Resonances/decaying states and the mathematics of quantum physics

There is sufficient experimental evidence that a Breit–Wigner scattering resonance of width Γ is the same physical entity as an exponentially decaying Gamow state of lifetime τ = ℏ / Γ . In order to derive a Gamow ket with exponential time evolution from the Breit–Wigner scattering amplitude of the S-matrix pole, one has to make assumptions about the mathematical properties of the energy wave function for the prepared in-state φ + and the detected out-“state” ψ − of a resonance scattering experiment. These mathematical properties identify the space of in-state energy-wave functions as { φ + ( E ) } = H 2 ¯ and of out-state wave functions as { ψ − ( E ) } = H + 2 as the Hardy function spaces of the lower and upper complex energy plane. The semigroup-time asymmetry t 0 = 0 < t < ∞ of causality is a consequence of the Paley-Wiener theorem for Hardy spaces. The experimental meaning of the begmning of time t 0 will be discussed.