Physics – Quantum Physics
Scientific paper
2008-03-21
J.Phys.A41:304019,2008
Physics
Quantum Physics
Plenary talk at the 5th International Symposium on Quantum Theory and Symmetries, July 22-28, 2007, Valladolid, Spain
Scientific paper
10.1088/1751-8113/41/30/304019
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.
Böhm Alexander
Bryant Paul
Sato Yuzuru
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