dorsal/arxiv
View SchemaRigged Hilbert Space Resonances and Time Asymmetric Quantum Mechanics
| Authors | A. Bohm, H. Kaldass |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9909081 |
| URL | https://arxiv.org/abs/quant-ph/9909081 |
Abstract
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states $\phi^+$ and the out-states (observables) $\psi^-$ of the S-matrix elements $(\psi^-,\phi^+)=(\psi^{out},S \phi^{in})$ any elements of the HS $\H$, $\{\psi^-\}=\{\phi^+\}(=\H)$, the RHS theory chooses the boundary conditions~: $\phi^+\in\Phi_-\subset\H\subset\Phi_-^\times$, $\psi^-\in\Phi_+\subset \H\subset \Phi_+^\times$, where $\Phi_-$ ($\Phi_+$) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states $\phi^+$ and observables $\psi^-$ provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets $|E^{\pm}\in\Phi_{\mp}^{\times}$ (solutions of the Lippmann-Schwinger equation with $\pm i\epsilon$ respectively) and the Gamow vectors $|E_R-i\Gamma/2^\pm\in\Phi_{\mp}^\times$. The Gamow vectors $|E_R-i\Gamma/2^-$ have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.
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"abstract": "The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is\nreviewed and contrasted with the standard Hilbert space (HS) theory of quantum\nmechanics. The main difference is in the choice of boundary conditions. Whereas\nthe conventional theory allows for the in-states $\\phi^+$ and the out-states\n(observables) $\\psi^-$ of the S-matrix elements $(\\psi^-,\\phi^+)=(\\psi^{out},S\n\\phi^{in})$ any elements of the HS $\\H$, $\\{\\psi^-\\}=\\{\\phi^+\\}(=\\H)$, the RHS\ntheory chooses the boundary conditions~:\n$\\phi^+\\in\\Phi_-\\subset\\H\\subset\\Phi_-^\\times$, $\\psi^-\\in\\Phi_+\\subset\n\\H\\subset \\Phi_+^\\times$, where $\\Phi_-$ ($\\Phi_+$) are Hardy class spaces\nassociated to the lower (upper) half-plane of the second sheet of the\nanalytically continued S-matrix. This can be phenomenologically justified by\ncausality. The two RHS\u0027s for states $\\phi^+$ and observables $\\psi^-$ provide\nnew vectors which are not in $\\H$, e.g. the Dirac-Lippmann-Schwinger kets\n$|E^{\\pm}\\in\\Phi_{\\mp}^{\\times}$ (solutions of the Lippmann-Schwinger equation\nwith $\\pm i\\epsilon$ respectively) and the Gamow vectors\n$|E_R-i\\Gamma/2^\\pm\\in\\Phi_{\\mp}^\\times$. The Gamow vectors $|E_R-i\\Gamma/2^-$\nhave all the properties that one heuristically needs for quasistable states. In\naddition, they give rise to asymmetric time evolution expressing\nirreversibility on the microphysical level.",
"arxiv_id": "quant-ph/9909081",
"authors": [
"A. Bohm",
"H. Kaldass"
],
"categories": [
"quant-ph",
"math-ph",
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],
"title": "Rigged Hilbert Space Resonances and Time Asymmetric Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9909081"
},
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