Quantal time asymmetry: mathematical foundation and physical interpretation
Abstract
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 τ is the lifetime of the decaying state and Γ is 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\mathcal{H} , and detected observables, using a space \Phi_+\subset\mathcal{H} , where -(+) refers to analyticity of the energy wavefunction 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 Φ+ and Φ- are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable \mathsf{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.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- August 2008
- DOI:
- 10.1088/1751-8113/41/30/304019
- arXiv:
- arXiv:0803.3233
- Bibcode:
- 2008JPhA...41D4019B
- Keywords:
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- Quantum Physics
- E-Print:
- Plenary talk at the 5th International Symposium on Quantum Theory and Symmetries, July 22-28, 2007, Valladolid, Spain