  For a quantum theory that includes exponentially decaying states and
Breit-Wigner resonances, which are related to each other by the lifetime-width
relation $\tau=\frac{\hbar}{\Gamma}$, where $\tau$ is the lifetime of the
decaying state and $\Gamma$ the width of the resonance, one has to go beyond
the Hilbert space and beyond the Schwartz-Rigged Hilbert Space
$\Phi\subset\mathcal{H}\subset\Phi^\times$ of the Dirac formalism. One has to
distinguish between prepared states, using a space $\Phi_-\subset\mat hcal{H}$,
and detected observables, using a space $\Phi_+\subset\mathcal{H}$, where
$-(+)$ refers to analyticity of the energy wave function in the lower (upper)
complex energy semiplane.
  This differentiation is also justified by causality: A state needs to be
prepared first, before an observable can be measured in it. The axiom that will
lead to the lifetime-width relation is that $\Phi_+$ and $\Phi_-$ are Hardy
spaces of the upper and lower semiplane, respectively. Applying this axiom to
the relativistic case for the variable $\s=p_\mu p^\mu$ leads to semigroup
transformations into the forward light cone (Einstein causality) and a precise
definition of resonance mass and width.
