# Time asymmetric quantum theory and its applications in non-relativistic and relativistic quantum systems

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## Abstract

The exact relation τ = ~/Γ between the width Γ of a resonance and the lifetime τ for a decay of this resonance could not be obtained in the conventional quantum theory based on the Hilbert space as well as on the Schwartz space in which the spaces of the states and of the observables are described by the same space. Furthermore, both dynamical evolution of the states (in Schrodinger picture) and observables (in Heisenberg picture) are symmetrically in time, given by an unitary group with time extending over −∞ < t < +∞. This time symmetric evolution is a mathematical consequence of Von Neumann theorem for the dynamical differential equations, Schrodinger equation for the state or Heisenberg equation for the observable, under either the Hilbert space or Schwartz space boundary condition. However, this unitary group evolution violates causality. In order to get a quantum theory in which the exact relation and causality are obtained, one has to replace the Hilbert space or the Schwartz space boundary condition by a new boundary condition based on the Hardy space axioms in which the space of the states and the space of the observables are described by two distinguishable Hardy spaces. As a consequence of the new Hardy space axiom, one obtains, instead of the time symmetric evolution for the states and the observables, time asymmetrical evolutions for the states and observables which are described by two semi-groups. The time asymmetrical evolution predicts an finite beginning of time t0(= 0) for quantum system which can be experimental observed by directly measuring the lifetime of the decaying state as the beginning of time of the ensemble in scattering experiments. A resonance obeying the exponential time evolution can then be described by a Gamow vector, which is defined as superposition of the exact out-plane wave states with exact Breit-Wigner energy distribution.