## Infinite infrared regularization and a state space for the Heisenberg algebra

We present a method for the construction of a Krein space completion for spaces of test functions, equipped with an indefinite inner product induced by a kernel which is more singular than a distribution of finite order.

We present a method for the construction of a Krein space completion for spaces of test functions, equipped with an indefinite inner product induced by a kernel which is more singular than a distribution of finite order. This generalizes a regularization method for infrared singularities in quantum field theory, introduced by G. Morchio and F. Strocchi, to the case of singularites of infinite order. We give conditions for the possibility of this procedure in terms of local differential operators and the Gelfand-Shilov test function spaces, as well as an abstract sufficient condition. As a model case we construct a maximally positive definite state space for the Heisenberg algebra in the presence of an infinite infrared singularity. See the corresponding paper: Schmidt, Andreas U.: "Mathematical Problems of Gauge Quantum Field Theory: A Survey of the Schwinger Model" and the presentation "Infinite Infrared Regularization in Krein Spaces"…

Author: | Andreas U. Schmidt |
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URN: | urn:nbn:de:hebis:30-11331 |

Document Type: | Article |

Language: | English |

Date of Publication (online): | 2005/06/21 |

Year of first Publication: | 2003 |

Publishing Institution: | Univ.-Bibliothek Frankfurt am Main |

Release Date: | 2005/06/21 |

Tag: | Gelfand-Shilov space; Heisenberg algebra; Infrared singularity; Krein space; indefinite inner product space |

Source: | Version of 03 March 2003, Orig. published in Journal of Mathematical Physics 36 no. 1 (January 2002) 243-259; Erratum:ibid. 43 no. 6 (June 2002) 3412 , http://www.math.uni-frankfurt.de/~aschmidt/#eprints |

HeBIS PPN: | 195731344 |

Institutes: | Mathematik |

Dewey Decimal Classification: | 510 Mathematik |

MSC-Classification: | 46C20 Spaces with indefinite inner product (Kreïn spaces, Pontryagin spaces, etc.) [See also 47B50] |

46F05 Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35] | |

47B50 Operators on spaces with an indefinite metric [See also 46C50] | |

81T05 Axiomatic quantum field theory; operator algebras | |

PACS-Classification: | 02.10.-v Logic, set theory, and algebra |

02.30.Tb Operator theory | |

11.10.-z Field theory (for gauge field theories, see 11.15.-q) | |

Sammlungen: | Universitätspublikationen |

Licence (German): | Veröffentlichungsvertrag für Publikationen |