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.
Categories: quant-ph