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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 細道和夫(Kazuo Hosomichi) | |
dc.contributor.author | Chen-Te Ma | en |
dc.contributor.author | 馬承德 | zh_TW |
dc.date.accessioned | 2021-06-17T06:15:10Z | - |
dc.date.available | 2018-09-17 | |
dc.date.copyright | 2018-09-17 | |
dc.date.issued | 2018 | |
dc.date.submitted | 2018-09-10 | |
dc.identifier.citation | [1] H. Casini, M. Huerta and J. A. Rosabal, “Remarks on entanglement
entropy for gauge fields,” Phys. Rev. D 89, no. 8, 085012 (2014) doi:10.1103/PhysRevD.89.085012 [arXiv:1312.1183 [hep-th]]. [2] C. T. Ma, “Entanglement with Centers,” JHEP 1601, 070 (2016) doi:10.1007/JHEP01(2016)070 [arXiv:1511.02671 [hep-th]]. [3] W. Donnelly, “Decomposition of entanglement entropy in lattice gauge theory,” Phys. Rev. D 85, 085004 (2012) doi:10.1103/PhysRevD.85.085004 [arXiv:1109.0036 [hep-th]]. [4] H. Araki and E. H. Lieb, “Entropy inequalities,” Commun. Math. Phys. 18, 160 (1970). doi:10.1007/BF01646092 E. H. Lieb and M. B. Ruskai, “Proof of the strong subadditivity of quantum-mechanical entropy,” J. Math. Phys. 14, 1938 (1973). doi:10.1063/1.1666274 [5] D. Harlow, “Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture,” JHEP 1601, 122 (2016) doi:10.1007/JHEP01(2016)122 [arXiv:1510.07911 [hep-th]]. [6] D. N. Kabat, “Black hole entropy and entropy of entanglement,” Nucl. Phys. B 453, 281 (1995) doi:10.1016/0550-3213(95)00443-V [hep-th/9503016]. [7] W. Donnelly and A. C. Wall, “Geometric entropy and edge modes of the electromagnetic field,” Phys. Rev. D 94, no. 10, 104053 (2016) doi:10.1103/PhysRevD.94.104053 [arXiv:1506.05792 [hep-th]]. W. Donnelly and A. C. Wall, “Entanglement entropy of electromagnetic edge modes,” Phys. Rev. Lett. 114, no. 11, 111603 (2015) doi:10.1103/PhysRevLett.114.111603 [arXiv:1412.1895 [hep-th]]. [8] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) doi:10.1103/PhysRevLett.96.181602 [hep-th/0603001]. [9] A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP 1308, 090 (2013) doi:10.1007/JHEP08(2013)090 [arXiv:1304.4926 [hep-th]]. [10] H. Casini, “Geometric entropy, area, and strong subadditivity,” Class. Quant. Grav. 21, 2351 (2004) doi:10.1088/0264-9381/21/9/011 [hep-th/0312238]. [11] X. Huang and C. T. Ma, “Analysis of the Entanglement with Centers,” arXiv:1607.06750 [hep-th]. [12] C. T. Ma, “Theoretical Properties of the Entanglement in a Strong Coupling Region,” arXiv:1609.04550 [hep-th]. [13] C. T. Ma, “Discussion of Entanglement Entropy in Quantum Gravity,” Fortsch. Phys. 66, no. 2, 1700095 (2018) doi:10.1002/prop.201700095 [arXiv:1609.03651 [hepth]]. A. Gromov and R. A. Santos, “Entanglement Entropy in 2D Non-abelian Pure Gauge Theory,” Phys. Lett. B 737, 60 (2014) doi:10.1016/j.physletb.2014.08.023 [arXiv:1403.5035 [hep-th]]. [14] J. Dixmier, ”Von Neumann algebras”, North Holland Publishing Company (1981). [15] H. Casini and M. Huerta, “Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice,” Phys. Rev. D 90, no. 10, 105013 (2014) doi:10.1103/PhysRevD.90.105013 [arXiv:1406.2991 [hep-th]]. [16] K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V. B. Scholz and F. Verstraete, “The entanglement of distillation for gauge theories,” Phys. Rev. Lett. 117, no. 13, 131602 (2016) doi:10.1103/PhysRevLett.117.131602 [arXiv:1511.04369 [quant-ph]]. [17] H. Casini and M. Huerta, “Entanglement entropy in free quantum field theory,” J. Phys. A 42, 504007 (2009) doi:10.1088/1751-8113/42/50/504007 [arXiv:0905.2562 [hep-th]]. [18] D. V. Fursaev and S. N. Solodukhin, “On the description of the Riemannian geometry in the presence of conical defects,” Phys. Rev. D 52, 2133 (1995) doi:10.1103/PhysRevD.52.2133 [hep-th/9501127]. [19] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] doi:10.1023/A:1026654312961 [hep-th/9711200]. [20] A. Gromov and R. A. Santos, “Entanglement Entropy in 2D Non-abelian Pure Gauge Theory,” Phys. Lett. B 737, 60 (2014) doi:10.1016/j.physletb.2014.08.023 [arXiv:1403.5035 [hep-th]]. [21] H. Casini, M. Huerta and R. C. Myers, “Towards a derivation of holographic entanglement entropy,” JHEP 1105, 036 (2011) doi:10.1007/JHEP05(2011)036 [arXiv:1102.0440 [hep-th]]. [22] H. Casini and M. Huerta, “Entanglement entropy of a Maxwell field on the sphere,” Phys. Rev. D 93, no. 10, 105031 (2016) doi:10.1103/PhysRevD.93.105031 [arXiv:1512.06182 [hep-th]]. [23] Y. N. Obukhov, “The Geometrical Approach To Antisymmetric Tensor Field Theory,” Phys. Lett. B 109, 195 (1982). doi:10.1016/0370-2693(82)90752-3 E. J. Copeland and D. J. Toms, “Quantized Antisymmetric Tensor Fields and Selfconsistent Dimensional Reduction in Higher Dimensional Space-times,” Nucl. Phys. B 255, 201 (1985). doi:10.1016/0550-3213(85)90134-8 [24] H. Casini and M. Huerta, “Entanglement entropy for the n-sphere,” Phys. Lett. B 694, 167 (2011) doi:10.1016/j.physletb.2010.09.054 [arXiv:1007.1813 [hep-th]]. [25] A. Cappelli and G. D’Appollonio, “On the trace anomaly as a measure of degrees of freedom,” Phys. Lett. B 487, 87 (2000) doi:10.1016/S0370-2693(00)00809-1 [hepth/0005115]. [26] R. Camporesi and A. Higuchi, ”The Plancherel measure for p-forms in real hyperbolic spaces,” J. Geom. Phys. 15, 57 (1994). [27] T. Faulkner, “The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT,” arXiv:1303.7221 [hep-th]. [28] K. Ohmori and Y. Tachikawa, “Physics at the entangling surface,” J. Stat. Mech. 1504, P04010 (2015) doi:10.1088/1742-5468/2015/04/P04010 [arXiv:1406.4167 [hep-th]]. [29] H. Casini and M. Huerta, “A Finite entanglement entropy and the c-theorem,” Phys. Lett. B 600, 142 (2004) doi:10.1016/j.physletb.2004.08.072 [hep-th/0405111]. [30] J. Polchinski, ” String theory. Vol. 1: An introduction to the bosonic string,” Cambridge University Press (2007). [31] M. Headrick, A. Lawrence and M. Roberts, “Bose-Fermi duality and entanglement entropies,” J. Stat. Mech. 1302, P02022 (2013) doi:10.1088/1742- 5468/2013/02/P02022 [arXiv:1209.2428 [hep-th]]. | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/71924 | - |
dc.description.abstract | 規範理論中的糾纏在希爾伯特空間的張量積分解中很難定義,所以我們有興趣從中心去理解希爾伯特空間的非張量積分解中的糾纏,該中心與希爾伯特空間中的所有運算符可交換。我們一開始討論部分跡的運算和不等式以及中心的數學特性,特別是強亞可加性。然後我們考慮哈密頓方法,證明在 2p + 2 維 p-型非相互作用理論中糾纏熵的電和磁的選擇是相同的,並且提出用拉格朗日公式計算中心的糾纏熵。我們使用複制技巧來計算愛因斯坦希爾伯特引力理論中的糾纏熵,並在 2p + 2 維中
p-形式的非相互作用理論中的糾纏熵的通用項重寫成偶數維度 0 形式的非交互作用理論的熵的通用項。特別地,我們討論了二維愛因斯坦 - 希爾伯特引力理論中糾纏熵中的余維數二的面,並討論了糾纏熵的非體積定律與平移不變性之間的關係。最後,我們證明了二維保形場理論在一些特殊情況下的糾纏熵的通用項和互信息不依賴於中心的選擇。 | zh_TW |
dc.description.abstract | Entanglement in gauge theories is hard define in a tensor product decomposition of a Hilbert space so we are interested in understanding the entanglement in a non-tensor product decomposition of a Hilbert space from centers, which commute with all operators in the Hilbert space. We begin with discussing mathematical properties of a partial trace operation and inequalities with the centers, especially the strong subadditivity. Then we consider the Hamiltonian method to show that the electric and magnetic choices of the entanglement entropy in non-interacting theories of the p-form in 2p+2 dimensions are the same and also propose the Lagrangian formulation to compute the entanglement entropy with centers. We use the replica trick to compute the entanglement entropy in the Einstein-Hilbert gravity theory and rewrite universal terms
of the entanglement entropy in non-interacting theory of the p-form in 2p+2 dimensions in terms of universal terms of the entanglement entropy in the non-interacting theory of the 0-form in even dimensions. Especially, we discuss a codimension two surface term in the entanglement entropy in the two dimensional Einstein-Hilbert gravity theory and also discuss a relation between a non-volume law of the entanglement entropy and the translational invariance. Finally, we prove that universal terms of the entanglement entropy and mutual information in some special cases of two dimensional conformal field theory do not depend on a choice of centers. | en |
dc.description.provenance | Made available in DSpace on 2021-06-17T06:15:10Z (GMT). No. of bitstreams: 1 ntu-107-D03222006-1.pdf: 999345 bytes, checksum: 7a3e40c2d1cf1d8ecd6a2749d276c6b1 (MD5) Previous issue date: 2018 | en |
dc.description.tableofcontents | 1. Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Entanglement Entropy with Centers . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Properties of the Entanglement with Centers . . . . . . . . . . . . . . . . 13 2.2.1 Partial Trace Operation . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Decomposition of a Hilbert Space . . . . . . . . . . . . . . . . . . 14 2.2.3 Entanglement Inequalities . . . . . . . . . . . . . . . . . . . . . . 16 3. Computation Metho ds in the Entanglement Entropy with Centers . . . . . . . 19 3.1 The Hamiltonian Formulation in the p-Form Non-Interacting Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 The p-form Abelian Gauge Theory . . . . . . . . . . . . . . . . . 23 3.2 The Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.2 Replica Trick and Conical Method . . . . . . . . . . . . . . . . . 27 3.3 The Einstein Gravity Theory . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Entanglement Entropy in the Einstein Gravity Theory . . . . . . 28 3.3.2 Comments on the Holographic Entanglement Entropy . . . . . . . 28 3.4 Two Dimensional Einstein-Hilbert Theory . . . . . . . . . . . . . . . . . 29 3.5 The Entanglement with Centers in the Non-Interacting Theory of the p-Form 31 3.5.1 Review of Boundary Entanglement Entropy in the Abelian OneForm Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5.2 Boundary Entanglement Entropy in the Massive Non-Interacting Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.3 Boundary Entanglement Entropy in the Ab elian p-Form Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5.4 Bulk Entanglement Entropy in the Massive Non-Interacting Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.5 Bulk Entanglement Entropy in the Abelian p-Form Gauge Theory 39 3.5.6 Universal Terms of the Entanglement Entropy . . . . . . . . . . . 44 4. Non-Volume Law of the Entanglement Entropy . . . . . . . . . . . . . . . . . 48 5. Two Dimensional Conformal Field Theory . . . . . . . . . . . . . . . . . . . . 50 5.1 Mutual Information for Single Interval . . . . . . . . . . . . . . . . . . . 50 5.2 Mutual Information for Multiple Intervals . . . . . . . . . . . . . . . . . 54 5.3 Universal Terms of the Entanglement Entropy . . . . . . . . . . . . . . . 55 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 App endix 61 A. Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 B. Review of the von Neumann Algebra . . . . . . . . . . . . . . . . . . . . . . . 64 B.1 Definition of the von Neumann Algebra . . . . . . . . . . . . . . . . . . . 64 B.2 Topology in the von Neumann Algebra . . . . . . . . . . . . . . . . . . . 66 B.3 The Borel Spaces and Measure . . . . . . . . . . . . . . . . . . . . . . . . 67 B.4 Some Useful Theorems of the von Neumann Algebra . . . . . . . . . . . 67 C. Details of the Strong Subadditivity . . . . . . . . . . . . . . . . . . . . . . . . 71 D. The Average Entanglement Entropy in Free Theory . . . . . . . . . . . . . . . 76 E. Details of the Entanglement Entropy in the Einstein Gravity Theory . . . . . 78 F. AdS5 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 G. Review of Two Dimensional Einstein-Hilbert Action . . . . . . . . . . . . . . . 83 G.1 Curves in the Plane and in the Space . . . . . . . . . . . . . . . . . . . . 83 G.2 Surfaces in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 85 G.3 The First Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . 87 G.4 Curvature of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 G.4.1 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . 88 G.4.2 The Gauss and Weingarten Maps . . . . . . . . . . . . . . . . . . 88 G.4.3 Normal and Geodesic Curvatures . . . . . . . . . . . . . . . . . . 90 G.4.4 The Gauss Equations . . . . . . . . . . . . . . . . . . . . . . . . . 90 G.4.5 Gaussian and Mean Curvatures . . . . . . . . . . . . . . . . . . . 91 G.5 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 93 G.5.1 The Gauss-Bonnet Theorem for Simple Closed Curves . . . . . . 93 G.5.2 The Gauss-Bonnet Theorem for Curvilinear Polygons . . . . . . . 96 G.5.3 The Gauss-Bonnet Theorem for Compact Surfaces . . . . . . . . . 97 G.6 The Gauss and Codazzi-Mainardi Equations . . . . . . . . . . . . . . . . 99 G.7 Two Dimensional Einstein-Hilbert Theory . . . . . . . . . . . . . . . . . 104 H. Review of Two Dimensional Finite Entropy . . . . . . . . . . . . . . . . . . . 106 I. Basics of Two Dimensional Conformal Field Theory . . . . . . . . . . . . . . . 109 J. R´enyi entropy of Multiple Intervals . . . . . . . . . . . . . . . . . . . . . . . . 118 | |
dc.language.iso | en | |
dc.title | 量子場論的糾纏 | zh_TW |
dc.title | Entanglement in Quantum Field Theory | en |
dc.type | Thesis | |
dc.date.schoolyear | 107-1 | |
dc.description.degree | 博士 | |
dc.contributor.oralexamcommittee | 黃宇廷(Yu-tin Huang),溫文鈺(Wen-Yu Wen),賀培銘(Pei-Ming Ho),陳丕燊(Pisin Chen) | |
dc.subject.keyword | 糾纏熵,中心,諾伊曼代數,保形場論, | zh_TW |
dc.subject.keyword | Entanglement Entropy,Centers,von Neumann Algebra,Conformal Field Theory, | en |
dc.relation.page | 125 | |
dc.identifier.doi | 10.6342/NTU201804099 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2018-09-10 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 物理學研究所 | zh_TW |
顯示於系所單位: | 物理學系 |
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