Phase space descriptions for simplicial 4d geometries

Abstract

Starting from the canonical phase space for discretised (4d) BF-theory, we implement a canonical version of the simplicity constraints and construct phase spaces for simplicial geometries. Our construction allows us to study the connection between different versions of Regge calculus and approaches using connection variables, such as loop quantum gravity. We find that on a fixed triangulation the (gauge invariant) phase space associated to loop quantum gravity is genuinely larger than the one for length and even area Regge calculus. Rather, it corresponds to the phase space of area-angle Regge calculus, as defined by Dittrich and Speziale in [arXiv:0802.0864] (prior to the imposition of gluing constraints, that ensure the metricity of the triangulation). We argue that this is due to the fact that the simplicity constraints are not fully implemented in canonical loop quantum gravity. Finally, we show that for a subclass of triangulations one can construct first class Hamiltonian and Diffeomorphism constraints leading to flat 4d space-times.

corrected structure constants, several references add

References in corpus (25)

• 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
• A "general boundary" formulation for quantum mechanics and quantum gravity
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Improved and Perfect Actions in Discrete Gravity
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• Area-angle variables for general relativity
• The length operator in Loop Quantum Gravity
• (Broken) Gauge Symmetries and Constraints in Regge Calculus
• Stepping out of Homogeneity in Loop Quantum Cosmology
• The complete LQG propagator: II. Asymptotic behavior of the vertex
• Regge calculus from a new angle
• Lorentzian LQG vertex amplitude
• Linearized dynamics from the 4-simplex Regge action
• Path integral representation of spin foam models of 4d gravity
• Breaking and restoring of diffeomorphism symmetry in discrete gravity
• Quantum Gravity, or The Art of Building Spacetime
• Simplicity and closure constraints in spin foam models of gravity
• Plebanski Theory and Covariant Canonical Formulation
• Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams
• Alternative symplectic structures for SO(3,1) and SO(4) four-dimensional BF theories

Cited by in corpus (101)

• The Spin Foam Approach to Quantum Gravity
• A Short Review of Loop Quantum Gravity
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• Polyhedra in loop quantum gravity
• On the semiclassical limit of 4d spin foam models
• Improved and Perfect Actions in Discrete Gravity
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• From twistors to twisted geometries
• Edge modes of gravity -- II: Corner metric and Lorentz charges
• Coarse graining methods for spin net and spin foam models
• Quantum geometry from phase space reduction
• From the discrete to the continuous - towards a cylindrically consistent dynamics
• On the geometry of loop quantum gravity on a graph
• Non-commutative flux representation for loop quantum gravity
• Edge modes of gravity. Part III. Corner simplicity constraints
• Regge calculus from a new angle
• Simplicity in simplicial phase space
• A new vacuum for Loop Quantum Gravity
• Effective Spin Foam Models for Four-Dimensional Quantum Gravity
• Effective Spin Foam Models for Lorentzian Quantum Gravity
• From covariant to canonical formulations of discrete gravity
• Towards a phase diagram for spin foams
• Path integral measure and triangulation independence in discrete gravity
• Canonical simplicial gravity
• Time evolution as refining, coarse graining and entangling
• Investigation of the Spinfoam Path integral with Quantum Cuboid Intertwiners
• SU(2) graph invariants, Regge actions and polytopes
• Operator Spin Foam Models
• Breaking and restoring of diffeomorphism symmetry in discrete gravity
• Quantum gravity asymptotics from the SU(2) 15j symbol
• Coarse graining free theories with gauge symmetries: the linearized case
• The Hamiltonian constraint in 3d Riemannian loop quantum gravity
• The twistorial structure of loop-gravity transition amplitudes
• Quantum group spin nets: refinement limit and relation to spin foams
• Quantum gravity kinematics from extended TQFTs
• Recurrence relations for spin foam vertices
• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• Critical Overview of Loops and Foams
• Discrete gravity dynamics from effective spin foams
• Asymptotics of lowest unitary SL(2,C) invariants on graphs
• Discretization independence implies non-locality in 4D discrete quantum gravity
• Twistor Networks and Covariant Twisted Geometries
• Asymptotic analysis of the EPRL model with timelike tetrahedra
• Dirac's discrete hypersurface deformation algebras
• On the role of the Barbero-Immirzi parameter in discrete quantum gravity
• Classical Setting and Effective Dynamics for Spinfoam Cosmology
• Classical general relativity as BF-Plebanski theory with linear constraints
• Flux formulation of loop quantum gravity: Classical framework
• A Discrete and Coherent Basis of Intertwiners
• Bubble divergences and gauge symmetries in spin foams
• Simple model for quantum general relativity from loop quantum gravity
• On the Implementation of the Canonical Quantum Simplicity Constraint
• The Holst Spin Foam Model via Cubulations
• A new Hamiltonian for the Topological BF phase with spinor networks
• Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex
• Spinning geometry = Twisted geometry
• From spin foams to area metric dynamics to gravitons
• Towards effective actions for the continuum limit of spin foams
• Semiclassical regime of Regge calculus and spin foams
• (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
• Bubble Networks: Framed Discrete Geometry for Quantum Gravity
• Twisted geometries, twistors and conformal transformations
• Discretisations, Constraints and Diffeomorphisms in Quantum Gravity
• New action for simplicial gravity in four dimensions
• A note on the secondary simplicity constraints in loop quantum gravity
• 2+1D Loop Quantum Gravity on the Edge
• Twisted Geometries Coherent States for Loop Quantum Gravity
• Diffeomorphism symmetry in quantum gravity models
• Feynman diagrammatic approach to spin foams
• The Plebanski sectors of the EPRL vertex
• Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators
• Quantum geometry from higher gauge theory
• Spinors and Twistors in Loop Gravity and Spin Foams
• Squeezed vacua in loop quantum gravity
• The Construction of Spin Foam Vertex Amplitudes
• Canonical formulation of Poincare BFCG theory and its quantization
• Holographic description of boundary gravitons in (3+1) dimensions
• Thermal quantum spacetime
• Corner symmetry and quantum geometry
• Lorentzian quantum cosmology from effective spin foams
• Twisted geometries are area-metric geometries
• Spinfoams: Foundations
• Continuous formulation of the Loop Quantum Gravity phase space
• Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity
• Area-metric gravity revisited
• Spikes and spines in 4D Lorentzian simplicial quantum gravity
• State Sums and Geometry
• Poincare-Plebanski formulation of GR and dual simplicity constraints
• Degenerate Plebanski Sector and Spin Foam Quantization
• A new realization of quantum geometry
• Spin foams
• Geometry from local flatness in Lorentzian spin foam theories
• Linking shape dynamics and loop quantum gravity
• From area metric backgrounds to the cosmological constant and corrections to the Polyakov action
• On Geometry and Symmetries in Classical and Quantum Theories of Gauge Gravity
• Classical GR as a topological theory with linear constraints
• Discrete Hamiltonian for General Relativity
• At the Corner of Space and Time
• New insights in quantum geometry
• Spin Networks and Cosmic Strings in 3+1 Dimensions
• Polyhedron phase space using 2-groups: kappa-Poincare as a Poisson 2-group