黑洞熵：彎曲時空量子場論，全息對偶與量子信息

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研究生：	連智華 Lien, Jr-Hua
論文名稱：	黑洞熵：彎曲時空量子場論，全息對偶與量子信息 Black Hole Entropy: Quantum Field in Curved Spacetime, Holographic Duality and Quantum Information
指導教授：	朱創新 Chu, Chong-Sun
口試委員:	陳江梅 Chen, Chiang-Mei 溫文鈺 Wen, Wen-Yu
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2019
畢業學年度：	107
語文別：	英文
論文頁數：	106
中文關鍵詞：	黑洞、黑洞熵、霍金輻射、彎曲時空量子場、全息對偶、量子信息
外文關鍵詞：	black hole, black hole entropy, Hawking radiation, quantum field in curved spacetime, holographic duality, quantum information
相關次數：	點閱：2 下載：0
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

理論物理學中最困難和最根本性的未解決問題之一是量子重力，許多人認為研究黑洞的量子性質是其中一條通往量子重力的道路，而黑洞熵是在黑洞的量子性質中最重要的研究方向之一。
雖然黑洞熵是一個宏觀物理量，但普遍認為它應該具有量子層面的起源，過去已有許多理論試圖給出黑洞熵的微觀解釋，我們也希望透過研究黑洞熵的量子起源，來幫助人們理解時空的量子結構以及量子重力。
本篇論文主要回顧了幾個關於理論物理學中黑洞熵的重要研究，包括廣義相對論中的黑洞力學、霍金-貝肯斯坦溫度及其相對應的黑洞熵、磚牆法計算出的黑洞熵，接著回顧近年來由弦論所啟發，關於全像對偶架構下的黑洞熵計算，如極端克爾黑洞和共形場對應、黑洞時空相關的反德西特時空與共形場對應、黑洞時空中的量子信息，並計算黑洞時空中量子場的糾纏熵，最後給出動態黑洞下量子糾纏熵隨黑洞的變化關係。
此外，熵在經典和量子信息理論中具有自然且漂亮的意義，並已長久用於電腦科學和通信技術，然而，熵在這些面向的意義似乎尚未具體的引入到黑洞物理學的研究中。我們將古典與量子信息理論中關於熵的內容作為黑洞熵回顧的一部分。

One of the most difficult and fundamental open problems in theoretical physics is quantum gravity, many people believe that the study of the quantum properties of the black hole is a road toward quantum gravity. The black hole entropy is one of the most important topics in this direction.
There were many theories to give the microscopic explanation of the black hole entropy, although the black hole entropy as a macroscopic physical quantity, we believe that it should have a quantum level origin so that we could understand the quantum structure of spacetime.
In this thesis, we essentially review several important research about the black hole entropy in the history of physics, including the classical black hole mechan- ics theory of general relativity, the Hawking temperature, the Hawking-Bekenstein entropy, the brick wall method, and then we review the related modern results of the black hole entropy in the context of holographic dualities, such as black hole and conformal field correspondence (Kerr/CFT), and AdS/CFT correspondence and quantum information related to the black hole spacetime. In the last chapter, we calculate the entanglement entropy of quantum field in the black hole spacetime and give the relation for dynamical black hole and entanglement entropy.
In addition, the entropy has nature explanation in classical and quantum in- formation theory and used in the computer science and communication technology. However, the interpretation of this aspect for entropy seems not to appear in the study of black hole physics yet. We put those materials about information theory in this thesis as a part of the review about the black hole entropy.

1 Introduction . . . . . ..7
2 Black Hole Temperature form Quantum Fields and Near Horizon Geometry. . . . . .. 9
2.1 BlackHoleMechanicsinGeneralRelativity . . . . . . . . . . . . . . 9
2.1.1 Bekenstein-Smarr Formula for Stationary Black Hole . . . . .9
2.1.2 BlackHoleMechanics ...................... 13
2.2 BlackHoleTemperature ......................... 15
2.2.1 Hawking Effect of Quantum Fields in Curved Spacetime . . . 15
2.2.2 Temperature for Near Horizon Geometry of Non-Extremal StaticBlackHole............... 21
2.2.3 Near Horizon Geometry of Extremal Reissner-Nordstrom Black Hole . . . .. . . . . . .  28
3 Black Hole Entropy and Holographic Duality . . . . . ..32
3.1 Bekenstein-Hawking Entropy From Classical Analogy . . . . . . . . . 32
3.2 Black Hole Entropy From t’Hooft Brick-Wall Method . . . . . . . . . 34
3.3 Kerr/CFTCorrespondence........................ 38
3.3.1 Thermodynamic Quantities of Kerr and Extremal Kerr Black Hole  . . . . . 39
3.3.2 Near-Horizon Geometry for Extremal Kerr Black Hole and warpedAdS3 ...... 42
3.3.3 EntropyforExtremalKerrBlackHole . . . . . . . . . . . . . 45
3.4 Ryu-Takayanagi Formula and Black Hole Microstates . . . . . . . . . 48
4 Quantum Information Near Black Hole 54
4.1 Entanglement Entropy Near Schwarzschild Black Hole . . . . . .. . 54
4.2 EntanglementEntropyforVaidyaBlackHole . . . . . . .. . . . . 60
5 Summary and Future Work . . . . . .. 64
A Anti-de Sitter Spacetime and Conformal Boundary . . . . . ..66
A.1 CoordinatesofAdS............................ 67
A.2 ConformalBoundaryofAdS....................... 72
B Warped Geometry 75
B.1 WarpedAdS3andAdS2×S1 ...................... 75
B.2 PoincareplaneandSL(2,R) ...................... 77
B.3 SL(2,C)→SO(1,3)andSL(2,R)→SO(1,2) . . . . . . . . . . . . . 78
B.4 IsometryGroupofAdSandItsHomomorphism . . . . . . . . . . 80
C Covariant Phase Space Formalism 81
C.1 CanonicalSurfaceCharges........................ 81
C.2 ChargeAlgebra .............................. 84
C.3 Noether-WaldSurfaceChargeinGravitation . . . . . . . . . . . 85
D Entropy in Classical and Quantum Information 87
D.1 Shannon Entropy, Classical Typicality and Shannon’s Source Coding Theorem.................................. 87
D.2 Algorithmic Entropy and Kolmogorov Complexity . . . . . . . . . . . 92
D.3 Von Neumann Entropy, Quantum Typicality and Schumacher’s Quantum CodingTheorem........................... 94
D.4 Information Entropy and Thermodynamic Entropy . . . . . . . . . . 100
D.5 Holevo Information............................101
Bibliography..................... 103

                                

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