### Abstract

Dynamical differential equations, like the Schrödinger equation for the states, or the Heisenbergequa tion for the observables, need to be solved under boundary conditions. The original boundary condition of von Neumann, the Hilbert space axiom, required that the allowed wave functions are Lebesgue square integrable. This leads by a mathematical theorem of Stone-von Neumann to the unitary group evolution meaning the time t extends over -∞ < t < +<. Physicists do not use Lebesgue integrals but followed a different path usinga lmost exclusively the Dirac formalism and well-behaved (Schwartz) functions. This led the mathematicians to Schwartz-Rigged Hilbert spaces (Gelfand triplets), which are the mathematical core of Dirac's bra-ket formalism. This is insufficient for a theory that includes resonance and decay phenomena, which requires analytic continuation in energy E in order to accommodate exponentially decayingG amow kets, Breit-Wigner (Lorentzian) resonances, and Lippmann-Schwinger kets. This leads to a pair of Rigged Hilbert Spaces of smooth Hardy functions, one representing the prepared states of scatteringe xperiments (preparation apparatus) and the other representingd etected observables (registration apparatus). A mathematical consequence of the Hardy space axiom is that the time evolution is asymmetric given by the semi-group, i.e., t0 ≤ t < +<, with a finite t0. What would the meaningo f that t0 be?

Original language | English (US) |
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Title of host publication | Geometric Methods in Physics |

Editors | Theodore Voronov, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, S. Twareque Ali |

Publisher | Springer International Publishing |

Pages | 211-228 |

Number of pages | 18 |

ISBN (Print) | 9783034804479 |

State | Published - Jan 1 2013 |

Externally published | Yes |

Event | 30th Workshop on Geometric Methods in Physics, 2011 - Bialowieza, Poland Duration: Jun 26 2011 → Jul 2 2011 |

### Publication series

Name | Trends in Mathematics |
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Volume | 59 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Conference

Conference | 30th Workshop on Geometric Methods in Physics, 2011 |
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Country | Poland |

City | Bialowieza |

Period | 6/26/11 → 7/2/11 |

### Keywords

- Hardy space
- Rigged Hilbert space
- Semigroup
- Time asymmetry
- Unitary group

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Geometric Methods in Physics*(pp. 211-228). (Trends in Mathematics; Vol. 59). Springer International Publishing.