An embedding of loop quantum cosmology in (b, v) variables into a full theory context

Loop quantum cosmology in (b, v) variables, which is governed by a unit step size difference equation, is embedded into a full theory context based on similar variables. A full theory context here means a theory of quantum gravity arrived at using the quantisation techniques used in loop quantum gravity, however based on a different choice of elementary variables and classical gauge fixing suggested by loop quantum cosmology. From the full theory perspective, the symmetry reduction is characterised by the vanishing of certain phase space functions which are implemented as operator equations in the quantum theory. The loop quantum cosmology dynamics arise as the action of the full theory Hamiltonian on maximally coarse states in the kernel of the reduction constraints. An application of this reduction procedure to spherical symmetry is also sketched, with similar results, but only one canonical pair in (b, v) form.Comment: 20 pages; v2: journal version, clarifications and comments adde

An elementary introduction to loop quantum gravity

An introduction to loop quantum gravity is given, focussing on the fundamental aspects of the theory, different approaches to the dynamics, as well as possible future directions. It is structured in five lectures, including exercises, and requires only little prior knowledge of quantum mechanics, gauge theory, and general relativity. The main aim of these lectures is to provide non-experts with an elementary understanding of loop quantum gravity and to evaluate the state of the art of the field. Technical details are avoided wherever possible.Comment: 48+14 pages, 8 figure

State refinements and coarse graining in a full theory embedding of loop quantum cosmology

Bridging between descriptions involving few large and many small quantum numbers is the main open problem in loop quantum gravity. In other words, one would like to be able to represent the same physical system in terms of a few "coarse"' quantum numbers, while the effective dynamics at the coarse level should agree with the one induced by a description involving many small quantum numbers. Efforts to understand this relationship face the problem of the enormous computational complexity involved in evolving a generic state containing many quanta. In a cosmological context however, certain symmetry assumptions on the quantum states allow to simplify the problem. In this paper, we will show how quantum states describing a spatially flat homogeneous and isotropic universe can be refined and coarse grained. Invariance of the dynamics of the coarse observables is shown to require a certain scaling property (familiar from loop quantum cosmology) of the quantum states if no running of parameters is taken into account. The involved states are solutions to the Hamiltonian constraint when terms coming from spatial derivatives are neglected, i.e. one works in the approximation of non-interacting FRW patches. The technical means to arrive at this result are a version of loop quantum gravity based on variables inspired by loop quantum cosmology, as well as an exact solution to the quantum dynamics of loop quantum cosmology which extends to the full theory in the chosen approximation.Comment: 12 pages; v2: journal version, many clarifications and minor correction

Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition

We recall some well and some less known results about the Kodama state and the related $\theta$ ambiguity in defining canonical variables. Based on them, we make some comments highlighting that the Kodama state for real connection variables can be given a precise meaning and that it implements a vacuum peaked on a (in a suitable sense) maximally symmetric geometry. We also highlight the similarity of this construction with the isolated horizon boundary condition $F \propto \Sigma$ and stress that it is, in agreement with earlier work, inadequate to define the notion of a quantum horizon.Comment: 6+3 pages; v2: journal version, slight shortening and added clarification

A note on quantum supergravity and AdS/CFT

We note that the non-perturbative quantisation of supergravity as recently investigated using loop quantum gravity techniques provides an opportunity to probe an interesting sector of the AdS/CFT correspondence, which is usually not considered in conventional treatments. In particular, assuming a certain amount of convergence between the quantum supergravity sector of string theory and quantum supergravity constructed via loop quantum gravity techniques, we argue that the large quantum number expansion in loop quantum supergravity corresponds to the $1/N^2_c$ expansion in the corresponding gauge theory. In order to argue that we are indeed dealing with an appropriate quantum supergravity sector of string theory, high energy ($\alpha'$) corrections are being neglected, leading to a gauge theory at strong coupling, yet finite $N_c$. The arguments given in this paper are mainly of qualitative nature, with the aim of serving as a starting point for a more in depth interaction between the string theory and loop quantum gravity communities.Comment: 8 pages, comments welcom

A note on conformally compactified connection dynamics tailored for anti-de Sitter space

A framework conceptually based on the conformal techniques employed to study the structure of the gravitational field at infinity is set up in the context of loop quantum gravity to describe asymptotically anti-de Sitter quantum spacetimes. A conformal compactification of the spatial slice is performed, which, in terms of the rescaled metric, has now finite volume, and can thus be conveniently described by spin networks states. The conformal factor used is a physical scalar field, which has the necessary asymptotics for many asymptotically AdS black hole solutions.Comment: 8 pages; v2: journal version, title changed, minor clarifications and comments adde

On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity

Building on a recent proposal for a quantum reduction to spherical symmetry from full loop quantum gravity, we investigate the relation between a quantisation of spherically symmetric general relativity and a reduction at the quantum level. To this end, we generalise the previously proposed quantum reduction by dropping the gauge fixing condition on the radial diffeomorphisms, thus allowing to make direct contact between previous work on reduced quantisation. A dictionary between spherically symmetric variables and observables with respect to the reduction constraints in the full theory is discussed, as well as an embedding of reduced quantum states to a sub sector of the quantum symmetry reduced full theory states. On this full theory sub sector, the quantum algebra of the mentioned observables is computed and shown to qualitatively reproduce the quantum algebra of the reduced variables in the large quantum number limit for a specific choice of regularisation. Insufficiencies in recovering the reduced algebra quantitatively from the full theory are attributed to the oversimplified full theory quantum states we use.Comment: 35 pages; v2: journal version, minor clarification

The Wald entropy formula and loop quantum gravity

We outline how the Wald entropy formula naturally arises in loop quantum gravity based on recently introduced dimension-independent connection variables. The key observation is that in a loop quantization of a generalized gravity theory, the analog of the area operator turns out to measure, morally speaking, the Wald entropy rather than the area. We discuss the explicit example of (higher-dimensional) Lanczos-Lovelock gravity and comment on recent work on finding the correct numerical prefactor of the entropy by comparing it to a semiclassical effective action.Comment: 23 pages. v2: journal version. Details added, discussion clarifie

A quantum reduction to Bianchi I models in loop quantum gravity

We propose a quantum symmetry reduction of loop quantum gravity to Bianchi I spacetimes. To this end, we choose the diagonal metric gauge for the spatial diffeomorphism constraint at the classical level, leading to an $\mathbb{R}_{\text{Bohr}}$ gauge theory, and quantise the resulting theory via loop quantum gravity methods. Constraints which lead classically to a suitable reduction are imposed at the quantum level. The dynamics of the resulting model turn out to be very simple and manifestly coincide with those of a polymer quantisation of a Bianchi I model for the simplest choice of full theory quantum states compatible with the Bianchi I reduction. In particular, the "improved" $\bar{\mu}$ dynamics of loop quantum cosmology can be obtained by modifying the regularisation of the Hamiltonian constraint with similar ideas, in turn yielding insights into the full theory dynamics.Comment: 5 pages. v2: partly rewritten to clarify derivation, gauge group enlarged from U(1) to R_Bohr, relation to old and new LQC dynamics discussed, previous results unchanged. v3: journal version, minor clarification

A note on entanglement entropy and quantum geometry

It has been argued that the entropy which one is computing in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field and that the calculation performed is not restricted to horizons. We recall existing work on this issue and explain how recent work on generalising these computations to arbitrary spacetime dimensions D+1>2 supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest. In a certain semiclassical regime in 3+1 dimensions, this entropy is given by the Bekenstein-Hawking formula.Comment: 14 pages. v2: journal version. Comment adde