Discreteness of area and volume in quantum gravity

Abstract

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.

36 pages, latex, 13 figures uuencoded

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• Relational Quantum Cosmology
• Scalar curvature operator for models of loop quantum gravity on a cubical graph
• Anisotropic loop quantum cosmology with self-dual variables
• A Gaussian Weave for Kinematical Loop Quantum Gravity
• Towards computational insights into the large-scale structure of spin foams
• Topspin Networks in Loop Quantum Gravity
• Generalized Quantization Principle in Canonical Quantum Gravity and Application to Quantum Cosmology
• Symmetry Reduced Loop Quantum Gravity: A Bird's Eye View
• Is Bianchi I a Bouncing Cosmology in the Wheeler-DeWitt picture?
• Obtaining the equation of motion for a fermionic particle in a generalized Lorentz-violating system framework
• Probing Planck Scale Spacetime By Cavity Opto-Atomic ${}^{87}$Rb Interferometry
• Spacetime at the Planck Scale: The Quantum Computer View
• Semiclassical and quantum behavior of the Mixmaster model in the polymer approach for the isotropic Misner variable
• Closing the gaps in quantum space and time: Conformally augmented gauge structure of gravitation
• Quantum Gravity Hamiltonian for Manifolds with Boundary
• Isotropic Loop Quantum Cosmology with Matter II: The Lorentzian Constraint
• On the perturbative expansion of a quantum field theory around a topological sector
• Eigenvalues of the volume operator in loop quantum gravity
• Diffeomorphism invariant cosmological sector in loop quantum gravity
• Deriving loop quantum cosmology dynamics from diffeomorphism invariance
• Propagating spin modes in canonical quantum gravity
• Black-hole horizons in modified space-time structures arising from canonical quantum gravity
• Long range correlations in quantum gravity
• Constraining Snyder and GUP models with low-mass stars
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Lorentz-covariant Hamiltonian analysis of BF gravity with the Immirzi parameter
• Collective motion in prolate γ-rigid nuclei within minimal length concept via a quantum perturbation method
• On the horizon area of effective loop quantum black holes
• Strong Coupling Quantum Gravity and Physics beyond the Planck Scale
• Lorentz-violating dilatations in the momentum space and some extensions on non-linear actions of Lorentz algebra-preserving systems
• On the viability of Planck scale cosmology with quartessence
• Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases
• Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant
• Immirzi parameter without Immirzi ambiguity: Conformal loop quantization of scalar-tensor gravity
• The generator of spatial diffeomorphisms in the Koslowski- Sahlmann representation
• Modified entropies as the origin of generalized uncertainty principles
• Scalar field in the Bianchi I: Non commutative classical and Quantum Cosmology
• Extension of Loop Quantum Gravity to Metric Theories beyond General Relativity
• Discrete quantum spectrum of black holes
• Compatibility of radial, Lorenz and harmonic gauges
• A novel mechanism for probing the Planck scale
• Polymer quantization versus the Snyder noncommutative space
• Imposing det E > 0 in discrete quantum gravity
• Lorentzian quantum cosmology from effective spin foams
• Vassiliev invariants: a new framework for quantum gravity
• Loop Quantum Gravity's Boundary Maps
• Membrane paradigm realized?
• Minimal Length, Nuclear Matter, and Neutron Stars
• Chern-Simons theory in 11 dimensions as a non-perturbative phase of M theory
• 4-dimensional BF Gravity on the Lattice
• Towards a unitary, renormalizable and ultraviolet-complete quantum theory of gravity
• Comparison of area spectra in loop quantum gravity
• Non-Gaussian features from the inverse volume corrections in loop quantum cosmology
• An example of Feynman-Jackson integral
• Could Planck Star Remnants be Dark Matter?
• On the counting of black hole states in loop quantum gravity
• Visions in Quantum Gravity
• Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models
• Twisted geometries are area-metric geometries
• Reduced Phase Space Approach to the $U(1{)}^3$ model for Euclidean Quantum Gravity
• Canonical loop quantization of the lowest-order projectable Horava gravity
• On the value of the Immirzi parameter and the horizon entropy
• Consistency check on the fundamental and alternative flux operators in loop quantum gravity
• Equivalence of models in loop quantum cosmology and group field theory
• From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier
• One vertex spin-foams with the Dipole Cosmology boundary
• Black hole entropy for the general area spectrum
• Spectrum of quantized black hole, correspondence principle, and holographic bound
• Time Uncertainty in Quantum Gravitational Systems
• Quantum Computation toward Quantum Gravity
• On Pairs of Difference Operators Satisfying: [P,Q] = Id
• Holographic entanglement in spin network states: a focused review
• Position-dependent mass in strong quantum gravitational background fields
• A Correction to the Immirizi Parameter of SU(2) Spin Networks
• The quantum clock: a critical discussion on space-time
• Covariant Origin of the $U(1{)}^3$ model for Euclidean Quantum Gravity
• Towards anisotropic cosmology in group field theory
• Quantum Big Bounce of the isotropic Universe via a relational time
• On the Microcanonical Entropy of a Black Hole
• Coherent 3j-symbol representation for the loop quantum gravity intertwiner space
• Area-metric gravity revisited
• Quantum clock: A critical discussion on spacetime
• The closure constraint for the hyperbolic tetrahedron as a Bianchi identity
• A Minimal Model for Quantum Gravity
• Loop quantum modified gravity and its cosmological application
• Quantized Black Holes, Their Spectrum and Radiation
• Corrections to flat-space particle dynamics arising from space granularity
• Perturbative calculation of quasi-normal modes of arbitrary spin in Schwarzschild spacetime
• Elementary charge and neutrino's mass from Planck length
• Minimal length and the flow of entropy from black holes
• Fundamental physics in space: a Quantum-Gravity perspective
• Shape in an Atom of Space: Exploring quantum geometry phenomenology
• Asymptotic quasinormal modes of scalar field in a gravity's rainbow
• Canonical Noncommutativity Algebra for the Tetrad Field in General Relativity
• On Loop States in Loop Quantum Gravity
• The Path Integral for the Loop Representation of Lattice Gauge Theories
• Tachyon field in loop quantum cosmology: An example of traversable singularity
• Piecewise linear loop quantum gravity
• Late time cosmic acceleration from natural infrared cutoff?
• Fermion spins in loop quantum gravity
• Strings as perturbations of evolving spin-networks
• Extended Uncertainty Principle: A Deeper Insight into the Hubble Tension?
• Combinatorial dynamics in quantum gravity
• A Brane World Model with Intersecting Branes
• Knot theory and a physical state of quantum gravity
• The large cosmological constant approximation to classical and quantum gravity: model examples
• Scalar Cosmological Perturbations from Quantum Entanglement within Lorentzian Quantum Gravity
• Matter in Loop Quantum Gravity without time gauge: a non-minimally coupled scalar field
• In Search of Quantum de Sitter Space: Generalizing the Kodama State
• The thermodynamics of quantum spacetime histories
• Quaternionic Quantization Principle in General Relativity and Supergravity
• How Is the Maximum Entropy of a Quantized Surface Related to Its Area?
• How many quanta are there in a quantum spacetime?
• Area Propagator and Boosted Spin Networks in Loop Quantum Gravity
• Entangled Space-Time
• Open-Flux Solutions to the Constraints for Plane Gravity Waves
• Relative entropy as a measure of inhomogeneity in general relativity
• Noncommutative spaces and covariant formulation of statistical mechanics
• Universe scenarios from loop quantum cosmology
• Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity
• Entropy and Area of Black Holes in Loop Quantum Gravity
• Quasilocal Energy for Spin-net Gravity
• Universal properties of the evolution of the Universe in modified loop quantum cosmology
• Massive motion in Brans-Dicke geometry and beyond
• Classical Geometry from a Physical State in Canonical Quantum Gravity
• A proposal for a Bohmian ontology of quantum gravity
• Canonical ``Loop'' Quantum Gravity and Spin Foam Models
• Quantum Geometry of Isolated Horizons and Black Hole Entropy
• Black holes quasinormal modes, Loop Quantum Gravity Immirzi parameter and nonextensive statistics
• $SU(1,1)$ Barbero-like variables derived from Holst action
• Second-order amplitudes in loop quantum gravity
• The cosmological implications of a fundamental length: a DSR inspired de-Sitter spacetime
• Noncommutative geometry induced by spin effects
• Quantum corrections to the Hawking radiation spectrum
• The Koslowski- Sahlmann representation: Gauge and diffeomorphism invariance
• The information-complete quantum theory
• A new realization of quantum geometry
• Coherent State Operators in Loop Quantum Gravity
• Canonical quantum gravity and consistent discretizations
• Area Operator in Loop Quantum Gravity
• Holostar thermodynamics
• On the distribution of the eigenvalues of the area operator in loop quantum gravity
• Loop quantum cosmology from an alternative Hamiltonian
• Bianchi type I, Schutz perfect fluid and evolutionary quantum cosmology
• Quantization of static space-times
• Real sector of the nonminimally coupled scalar field to self-dual gravity
• The damped harmonic oscillator at the classical limit of the Snyder-de Sitter space
• Exotic fermionic fields and minimal length
• Revisiting the solution of the second-class constraints of the Holst action
• Holographic relations in loop quantum gravity
• Energy and directional signatures for plane quantized gravity waves
• Black hole evaporation within a momentum-dependent metric
• The status of Quantum Geometry in the dynamical sector of Loop Quantum Cosmology
• An analytical computation of asymptotic Schwarzschild quasinormal frequencies
• Asymptotic black hole quasinormal frequencies
• Graphical method in loop quantum gravity: I. Derivation of the closed formula for the matrix element of the volume operator
• The Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator
• Considering boundary conditions for black hole entropy in loop quantum gravity
• Three-geometry and reformulation of the Wheeler-DeWitt equation
• The holographic solution - Why general relativity must be understood in terms of strings
• Formation and Stability of Area Quantized Black Holes
• A one-particle time of arrival operator for a free relativistic spin-0 charged particle in (1+1) dimensions
• Barbero's formulation from a $BF$-type action with the Immirzi parameter
• Entropy bound for the photon gas in noncommutative spacetime
• Bibliography of publications related to Classical and Quantum Gravity in terms of Connection and Loop Variables
• Intertwiner Entanglement Excitation and Holonomy Operator
• Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity
• The geometry and entanglement entropy of surfaces in loop quantum gravity
• Integrable maps from Galois differential algebras, Borel transforms and number sequences
• $\mathcal{N}=1$ Supergravity with LQG methods and quantization of the SUSY constraint
• $\kappa$-Minkowski and Snyder algebra from reparametrisation symmetry
• Quantum Gravity: physics from supergeometries
• ESA Voyage 2050 white paper -- GrailQuest: hunting for Atoms of Space and Time hidden in the wrinkle of Space-Time
• An experiment to test the discreteness of time
• Fermion coupling to loop quantum gravity: canonical formulation
• Is the diagonal case a general picture for Loop Quantum Cosmology?
• Loop Quantum Gravity effects on electromagnetic properties of charged leptons
• Compact anisotropic stars with membrane - a new class of exact solutions to the Einstein field equations
• From loop quantum gravity to cosmology: the two-vertex model
• Generalized uncertainty principle and neutrino phenomenology
• Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit
• Gauge symmetries in Ashtekar's formulation of general relativity
• Linear-Quadratic GUP and Thermodynamic Dimensional Reduction
• Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?
• Probing the Shape of Quantum Surfaces: the Quadrupole Moment Operator
• Asymptotics of classical spin networks
• Generalized Uncertainty Principle and Quantum Gravity Phenomenology
• A discrete, unitary, causal theory of quantum gravity
• A "Helium Atom" of Space: Dynamical Instability of the Isochoric Pentahedron
• Comments on area spectra in Loop Quantum Gravity
• A spin network generalization of the Jones Polynomial and Vassiliev invariants
• Dynamics in canonical models of loop quantum gravity
• Quantum Computing Spacetime
• Quantum geometry with a nondegenerate vacuum: a toy model
• Loop quantum cosmology from an alternative Hamiltonian. II. Including the Lorentzian term
• Quantum Geometry and Black Holes
• Decoding infrared imprints of quantum origins of black holes
• Early universe thermostatistics in curved momentum spaces
• On the Support of Physical Measures in Gauge Theories
• The fermionic contribution to the spectrum of the area operator in nonperturbative quantum gravity
• Towards a background independent approach to M theory
• Angle and Volume Studies in Quantized Space
• Relating spin-foam to canonical loop quantum gravity by graphical calculus
• Circular orbits and accretion disk around a deformed-Schwarzschild black hole in loop quantum gravity
• Phenomenological implications of an alternative Hamiltonian constraint for quantum cosmology
• Immirzi Ambiguity, Boosts and Conformal Frames for Black Holes
• Quantized Curvature in Loop Quantum Gravity
• A review on Loop Quantum Gravity
• Quasinormal modes and horizon area quantisation in Loop Quantum Gravity
• Symplectic quantization II: dynamics of space-time quantum fluctuations and the cosmological constant
• Alternative $k=-1$ loop quantum cosmology
• Basis states for gravitons in non-perturbative loop representation space
• The Time Evolution of Quantum Universe in The Quantum Potential Picture
• Loop Quantum Gravity and the The Planck Regime of Cosmology
• Thermodynamics of Quantum Isolated Horizons with model Hamiltonians
• Classical and Quantum Polyhedra
• Bouncing scalar field cosmology in the polymeric minisuperspace picture
• Divergences in spinfoam quantum gravity
• Generating Functionals for Spin Foam Amplitudes
• Five-Dimensional Mechanics as the Starting Point for the Magueijo-Smolin Doubly Special Relativity
• `Third' Quantization of Vacuum Einstein Gravity and Free Yang-Mills Theories
• Small-Scale Structure of Spacetime: Bounds and Conjectures
• Planck scale effect on the thermodynamics of photon gas
• Random Invariant Tensors
• Photon gas thermodynamics in dS and AdS momentum spaces
• The kinematical Setup of Quantum Geometry: A Brief Review
• Aspects of quantum gravity
• Gauge Fixing and Observables in General Relativity
• Note on quantum entanglement and quantum geometry
• Off-shell Noether currents and potentials for first-order general relativity
• A discrete curvature on a planar graph
• Relativistic Generalized Uncertainty Principle and Its Implications
• Cosmological states in loop quantum gravity on homogeneous graphs
• Renormalizing Spacetime
• New Operators for Spin Net Gravity: Definitions and Consequences
• Does loop quantum gravity imply Lambda=0?
• Conformal Cores of Quantum Black Holes in Quadratic Gravity
• Quantum fate of timelike naked singularity with scalar hair
• An overview of canonical quantum gravity
• Embedded graph invariants in Chern-Simons theory
• Classical gauge principle -- From field theories to classical mechanics
• Planck-scale models of the Universe
• Quantum Entanglement Dynamics of Spacetime and Matter
• Quantum Space-times: Beyond the Continuum of Minkowski and Einstein
• Master constraint approach to quantum-reduced loop gravity
• Self-organized critical behavior: the evolution of frozen spin networks model in quantum gravity
• Physical Diffeomorphisms in Loop Quantum Gravity
• Introduction to supersymmetric spin networks
• Gravity in presence of fermions as a SU(2) gauge theory
• A note on Penrose's Spin-Geometry Theorem and the geometry of `empirical quantum angles'
• Radiative properties of a nonsingular black hole: Hawking radiation and gray-body factor
• Does matter differ from vacuum?
• Fermion mass gap in the loop representation of quantum gravity
• Reconstructing the metric in group field theory
• Fiber bundle structure in Ashtekar-Barbero-Immirzi formulation of General Relativity
• Topology and Volume Effects in Quantum Gravity: Wheeler-DeWitt Theory
• Evidence for Planck Luminosity Bound in Quantum Gravity
• A dynamical evolution model on the black hole horizon
• Quantum Big-Bounce as a phenomenology of RQM in the Mini-superspace
• Superradiant instability of area quantized Kerr black hole with discrete reflectivity
• Spectroscopy of an AdS Reissner-Nordstrom black hole
• Effective Quantum Gravitational Collapse in a Polymer Framework
• Quantum cosmology as a lattice in a box
• Potential of Neutrino Telescopes to Detect Quantum Gravity-Induced Decoherence in the Presence of Dark Fermions
• Computing the graph-changing dynamics of loop quantum gravity
• Asymptotic Flatness and Quantum Geometry
• Topological Classification of a 4D AdS Black Hole with Non-Minimal Maxwell Coupling
• Purely geometric path integral for spin foams
• On the fundamental length of quantum geometry and the black hole entropy
• Semiclassical states in quantum gravity: Curvature associated to a Voronoi graph
• 5-Dimensional $SO(1,4)$-Invariant Action as an Origin to the Magueijo-Smolin Doubly Special Relativity Proposal
• Can Effects of a Generalized Uncertainty Principle Appear in Compact Stars?
• Quantum Gravity Effects on the Tachyon Inflation from Thermodynamic Perspective
• Black hole entropy predictions without the Immirzi parameter and Hawking radiation of a single-partition black hole
• Loops versus strings
• Maxwell-Boltzmann type Hawking radiation
• Remarks on the black hole entropy and Hawking spectrum in Loop Quantum Gravity
• Spin network coherent states for planar gravitational waves. I
• Deformations of the Canonical Commutation Relations and Metric Structures
• Quantum Gravity
• A possible solution to the problem of time in quantum cosmology
• A Curvature Operator for a Regular Tetrahedron Shape in LQG
• Quantum gravity, space-time structure, and cosmology
• Mathematical structure of loop quantum cosmology
• Much ado about nothing: a treatise on empty and not-so-empty spacetimes
• Quantum gravity in the very early universe
• Woven Geometries: Black Holes
• The d-Dimensional Cosmological Constant and the Holographic Horizons
• Effective geometry of Bell-network states on a dipole graph
• Deparametrization and quantization of scalar-tensor gravity and its cosmological model
• Loop quantum gravity - a short review
• Properties of the Rovelli-Smolin-DePietri volume operator in the spaces of monochromatic intertwiners
• Natural discrete differential calculus in physics
• Black Hole Statistical Physics: Entropy
• Cosmological Natural Selection Revisited. Some Remarks on the Conceptual Conundrum and Possible Alleys
• Entanglement in simple spin networks with a boundary
• Quantum Hairy Black Hole Formation and Horizon Quantum Mechanics
• Cosmology in Loop Quantum Gravity: symmetry reduction preserving gauge degrees of freedom
• Proceedings to the 'Euroconference on Symmetries Beyond the Standard Model', 12. - 17. July 2003, Portoroz, Slovenia (Part 1 of 2)
• Hamiltonian gravity in tetrad-connection variables
• Approximation of the naive black hole degeneracy
• A note on the Hamiltonian constraint in canonical GR
• Exotic Smoothness and Quantum Gravity II: exotic R^4, singularities and cosmology
• Spacetime Quanta? : The Discrete Spectrum of a Quantum Spacetime Four-Volume Operator in Unimodular Loop Quantum Cosmology
• Making quantum gravity calculable
• Modeling M-Theory Vacua via Gauged S-Duality
• The Quantum Trihedron
• Three Dimensional Strings as Collective Coordinates of Four Dimensional Non-Perturbative Quantum Gravity
• Condensation of area quanta ensembles with quantum statistics in Schwarzschild spacetimes
• Doubly Quantum Mechanics
• A Loop Quantum Gravity Inspired Action for the Bosonic String and Emergent Dimensions at Large Scales
• Coherent states for fractional powers of the harmonic oscillator Hamiltonian
• Finite Hilbert space and maximum mass of Schwarzschild black holes from a Generalized Uncertainty Principle
• A geometric procedure for the reduced-state-space quantisation of constrained systems
• Combinatorial Spacetime from Loop Quantum Gravity
• Polymer Quantization of the Isotropic Universe: comparison with the Bounce of Loop Quantum Cosmology
• Cosmological implications of the hydrodynamical phase of group field theory
• Scaling-Based Quantization of Spacetime Microstructure
• A novel mechanism for probing the Planck scale with wavepackets following general distributions
• Phonons mimicking doubly special relativity kinematics
• Radiation of Quantized Black Hole
• Quantum Gravity and the Big Bang
• Discrete Hamiltonian evolution and quantum gravity
• Quantization of Black Holes in the Shielded Strong Gravity Scenario (I. Neutral Scalar States)
• New Hints from General Relativity
• Relational Approach to Spin Networks
• Finitary-Algebraic `Resolution' of the Inner Schwarzschild Singularity
• A Scenario for Strong Gravity in Particle Physics: An alternative mechanism for black holes to appear at accelerator experiments
• Quantum Dyanmics of Loop Quantum Gravity
• Die einheitliche Beschreibung der fundamentalen Objekte und Wechselwirkungen der Natur in der Quantentheorie der Ur-Alternativen
• Quantum time uncertainty in Schwarzschild-anti-de Sitter black holes
• Motion in Quantum Gravity
• Revisiting Quantum Volume Operator
• Quantum Gravity for Dummies
• A rotating GUP black hole: metric, shadow, and bounds on quantum parameters
• A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics
• Finite-Resolution Limits on Quantum Uniqueness
• A note on cutting spin networks and the area spectrum in loop quantum gravity
• Dimensional Phenomenology in Polymeric Quantization Framework
• Categorical Quantum Volume Operator
• Closure of Constraints for Plane Gravity Waves
• Statistical physics on Euclidean Snyder space: connections with the GUP and cosmological implications
• Hamiltonian Theory: generalizations to higher dimensions, supersymmetry and modified gravity
• The holostar - a self-consistent model for a compact self-gravitating object
• New insights in quantum geometry
• Hidden Symmetries in Gravity : Black holes and other minisuperspaces
• Space-like Singularities of General Relativity: A Phantom menace?
• Thermodynamics of spacetime from minimal area
• Effective dynamics of Janis-Newman-Winicour spacetime
• Internal structure of Hayward black holes
• Geodesics in Quantum Gravity
• Bounds on Large Extra Dimensions from the Generalized Uncertainty Principle
• Non singular M theory Universe in Loop Quantum Cosmology -- inspired Models
• Gaussian Planck Relics are Ruled-Out as Dark Matter by LIGO
• Deformed Heisenberg Algebra with a minimal length: Application to some molecular potentials
• Quantum volume and length fluctuations in a midi-superspace model of Minkowski space
• Thermal Time and Kepler's Second Law
• Holographic bound in covariant loop quantum gravity
• Time and a Temporally Statistical Quantum Geometrodynamics
• The phenomenology of Quantum Bounce in the Klein-Gordon, Wheeler-DeWitt and Dirac formalisms
• Relativistic Kinematics of Two-Parametric Riemann Surface in Genus Two
• Proof of Bekenstein-Mukhanov ansatz in loop quantum gravity
• Probing Planck scale effects on absolute mass limit in neutrino flavor evolution
• Length versus time-interval discreteness in 2+1 Lorentzian canonical quantum gravity
• Einstein-Hilbert Path Integrals in ${\mathbb{R}}^4$
• Combinatorial solutions to the Hamiltonian constraint in (2+1)-dimensional Ashtekar gravity
• A note on the fuzzy sphere area spectrum, black hole luminosity, and the quantum nature of spacetime
• Structures associated with the Borromean rings complement in the Poincaré ball
• Deriving Spin within a discrete-time theory
• Quantization of Length in Spaces with Position-Dependent Noncommutativity
• On the Hamiltonian formalism of the Tetrad-Gravity with fermions
• The effect of loop quantum gravitational rainbow functions on the formation of naked singularities
• Quantum Hall Effect and Black Hole Entropy in Loop Quantum Gravity
• Gravitational quantum states as finite representations of the Lorentz group
• Examination of the nature of the Bianchi type cosmological singularities
• Raum, Zeit und Wechselwirkung in der Quantentheorie der Ur-Alternativen
• Hilbert space representation for quasi-Hermitian position-deformed Heisenberg algebra and Path integral formulation
• Hermiticity of the Volume Operators in Loop Quantum Gravity
• Canonical quantum gravity with new variables and loops: a report
• A class of entangled and diffeomorphism-invariant states in loop quantum gravity: Bell-network states
• Loop Special Relativity: Kaluza-Klein area metric as a line element for stringy events