asymmetric boundary condition is formulated as a pair of Hardy rigged Hilbert spaces. They are developed to incorporate Einstein's causality. Within the framework of Hardy rigged Hilbert space, decaying states are described by Gamow vectors, and they are associated to S-matrix poles in the lower-half complex energy plane. This framework provides one a non-perturbative description of a decaying particle. From the Gamow vectors, exponential decay law of a relativistic particle is derived. Finally the neutral kaon decay experiment and the Z-boson resonance are discussed as applications.