A Lagrangian approach to the Barrett-Crane spin foam model

Abstract

We provide the Barrett-Crane spin foam model for quantum gravity with a discrete action principle, consisting in the usual BF term with discretized simplicity constraints which in the continuum turn topological BF theory into gravity. The setting is the same as usually considered in the literature: space-time is cut into 4-simplices, the connection describes how to glue these 4-simplices together and the action is a sum of terms depending on the holonomies around each triangle. We impose the discretized simplicity constraints on disjoint tetrahedra and we show how the Lagrange multipliers for the simplicity constraints distort the parallel transport and the correlations between neighbouring 4-simplices. We then construct the discretized BF action using a non-commutative product between $\SU(2)$ plane waves. We show how this naturally leads to the Barrett-Crane model. This clears up the geometrical meaning of the model. We discuss the natural generalization of this action principle and the spin foam models it leads to. We show how the recently introduced spinfoam fusion coefficients emerge with a non-trivial measure. In particular, we recover the Engle-Pereira-Rovelli spinfoam model by weakening the discretized simplicity constraints. Finally, we identify the two sectors of Plebanski's theory and we give the analog of the Barrett-Crane model in the non-geometric sector.

arXiv comment25 pages, revtex4, 4 figures

References in corpus (33)

• LQG vertex with finite Immirzi parameter
• Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action
• Relativistic spin networks and quantum gravity
• A new spinfoam vertex for quantum gravity
• Topological Lattice Models in Four Dimensions
• ``Sum over Surfaces'' form of Loop Quantum Gravity
• A New Spin Foam Model for 4d Gravity
• The loop-quantum-gravity vertex-amplitude
• 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory
• Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory
• Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space
• Flipped spinfoam vertex and loop gravity
• On the semiclassical limit of 4d spin foam models
• Graviton propagator from background-independent quantum gravity
• Three dimensional loop quantum gravity: physical scalar product and spin foam models
• so(4) Plebanski Action and Relativistic Spin Foam Model
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• A spin foam model without bubble divergences
• Quantum tetrahedra and simplicial spin networks
• The complete LQG propagator: II. Asymptotic behavior of the vertex
• Group Integral Techniques for the Spinfoam Graviton Propagator
• Lorentzian LQG vertex amplitude
• Path integral representation of spin foam models of 4d gravity
• Worldsheet formulations of gauge theories and gravity
• Spin foam model from canonical quantization
• Simplicity and closure constraints in spin foam models of gravity
• Renormalizable Non-Metric Quantum Gravity?
• Numerical evidence of regularized correlations in spin foam gravity
• The perturbative Regge-calculus regime of Loop Quantum Gravity
• Coherent states, constraint classes, and area operators in the new spin-foam models
• Boundary terms in the Barrett-Crane spin foam model and consistent gluing

Cited by in corpus (28)

• The Spin Foam Approach to Quantum Gravity
• Group field theory with non-commutative metric variables
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• Spin foam models for quantum gravity from lattice path integrals
• Holonomy Spin Foam Models: Definition and Coarse Graining
• Lifting SU(2) Spin Networks to Projected Spin Networks
• Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds
• Operator Spin Foam Models
• Coarse graining of spin net models: dynamics of intertwiners
• The twistorial structure of loop-gravity transition amplitudes
• Holomorphic Simplicity Constraints for 4d Spinfoam Models
• From lattice BF gauge theory to area-angle Regge calculus
• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• On the role of the Barbero-Immirzi parameter in discrete quantum gravity
• Effective Hamiltonian Constraint from Group Field Theory
• Bubble divergences and gauge symmetries in spin foams
• The Holst Spin Foam Model via Cubulations
• Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex
• Testing the imposition of the Spin Foam Simplicity Constraints
• Sub-leading asymptotic behaviour of area correlations in the Barrett-Crane model
• Feynman diagrammatic approach to spin foams
• Renormalization of a tensorial field theory on the homogeneous space SU(2)/U(1)
• Encoding simplicial quantum geometry in group field theories
• Spinfoam Fermions: PCT Symmetry, Dirac Determinant, and Correlation Functions
• The Barrett-Crane model: asymptotic measure factor
• The new spin foam models and quantum gravity
• A Short and Subjective Introduction to the Spinfoam Framework for Quantum Gravity
• Commuting Simplicity and Closure Constraints for 4D Spin Foam Models