Constrained HRT Surfaces and their Entropic Interpretation
Abstract Consider two boundary subregions A and B that lie in a common boundary Cauchy surface, and consider also the associated HRT surface γ B for B. In that context, the constrained HRT surface γ A:B can be defined as the codimension-2 bulk surface anchored to A that is obtained by a maximin cons...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-02-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP02(2024)151 |
| _version_ | 1797276054434349056 |
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| author | Xi Dong Donald Marolf Pratik Rath |
| author_facet | Xi Dong Donald Marolf Pratik Rath |
| author_sort | Xi Dong |
| collection | DOAJ |
| description | Abstract Consider two boundary subregions A and B that lie in a common boundary Cauchy surface, and consider also the associated HRT surface γ B for B. In that context, the constrained HRT surface γ A:B can be defined as the codimension-2 bulk surface anchored to A that is obtained by a maximin construction restricted to Cauchy slices containing γ B . As a result, γ A:B is the union of two pieces, γ A : B B $$ {\gamma}_{A:B}^B $$ and γ A : B B ¯ $$ {\gamma}_{A:B}^{\overline{B}} $$ lying respectively in the entanglement wedges of B and its complement B ¯ $$ \overline{B} $$ . Unlike the area A γ A $$ \mathcal{A}\left({\gamma}_A\right) $$ of the HRT surface γ A , at least in the semiclassical limit, the area A γ A : B $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ of γ A:B commutes with the area A γ B $$ \mathcal{A}\left({\gamma}_B\right) $$ of γ B . To study the entropic interpretation of A γ A : B $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ , we analyze the Rényi entropies of subregion A in a fixed-area state of subregion B. We use the gravitational path integral to show that the n ≈ 1 Rényi entropies are then computed by minimizing A γ A $$ \mathcal{A}\left({\gamma}_A\right) $$ over spacetimes defined by a boost angle conjugate to A γ B $$ \mathcal{A}\left({\gamma}_B\right) $$ . In the case where the pieces γ A : B B $$ {\gamma}_{A:B}^B $$ and γ A : B B ¯ $$ {\gamma}_{A:B}^{\overline{B}} $$ intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by A γ A : B 4 G $$ \frac{\mathcal{A}\left({\gamma}_{A:B}\right)}{4G} $$ . We discuss how the n ≈ 1 Rényi entropy differs from the von Neumann entropy due to a lack of commutativity of the n → 1 and G → 0 limits. We also discuss how the behaviour changes as a function of the width of the fixed-area state. Our results are relevant to some of the issues associated with attempts to use standard random tensor networks to describe time dependent geometries. |
| first_indexed | 2024-03-07T15:22:49Z |
| format | Article |
| id | doaj.art-4d0dfd0d98ba4591be9b9a92451c013f |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-07T15:22:49Z |
| publishDate | 2024-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-4d0dfd0d98ba4591be9b9a92451c013f2024-03-05T17:29:02ZengSpringerOpenJournal of High Energy Physics1029-84792024-02-012024212110.1007/JHEP02(2024)151Constrained HRT Surfaces and their Entropic InterpretationXi Dong0Donald Marolf1Pratik Rath2Department of Physics, University of CaliforniaDepartment of Physics, University of CaliforniaDepartment of Physics, University of CaliforniaAbstract Consider two boundary subregions A and B that lie in a common boundary Cauchy surface, and consider also the associated HRT surface γ B for B. In that context, the constrained HRT surface γ A:B can be defined as the codimension-2 bulk surface anchored to A that is obtained by a maximin construction restricted to Cauchy slices containing γ B . As a result, γ A:B is the union of two pieces, γ A : B B $$ {\gamma}_{A:B}^B $$ and γ A : B B ¯ $$ {\gamma}_{A:B}^{\overline{B}} $$ lying respectively in the entanglement wedges of B and its complement B ¯ $$ \overline{B} $$ . Unlike the area A γ A $$ \mathcal{A}\left({\gamma}_A\right) $$ of the HRT surface γ A , at least in the semiclassical limit, the area A γ A : B $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ of γ A:B commutes with the area A γ B $$ \mathcal{A}\left({\gamma}_B\right) $$ of γ B . To study the entropic interpretation of A γ A : B $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ , we analyze the Rényi entropies of subregion A in a fixed-area state of subregion B. We use the gravitational path integral to show that the n ≈ 1 Rényi entropies are then computed by minimizing A γ A $$ \mathcal{A}\left({\gamma}_A\right) $$ over spacetimes defined by a boost angle conjugate to A γ B $$ \mathcal{A}\left({\gamma}_B\right) $$ . In the case where the pieces γ A : B B $$ {\gamma}_{A:B}^B $$ and γ A : B B ¯ $$ {\gamma}_{A:B}^{\overline{B}} $$ intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by A γ A : B 4 G $$ \frac{\mathcal{A}\left({\gamma}_{A:B}\right)}{4G} $$ . We discuss how the n ≈ 1 Rényi entropy differs from the von Neumann entropy due to a lack of commutativity of the n → 1 and G → 0 limits. We also discuss how the behaviour changes as a function of the width of the fixed-area state. Our results are relevant to some of the issues associated with attempts to use standard random tensor networks to describe time dependent geometries.https://doi.org/10.1007/JHEP02(2024)151AdS-CFT CorrespondenceGauge-Gravity Correspondence |
| spellingShingle | Xi Dong Donald Marolf Pratik Rath Constrained HRT Surfaces and their Entropic Interpretation Journal of High Energy Physics AdS-CFT Correspondence Gauge-Gravity Correspondence |
| title | Constrained HRT Surfaces and their Entropic Interpretation |
| title_full | Constrained HRT Surfaces and their Entropic Interpretation |
| title_fullStr | Constrained HRT Surfaces and their Entropic Interpretation |
| title_full_unstemmed | Constrained HRT Surfaces and their Entropic Interpretation |
| title_short | Constrained HRT Surfaces and their Entropic Interpretation |
| title_sort | constrained hrt surfaces and their entropic interpretation |
| topic | AdS-CFT Correspondence Gauge-Gravity Correspondence |
| url | https://doi.org/10.1007/JHEP02(2024)151 |
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