The problem of describing the quantum behavior of gravity, and thus understanding quantum spacetime, is still open. Loop quantum gravity is a well-developed approach to this problem. It is a mathematically well-defined background-independent quantization of general relativity, with its conventional matter couplings. Today research in loop quantum gravity forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained so far are: (i) The computation of the spectra of geometrical quantities such as area and volume, which yield tentative quantitative predictions for Planck-scale physics. (ii) A physical picture of the microstructure of quantum spacetime, characterized by Planck-scale discreteness. Discreteness emerges as a standard quantum effect from the discrete spectra, and provides a mathematical realization of Wheeler’s “spacetime foam” intuition. (iii) Control of spacetime singularities, such as those in the interior of black holes and the cosmological one. This, in particular, has opened up the possibility of a theoretical investigation into the very early universe and the spacetime regions beyond the Big Bang. (iv) A derivation of the Bekenstein-Hawking black-hole entropy. (v) Low-energy calculations, yielding n-point functions well defined in a background-independent context. The theory is at the roots of, or strictly related to, a number of formalisms that have been developed for describing background-independent quantum field theory, such as spin foams, group field theory, causal spin networks, and others. I give here a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.

Loop Quantum Gravity

This article reviews the present status of the spin-foam approach to the quantization of gravity. Special attention is payed to the pedagogical presentation of the recently-introduced new models for four-dimensional quantum gravity. The models are motivated by a suitable implementation of the path integral quantization of the Plebanski formulation of gravity on a simplicial regularization. The article also includes a self-contained treatment of 2+1 gravity. The simple nature of the latter provides the basis and a perspective for the analysis of both conceptual and technical issues that remain open in four dimensions.

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The Spin-Foam Approach to Quantum Gravity

https://dspace.library.uu.nl/bitstream/handle/1874/7992/Loll_98_non_perturbative_lorentzian_quantum-gravity.pdf;sequence=1

Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change

Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory in which the pullback of the curvature to the boundary is self‐dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all SU(2) Chern–Simon theories defined by all choices of punctures and representations on the spatial boundary S. The integer level k of Chern–Simons theory is found to be given by k=6π/G2Λ+α, where Λ is the cosmological constant and α is a CP breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. Given these results, it is natural to make the conjecture that the quantum states of the system are completely determined by measurements mad...

Linking topological quantum field theory and nonperturbative quantum gravity

A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are “isolated” near future time-like infinity or for a finite time interval The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration A precise notion of non-rotating isolated horizons is formulated to capture these ideas With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon The required term turns out to be precisely the Chern-Simons action for the self-dual connection The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions Typeset using REVTEX

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1999/0003/0003/ATMP-1999-0003-0003-a001.pdf

Isolated horizons: The Classical phase space

We study in this paper a new approach to the problem of relating solutions to the Einstein field equations with Riemannian and Lorentzian signatures. The procedure can be thought of as a “real Wick rotation”. We give a modified action for general relativity, depending on two real parameters, that can be used to control the signature of the solutions to the field equations. We show how this procedure works for the Schwarzschild metric and discuss some possible applications of the formalism in the context of signature change, the problem of time, black hole thermodynamics... PACS 04.20Cv, 04.20Fy. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by CERN Document Server

From Euclidean to Lorentzian general relativity: The real way.

I show in this letter that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3)$ connections. The constraints are simple polynomials in the basic variables. The present framework gives us a new formulation of General Relativity that keeps some of the interesting features of the Ashtekar formulation without the complications associated with the complex character of the latter.

Barbero G Jf

Papers

From Euclidean to Lorentzian general relativity: The real way.

Real-polynomial formulation of general relativity in terms of connections.