States In Quantum Gravity Research Articles

Semiclassical states in loop quantum gravity: an introduction

Unifying general relativity and quantum mechanics is a great challenge left to us by Einstein. To face this challenge, considerable progress has been made in non-perturbative canonical (loop) quantum gravity during the past 20 years. The kinematical Hilbert space of the quantum theory is constructed rigorously. However, the semiclassical analysis of the theory is still a crucial and open issue. In this review, we first introduce our work on constructing a semiclassical weave state, using the \(\hat Q\)[ω] operator on the kinematical Hilbert space of loop quantum gravity. Then we give an introduction to the two different approaches currently investigated for constructing coherent states in the kinematical Hilbert space. The current status of semiclassical analysis in loop quantum gravity is then summarized.

Semi-classical quantization of spacetimes with apparent horizons

Coherent or semi-classical states in canonical quantum gravity describe the classical Schwarzschild spacetime. By tracing over the coherent state wavefunction inside the horizon, a density matrix is derived. Bekenstein–Hawking entropy is obtained from the density matrix, modulo the Immirzi parameter. The expectation value of the area and curvature operator is evaluated in these states. The behaviour near the singularity of the curvature operator shows that the singularity is resolved. We then generalize the results to spacetimes with spherically symmetric apparent horizons.

Holographic description of AdS cosmologies

To gain insight in the quantum nature of the big bang, we study the dual field theory description of asymptotically anti-de Sitter solutions of supergravity that have cosmological singularities. The dual theories do not appear to have a stable ground state. One regularization of the theory causes the cosmological singularities in the bulk to turn into giant black holes with scalar hair. We interpret these hairy black holes in the dual field theory and use them to compute a finite temperature effective potential. In our study of the field theory evolution, we find no evidence for a "bounce" from a big crunch to a big bang. Instead, it appears that the big bang is a rare fluctuation from a generic equilibrium quantum gravity state.

Feynman Propagator for Spin Foam Quantum Gravity

We link the notion causality with the orientation of the spin foam 2-complex. We show that all current spin foam models are orientation independent. Using the technology of evolution kernels for quantum fields on Lie groups, we construct a generalized version of spin foam models, introducing an extra proper time variable. We prove that different ranges of integration for this variable lead to different classes of spin foam models: the usual ones, interpreted as the quantum gravity analogue of the Hadamard function of quantum field theory (QFT) or as inner products between quantum gravity states; and a new class of causal models, the quantum gravity analogue of the Feynman propagator in QFT, nontrivial function of the orientation data, and implying a notion of "timeless ordering".

The graviton vacuum as a distributional state in kinematic loop quantum gravity

The quantum behaviour of weak gravitational fields admits an adequate, albeit approximate, description by those graviton states in which the expectation values and fluctuations of the linearized gravitational field are small. Such states must approximate corresponding states in full quantum gravity. We analyse the nature of this approximation for the graviton vacuum state in the context of kinematical loop quantum gravity (LQG) wherein the constraints are ignored. We identify the graviton vacuum state with kinematically non-normalizable, distributional states in LQG by demanding that relations between linearized operator actions on the former are mirrored by those of their nonlinear counterparts on the latter. We define a semi-norm on the space of kinematical distributions and show that the identification is approximate up to distributions which are small in this semi-norm. We argue that our candidate states are annihilated by the linearized constraints (expressed as operators in the full theory) to leading order in the parameter characterizing the approximation. This suggests the possibility, in a scheme such as ours, of solving the full constraints order by order in this parameter. The main drawback of our considerations is that they depend on certain auxilliary constructions which, though mathematically well defined, do not arise from physical insight. Our work is an attempt to implement an earlier proposal of Iwasaki and Rovelli.

Implementing causality in the spin foam quantum geometry

We analyze the classical and quantum geometry of the Barrett–Crane spin foam model for four-dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transition amplitudes between quantum gravity states, i.e., realizing in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories).

Gravitons from a loop representation of linearized gravity

Loop quantum gravity is based on a classical formulation of $3+1$ gravity in terms of a real $\mathrm{SU}(2)$ connection. Linearization of this classical formulation about a flat background yields a description of linearized gravity in terms of a real $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ connection. A ``loop'' representation, in which holonomies of this connection are unitary operators, can be constructed. These holonomies are not well defined operators in the standard graviton Fock representation. We generalize our recent work on photons and $U(1)$ holonomies to show that Fock space gravitons are associated with distributional states in the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ loop representation. Our results may illuminate certain aspects of the much deeper (and as yet unkown) relation between gravitons and states in nonperturbative loop quantum gravity. This work leans heavily on earlier seminal work by Ashtekar, Rovelli and Smolin (ARS) on the loop representation of linearized gravity using complex connections. In the last part of this work we show that the loop representation based on the real $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ connection also provides a useful kinematic arena in which it is possible to express the ARS complex connection-based results in the mathematically precise language currently used in the field.

Resummation and Higher Order Renormalization in 4D Quantum Gravity

Higher order renormalization in 4D quantum gravity is carried out using dimensional regularization with great care concerning the conformal-mode dependence. In this regularization, resummation can be automatically carried out without making an assumption like that of David, Distler and Kawai. In this paper we consider a model of 4D quantum gravity coupled to QED. Resummation inevitably implies a four-derivative quantum gravity. The renormalizability is directly checked up to O(er6) and O(tr2), where er and tr are the running coupling constants of QED and the traceless gravitational mode. There is no other running coupling constant in our model. The conformal mode is treated exactly, which means it is unrenormalized. It is found that Hathrell's results are included in our results. As a by-product, it is found that a higher-order gravitational correction to the beta function of QED is negative. An advantage of our model is that in the very high-energy regime, it closely resembles exactly solvable 2D quantum gravity. Thus, we can study physical states of 4D quantum gravity in this regime in parallel to those of 2D quantum gravity, which can be described with diffeomorphism invariant composite fields.

Photons from quantized electric flux representations

The quantum theory of U(1) connections admits a diffeomorphism invariant representation in which the electric flux through any surface is quantized. This representation is the analog of the representation of quantum SU(2) theory used in loop quantum gravity. We investigate the relation between this representation, in which the basic excitations are `polymer-like', and the Fock representation, in which the basic excitations are wave-like photons. We show that normalizable states in the Fock space are associated with `distributional' states in the quantized electric flux representation. This work is motivated by the question of how wave-like gravitons in linearised gravity arise from polymer-like states in non-perturbative loop quantum gravity.

Triad representation of the Chern-Simons state in quantum gravity

We investigate a triad representation of the Chern-Simons state of quantum gravity with a non-vanishing cosmological constant. It is shown that the Chern-Simons state, which is a well-known exact wavefunctional within the Ashtekar theory, can be transformed to the real triad representation by means of a suitably generalized Fourier transformation, yielding a complex integral representation for the corresponding state in the triad variables. It is found that topologically inequivalent choices for the complex integration contour give rise to linearly independent wavefunctionals in the triad representation, which all arise from the one Chern-Simons state in the Ashtekar variables. For a suitable choice of the normalization factor, these states turn out to be gauge-invariant under arbitrary, even topologically non-trivial gauge-transformations. Explicit analytical expressions for the wavefunctionals in the triad representation can be obtained in several interesting asymptotic parameter regimes, and the associated semiclassical 4-geometries are discussed. In restriction to Bianchi-type homogeneous 3-metrics, we compare our results with earlier discussions of homogeneous cosmological models. Moreover, we define an inner product on the Hilbert space of quantum gravity, and choose a natural gauge-condition fixing the time-gauge. With respect to this particular inner product, the Chern-Simons state of quantum gravity turns out to be a non-normalizable wavefunctional.

Review: Gas of Wormholes: A Possible Ground State of Quantum Gravity

In order to gain insight into the possible Ground State of Quantized Einstein's Gravity, we have derived a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat space-time. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat space-time but around a ``gas'' of wormholes of mass m_p, the Planck mass (m_p ~= 10^{19}GeV) and average distance l_p, the Planck length a_p(a_p ~= 10^{-33}cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, space-time, the arena of physical reality, turns out to be well described by Wheeler's quantum foam and adequately modeled by a space-time lattice with lattice constant l_p, the Planck lattice.

Classical geometry from a physical state in canonical quantum gravity

We construct a weave state which approximates a degenerate 3-metric of rank 2 at large scales. It turns out that a non-degenerate metric region can be evolved from this degenerate metric by the classical Ashtekar equations, hence the degeneracy of 3-metrics is not preserved by the evolution of Ashtekar's equations. As the s-knot state corresponding to this weave is shown to solve all the quantum constraints in loop quantum gravity, a physical state in canonical quantum gravity is related to the familiar classical geometry.

Existence of spinorial states in pure loop quantum gravity

We demonstrate the existence of spinorial states in a kinematical theory of canonical quantum gravity without matter. This should be regarded as evidence in support of the conjecture that bound states with particle properties appear in association with spatial regions of non-trivial topology. In asymptotically trivial general relativity the momentum constraint generates only a subgroup of the spatial diffeomorphisms. The remaining diffeomorphisms give rise to the mapping class group, which acts as a symmetry group on the phase space. This action induces a unitary representation on the loop state space of the Ashtekar formalism. Certain elements of the diffeomorphism group can be regarded as asymptotic rotations of space relative to its surroundings. We construct states that transform non-trivially under a 2-rotation: gravitational quantum states with fractional spin.

Projector on physical states in loop quantum gravity

We construct the operator that projects on the physical states in loop quantum gravity. To this aim, we consider a diffeomorphism invariant functional integral over scalar functions. The construction defines a covariant, Feynman-like, spacetime formalism for quantum gravity and relates this theory to the spin foam models. We also discuss how expectation values of physical quantity can be computed.

On the ground state of quantum gravity

In order to gain insight into the possible Ground State of Quantized Einstein's Gravity, we have devised a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat space-time. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat space-time but around a ``gas'' of wormholes, whose size is the Planck length $a_p$ ($a_p\simeq 10^{-33}$cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, space-time, the arena of physical reality, turns out to be well described by Wheeler's Quantum Foam and adequately modeled by a space-time lattice with lattice constant $a_p$, the Planck lattice.

Chern-Simons states in spin-network quantum gravity

In the context of canonical quantum gravity in terms of Ashtekar's new variables, it is known that there exists a state that is annihilated by all the quantum constraints and that is given by the exponential of the Chern--Simons form constructed with the Asthekar connection. We make a first exploration of the transform of this state into the spin-network representation of quantum gravity. The discussion is limited to trivalent nets with planar intersections. We adapt an invariant of tangles to construct the transform and study the action of the Hamiltonian constraint on it. We show that the first two coefficients of the expansion of the invariant in terms of the inverse cosmological constant are annihilated by the Hamiltonian constraint. We also discuss issues of framing that arise in the construction.

Graphical evolution of spin network states

The evolution of spin network states in loop quantum gravity can be described by introducing a time variable, defined by the surfaces of constant value of an auxiliary scalar field. We regulate the Hamiltonian, generating such an evolution, and evaluate its action both on edges and on vertices of the spin network states. The analytical computations are carried out completely to yield a finite, diffeomorphism invariant result. We use techniques from the recoupling theory of colored graphs with trivalent vertices to evaluate the graphical part of the Hamiltonian action. We show that the action on edges is equivalent to a diffeomorphism transformation, while the action on vertices adds new edges and re-routes the loops through the vertices.

Semiclassical black hole states and entropy

We discuss semiclassical states in quantum gravity corresponding to Schwarzschild as well as Reissner Nordstr\"om black holes. We show that reduced quantisation of these models is equivalent to Wheeler-DeWitt quantisation with a particular factor ordering. We then demonstrate how the entropy of black holes can be consistently calculated from these states. While this leads to the Bekenstein-Hawking entropy in the Schwarzschild and non-extreme Reissner-Nordstr\"om cases, the entropy for the extreme Reissner-Nordstr\"om case turns out to be zero.

2d quantum gravity and boundary state in noncritical string theory

2d quantum gravity and boundary state in noncritical string theory

Extended knots and the space of states of quantum gravity

In the loop representation the quantum constraints of gravity can be solved. This fact allowed significant progress in the understanding of the space of states of the theory. The analysis of the constraints over loop-dependent wavefunctions has been traditionally based upon geometric (in contrast to analytic) properties of the loops. The reason for this preferred way is twofold: on the one hand, the inherent difficulties associated with the analytic loop calculus, and on the other hand, our limited knowledge about the analytic properties of knots invariants. Extended loops provide a way to overcome the difficulties at both levels. On the one hand, a systematic method to construct analytic expressions of diffeomorphism invariants (the extended knots) in terms of the Chern-Simons propagators can be developed. Extended knots are simply related to ordinary knots (at least formally). The analytic expressions of knot invariants could be produced then in a generic way. On the other hand, the evaluation of the Hamiltonian over extended loop wavefunctions can be thoroughly accomplished in the extended loop framework. These two ingredients promote extended loops as a potential resort for answering important questions about quantum gravity.