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LOOP QUANTUM GRAVITY with COSMOLOGICAL CONSTANT By LOOP QUANTUM GRAVITY WITH COSMOLOGICAL CONSTANT by Zichang Huang A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, FL May 2019 Copyright 2019 by Zichang Huang ii Loop Quantum Gravity with Cosmological Constant by Zichang Huang This dissertation was prepared under the direction of the candidate's dissertation advisor, Dr. Muxin Han, Department of Physics, and has been approved by all members of the supervisory committee. It was submitted to the faculty of the Charles E. Schmidt College of Science and was accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Muxin Han, Ph. Dissertatio Ad isor uc T. Wille, Ph.D. Chair, Department of Physics J~h.D: ~::;...=_~~;;,...oz:;___.__ ~___ -"_~ ____ Erik~~ Lundberg, .D. At · . ~edini, Ph.D. ean, Charles E. Schmtdt College Of Science Khaled Sobhan, Ph.D. Interim Dean, Graduate College 111 ACKNOWLEDGEMENTS The dissertation here has been supported by many people from different continents all over the world. I would use out of my word to thank all of them. But at first, I would like to show my graceful thanks to my advisor Dr. Muxin Han and my collaborator Dr. Antonia Zipfel. I have learned a lot from them. They have shown me a attractive world about the quantum gravity, and widen my horizon. Then I would like to thank the faculties in Florida Atlantic University physics de- partment who gave there fully support to my dissertation. Specially thanks to Dr. Jonathan Engle, Dr. Chirs Beetle, Dr. Warner Miller, Dr.Wolfgang Tichy, Dr. Luc T. Wille and Dr. Armin Fuchs, etc. Also thanks to Dr. Eric Lundberg from Math department of Florida Atlantic University for being my committee. During my stay in Florida Atlantic University, the Carlo Rovelli’s group in CPT- Marseille has also provided a great help for my research. Here is my heartfelt thanks to Prof. Carlo Rovelli, Prof. Simone Speziale, Prof. Alejandro Perez and my friend Hong- guang Liu. I would also like to thank the Prof. Canbin Liang and Prof. Yongge Ma from Beijing Normal University. Their patient guide lead me to the realm of gravity physics. Finally, I would like to thank my father Weiruo Huang and my mother Ni Dong. Their love and encouragement has always been an important support of my live. iv ABSTRACT Author: Zichang Huang Title: Loop Quantum Gravity with Cosmological Constant Institution: Florida Atlantic University Dissertation Advisor: Dr. Muxin Han Degree: Doctor of Philosophy Year: 2019 The spin-foam is a covariant path-integral style approaching to the quantization of the gravity. There exist several spin-foam models of which the most successful one is the Engle-Pereira-Rovelli-Levine/Freidel-Krasnov (EPRL-FK) model. Using the EPRL- FK model people are able to calculate the transition amplitude and the n-point functions of 4D geometry(both Euclidean and Lorentzian) surrounding by a given triangulated 3D ge- ometry. The semi-classical limit of the EPRL-FK amplitude reproduces discrete classical gravity under certain assumptions, which shows that the EPRLFK model can be under- stood as UV completion of general relativity. On the other hand, it is very hard to dene a continuum limit and couple a cosmological constant to the EPRL-FK model. In this dissertation, we addressed the problems about continuum limit and coupling a cosmological constant to the EPRL-FK model. Followed by chapter one as a brief in- troduction of the loop quantum gravity and EPRL-FK model, chapter two introduces our work about demonstrating (for the first time) that smooth curved spacetime geometries satisfying Einstein equation can emerge from discrete spin-foam models under an appro- priate low energy limit, which corresponds to a semi-classical continuum limit of spin-foam models. In chapter three, we bring in the cosmological constant into the spin-foam model v by coupling the SL(2; C) Chern-Simons action with the EPRL action, and find that the quantum simplicity constraint is realized as the 2d surface defect in SL(2; C)Chern-Simons theory in the construction of spin-foam amplitudes. In chapter four, we present a way to describe the twisted geometry with cosmological constant whose corresponding quantum states can forms the Hilbert space of the loop quantum gravity with cosmological constant. In chapter five, we introduced a new definition of the graviton propagator, and calculate its semi-classical limit in the contents of spin-foam model with the cosmological constant. Finally the chapter six will be a outlook for my future work. vi To my parents, and to my friends LOOP QUANTUM GRAVITY WITH COSMOLOGICAL CONSTANT List of Figures ............................... xi 1 Introduction ................................ 1 2 Semi-Classical Continuum Limit and Emergent Gravity from Spin-Foam model .................................... 6 2.1 Introduction..................................6 2.2 Spin foam models:..............................7 2.3 Large spin analysis:..............................8 2.4 Semiclassical continuum limit:........................ 13 2.5 Emergent Linearized gravity:......................... 15 2.6 Conclusion.................................. 16 3 Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Com- plex Chern-Simons Theory ......................... 17 3.1 Introduction.................................. 17 3.2 Simplicity Constraint and Curved Tetrahedron................ 21 3.3 Quantization of Flat Connection and Simplicity Constraint......... 24 3 3.4 SL(2; C) Chern-Simons Theory on S n Γ5 .................. 26 3.5 SL(2; C) Chern-Simons theory on M3 with Surface Defect......... 30 3.6 A Field-Theoretic Description of the Surface Defect............. 37 3.7 Surface Degree of Freedom.......................... 41 3.8 Conclusion.................................. 43 viii 4 SU(2) Flat Connection on Riemann Surface and 3D Twisted Geometry with Cosmological Constant ........................... 45 4.1 Introduction.................................. 45 4.2 Flat Connection on Riemann Surface and Twisted Geometry........ 48 4.2.1 From Graph to Riemann Surface................... 48 4.2.2 Flat Connection on Riemann Surface and Curved Tetrahedron... 51 4.2.3 Gluing 4-holed Spheres........................ 53 4.2.4 Relation with Twisted Geometry.................. 54 4.3 Geometric interpretation of Twisted Geometry................ 56 4.3.1 Exponentiated Flux.......................... 58 4.3.2 Twist Angle and Extrinsic Curvature................. 59 4.3.3 More General Choices of base points................ 62 4.4 Symplectic Structure............................. 68 4.5 Quantization.................................. 69 5 Spin foam propagator: A new perspective to include the cosmological constant 71 5.1 Introduction.................................. 71 5.2 Different proposals for a graviton propagator................. 74 5.2.1 Standard proposal and conflicts with a cosmological constant... 74 5.2.2 Perturbative truncated metric..................... 76 5.3 Spin foam model with cosmological constant................ 79 5.3.1 2-point function............................ 82 5.4 Asymptotic Limit of the 2-point function................... 84 5.4.1 Calculation scheme.......................... 85 5.4.2 First non-vanishing order of the graviton propagator........ 86 5.5 Summary................................... 88 6 Outlooks .................................. 91 6.1 Twistor Theory and Twisted Geometry with Cosmological Constant.... 91 ix 6.2 Emergent Graviton Propagator form Spin Foam............... 92 6.3 More Applications of the Semi-Classical Continuum Limit of the Spin- Foam model.................................. 92 6.4 Numerical Spinfoam and Tensor Network.................. 93 Appendices ................................. 95 A Appendices For Chapter Two......................... 96 A.1 Spin Foam Models (SFMs) and Tensor Networks.......... 96 A.2 Large Spin Analysis......................... 99 A.3 Expansion of the linearized theory.................. 106 A.4 Semiclassical Continuum Limit (SCL)................ 108 A.5 Convergence to Smooth Geometry.................. 111 A.6 Some Topological Properties of the Triangulation.......... 115 B Appendices for Chapter Four......................... 119 B.1 Complex Fenchel-Nielsen (FN) Coordinate............. 119 C Appendices for Chapter Five......................... 123 C.1 The area of the triangle in the geometry with a cosmological constant123 C.2 Calculation of the saddle point and the Hessian........... 124 C.3 Hessians............................... 127 C.4 Chern-Simons Propagator for Non-Compact Gauge Group..... 128 Bibliography ................................ 131 x LIST OF FIGURES 1.1 Cube of theoretical physics..........................1 2.1 (a) The 5-valent vertex in a 4-simplex illustrates a rank-5 tensor jAσi. Glu- ing 4-simplices σ in K gives a tensor network TN(K; J~), where each link associates to a maximally entangled state of a pair of iτs. (b) A triangu- lation of the hypercube. The 4d hypercubic lattice with the triangulated hypercube makes K. (c) An illustration of the neighborhood N (the re- gion bounded by blue dashed lines) in the space of J~. The red curve il- ~ ~ lustrates MRegge, including J(`) as the perturbation of J(`˚). The black and blue arrows are basis vectorse ˆi(`) and @J~(`)=@`, transverse and tangent to MRegge.....................................9 3 3.1 Γ5-graph embedded in S ..........................
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