arXiv subjects

Norbert Bodendorfer

Publications and source records attributed to Norbert Bodendorfer.

At least 37 records · Page 2Linked to original sources

arXiv2019

Lattice study of Rényi entanglement entropy in $SU(N_c)$ lattice Yang-Mills theory with $N_c = 2, 3, 4$

We consider the second Rényi entropy $S^{(2)}$ in pure lattice gauge theory with $SU(2)$, $SU(3)$ and $SU(4)$ gauge groups, which serves as a first approximation for the entanglement entropy and the entropic $C$-function. We compare the results for different gauge groups using scale setting via the string tension. We confirm that at small distances $l$ our approximation for the entropic $C$-function $C(l)$, calculated for the slab-shaped entangled region of width $l$, scales as $N_c^2 - 1$ in accordance with its interpretation in terms of free gluons. At larger distances $l$ $C(l)$ is found to approach zero for $N_c = 3, 4$, somewhat more rapidly for $N_c = 4$ than for $N_c = 3$. This finding supports the conjectured discontinuity of the entropic $C$-function in the large-$N$ limit, which was found in the context of AdS/CFT correspondence and which can be interpreted as transition between colorful quarks and gluons at small distances and colorless confined states at long distances. On the other hand, for $SU(2)$ gauge group the long-distance behavior of the entropic $C$-function is inconclusive so far. There exists a small region of lattice spacings yielding results consistent with $N_c=3,4$, while results from other lattice spacings deviate without clear systematics. We discuss several possible causes for discrepancies between our results and the behavior of entanglement entropy in holographic models.

hep-lat↗

arXiv2019

Coarse graining as a representation change

We discuss how SU$(1,1)$ coherent states from the discrete series allow for a natural coarse graining operation. The physical application are quantum theories based on a set of three extensive observables whose Poisson algebra is isomorphic to su$(1,1)$. In particular, we show that a Perelomov coherent state with representation label $N j_0$ and spinor label $z$ encodes the physics of $N$ independent subsystems with labels $j_0, z$. This property is suggested by existing results for the expectation values and variances of the observables. We prove that it holds for all higher moments. Our results in particular apply to a recent quantum cosmology model that has been derived using SU(1,1) group theoretic quantisation techniques. For it, it follows that a certain notion of fiducial cell independence holds exactly at the quantum level when using the coherent states.

gr-qc↗

arXiv2019

A note on the scalar products in sLQC

Solvable loop quantum cosmology provides a simple model of spatially flat, homogeneous, and isotropic quantum cosmology where the initial singularity is resolved. A close inspection of the literature reveals that there exist two different proposals for a scalar product that are usually assumed to be identical, but agree only in the large volume limit. The small volume limit, and thus the reported difference, turns out to be important for questions such as coarse graining fundamental quantum states in full theory embeddings of these cosmological models.

gr-qc↗

arXiv2018

On the canonical structure of general relativity with a limiting curvature and its relation to loop quantum gravity

Chamseddine and Mukhanov recently proposed a modified version of general relativity that implements the idea of a limiting curvature. In the spatially flat, homogeneous, and isotropic sector, their theory turns out to agree with the effective dynamics of the simplest version of loop quantum gravity if one identifies their limiting curvature with a multiple of the Planck curvature. At the same time, it extends to full general relativity without any symmetry assumptions and thus provides an ideal toy model for full loop quantum gravity in the form of a generally covariant effective action known to all orders. In this paper, we study the canonical structure of this theory and point out some interesting lessons for loop quantum gravity. We also highlight in detail how the two theories are connected in the spatially flat, homogeneous, and isotropic sector.

gr-qc↗

arXiv2018

Holographic Signatures of Resolved Cosmological Singularities II: Numerical Investigations

A common strategy to investigate the fate of gravitational singularities in asymptotically AdS spacetimes is to translate the question from the gravitational side to a dual field theory using the gauge/gravity correspondence and to do a field theory computation. Given recent progress in singularity resolution via non-perturbative quantum gravity, it is natural to now turn the question around and to ask about field theory signatures of resolved singularities. An investigation along this line has been initiated in a companion paper, where a finite-distance pole exhibited by the two-point correlator in the dual field theory, which has previously been linked directly to the gravitational bulk singularity, has been resolved in this way. In order to perform analytic computations, some simplifications were necessary. In this paper, we lift these restrictions by tackling the problem numerically. Our analysis shows that the pole in the two-point correlator gets resolved in the same manner as before.

hep-th↗

arXiv2017

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.

gr-qc↗

arXiv2016

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.

gr-qc↗

arXiv2016

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.

gr-qc↗

arXiv2016

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.

gr-qc↗

arXiv2016

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 $θ$ 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 $F \propto Σ$ and stress that it is, in agreement with earlier work, inadequate to define the notion of a quantum horizon.

gr-qc↗

arXiv2016

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.

gr-qc↗

arXiv2016

The algebra of observables in Gaußian normal spacetime coordinates

We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gaußian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a "local" algebra of observables can be constructed, it turns out that this is not possible for the spacetime version. The technical reason for this observation is that the new gauge condition needed to upgrade the spatial to a spacetime radial gauge does not Poisson-commute with the previous gauge conditions. It follows that the involved Dirac bracket is inherently non-local in the sense that no complete set of observables can be found which is constructed locally and at the same time has local Dirac brackets. A locally constructed observable here is defined as a finite polynomial of the canonical variables at a given physical point specified by the Gaußian normal spacetime coordinates.

gr-qc↗

arXiv2015

General relativity in the radial gauge: Reduced phase space and canonical structure

Firstly, we present a reformulation of the standard canonical approach to spherically symmetric systems in which the radial gauge is imposed. This is done via the gauge unfixing technique, which serves as the exposition in the context of the radial gauge. Secondly, we apply the same techniques to the full theory, without assuming spherical symmetry, resulting in a reduced phase space description of general relativity. The canonical structure of the theory is analyzed.

gr-qc↗

arXiv2015

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 ($α'$) 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.

hep-th↗

arXiv2015

A quantum reduction to spherical symmetry in loop quantum gravity

Based on a recent purely geometric construction of observables for the spatial diffeomorphism constraint, we propose two distinct quantum reductions to spherical symmetry within full 3+1-dimensional loop quantum gravity. The construction of observables corresponds to using the radial gauge for the spatial metric and allows to identify rotations around a central observer as unitary transformations in the quantum theory. Group averaging over these rotations yields our first proposal for spherical symmetry. Hamiltonians of the full theory with angle-independent lapse preserve this spherically symmetric subsector of the full Hilbert space. A second proposal consists in implementing the vanishing of a certain vector field in spherical symmetry as a constraint on the full Hilbert space, leading to a close analogue of diffeomorphisms invariant states. While this second set of spherically symmetric states does not allow for using the full Hamiltonian, it is naturally suited to implement the spherically symmetric midisuperspace Hamiltonian, as an operator in the full theory, on it. Due to the canonical structure of the reduced variables, the holonomy-flux algebra behaves effectively as a one parameter family of 2+1-dimensional algebras along the radial coordinate, leading to a diagonal non-vanishing volume operator on 3-valent vertices. The quantum dynamics thus becomes tractable, including scenarios like spherically symmetric dust collapse.

gr-qc↗

arXiv2015

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" $\barμ$ 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.

gr-qc↗

arXiv2014

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.

gr-qc↗

arXiv2014

New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom

In this paper, we generalise the treatment of isolated horizons in loop quantum gravity, resulting in a Chern-Simons theory on the boundary in the four-dimensional case, to non-distorted isolated horizons in 2(n+1)-dimensional spacetimes. The key idea is to generalise the four-dimensional isolated horizon boundary condition by using the Euler topological density of a spatial slice of the black hole horizon as a measure of distortion. The resulting symplectic structure on the horizon coincides with the one of higher-dimensional SO(2(n+1))-Chern-Simons theory in terms of a Peldan-type hybrid connection and resembles closely the usual treatment in 3+1 dimensions. We comment briefly on a possible quantisation of the horizon theory. Here, some subtleties arise since higher-dimensional non-Abelian Chern-Simons theory has local degrees of freedom. However, when replacing the natural generalisation to higher dimensions of the usual boundary condition by an equally natural stronger one, it is conceivable that the problems originating from the local degrees of freedom are avoided, thus possibly resulting in a finite entropy.

gr-qc↗