Hessian in the spinfoam models with cosmological constant

Abstract

In this paper, we introduce a general method to prove the non-degeneracy of the Hessian in the spinfoam vertex amplitude for quantum gravity and apply it to the spinfoam models with a cosmological constant ($\Lambda$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat $\mathrm{SL}(2,\mathbb{C})$ connections, we demonstrate that the Hessian is non-degenerate for critical points corresponding to non-degenerate, geometric 4-simplices in de Sitter or anti-de Sitter space. Non-degeneracy of the Hessian is an important necessary condition for the stationary phase method to be applicable. With a non-degenerate Hessian, this method not only confirms the connection of the $\Lambda$-SF model to semiclassical gravity, but also shows that there are no dominant contributions from exceptional configurations as in the Barrett-Crane model. Given its general nature, we expect our criterion to be applicable to other spinfoam models under mild adjustments.

33 pages, 3 figures; published version

References in corpus (49)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• The Spin Foam Approach to Quantum Gravity
• LQG vertex with finite Immirzi parameter
• Relativistic spin networks and quantum gravity
• A new spinfoam vertex for quantum gravity
• A New Spin Foam Model for 4d Gravity
• Group Field Theory: An overview
• A Lorentzian Signature Model for Quantum General Relativity
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space
• Lorentzian spin foam amplitudes: graphical calculus and asymptotics
• Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity
• Spacetime as a Feynman diagram: the connection formulation
• Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• LQG propagator from the new spin foams
• From twistors to twisted geometries
• Asymptotics of Relativistic Spin Networks
• The Ponzano-Regge model
• 4-dimensional Spin-foam Model with Quantum Lorentz Group
• Quantum deformation of two four-dimensional spin foam models
• Topological BF Theories in 3 and 4 Dimensions
• Asymptotics of 6j and 10j symbols
• Classical 6j-symbols and the tetrahedron
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Asymptotics of 10j symbols
• Cosmological Deformation of Lorentzian Spin Foam Models
• Linearized dynamics from the 4-simplex Regge action
• Lorentzian spinfoam propagator
• Semi-Classical Limits of Simplicial Quantum Gravity
• Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks
• Spin Foam Models of Riemannian Quantum Gravity
• Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity
• In quantum gravity, summing is refining
• Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps
• Recurrence relations for spin foam vertices
• Numerical evidence of regularized correlations in spin foam gravity
• LQG propagator: III. The new vertex
• Gauge symmetries in spinfoam gravity: the case for "cellular quantization"
• Four-dimensional Spinfoam Quantum Gravity with Cosmological Constant: Finiteness and Semiclassical Limit
• 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
• SL(2,C) Chern-Simons Theory, Flat Connections, and Four-dimensional Quantum Geometry
• Non-Perturbative 3D Quantum Gravity: Quantum Boundary States and Exact Partition Function
• Complex critical points in Lorentzian spinfoam quantum gravity: 4-simplex amplitude and effective dynamics on double-${\Delta }_3$ complex
• The Barrett-Crane model: asymptotic measure factor
• Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
• The hessian in spin foam models