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There is an equivalent formulation of the horizon degrees of freedom in terms of a U (1) gauge theory which is just a gauged fixed version of the SU (2) theory. These developments will be surveyed here. Quantum theory based on either formulation can be used to count the horizon micro-states associated with quantum geometry fluctuations and from this the micro-canonical entropy can be obtained. We shall review the computation in SU (2) formulation. Leading term in the entropy is proportional to horizon area with a coefficient depending on the Barbero-Immirzi parameter which is fixed by matching this result with the Bekenstein-Hawking formula. Remarkably there are corrections beyond the area term, the leading one is logarithm of the horizon area with a definite coefficient −3/2, a result which is more than a decade old now. How the same results are obtained in the equivalent U (1) framework will also be indicated. Over years, this entropy formula has also been arrived at from a variety of other perspectives. In particular, entropy of BTZ black holes in three dimensional gravity exhibits the same logarithmic correction. Even in the String Theory, many black hole models are known to possess such properties. This suggests a possible universal nature of this logarithmic correction.","publication_date":"2012,,","publication_name":"Symmetry, Integrability and Geometry: Methods and Applications","grobid_abstract_attachment_id":"44830242"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Entropy of Quantum Black Holes","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [47139266]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":44830242,"attachmentType":"pdf"}"><img alt="First page of “Entropy of Quantum Black Holes”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/44830242/mini_magick20190213-29118-19vcx9p.png?1550103276" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Entropy of Quantum Black Holes</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="47139266" href="https://independent.academia.edu/RomeshKaul"><img alt="Profile image of Romesh Kaul" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Romesh Kaul</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2012, Symmetry, Integrability and Geometry: Methods and Applications</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">30 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 24498090; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">In the Loop Quantum Gravity, black holes (or even more general Isolated Horizons) are described by a SU (2) Chern-Simons theory. There is an equivalent formulation of the horizon degrees of freedom in terms of a U (1) gauge theory which is just a gauged fixed version of the SU (2) theory. These developments will be surveyed here. Quantum theory based on either formulation can be used to count the horizon micro-states associated with quantum geometry fluctuations and from this the micro-canonical entropy can be obtained. We shall review the computation in SU (2) formulation. Leading term in the entropy is proportional to horizon area with a coefficient depending on the Barbero-Immirzi parameter which is fixed by matching this result with the Bekenstein-Hawking formula. Remarkably there are corrections beyond the area term, the leading one is logarithm of the horizon area with a definite coefficient −3/2, a result which is more than a decade old now. How the same results are obtained in the equivalent U (1) framework will also be indicated. Over years, this entropy formula has also been arrived at from a variety of other perspectives. In particular, entropy of BTZ black holes in three dimensional gravity exhibits the same logarithmic correction. Even in the String Theory, many black hole models are known to possess such properties. 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Borja</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics: Conference Series, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We calculate the black hole entropy in Loop Quantum Gravity as a function of the horizon area and provide the exact formula for the leading and sub-leading terms. 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More precisely, we compute the leading term and the logarithmic correction of both the spherically symmetric and the distorted SU (2) black holes. Contrary to what has been done in previous works, we have to take into account "quantum corrections" in our framework in the sense that the level k of the Chern-Simons theory which describes the black hole is finite and not sent to infinity. Therefore, the new results presented here allow for the computation of the entropy in models where the quantum group corrections are important. * Fédération Denis Poisson Orléans-Tours, CNRS/UMR 6083 † Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var; laboratoire afiliéà la FRUMAM (FR 2291)</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The SU(2) black hole entropy revisited","attachmentId":45695390,"attachmentType":"pdf","work_url":"https://www.academia.edu/25380075/The_SU_2_black_hole_entropy_revisited","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/25380075/The_SU_2_black_hole_entropy_revisited"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="7011515" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/7011515/Quantum_geometry_and_microscopic_black_hole_entropy">Quantum geometry and microscopic black hole entropy</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="11935389" href="https://unam.academia.edu/AlejandroCorichi">Alejandro Corichi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Classical and Quantum Gravity, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">Quantum black holes within the loop quantum gravity (LQG) framework are considered. The number of microscopic states that are consistent with a black hole of a given horizon area $A_0$ are counted and the statistical entropy, as a function of the area, is obtained for $A_0$ up to $550 l^2_{\rm Pl}$. The results are consistent with an asymptotic linear relation and a logarithmic correction with a coefficient equal to -1/2. The Barbero-Immirzi parameter that yields the asymptotic linear relation compatible with the Bekenstein-Hawking entropy is shown to coincide with a value close to $\gamma=0.274$, which has been previously obtained analytically. However, a new and oscillatory functional form for the entropy is found for small, Planck size, black holes that calls for a physical interpretation.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Quantum geometry and microscopic black hole entropy","attachmentId":33672474,"attachmentType":"pdf","work_url":"https://www.academia.edu/7011515/Quantum_geometry_and_microscopic_black_hole_entropy","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/7011515/Quantum_geometry_and_microscopic_black_hole_entropy"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="24498086" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/24498086/Quantum_black_hole_entropy">Quantum black hole entropy</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="47139266" href="https://independent.academia.edu/RomeshKaul">Romesh Kaul</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physics Letters B, 1998</p><p class="ds-related-work--abstract ds2-5-body-sm">We derive an exact formula for the dimensionality of the Hilbert space of the boundary states of SU (2) Chern-Simons theory, which, according to the recent work of Ashtekar et al, leads to the Bekenstein-Hawking entropy of a four dimensional Schwarzschild black hole. Our result stems from the relation between the (boundary) Hilbert space of the Chern-Simons theory with the space of conformal blocks of the Wess-Zumino model on the boundary 2sphere. The issue of the Bekenstein-Hawking (B-H) [1], [2] entropy of black holes has been under intensive scrutiny for the last couple of years, following the derivation of the entropy of certain extremal charged black hole solutions of toroidally compactified heterotic string and also type IIB superstring from the underlying string theories [3], [4]. In the former case of the heterotic string, the entropy was shown to be proportional to the area of the 'stretched' horizon of the corresponding extremal black hole, while in the latter case it turned out to be precisely the B-H result. The latter result was soon generalized to a large number of four and five dimensional black holes of type II string theory and M-theory (see [5] for a review), all of which could be realized as certain D-brane configurations and hence saturated the BPS bound. Unfortunately, the simplest black hole of all, the four *</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Quantum black hole entropy","attachmentId":44830244,"attachmentType":"pdf","work_url":"https://www.academia.edu/24498086/Quantum_black_hole_entropy","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/24498086/Quantum_black_hole_entropy"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="60494778" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/60494778/Black_Hole_Entropy_and_Quantum_Gravity">Black Hole Entropy and Quantum Gravity</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="193269250" href="https://independent.academia.edu/ParthasarathiMajumdar2">Parthasarathi Majumdar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Indian J Phys, 1998</p><p class="ds-related-work--abstract ds2-5-body-sm">An elementary introduction is given to the problem of black hole entropy as formulated by Bekenstein and Hawking. The information theoretic basis of Bekenstein's formulation is briefly reviewed and compared with Hawking's approach. The issue of calculating the entropy by actual counting of microstates is taken up next within two currently popular approaches to quantum gravity, viz., string theory and canonical quantum gravity. The treatment of the former assay is confined to a few remarks, mainly of a critical nature, while some of the computational techniques of the latter approach are elaborated. We conclude by trying to find commonalities between these two rather disparate directions of work.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Black Hole Entropy and Quantum Gravity","attachmentId":73914720,"attachmentType":"pdf","work_url":"https://www.academia.edu/60494778/Black_Hole_Entropy_and_Quantum_Gravity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/60494778/Black_Hole_Entropy_and_Quantum_Gravity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="22225859" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22225859/Statistics_holography_and_black_hole_entropy_in_loop_quantum_gravity">Statistics, holography, and black hole entropy in loop quantum gravity</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32309944" href="https://independent.academia.edu/AmitavaGhosh7">Amitava Ghosh</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review D, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">In loop quantum gravity the quantum states of a black hole horizon are produced by point-like discrete quantum geometry excitations (or punctures) labelled by spin j. The excitations possibly carry other internal degrees of freedom also, and the associated quantum states are eigenstates of the area A operator. On the other hand, the appropriately scaled area operator A/(8π ) is also the physical Hamiltonian associated with the quasilocal stationary observers located at a small distance from the horizon. Thus, the local energy is entirely accounted for by the geometric operator A. We assume that:</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Statistics, holography, and black hole entropy in loop quantum gravity","attachmentId":42877095,"attachmentType":"pdf","work_url":"https://www.academia.edu/22225859/Statistics_holography_and_black_hole_entropy_in_loop_quantum_gravity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/22225859/Statistics_holography_and_black_hole_entropy_in_loop_quantum_gravity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="27223463" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/27223463/Entropy_and_area_of_black_holes_in_loop_quantum_gravity">Entropy and area of black holes in loop quantum gravity</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51272579" href="https://independent.academia.edu/IosifKhriplovich">Iosif Khriplovich</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physics Letters B, 2002</p><p class="ds-related-work--abstract ds2-5-body-sm">Simple arguments related to the entropy of black holes strongly constrain the spectrum of the area operator for a Schwarzschild black hole in loop quantum gravity. In particular, this spectrum is fixed completely by the assumption that the black hole entropy is maximum. Within the approach discussed, one arrives in loop quantum gravity at a quantization rule with integer quantum numbers n for the entropy and area of a black hole.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Entropy and area of black holes in loop quantum gravity","attachmentId":47481046,"attachmentType":"pdf","work_url":"https://www.academia.edu/27223463/Entropy_and_area_of_black_holes_in_loop_quantum_gravity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/27223463/Entropy_and_area_of_black_holes_in_loop_quantum_gravity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":44830242,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":44830242,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_44830242" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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data-entity-id="60494787" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/60494787/Black_Hole_Entropy_Certain_Quantum_Features">Black Hole Entropy: Certain Quantum Features</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="193269250" href="https://independent.academia.edu/ParthasarathiMajumdar2">Parthasarathi Majumdar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (In 3 Volumes), 2002</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Black Hole Entropy: Certain Quantum 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ds2-5-body-sm ds2-5-body-link" data-author-id="3849846" href="https://univ-amu.academia.edu/CarloRovelli">Carlo Rovelli</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Black hole entropy from loop quantum gravity","attachmentId":32568366,"attachmentType":"pdf","work_url":"https://www.academia.edu/5444268/Black_hole_entropy_from_loop_quantum_gravity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/5444268/Black_hole_entropy_from_loop_quantum_gravity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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