Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model

Abstract

We consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving integrals over SU(2), and use it to write the two-point function of 3d gravity on a single tetrahedron as a group integral. The perturbative expansion of this expression is then performed with respect to the boundary geometry using a simple saddle-point analysis. We derive the complete expansion in inverse powers of the length scale and evaluate explicitly the quantum corrections up to second order. Finally, we use the same method to provide the complete expansion of the isosceles 6j-symbol with the explicit phases at all orders and the next-to-leading correction to the Ponzano-Regge asymptotics.

arXiv comment16 pages, 3 figs, revtex4, a numerical coefficient corrected in eq (60)

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Cited by in corpus (24)

• Colored Group Field Theory
• Topological Graph Polynomials in Colored Group Field Theory
• LQG propagator from the new spin foams
• Lost in Translation: Topological Singularities in Group Field Theory
• SU(2) graph invariants, Regge actions and polytopes
• Recurrence relations for spin foam vertices
• A new Hamiltonian for the Topological BF phase with spinor networks
• Pushing Further the Asymptotics of the 6j-symbol
• Generating Functions for Coherent Intertwiners
• Semiclassical regime of Regge calculus and spin foams
• Numerical computations of next-to-leading order corrections in spinfoam large-$j$ asymptotics
• Sub-leading asymptotic behaviour of area correlations in the Barrett-Crane model
• 2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions
• The 6j-symbol: Recursion, Correlations and Asymptotics
• Asymptotics of Wigner 3nj-symbols with Small and Large Angular Momenta: an Elementary Method
• Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
• $SO(3)$-invariant phase of the $O(N{)}^3$ tensor model
• Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity
• A Short and Subjective Introduction to the Spinfoam Framework for Quantum Gravity
• Second-order amplitudes in loop quantum gravity
• A New Recursion Relation for the 6j-Symbol
• Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
• Quasi-Local 3D Quantum Gravity : Exact Amplitude and Holography
• Boundary State Stability under Spinfoam Evolution for the Quantum 4-Simplex