How to Count States in Gravity
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912439537238016 |
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| author | Balasubramanian, Vijay Yildirim, Tom |
| author_facet | Balasubramanian, Vijay Yildirim, Tom |
| contents | Gibbons and Hawking proposed that the Euclidean gravity path integral with periodic boundary conditions in time computes the thermal partition sum of gravity. As a corollary, they argued that a derivative of the associated free energy with respect to the Euclidean time period computes gravitational entropy. Why is this interpretation is correct? That is, why does this path integral compute a trace over the Hilbert space? Here, we show that the quantity computed by the Gibbons-Hawking path integral is equal to an {\it a priori} different object -- an explicit thermal trace over the Hilbert space spanned by states produced by the Euclidean gravity path integral. This follows in two ways: (a) if the Hilbert space with two boundaries factorizes into a product of two single boundary Hilbert spaces, as we have previously shown; and (b) via explicit resolution of the trace by a spanning basis of states. We similarly show how a replicated Euclidean gravity path integral with a single periodic boundary computes a Hilbert space trace of powers of the density matrix, explaining why this approach computes the entropy of states entangled between two universes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How to Count States in Gravity Balasubramanian, Vijay Yildirim, Tom High Energy Physics - Theory General Relativity and Quantum Cosmology High Energy Physics - Phenomenology Gibbons and Hawking proposed that the Euclidean gravity path integral with periodic boundary conditions in time computes the thermal partition sum of gravity. As a corollary, they argued that a derivative of the associated free energy with respect to the Euclidean time period computes gravitational entropy. Why is this interpretation is correct? That is, why does this path integral compute a trace over the Hilbert space? Here, we show that the quantity computed by the Gibbons-Hawking path integral is equal to an {\it a priori} different object -- an explicit thermal trace over the Hilbert space spanned by states produced by the Euclidean gravity path integral. This follows in two ways: (a) if the Hilbert space with two boundaries factorizes into a product of two single boundary Hilbert spaces, as we have previously shown; and (b) via explicit resolution of the trace by a spanning basis of states. We similarly show how a replicated Euclidean gravity path integral with a single periodic boundary computes a Hilbert space trace of powers of the density matrix, explaining why this approach computes the entropy of states entangled between two universes. |
| title | How to Count States in Gravity |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2506.15767 |