A lattice worldsheet sum for 4-d Euclidean general relativity

Abstract

A lattice model for four dimensional Euclidean quantum general relativity is proposed for a simplicial spacetime. It is shown how this model can be expressed in terms of a sum over worldsheets of spin networks, and an interpretation of these worldsheets as spacetime geometries is given, based on the geometry defined by spin networks in canonical loop quantized GR. The spacetime geometry has a Planck scale discreteness which arises "naturally" from the discrete spectrum of spins of SU(2) representations (and not from the use of a spacetime lattice). The lattice model of the dynamics is a formal quantization of the classical lattice model of \cite{Rei97a}, which reproduces, in a continuum limit, Euclidean general relativity.

50 pages latex. 8 postscript figures. Uses psfig.sty

References in corpus (22)

• Quantum Theory of Gravity I: Area Operators
• Real Ashtekar Variables for Lorentzian Signature Space-times
• Quantum Spin Dynamics (QSD)
• Spin Networks and Quantum Gravity
• Weaving a classical geometry with quantum threads
• Topological Lattice Models in Four Dimensions
• ``Sum over Surfaces'' form of Loop Quantum Gravity
• Spin Network States in Gauge Theory
• Geometry Eigenvalues and Scalar Product from Recoupling Theory in Loop Quantum Gravity
• The Immirzi parameter in quantum general relativity
• Closed formula for the matrix elements of the volume operator in canonical quantum gravity
• The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis
• Representation Theory of Analytic Holonomy C* Algebras
• Reality Conditions and Ashtekar Variables: a Different Perspective
• Conformal Invariance and Cosmic Background Radiation
• A left-handed simplicial action for euclidean general relativity
• From Euclidean to Lorentzian General Relativity: The Real Way
• Semi-Classical Limits of Simplicial Quantum Gravity
• A Real Polynomial Formulation of General Relativity in terms of Connections
• Dual formulation of spin network evolution
• A definition of the Ponzano-Regge quantum gravity model in terms of surfaces
• Combinatorial space from loop quantum gravity

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• Discrete approaches to quantum gravity in four dimensions
• Spacetime as a Feynman diagram: the connection formulation
• so(4) Plebanski Action and Relativistic Spin Foam Model
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Phase space descriptions for simplicial 4d geometries
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• An invitation to loop quantum gravity
• Quantum geometry with intrinsic local causality
• The complete LQG propagator: II. Asymptotic behavior of the vertex
• Simplicity in simplicial phase space
• Lorentzian LQG vertex amplitude
• On relativistic spin network vertices
• Lifting SU(2) Spin Networks to Projected Spin Networks
• Regularization and finiteness of the Lorentzian LQG vertices
• Supersymmetric Spin Networks and Quantum Supergravity
• Boosting Wigner's nj-symbols
• A spin foam model for pure gauge theory coupled to quantum gravity
• Barrett-Crane spin foam model from generalized BF-type action for gravity
• A finiteness proof for the Lorentzian state sum spinfoam model for quantum general relativity
• Discrete Riemannian Geometry
• Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
• The century of the incomplete revolution: searching for general relativistic quantum field theory
• Spin Foam Diagrammatics and Topological Invariance
• Classical general relativity as BF-Plebanski theory with linear constraints
• Coupling gauge theory to spinfoam 3d quantum gravity
• Eleven dimensional supergravity as a constrained topological field theory
• Emergent Cosmology from Quantum Gravity in the Lorentzian Barrett-Crane Tensorial Group Field Theory Model
• An algebraic approach to coarse graining
• Pushing Further the Asymptotics of the 6j-symbol
• Continuum spin foam model for 3d gravity
• Boundary terms in the Barrett-Crane spin foam model and consistent gluing
• Nonperturbative dynamics for abstract (p,q) string networks
• A proposal for analyzing the classical limit of kinematic loop gravity
• Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators
• Chern-Simons Term for BF Theory and Gravity as a Generalized Topological Field Theory in Four Dimensions
• A candidate for a background independent formulation of M theory
• The Construction of Spin Foam Vertex Amplitudes
• Loop quantization from a lattice gauge theory perspective
• Conserved Quantities in Background Independent Theories
• Knot theory and a physical state of quantum gravity
• Piecewise linear loop quantum gravity
• Degenerate Plebanski Sector and Spin Foam Quantization
• Poincare-Plebanski formulation of GR and dual simplicity constraints
• Spin foam quantization of SO(4) Plebanski's action
• Rigorous construction of a Spin Foam Model for Lorentzian BF theory and Gravity: The foundations
• Introduction to supersymmetric spin networks
• Topological quantum field theory and quantum gravity