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For a quantum theory that includes exponentially decaying states andBreit-Wigner resonances, which are related to each other by the lifetime-widthrelation $\tau=\frac{\hbar}{\Gamma}$, where $\tau$ is the lifetime of thedecaying state and $\Gamma$ the width of the resonance, one has to go beyondthe Hilbert space and beyond the Schwartz-Rigged Hilbert Space$\Phi\subset\mathcal{H}\subset\Phi^\times$ of the Dirac formalism. One has todistinguish 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 beprepared first, before an observable can be measured in it. The axiom that willlead to the lifetime-width relation is that $\Phi_+$ and $\Phi_-$ are Hardyspaces of the upper and lower semiplane, respectively. Applying this axiom tothe relativistic case for the variable $\s=p_\mu p^\mu$ leads to semigrouptransformations into the forward light cone (Einstein causality) and a precisedefinition of resonance mass and width.

Quantal time asymmetry: mathematical foundation and physical interpretation