Hessian in the spinfoam models with cosmological constant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914389397864448 |
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| author | Kamiński, Wojciech Pan, Qiaoyin |
| author_facet | Kamiński, Wojciech Pan, Qiaoyin |
| contents | In this paper, we introduce a general method to prove the non-degeneracy of the Hessian in the spinfoam vertex amplitude for quantum gravity and apply it to the spinfoam models with a cosmological constant ($Λ$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat ${\rm SL}(2,\mathbb{C})$ connections, we demonstrate that the Hessian is non-degenerate for critical points corresponding to non-degenerate, geometric 4-simplices in de Sitter or anti-de Sitter space. Non-degeneracy of the Hessian is an important necessary condition for the stationary phase method to be applicable. With a non-degenerate Hessian, this method not only confirms the connection of the $Λ$-SF model to semiclassical gravity, but also shows that there are no dominant contributions from exceptional configurations as in the Barrett-Crane model. Given its general nature, we expect our criterion to be applicable to other spinfoam models under mild adjustments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12755 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hessian in the spinfoam models with cosmological constant Kamiński, Wojciech Pan, Qiaoyin General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics In this paper, we introduce a general method to prove the non-degeneracy of the Hessian in the spinfoam vertex amplitude for quantum gravity and apply it to the spinfoam models with a cosmological constant ($Λ$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat ${\rm SL}(2,\mathbb{C})$ connections, we demonstrate that the Hessian is non-degenerate for critical points corresponding to non-degenerate, geometric 4-simplices in de Sitter or anti-de Sitter space. Non-degeneracy of the Hessian is an important necessary condition for the stationary phase method to be applicable. With a non-degenerate Hessian, this method not only confirms the connection of the $Λ$-SF model to semiclassical gravity, but also shows that there are no dominant contributions from exceptional configurations as in the Barrett-Crane model. Given its general nature, we expect our criterion to be applicable to other spinfoam models under mild adjustments. |
| title | Hessian in the spinfoam models with cosmological constant |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2510.12755 |