## Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes

- According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted suchAccording to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.…

Author details: | Max LewandowskiORCiDGND |
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DOI: | https://doi.org/10.1063/5.0055753 |

ISSN: | 0022-2488 |

ISSN: | 1089-7658 |

Title of parent work (English): | Journal of mathematical physics |

Publisher: | American Institute of Physics |

Place of publishing: | Melville |

Publication type: | Article |

Language: | English |

Date of first publication: | 2022/01/03 |

Publication year: | 2022 |

Release date: | 2023/01/25 |

Volume: | 63 |

Issue: | 1 |

Article number: | 013501 |

Number of pages: | 34 |

Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |

DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik | |

Peer review: | Referiert |