Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866913492411351040 |
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| author | Moretti, Valter Murro, Simone Volpe, Daniele |
| author_facet | Moretti, Valter Murro, Simone Volpe, Daniele |
| contents | Given a pair of normally hyperbolic operators over (possibily different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called {\em Møller operator}, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called {\em paracausal relation}. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a $*$-isomorphism between (generally off-shell) $CCR$-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this $*$-isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_06685 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory Moretti, Valter Murro, Simone Volpe, Daniele Mathematical Physics General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Operator Algebras 53C50, 81T05 (Primary), 35L52, 58J45 (Secondary) Given a pair of normally hyperbolic operators over (possibily different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called {\em Møller operator}, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called {\em paracausal relation}. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a $*$-isomorphism between (generally off-shell) $CCR$-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this $*$-isomorphism. |
| title | Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory |
| topic | Mathematical Physics General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Operator Algebras 53C50, 81T05 (Primary), 35L52, 58J45 (Secondary) |
| url | https://arxiv.org/abs/2109.06685 |