SU(2) graph invariants, Regge actions and polytopes

Abstract

We revisit the the large spin asymptotics of 15j symbols in terms of cosines of the 4d Euclidean Regge action, as derived by Barrett and collaborators using a saddle point approximation. We bring it closer to the perspective of area-angle Regge calculus and twisted geometries, and compute explicitly the Hessian and phase offsets. We then extend it to more general SU(2) graph invariants, showing that saddle points still exist and have a similar structure. For graphs dual to 4d polytopes we find again two distinct saddle points leading to a cosine asymptotic formula, however a conformal shape-mismatch is allowed by these configurations, and the asymptotic action is thus a generalisation of the Regge action. The allowed mismatch correspond to angle-matched twisted geometries, 3d polyhedral tessellations with adjacent faces matching areas and 2d angles, but not their diagonals. We study these geometries, identify the relevant subsets corresponding to 3d Regge data and flat polytope data, and discuss the corresponding Regge actions emerging in the asymptotics. Finally, we also provide the first numerical confirmation of the large spin asymptotics of the 15j symbol. We show that the agreement is accurate to the per cent level already at spins of order 10, and the next-to-leading order oscillates with the same frequency and same global phase.

v2: Section added on the implications of our results for spin foam models of quantum gravity, few amendments, references updated. 36 pages, many figures and many footnotes v3: Comparison with more recent work added at the end of the conclusions

References in corpus (18)

• LQG vertex with finite Immirzi parameter
• A new spinfoam vertex for quantum gravity
• The loop-quantum-gravity vertex-amplitude
• Polyhedra in loop quantum gravity
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• From twistors to twisted geometries
• Area-angle variables for general relativity
• Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
• Generalized Spinfoams
• Simplicity and closure constraints in spin foam models of gravity
• Numerical evidence of regularized correlations in spin foam gravity
• Asymptotics of lowest unitary SL(2,C) invariants on graphs
• Discretization independence implies non-locality in 4D discrete quantum gravity
• Asymptotic analysis of the EPRL model with timelike tetrahedra
• Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model
• Towards a Cosmological subsector of Spin Foam Quantum Gravity
• On the volume simplicity constraint in the EPRL spin foam model
• Discretizing parametrized systems: the magic of Ditt-invariance

Cited by in corpus (45)

• Effective Spin Foam Models for Four-Dimensional Quantum Gravity
• Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
• Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory
• Gluing polyhedra with entanglement in loop quantum gravity
• A high-performance code for EPRL spin foam amplitudes
• Renormalization of symmetry restricted spin foam models with curvature in the asymptotic regime
• Asymptotics of lowest unitary SL(2,C) invariants on graphs
• Numerical analysis of spin foam dynamics and the flatness problem
• Quasi-local holographic dualities in non-perturbative 3d quantum gravity II - From coherent quantum boundaries to BMS3 characters
• How-To compute EPRL spin foam amplitudes
• Primordial fluctuations from quantum gravity
• Emergence of Spacetime in a restricted Spin-foam model
• Searching for classical geometries in spin foam amplitudes: a numerical method
• Entanglement entropy of Bell-network states in LQG: Analytical and numerical results
• Emergent 4-dimensional linearized gravity from spin foam model
• Twisted Geometries Coherent States for Loop Quantum Gravity
• Asymptotic analysis of spin-foams with time-like faces in a new parameterisation
• Numerical analysis of the self-energy in covariant Loop Quantum Gravity
• Spin-foams as semi-classical vertices: gluing constraints and a hybrid algorithm
• Quantum Gravity on the computer: Impressions of a workshop
• Renormalization of group field theories for quantum gravity: new scaling results and some suggestions
• On the volume simplicity constraint in the EPRL spin foam model
• Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
• Numerical evaluation of spin foam amplitudes beyond simplices
• Remarks on a melonic field theory with cubic interaction
• Hopf link volume simplicity constraints in spin foam models
• $SO(3)$-invariant phase of the $O(N{)}^3$ tensor model
• Deformation of the EPRL spin foam model by a cosmological constant
• Spinfoams: Foundations
• Tullio Regge's legacy: Regge calculus and discrete gravity
• Poincare-Plebanski formulation of GR and dual simplicity constraints
• Measuring the last burst of non-singular black holes
• Semi-Classical Holomorphic Transition Amplitudes in Covariant Loop Quantum Gravity
• Toward matter dynamics in spin foam quantum gravity
• Curvature effects in the spectral dimension of spin foams
• A Monte Carlo algorithm for spin foam intertwiners
• Geometry from local flatness in Lorentzian spin foam theories
• Tunneling of quantum geometries in spinfoams
• Studying the EPRL spinfoam self-energy
• Cosmological states in loop quantum gravity on homogeneous graphs
• Partial absence of cosine problem in 3d Lorentzian spin foams
• Spinfoam tunneling of quantum geometries in angle variables
• Non-convex 4d polytopes in Spin Foam Models
• Entanglement entropy of physical states in hypercuboidally truncated spin foam quantum gravity
• (2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality