The length operator in Loop Quantum Gravity

Abstract

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity - the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.

33 pages, 12 figures; NPB version

References in corpus (46)

• Background Independent Quantum Gravity: A Status Report
• Discreteness of area and volume in quantum gravity
• Quantum Theory of Gravity I: Area Operators
• Real Ashtekar Variables for Lorentzian Signature Space-times
• Quantum Spin Dynamics (QSD)
• Loop Quantum Gravity
• Spin Networks and Quantum Gravity
• Weaving a classical geometry with quantum threads
• Real and complex connections for canonical gravity
• Spin Foam Models
• Partial observables
• Differential Geometry on the Space of Connections via Graphs and Projective Limits
• Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
• Quantum Gravity and Regge Calculus
• Spin Network States in Gauge Theory
• Quantum Spin Dynamics (QSD) II
• Graviton propagator from background-independent quantum gravity
• Geometry Eigenvalues and Scalar Product from Recoupling Theory in Loop Quantum Gravity
• Representations of the Weyl Algebra in Quantum Geometry
• QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• A length operator for canonical quantum gravity
• Coherent States for Canonical Quantum General Relativity and the Infinite Tensor Product Extension
• Closed formula for the matrix elements of the volume operator in canonical quantum gravity
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures
• Representation Theory of Analytic Holonomy C* Algebras
• Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• Quantum tetrahedra and simplicial spin networks
• Volume and Quantizations
• Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• The complete LQG propagator: II. Asymptotic behavior of the vertex
• Group Integral Techniques for the Spinfoam Graviton Propagator
• Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity II
• Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity I
• Spectrum of the Volume Operator in Quantum Gravity
• A left-handed simplicial action for euclidean general relativity
• Separable Hilbert space in Loop Quantum Gravity
• Operators for quantized directions
• Properties of the Volume Operator in Loop Quantum Gravity I: Results
• A semiclassical tetrahedron
• The perturbative Regge-calculus regime of Loop Quantum Gravity
• Area Regge Calculus and Discontinuous Metrics
• Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann
• Modelling Space with an Atom of Quantum Geometry
• Refractive gravitational waves and quantum fluctuations

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• Phase space descriptions for simplicial 4d geometries
• LQG propagator from the new spin foams
• Self-dual Black Holes in LQG: Theory and Phenomenology
• Spin Foams and Canonical Quantization
• Effective Spin Foam Models for Four-Dimensional Quantum Gravity
• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• Lorentzian spinfoam propagator
• Testing loop quantum cosmology
• New length operator for loop quantum gravity
• Effective Dynamics from Coherent State Path Integral of Full Loop Quantum Gravity
• Gluing polyhedra with entanglement in loop quantum gravity
• Turning Big Bang into Big Bounce: I. Classical Dynamics
• Spinfoams in the holomorphic representation
• Preinflationary dynamics in loop quantum cosmology: Monodromy Potential
• Critical Overview of Loops and Foams
• Discrete gravity dynamics from effective spin foams
• A curvature operator for LQG
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• From Classical To Quantum Gravity: Introduction to Loop Quantum Gravity
• The Holst Spin Foam Model via Cubulations
• Loop Quantum Gravity a la Aharonov-Bohm
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• Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
• Spinning geometry = Twisted geometry
• Time evolution in deparametrized models of loop quantum gravity
• Semiclassical regime of Regge calculus and spin foams
• Entanglement entropy of Bell-network states in LQG: Analytical and numerical results
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• Background dynamics of pre-inflationary scenario in Brans-Dicke loop quantum cosmology
• Towards Loop Quantization of Plane Gravitational Waves
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• Symmetry Reduced Loop Quantum Gravity: A Bird's Eye View
• Anisotropic loop quantum cosmology with self-dual variables
• Twisted geometries are area-metric geometries
• Shape in an Atom of Space: Exploring quantum geometry phenomenology
• Fractal Quantum Space-Time
• Length and distance on a quantum space
• Reformulation of Boundary BF Theory Approach to Statistical Explanation of the Entropy of Isolated Horizons
• Graphical method in loop quantum gravity: I. Derivation of the closed formula for the matrix element of the volume operator
• An experiment to test the discreteness of time
• The geometry and entanglement entropy of surfaces in loop quantum gravity
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• Dynamics in canonical models of loop quantum gravity
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• Loop Quantum Gravity and the The Planck Regime of Cosmology
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• Quantization of Spacetime Based on Spacetime Interval Operator
• Relativistic Generalized Uncertainty Principle and Its Implications
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• Evidence for Planck Luminosity Bound in Quantum Gravity
• Expansion operators in spherically symmetric loop quantum gravity
• A Curvature Operator for a Regular Tetrahedron Shape in LQG
• Effective geometry of Bell-network states on a dipole graph
• Space-like Singularities of General Relativity: A Phantom menace?
• Thermodynamics of spacetime from minimal area
• A Quantized Spacetime Based on $Spin(3,1)$ Symmetry
• Hermiticity of the Volume Operators in Loop Quantum Gravity
• Quantum volume and length fluctuations in a midi-superspace model of Minkowski space
• Semiclassical Length Measure from a Quantum-Gravity Wave Function