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      On the UV dimensions of Loop Quantum Gravity

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          Abstract

          Planck-scale dynamical dimensional reduction is attracting more and more interest in the quantum-gravity literature since it seems to be a model independent effect. However different studies base their results on different concepts of spacetime dimensionality. Most of them rely on the \textit{spectral} dimension, others refer to the \textit{Hausdorff} dimension and, very recently, it has been introduced also the \textit{thermal} dimension. We here show that all these distinct definitions of dimension give the same outcome in the case of the effective regime of Loop Quantum Gravity (LQG). This is achieved by deriving a modified dispersion relation from the hypersurface-deformation algebra with quantum corrections. Moreover we also observe that the number of UV dimensions can be used to constrain the ambiguities in the choice of these LQG-based modifications of the Dirac spacetime algebra. In this regard, introducing the \textit{polymerization} of connections i.e. \(K \rightarrow \frac{\sin(\delta K)}{\delta}\), we find that the leading quantum correction gives \(d_{UV} = 2.5\). This result may indicate that the running to the expected value of two dimensions is ongoing, but it has not been completed yet. Finding \(d_{UV}\) at ultra-short distances would require to go beyond the effective approach we here present.

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          Quasilocal Energy and Conserved Charges Derived from the Gravitational Action

          The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on \({}^3B\), the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate \({}^3B\). The integral of the energy density over such a two-surface \(B\) is the quasilocal energy associated with a spacelike three-surface \(\Sigma\) whose intersection with \({}^3B\) is the boundary \(B\). The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary \(B\) as a surface embedded in \(\Sigma\). The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on \({}^3B\) in the direction orthogonal to \(B\). Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on \({}^3B\). For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.
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            Background Independent Quantum Gravity: A Status Report

            , (2004)
            The goal of this article is to present an introduction to loop quantum gravity -a background independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the article should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the article is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the article to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.
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              The Asymptotic Safety Scenario in Quantum Gravity

              The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. The evidence from symmetry truncations and from the truncated flow of the effective average action is presented in detail. A dimensional reduction phenomenon for the residual interactions in the extreme ultraviolet links both results. For practical reasons the background effective action is used as the central object in the quantum theory. In terms of it criteria for a continuum limit are formulated and the notion of a background geometry self-consistently determined by the quantum dynamics is presented. Self-contained appendices provide prerequisites on the background effective action, the effective average action, and their respective renormalization flows.
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                Author and article information

                Journal
                2016-05-19
                2016-07-28
                Article
                10.1155/2016/9897051
                1605.05979
                d3a24b49-8d1f-4c1c-8ee9-a49a9468321a

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                Article ID 9897051, 7 pages. Advances in High Energy Physics (2016)
                gr-qc

                General relativity & Quantum cosmology
                General relativity & Quantum cosmology

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