Quantum geometry from phase space reduction

Abstract

In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

33 pages, 1 figure

References in corpus (11)

• LQG vertex with finite Immirzi parameter
• A new spinfoam vertex for quantum gravity
• A New Spin Foam Model for 4d Gravity
• The loop-quantum-gravity vertex-amplitude
• Flipped spinfoam vertex and loop gravity
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• Holomorphic Factorization for a Quantum Tetrahedron
• Path integral representation of spin foam models of 4d gravity
• Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups
• A semiclassical tetrahedron

Cited by in corpus (74)

• The Spin Foam Approach to Quantum Gravity
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• Polyhedra in loop quantum gravity
• Towards Spinfoam Cosmology
• LQG propagator from the new spin foams
• From twistors to twisted geometries
• Self-dual Black Holes in LQG: Theory and Phenomenology
• U(N) Coherent States for Loop Quantum Gravity
• Spin foam models for quantum gravity from lattice path integrals
• The Fine Structure of SU(2) Intertwiners from U(N) Representations
• Coherent spin-networks
• Loop quantum gravity: the first twenty five years
• On the geometry of loop quantum gravity on a graph
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Effective Spin Foam Models for Four-Dimensional Quantum Gravity
• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• Effective Spin Foam Models for Lorentzian Quantum Gravity
• Holomorphic Factorization for a Quantum Tetrahedron
• Lorentzian spinfoam propagator
• Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory
• Discreteness of the volume of space from Bohr-Sommerfeld quantization
• SU(2) graph invariants, Regge actions and polytopes
• Investigation of the Spinfoam Path integral with Quantum Cuboid Intertwiners
• Quantum gravity asymptotics from the SU(2) 15j symbol
• Holonomy spin foam models: Asymptotic geometry of the partition function
• Generalized Spinfoams
• Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory
• A spin foam model for general Lorentzian 4-geometries
• The twistorial structure of loop-gravity transition amplitudes
• Gluing polyhedra with entanglement in loop quantum gravity
• Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals
• Holomorphic Simplicity Constraints for 4d Spinfoam Models
• Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps
• Spinfoams in the holomorphic representation
• Deformation Operators of Spin Networks and Coarse-Graining
• Many-nodes/many-links spinfoam: the homogeneous and isotropic case
• Bohr-Sommerfeld Quantization of Space
• A curvature operator for LQG
• Quasi-local holographic dualities in non-perturbative 3d quantum gravity II - From coherent quantum boundaries to BMS3 characters
• Black Hole Entropy, Loop Gravity, and Polymer Physics
• A Discrete and Coherent Basis of Intertwiners
• Deformations of Polyhedra and Polygons by the Unitary Group
• Simple model for quantum general relativity from loop quantum gravity
• Spin foams with timelike surfaces
• A new Hamiltonian for the Topological BF phase with spinor networks
• On the exact evaluation of spin networks
• Revisiting the Simplicity Constraints and Coherent Intertwiners
• Four-dimensional Spinfoam Quantum Gravity with Cosmological Constant: Finiteness and Semiclassical Limit
• Sub-leading asymptotic behaviour of area correlations in the Barrett-Crane model
• Analytic Continuation of Spin foam Models
• Unitary irreducible representations of SL(2,C) in discrete and continuous SU(1,1) bases
• Learning about Quantum Gravity with a Couple of Nodes
• Inequivalent coherent state representations in group field theory
• Polytopes in all dimensional loop quantum gravity
• Spinfoam Fermions: PCT Symmetry, Dirac Determinant, and Correlation Functions
• Entropy in the Classical and Quantum Polymer Black Hole Models
• Geometric parametrization of $SO(D+1)$ phase space of all dimensional loop quantum gravity
• Coherent 3j-symbol representation for the loop quantum gravity intertwiner space
• Tullio Regge's legacy: Regge calculus and discrete gravity
• The new spin foam models and quantum gravity
• Quantum edge modes in 3d gravity and 2+1d topological phases of matter
• Semiclassical Behavior of Spinfoam Amplitude with Small Spins and Entanglement Entropy
• Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity
• Dynamics in canonical models of loop quantum gravity
• A simpler way of imposing simplicity constraints
• Random Invariant Tensors
• Generating Functionals for Spin Foam Amplitudes
• Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?
• Quasi-Local 3D Quantum Gravity : Exact Amplitude and Holography
• Entanglement entropy of physical states in hypercuboidally truncated spin foam quantum gravity
• Polyhedra in spacetime from null vectors
• On Geometry and Symmetries in Classical and Quantum Theories of Gauge Gravity
• Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
• Hermiticity of the Volume Operators in Loop Quantum Gravity