Quantum gravity is perhaps the most important open problem in fundamental physics. It is the problem of merging quantum mechanics and general relativity, the two great conceptual revolutions in the physics of the twentieth century. The loop and spinfoam approach, presented in this 2004 book, is one of the leading research programs in the field. The first part of the book discusses the reformulation of the basis of classical and quantum Hamiltonian physics required by general relativity. The second part covers the basic technical research directions. Appendices include a detailed history of the subject of quantum gravity, hard-to-find mathematical material, and a discussion of some philosophical issues raised by the subject. This fascinating text is ideal for graduate students entering the field, as well as researchers already working in quantum gravity. It will also appeal to philosophers and other scholars interested in the nature of space and time.

We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.Comment: 36 pages, latex, 13 figures uuencode

We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Immirzi parameter. We cover the Euclidean as well as the Lorentzian case. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on the Immirzi parameter, and, remarkably, holds in the Lorentzian case as well. The ad hoc flip of the symplectic structure that was initially required to derive the flipped vertex is not anymore needed for finite Immirzi parameter. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions

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The problem of finding the quantum theory of the gravitational field, and thus understanding what is quantum spacetime, is still open. One of the most active of the current approaches is loop quantum gravity. Loop quantum gravity is a mathematically well-defined, non-perturbative and background independent quantization of general relativity, with its conventional matter couplings. Research in loop quantum gravity today forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained are: The computation of the physical spectra of geometrical quantities such as area and volume, which yields quantitative predictions on Planck-scale physics.A derivation of the Bekenstein-Hawking black hole entropy formula.An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler’s intuition of a spacetime “foam”. Long standing open problems within the approach (lack of a scalar product, over-completeness of the loop basis, implementation of reality conditions) have been fully solved. The weak part of the approach is the treatment of the dynamics: at present there exist several proposals, which are intensely debated. Here, I provide a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.

We introduce a new basis on the state space of non-perturbative quantum gravity. The states of this basis are linearly independent, are well de ned in both the loop representation and the connection representation, and are labeled by a generalization of Penrose's spin networks. The new basis fully reduces the spinor identities (SU(2) Mandelstam identities) and simpli es calculations in non-perturbative quantum gravity. In particular, it allows a simple expression for the exact solutions of the Hamiltonian constraint (Wheeler-DeWitt equation) that have been discovered in the loop representation. Since the states in this basis diagonalize operators that represent the three geometry of space, such as the area and volumes of arbitrary surfaces and regions, these states provide a discrete picture of quantum geometry at the Planck scale.rovelli@vms.cis.pitt.edu, y smolin@phys.psu.edu

We argue that the statistical entropy relevant for the thermal interactions of a black hole with its surroundings is (the logarithm of) the number of quantum microstates of the hole which are distinguishable from the hole's exterior, and which correspond to a given hole's macroscopic configuration. We compute this number explicitly from first principles, for a Schwarzschild black hole, using nonperturbative quantum gravity in the loop representation. We obtain a black hole entropy proportional to the area, as in the Bekenstein-Hawking formula.Comment: 5 pages, latex-revtex, no figure

Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model-does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy n-point functions. These difficulties can be traced to the SO(4) → SU (2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity.The kinematics of loop quantum gravity (LQG) provides a well understood backgroundindependent language for a quantum theory of physical space [1,2,3]. The dynamics of the theory is not understood as cleanly. Dynamics is studied along two lines: hamiltonian (as in the Schrödinger equation) [4] or covariant (as in Feynman's covariant quantum field theory). We focus on the second. The key object that defines the dynamics in this language is the vertex amplitude, like the vertex eγ µ ∼∼ r < that defines the dynamics of perturbative QED. What is the vertex of LQG?The spinfoam formalism [5] is viewed as a possible tool for answering this question. It can be derived in a remarkable number of distinct ways, which converge to the definition of transition amplitudes as a Feynman sum over spinfoams. A spinfoam is a two-complex (union of faces, edges and vertices) colored with quantum numbers (spins associated to faces and intertwiners associated to edges); it can be loosely interpreted as a history of a spin network (a colored graph). Its amplitude contains the product of the amplitudes of each vertex, and thus the vertices play a role similar to the vertices of Feynman's covariant QFT [6,7]. This picture is nicely implemented in three dimensions (3d) by the Ponzano-Regge model [8], where the vertex amplitude is given by the 6j Wigner symbol, which can be obtained as a matrix element of the hamiltonian of 3d gravity [9].Compelling and popular as it is, however, this picture has never been fully implemented in 4d. The best studied model in the 4d euclidean context is the Barrett-Crane (BC) model [10]. This is simple and elegant, has remarkable finiteness properties [11], and can be considered a modification of a topological BF quantum field theory, by means of constraints -called simplicity constraints-whose classical limit yields precisely the constraints that change BF theory into general relativity (GR). Furthermore, in the lowenergy limit some of its n-point functions appear to agree with those computed from perturbative quantum GR [12]. However, the suspicion that something is wrong with the BC model has long...