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Quantum Dynamics of Loop Quantum Gravity Muxin Han Louisiana State University and Agricultural and Mechanical College, [email protected]

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Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2007 Quantum dynamics of loop quantum gravity Muxin Han Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Physical Sciences and Mathematics Commons Recommended Citation Han, Muxin, "Quantum dynamics of loop quantum gravity" (2007). LSU Master's Theses. 3521. https://digitalcommons.lsu.edu/gradschool_theses/3521 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. QUANTUM DYNAMICS OF LOOP QUANTUM GRAVITY A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Master of Science in The Department of Physics and Astronomy by Muxin Han B.S., Beijing Normal University, Beijing, 2005 May 2007 Acknowledgements First of all, I am grateful to Dr. Jorge Pullin for his advise and many corrections of this thesis, and to Dr. Jonathan Dowling for all his kind help in these two years. I also would like to thank Dr. Hwang Lee for his kind support and being a member of my committee. I would like to thank all the people who have discussed issues with me concerning the subject in the thesis. They are: Dr. Abhay Ashtekar, Dr. Lai-Him Chan, Dr. Weiming Huang, Dr. Jerzy Lewandowski, Dr. Yongge Ma, Dr. Andrzej Okolow, Dr. Jorge Pullin, Dr. Carlo Rovelli, Dr. Thomas Thiemann, Dr. Dmitry Uskov, Dr. Robert M. Wald, and Dr. Hongbao Zhang. This work is supported by the assistantship of LSU, the Horace Hearne Institute for Theo- retical Physics at LSU, and funding from Advanced Research and Development Activity. ii Table of Contents Acknowledgements :::::::::::::::::::::::::::::::::::: ii Abstract ::::::::::::::::::::::::::::::::::::::::::: iv 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Motivation of Quantum Gravity . 1 1.2 Purpose of Loop Quantum Gravity . 2 2 Classical Framework ::::::::::::::::::::::::::::::::: 4 2.1 Lagrangian Formalism . 4 2.2 Hamiltonian Formalism . 5 3 Foundations of Loop Quantum Gravity ::::::::::::::::::::::: 11 3.1 General Programme for Algebraic Quantization . 11 3.2 Quantum Configuration Space . 17 3.3 Cylindrical Functions on Quantum Configuration Space . 20 3.4 Loop Quantum Kinematics . 21 3.5 Spin-network Decomposition of Kinematical Hilbert Space . 25 3.6 Quantum Riemannian Geometry . 31 4 Implementation of Quantum Constraints :::::::::::::::::::::: 34 4.1 Solutions of Quantum Gaussian Constraint . 34 4.2 Solutions of Quantum Diffeomorphism Constraint . 35 4.3 Hamiltonian Constraint Operator . 37 4.4 Master Constraint Programme . 44 5 Quantum Matter Field on a Quantum Background ::::::::::::::::: 51 5.1 Polymer-like Representation of a Scalar Field . 51 5.2 Diffeomorphism Invariant Hamiltonian of a Scalar Field . 54 5.3 Hamiltonian Constraint Equation for the Coupled System . 61 5.4 Master Constraint for the Coupled System . 63 6 The Semiclassical Limit of Quantum Dynamics ::::::::::::::::::: 68 6.1 The Construction of Coherent States . 68 6.2 Algebraic Quantum Gravity Approach . 74 7 Conclusion and Discussion :::::::::::::::::::::::::::::: 79 References ::::::::::::::::::::::::::::::::::::::::: 81 Vita ::::::::::::::::::::::::::::::::::::::::::::: 91 iii Abstract In the last 20 years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. The aim of loop quantum gravity is to construct a mathematically rigorous, background independent, nonperturbative quantum theory for the Lorentzian gravitational field on a four-dimensional manifold. In this approach, the principles of quantum mechanics are combined with those of general relativity naturally. Such a combination provides us a picture of ”quantum Riemannian geometry”, which is discrete at a fundamental scale. In the investigation of quantum dynamics, the classical expressions of constraints are quantized as operators. The quantum evolution is contained in the solutions of the quantum constraint equations. On the other hand, the semi-classical analysis has to be carried out in order to test the semiclassical limit of the quantum dynamics. In this thesis, the structure of the dynamical theory in loop quantum gravity is presented ped- agogically. The outline is as follows: first we review the classical formalism of general relativity as a dynamical theory of connections. Then the kinematical Ashtekar-Isham-Lewandowski rep- resentation is introduced as a foundation of loop quantum gravity. We discuss the construction of a Hamiltonian constraint operator and the master constraint programme, for both the cases of pure gravity and matter field coupling. Finally, some strategies are discussed concerning testing the semiclassical limit of the quantum dynamics. iv 1 Introduction 1.1 Motivation of Quantum Gravity The current view of physics is that there exist four fundamental interactions: strong interaction, weak interaction, electromagnetic interaction and gravitational interaction. The description for the former three kinds of forces is quantized in the well-known standard model. The interactions are transmitted via the exchange of particles. However, the last kind of interaction, gravitational interaction, is described by Einstein’s theory of general relativity, which is a classical theory which describes the gravitational field as a smooth metric tensor field on a manifold, i.e., a 4- dimensional spacetime geometry. There is no ~ and hence no quantum structure of spacetime. Thus there is a fundamental inconsistency in our current description of the whole physical world. Physicists widely accept the assumption that our world is quantized at fundamental level. So all interactions should be brought into the framework of quantum mechanics fundamentally. As a result, the gravitational field should also have ”quantum structure”. Throughout the last century, our understanding of nature has considerably improved from macroscale to microscale, including the phenomena at molecule, atom, sub-atom, and elemen- tary particle scale. The standard model of particle physics agrees with all present experimental tests in laboratory (see e.g. [158]). However, because unimaginably large amount of energy would be needed, no experimental tests exist for processes that happen near the Planck scale 3 1=2 −33 5 1=2 −43 `p ≡ (G~=c ) ∼ 10 cm and tp ≡ (G~=c ) ∼ 10 s, which are viewed as the most fundamental scales. The Planck scale arises naturally in attempts to formulate a quantum the- ory of gravity, since `p and tp are unique combinations of speed of light c, Planck constant ~, and gravitational constant G, which have the dimensions of length and time respectively. The dimensional arguments suggest that at Planck scale the smooth structure of spacetime should break down, and therefore the well-known quantum field theory is invalid since it depends on a fixed smooth background spacetime. Hence we believe that physicists should go beyond the successful standard model to explore the new physics near Planck scale, which is, perhaps, a quantum field theory without a background spacetime, and this quantum field theory should include the quantum theory of gravity. Moreover, current theoretical physics is thirsting for a quantum theory of gravity to solve at least the following fundamental difficulties. • Classical Gravity - Quantum Matter Inconsistency The equation relating matter and the gravitational field is the famous Einstein field equa- tion: 1 R [g] − R[g]g = κT [g]; (1) αβ 2 αβ αβ where the left hand side of the equation concerns spacetime geometry which has classical smooth structure, while the right hand side concerns also matter field which is funda- mentally quantum mechanical in standard model. In quantum field theory the energy- momentum tensor of matter field should be an operator-valued tensor Tˆαβ. One possible way to keep classical geometry consistent with quantum matter is to replace Tαβ[g] by the expectation value < Tˆαβ[g] > with respect to some quantum state of the matter on a fixed spacetime. However, in the solution of this equation the background gαβ has to be changed due to the non-vanishing of < Tˆαβ[g] >. So one has to feed back the new metric 1 into the definition of the vacuum expectation value etc. The result of the iterations does not converge in general [70]. On the other hand, some other arguments show that such a semiclassical treatment may violate the principle of superposition in quantum mechanics [55]. This inconsistency motivates us to quantize the background geometry to arrive at an operator formula also on the left hand side of Eq.(1). • Singularities in General Relativity Einstein’s theory of General Relativity is considered as one of the most elegant theories in the 20th century. Many experimental tests confirm the theory in the classical domain [159]. However, Penrose and Hawking proved that singularities are inevitable in general spacetimes with matter satisfying certain conditions in, by now well known, singularity theorems (for a summary, see [91][156]). Thus general relativity as a classical theory breaks down in certain regions of spacetime in a generic way. One naturally expects that, in extra strong gravitational field domains near
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