Reflection positivity and its relation to disc, half plane and the strip
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929443968122880 |
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| author | Adamo, Maria Stella Neeb, Karl-Hermann Schober, Jonas |
| author_facet | Adamo, Maria Stella Neeb, Karl-Hermann Schober, Jonas |
| contents | We develop a novel perspective on reflection positivity (RP) on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on $H^\infty$ for a suitable involution. For the strip, reflection positivity naturally connects with Kubo--Martin--Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reflection positivity and its relation to disc, half plane and the strip Adamo, Maria Stella Neeb, Karl-Hermann Schober, Jonas Functional Analysis Operator Algebras Representation Theory 22E45, 43A35, 47B32, 47B91 We develop a novel perspective on reflection positivity (RP) on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on $H^\infty$ for a suitable involution. For the strip, reflection positivity naturally connects with Kubo--Martin--Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process. |
| title | Reflection positivity and its relation to disc, half plane and the strip |
| topic | Functional Analysis Operator Algebras Representation Theory 22E45, 43A35, 47B32, 47B91 |
| url | https://arxiv.org/abs/2407.21123 |