On the volume simplicity constraint in the EPRL spin foam model

Abstract

We propose a quantum version of the quadratic volume simplicity constraint for the EPRL spin foam model. It relies on a formula for the volume of 4-dimensional polyhedra, depending on its bivectors and the knotting class of its boundary graph. While this leads to no further condition for the 4-simplex, the constraint becomes non-trivial for more complicated boundary graphs. We show that, in the semi-classical limit of the hypercuboidal graph, the constraint turns into the geometricity condition observed recently by several authors.

arXiv comment14 pages, 5 figures

References in corpus (24)

• The Spin Foam Approach to Quantum Gravity
• 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
• A New Spin Foam Model for 4d Gravity
• Polyhedra in loop quantum gravity
• Asymptotic analysis of the EPRL four-simplex amplitude
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• Spin-Foams for All Loop Quantum Gravity
• The Ponzano-Regge model
• Numerical evidence for a phase transition in 4d spin foam quantum gravity
• Hypercuboidal renormalization in spin foam quantum gravity
• SU(2) graph invariants, Regge actions and polytopes
• Cosmological Constant in LQG Vertex Amplitude
• Investigation of the Spinfoam Path integral with Quantum Cuboid Intertwiners
• Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
• A proposed proper EPRL vertex amplitude
• Classical general relativity as BF-Plebanski theory with linear constraints
• The Holst Spin Foam Model via Cubulations
• Towards a Cosmological subsector of Spin Foam Quantum Gravity
• On knottings in the physical Hilbert space of LQG as given by the EPRL model
• The Lorentzian proper vertex amplitude: Classical analysis and quantum derivation
• Poincare-Plebanski formulation of GR and dual simplicity constraints

Cited by in corpus (13)

• SU(2) graph invariants, Regge actions and polytopes
• Renormalization of symmetry restricted spin foam models with curvature in the asymptotic regime
• Asymptotics of lowest unitary SL(2,C) invariants on graphs
• Numerical evaluation of spin foam amplitudes beyond simplices
• Hopf link volume simplicity constraints in spin foam models
• Deformation of the EPRL spin foam model by a cosmological constant
• Poincare-Plebanski formulation of GR and dual simplicity constraints
• Toward matter dynamics in spin foam quantum gravity
• Curvature effects in the spectral dimension of spin foams
• Entanglement entropy of physical states in hypercuboidally truncated spin foam quantum gravity
• Non-convex 4d polytopes in Spin Foam Models
• Volume of 4-polytopes from bivectors
• On Geometry and Symmetries in Classical and Quantum Theories of Gauge Gravity