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

##### 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.