States In Loop Quantum Gravity Research Articles

Cosmological states in loop quantum gravity on homogeneous graphs

We introduce a class of states characterized by proposed conditions of homogeneity and isotropy in loop quantum gravity and construct concrete examples given by Bell-network states on a special class of homogeneous graphs. Such states provide new representations of cosmological spaces that can be explored for the formulation of cosmological models in the context of loop quantum gravity. We show that their local geometry is described in an automorphism-invariant manner by one-node observables analogous to the one-body observables used in many-body quantum mechanics, and compute the density matrix representing the restriction of global states to the algebra of one-node observables. The von Neumann entropy of this density matrix provides a notion of entanglement entropy of a local region which respects automorphism-invariance and can be applied to states involving superpositions of distinct graphs.

The Barbero–Immirzi Parameter: An Enigmatic Parameter of Loop Quantum Gravity

The Barbero–Immirzi parameter, (γ), is introduced in loop quantum gravity (LQG), whose physical significance is still the biggest open question because of its profound traits. In some cases, it is real valued, while it is complex valued in other cases. This parameter emerges in the process of denoting a Lorentz connection with a non-compact group SO(3,1) in the form of a complex connection with values in a compact group of rotations, either SO(3) or SU(2). Initially, it appeared in the Ashtekar variables. Fernando Barbero proposed its possibility for inclusion within formalism. Its present value is fixed by counting micro states in loop quantum gravity and matching with the semi-classical black hole entropy computed by Stephen Hawking. This parameter is used to count the size of the quantum of area in Planck units. Until the discovery of the spectrum of the area operator in LQG, its significance remained unknown. However, its complete physical significance is yet to be explored. In the present paper, an introduction to the Barbero–Immirzi parameter in LQG, a timeline of this research area, and various proposals regarding its physical significance are given.

Quantum speed limit and stability of coherent states in quantum gravity

Utilizing the program of expectation values in coherent states and its recently developed algorithmic tools, this letter investigates the dynamical properties of cosmological coherent states for loop quantum gravity. To this end, the quantum speed limit (QSL) is adapted to quantum gravity, yielding necessary consistency checks for any proposal of stable families of states. To showcase the strength of the developed tools, they are applied to a prominent model: the Euclidean part of the quantum scalar constraint. We report the variance of this constraint evaluated on a family of coherent states showing that, for short times, this family passes the QSL test, allowing the transition from one coherent state to another one.

Twisted geometries coherent states for loop quantum gravity

We introduce a new family of coherent states for loop quantum gravity, inspired by the twisted geometry parametrization. We compute their peakedness properties and compare them with the heat-kernel coherent states. They show similar features for the area and the holonomy operators, but improved peakedness in the direction of the flux. At the gauge-invariant level, the new family is built from tensor products of coherent intertwiners. To study the peakedness of the holonomy operator, we introduce a new shift operator based on the harmonic oscillator representation associated with the twisted geometry parametrization. The new shift operator captures the components of the holonomy relevant to disentangle its action into a simple positive shift of the spins.

Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space

We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1, 1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1, 1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra . We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. We explicitly discuss how the representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the representation label sets a lower cut-off for the loop quantum gravity spins (=U(1) representation labels in LQC). Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the representation label and interpreted as the minimally resolved geometric scale.

Graph coherent states for loop quantum gravity

In this article we further investigate the construction of graph coherent states, first introduced in [1], in the context of loop quantum gravity. We specifically investigate the possibility of defining a family of graph coherent states adapted to the canonical loop quantum gravity Hamiltonian. After discussing various aspects of the general framework and the choice of operators, we use the Euclidean part of the Hamiltonian operator to propose a generator of the generalized canonical structure, necessary to define the coherent states. We then apply the construction procedure, leading to a new family of graph coherent states partially adapted to the gravity Hamiltonian in loop quantum gravity.

Towards Gaussian states for loop quantum gravity

An important challenge in loop quantum gravity is to find semiclassical states - states that are as close to classical as quantum theory allows. This is difficult because the states in the Hilbert space used in LQG are excitations over a vacuum in which geometry is highly degenerate. Additionally, fluctuations are distributed very unevenly between configuration and momentum variables. Coherent states that have been proposed to balance the uncertainties more evenly can, up to now, only do this for finitely many degrees of freedom. Our work is motivated by the desire to obtain Gaussian states that encompass all degrees of freedom. To obtain a toy-model we reformulate the U(1) holonomy-flux algebra in any dimension as a Weyl algebra, and discuss generalisations to SU(2). We then define and investigate a new class of states on these algebras which behave like quasifree states on the momentum variables.

Self-dual formulation of gravity in topological M-theory

Inspired by the low wave-length limit of topological M-theory, which re-constructs the theory of 3 + 1D gravity in the self-dual variables’ formulation, and by the realization that in Loop Quantum Gravity (LQG) the holonomy of a flat connection can be non-trivial if and only if a non-trivial (space-like) line defect is localized inside the loop, we argue that non-trivial gravitational holonomies can be put in correspondence with space-like M-branes. This suggests the existence of a new duality, which we call [Formula: see text] duality, interconnecting topological M-theory with LQG. We spell some arguments to show that fundamental S-strings are serious candidates to be considered in order to instantiate this correspondence to classes of LQG states. In particular, we consider the case of the holonomy flowers in LQG, and show that for this type of states the action of the Hamiltonian constraint, from the M-theory side, corresponds to a linear combination of appearance and disappearance of a SNS1-strings. Consequently, these processes can be reinterpreted, respectively, as enucleations or decays into open or closed strings.

Entanglement entropy of Bell-network states in loop quantum gravity: Analytical and numerical results

Bell-network states are loop-quantum-gravity states that glue quantum polyhedra with entanglement. We present an algorithm and a code that evaluates the reduced density matrix of a Bell-network state and computes its entanglement entropy. In particular, we use our code for simple graphs to study properties of Bell-network states and to show that they are non-typical in the Hilbert space. Moreover, we investigate analytically Bell-network states on arbitrary finite graphs. We develop methods to compute the R\'enyi entropy of order p for a restriction of the state to an arbitrary region. In the uniform large-spin regime, we determine bounds on the entanglement entropy and show that it obeys an area law. Finally, we discuss the implications of our results for correlations of geometric observables.

Hamiltonian flows of Lorentzian polyhedra: Kapovich-Millson phase space and SU(1, 1) intertwiners

We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N faces. Starting with the Schwinger representation of the su(1,1) Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for space-like vectors in the three-dimensional Minkowski space R1,2. Considering N copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with N faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of SU(1, 1)-invariant observables, whose Hamiltonian flows generate geometrical deformations of polyhedra. We distinguish area-preserving and area-changing deformations. We then show that the area-preserving observables form a glN(R) Lie algebra and that they generate a GLN(R) action on Lorentzian polyhedra at fixed total area. This action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a GLN(R) transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as SU(1, 1) intertwiners between unitary SU(1, 1)-representations from the principal continuous series. Those SU(1, 1)-intertwiners are the building blocks of spin network states in loop quantum gravity in 3 + 1 dimensions for time-like slicing, and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.

Cosmological coherent state expectation values in loop quantum gravity I. Isotropic kinematics

This is the first paper of a series dedicated to loop quantum gravity (LQG) coherent states and cosmology. The concept is based on the effective dynamics program of Loop Quantum Cosmology, where the classical dynamics generated by the expectation value of the Hamiltonian on semiclassical states is found to be in agreement with the quantum evolution of such states. We ask the question of whether this expectation value agrees with the one obtained in the full theory. The answer is in the negative, Dapor and Liegener (2017 arXiv:1706.09833). This series of papers is dedicated to detailing the computations that lead to that surprising result. In the current paper, we construct the family of coherent states in LQG which represent flat (k = 0) Robertson–Walker spacetimes, and present the tools needed to compute expectation values of polynomial operators in holonomy and flux on such states. These tools will be applied to the LQG Hamiltonian operator (in Thiemann regularization) in the second paper of the series. The third paper will present an extension to cosmologies and a comparison with alternative regularizations of the Hamiltonian.

Transition probability spaces in loop quantum gravity

We study the (generalized) transition probability spaces, in the sense of Mielnik and Cantoni, for spacetime quantum states in loop quantum gravity. First, we show that loop quantum gravity admits the structures of transition probability spaces. This is exemplified by first checking such structures in covariant quantum mechanics and then identifying the transition probability spaces in spin foam models via a simplified version of general boundary formulation. The transition probability space thus defined gives a simple way to reconstruct the discrete analog of the Hilbert space of the canonical theory and the relevant quantum logical structures. Second, we show that the transition probability space and in particular the spin foam model are 2-categories. Then we discuss how to realize in spin foam models two proposals by Crane about the mathematical structures of quantum gravity, namely, the quantum topos and causal sites. We conclude that transition probability spaces provide us with an alternative framework to understand various foundational questions of loop quantum gravity.

Kinematic projective quantum states for loop quantum gravity coupled to tensor fields

We present a construction of kinematic quantum states for theories of tensor fields of an arbitrary sort. The construction is based on projective techniques by Kijowski. Applying projective quantum states for Loop Quantum Gravity (LQG) obtained by Lanéry and Thiemann we construct quantum states for LQG coupled to tensor fields.

Ising spin network states for loop quantum gravity: a toy model for phase transitions

Non-perturbative approaches to quantum gravity call for a deep understanding of the emergence of geometry and locality from the quantum state of the gravitational field. Without background geometry, the notion of distance should emerge entirely from the correlations between the gravity fluctuations. In the context of loop quantum gravity, quantum states of geometry are defined as spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. Here, we develop the condensed-matter point of view on extracting the physical and geometrical information from spin network states: we introduce new Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. We construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints that entirely characterize our states. We discuss their phase diagram and show how the distance can be reconstructed from the correlations in the various phases. Finally, we propose generalizations of these Ising states, which open the perspective to study the coarse-graining and dynamics of spin network states using well-known condensed-matter techniques and results.

Flux formulation of loop quantum gravity: classical framework

We recently introduced a new representation for loop quantum gravity (LQG), which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in LQG, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.

Dimensional flow in discrete quantum geometries

In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension $d$ at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number $0<\ensuremath{\alpha}<d$, we find that the spatial spectral dimension reduces to ${d}_{\mathrm{S}}\ensuremath{\simeq}\ensuremath{\alpha}$ at small scales. The spatial Hausdorff dimension of such class of states varies between 1 and $d$, while the walk dimension takes the usual value ${d}_{\mathrm{W}}=2$. Therefore, these quantum geometries may be considered as fractal only when $\ensuremath{\alpha}=1$, where the ``magic number'' ${D}_{\mathrm{S}}\ensuremath{\simeq}2$ for the spectral dimension of spacetime, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.

Thermally correlated states in loop quantum gravity

We study a class of loop-quantum-gravity states characterized by (ultra-local) thermal correlations that reproduce some features of the ultraviolet structure of the perturbative quantum field theory vacuum. In particular, they satisfy an analog of the Bisognano–Wichmann theorem. These states are peaked on the intrinsic geometry and admit a semiclassical interpretation. We study how the correlations extend on the spin network beyond the ultra local limit.

Detailed black hole state counting in loop quantum gravity

The combinatorial problems associated with the counting of black hole states in loop quantum gravity can be analyzed by using suitable generating functions. These can be used to perform a statistical analysis of the black hole degeneracy spectrum, study the interesting sub-structure found in the entropy of microscopic black holes, and its asymptotic behaviour for large horizon areas. They are also relevant for the discussion of the thermodynamic limit for black holes, and the understanding of sub-leading corrections to the Bekenstein-Hawking law.

Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics

In this article we explore the origin of black hole thermodynamics using semiclassical states in loop quantum gravity. We re-examine the case of entropy using a density matrix for a coherent state and describe correlations across the horizon due to SU(2) intertwiners. We further show that Hawking radiation is a consequence of a non-Hermitian term in the evolution operator, which is necessary for entropy production or depletion at the horizon. This non-unitary evolution is also rooted in formulations of irreversible physics.

The kinematical Hilbert space of loop quantum gravity from BF theories

In this work, it is demonstrated how the kinematical Hilbert space of loop quantum gravity (LQG) can be inferred from the configuration space of BF theories via the imposition of the Hamiltonian constraints. In particular, it is outlined how the projection to the representations associated with Ashtekar–Barbero connections provides the correct procedure to implement second-class constraints and the corresponding nontrivial induced symplectic structure. Then, the reduction to SU(2) invariant intertwiners is analyzed and the properties of LQG states under Lorentz transformations are discussed.