{"title":"黑洞的纠缠熵","authors":"Sergey N. Solodukhin","doi":"10.12942/lrr-2011-8","DOIUrl":null,"url":null,"abstract":"<p>The entanglement entropy is a fundamental quantity, which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff, which regulates the short-distance correlations. The geometrical nature of entanglement-entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black-hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in four and six dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s <i>brick-wall</i> model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields, which non-minimally couple to gravity, is emphasized. The holographic description of the entanglement entropy of the blackhole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.</p>","PeriodicalId":686,"journal":{"name":"Living Reviews in Relativity","volume":"14 1","pages":""},"PeriodicalIF":26.3000,"publicationDate":"2011-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.12942/lrr-2011-8","citationCount":"461","resultStr":"{\"title\":\"Entanglement Entropy of Black Holes\",\"authors\":\"Sergey N. Solodukhin\",\"doi\":\"10.12942/lrr-2011-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The entanglement entropy is a fundamental quantity, which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff, which regulates the short-distance correlations. The geometrical nature of entanglement-entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black-hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in four and six dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s <i>brick-wall</i> model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields, which non-minimally couple to gravity, is emphasized. The holographic description of the entanglement entropy of the blackhole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.</p>\",\"PeriodicalId\":686,\"journal\":{\"name\":\"Living Reviews in Relativity\",\"volume\":\"14 1\",\"pages\":\"\"},\"PeriodicalIF\":26.3000,\"publicationDate\":\"2011-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.12942/lrr-2011-8\",\"citationCount\":\"461\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Living Reviews in Relativity\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.12942/lrr-2011-8\",\"RegionNum\":2,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, PARTICLES & FIELDS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Living Reviews in Relativity","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.12942/lrr-2011-8","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, PARTICLES & FIELDS","Score":null,"Total":0}
The entanglement entropy is a fundamental quantity, which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff, which regulates the short-distance correlations. The geometrical nature of entanglement-entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black-hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in four and six dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s brick-wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields, which non-minimally couple to gravity, is emphasized. The holographic description of the entanglement entropy of the blackhole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.
期刊介绍:
Living Reviews in Relativity is a peer-reviewed, platinum open-access journal that publishes reviews of research across all areas of relativity. Directed towards the scientific community at or above the graduate-student level, articles are solicited from leading authorities and provide critical assessments of current research. They offer annotated insights into key literature and describe available resources, maintaining an up-to-date suite of high-quality reviews, thus embodying the "living" aspect of the journal's title.
Serving as a valuable tool for the scientific community, Living Reviews in Relativity is often the first stop for researchers seeking information on current work in relativity. Written by experts, the reviews cite, explain, and assess the most relevant resources in a given field, evaluating existing work and suggesting areas for further research.
Attracting readers from the entire relativity community, the journal is useful for graduate students conducting literature surveys, researchers seeking the latest results in unfamiliar fields, and lecturers in need of information and visual materials for presentations at all levels.