Complexity and the Hilbert space dimension of 3D gravity
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911417663225856 |
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| author | Balasubramanian, Vijay Das, Rathindra Nath Erdmenger, Johanna Karl, Jonathan Verlinde, Herman |
| author_facet | Balasubramanian, Vijay Das, Rathindra Nath Erdmenger, Johanna Karl, Jonathan Verlinde, Herman |
| contents | A central problem in formulating a theory of quantum gravity is to determine the size and structure of the Hilbert space of black holes. Here we use a quantum dynamical Krylov complexity approach to calculate the Hilbert space dimension of a black hole in 2+1-dimensional Anti-de Sitter space. We achieve this by obtaining the spread of an initial thermofield double state over the Krylov basis. The associated Lanczos coefficients match those for chaotic motion on the $SL(2,\mathbb{R})$ group. By including non-perturbative effects in the path integral, which computes coarse-grained ensemble averages, we find that the complexity saturates at late times. The saturation value is given by the exponential of the Bekenstein-Hawking entropy. Our results introduce a new way to compute the Hilbert space dimension of complex interacting systems from the saturating value of spread complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complexity and the Hilbert space dimension of 3D gravity Balasubramanian, Vijay Das, Rathindra Nath Erdmenger, Johanna Karl, Jonathan Verlinde, Herman High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics Quantum Physics A central problem in formulating a theory of quantum gravity is to determine the size and structure of the Hilbert space of black holes. Here we use a quantum dynamical Krylov complexity approach to calculate the Hilbert space dimension of a black hole in 2+1-dimensional Anti-de Sitter space. We achieve this by obtaining the spread of an initial thermofield double state over the Krylov basis. The associated Lanczos coefficients match those for chaotic motion on the $SL(2,\mathbb{R})$ group. By including non-perturbative effects in the path integral, which computes coarse-grained ensemble averages, we find that the complexity saturates at late times. The saturation value is given by the exponential of the Bekenstein-Hawking entropy. Our results introduce a new way to compute the Hilbert space dimension of complex interacting systems from the saturating value of spread complexity. |
| title | Complexity and the Hilbert space dimension of 3D gravity |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2602.02645 |