Local rotational symmetry Gowdy model in loop quantum gravity
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In recent years, loop quantum gravity (LQG) has become a promising approach to Quantum Gravity (see e.g. for reviews). It has produced concrete results such as a rigorous derivation of the kinematical Hilbert space with discrete spectra for areas and volumes, the resulting finite isolated horizon entropy counting and regularization of black hole singularities, a well-defined framework for a (loop) quantum cosmology, and so on. Nevertheless, the model still has to face several key issues: a well-defined dynamics with a semi-classical regime described by Newton's gravity law and General Relativity, the existence of a physical semi-classical state corresponding to an approximately flat space-time, a proof that the no-gravity limit of LQG coupled to matter is standard quantum field theory, the Immirzi ambiguity, etc. Here, we address a fundamental issue at the root of LQG, which is necessarily related to these questions: why the SU(2) gauge group of loop quantum gravity? Indeed, the compactness of the SU(2) gauge group is directly responsible for the discrete spectra of areas and volumes, and therefore is at the origin of most of the successes of LQG: what happens if we drop this assumption?
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The spinfoam framework is a proposal for a regularized path integral for quantum gravity. Spinfoams define quantum space-time structures describing the evolution in time of the spin network states for quantum geometry derived from Loop Quantum Gravity (LQG). The construction of this covariant approach is based on the formulation of General Relativity as a topological theory plus the so-called simplicity constraints which introduce local degrees of freedom. The simplicity constraints are essential in turning the non-physical topological theory into 4d gravity. In this PhD manuscript, an original way to impose the simplicity constraints in 4d Euclidean gravity using harmonic oscillators is proposed and new coherent states, solutions of the constraints, are given. Moreover, a consistent spinfoam model for quantum gravity has to be connected to LQG and must have the right semi-classical limit. An explicit map between the spin network states of LQG and the boundary states of spinfoam models is given connecting the canonical and the covariant approaches. Finally, new techniques to compute semiclassical asymptotic expressions for the transition amplitudes of 3d quantum gravity and to extract semi-classical information from a spinfoam model are introduced. Explicit computations based on approximation methods and on the use of recurrence relations on spinfoam amplitudes have been performed. The results are relevant to derive quantum corrections to the dynamics of the gravitational field.
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This research paper basically tries to combine string theory and Loop Quantum Gravity to solve all the problems regarding quantum gravity and forms an unified theory of gravity which explains gravity at every scale also, gravity in interior of a black hole.
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Preface Part I. Foundations: 1. Spacetime as a quantum object 2. Physics without time 3. Gravity 4. Classical discretization Part II. The 3D Theory: 5. 3D Euclidean theory 6. Bubbles and cosmological constant Part III. The Real World: 7. The real world: 4D Lorentzian theory 8. Classical limit 9. Matter Part IV. Physical Applications: 10. Black holes 11. Cosmology 12. Scattering 13. Final remarks References Index.
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Loop quantum gravity is a physical theory which aims at unifying general relativity and quantum mechanics. It takes general relativity very seriously and modifies it via a quantisation. General relativity describes gravity in terms of geometry. Therefore, quantising such theory must be equivalent to quantising geometry and that is what loop quantum gravity does. This sounds like a mathematical task as well. This is why in this paper we will present the mathematics of loop quantum gravity. We will do it from a mathematician point of view. This paper is intended to be an introduction to loop quantum gravity for postgraduate students of physics and mathematics. In this work we will restrict ourselves to the three dimensional case.
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A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity.In particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity.Here we discuss the volume observable.We derive the volume operator in the covariant theory, and show that it matches the one of loop quantum gravity, as does the area.We also reconsider the implementation of the constraints that defines the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints, and show directly that all constraints vanish weakly on this space.
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We give a standard introduction to loop quantum gravity, from the ADM variables to spin network states. We include a discussion on quantum geometry on a fixed graph and its relation to a discrete approximation of general relativity.
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These notes are a didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity. The definition of real connection variables for general relativity, used as a starting point in the program, is described in a simple manner. The main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described. The main predictions of the theory such as the discovery of Planck scale discreteness of geometry and the computation of black hole entropy are reviewed. The quantization and solution of the constraints is explained by drawing analogies with simpler systems. Difficulties associated with the quantization of the scalar constraint are discussed.In a second part of the notes, the basic ideas behind the spin foam approach are presented in detail for the simple solvable case of 2+1 gravity. Some results and ideas for four dimensional spin foams are reviewed.
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These notes are a didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity. The definition of real connection variables for general relativity, used as a starting point in the program, is described in a simple manner. The main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described. The main predictions of the theory such as the discovery of Planck scale discreteness of geometry and the computation of black hole entropy are reviewed. The quantization and solution of the constraints is explained by drawing analogies with simpler systems. Difficulties associated with the quantization of the scalar constraint are this http URL a second part of the notes, the basic ideas behind the spin foam approach are presented in detail for the simple solvable case of 2+1 gravity. Some results and ideas for four dimensional spin foams are reviewed.
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Spin foam
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