Operators for quantized directions

Abstract

Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two ``bundles'' of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles.

v1: 20 pages, 23 figures v2: changes in presentation and regularization (final results unchanged). This is an expanded version of the one to be published in Class. Quant. Grav

Cited by in corpus (48)

• Background Independent Quantum Gravity: A Status Report
• Polyhedra in loop quantum gravity
• An invitation to loop quantum gravity
• LQG propagator from the new spin foams
• The length operator in Loop Quantum Gravity
• Loop quantum gravity: the first twenty five years
• How far are we from the quantum theory of gravity?
• The complete LQG propagator: II. Asymptotic behavior of the vertex
• Lorentzian spinfoam propagator
• New length operator for loop quantum gravity
• Observables in Loop Quantum Gravity with a cosmological constant
• Gluing polyhedra with entanglement in loop quantum gravity
• From lattice BF gauge theory to area-angle Regge calculus
• Bohr-Sommerfeld Quantization of Space
• A curvature operator for LQG
• Black Hole Entropy, Loop Gravity, and Polymer Physics
• LQG propagator: III. The new vertex
• Simple model for quantum general relativity from loop quantum gravity
• Loop Quantum Gravity a la Aharonov-Bohm
• The perturbative Regge-calculus regime of Loop Quantum Gravity
• Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
• A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
• Gravitational Statistical Mechanics: A model
• Semiclassical regime of Regge calculus and spin foams
• Entanglement entropy of Bell-network states in LQG: Analytical and numerical results
• Background independent quantizations: the scalar field II
• General geometric operators in all dimensional loop quantum gravity
• Modelling Space with an Atom of Quantum Geometry
• Towards Loop Quantization of Plane Gravitational Waves
• Asymptotics of LQG fusion coefficients
• Hamiltonian spinfoam gravity
• Scalar curvature operator for models of loop quantum gravity on a cubical graph
• Isotropic Loop Quantum Cosmology with Matter II: The Lorentzian Constraint
• Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity
• Shape in an Atom of Space: Exploring quantum geometry phenomenology
• Quasilocal Energy for Spin-net Gravity
• Coherent State Operators in Loop Quantum Gravity
• Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity
• Angle and Volume Studies in Quantized Space
• Dynamics in canonical models of loop quantum gravity
• Quantum Geometry Phenomenology: Angle and Semiclassical States
• Aspects of quantum gravity
• New Operators for Spin Net Gravity: Definitions and Consequences
• A note on Penrose's Spin-Geometry Theorem and the geometry of `empirical quantum angles'
• Infinities as a measure of our ignorance
• Condensation of area quanta ensembles with quantum statistics in Schwarzschild spacetimes
• Quantum volume and length fluctuations in a midi-superspace model of Minkowski space
• Semiclassical Length Measure from a Quantum-Gravity Wave Function