Path Integral Representation of Lorentzian Spinfoam Model, Asymptotics, and Simplicial Geometries

Abstract

A new path integral representation of Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model is derived by employing the theory of unitary representation of SL(2,$\mathbb{C}$). The path integral representation is taken as a starting point of semiclassical analysis. The relation between the spinfoam model and classical simplicial geometry is studied via the large spin asymptotic expansion of the spinfoam amplitude with all spins uniformaly large. More precisely in the large spin regime, there is an equivalence between the spinfoam critical configuration (with certain nondegeneracy assumption) and a classical Lorentzian simplicial geometry. Such an equivalence relation allows us to classify the spinfoam critical configurations by their geometrical interpretations, via two types of solution-generating maps. The equivalence between spinfoam critical configuration and simplical geometry also allows us to define the notion of globally oriented and time-oriented spinfoam critical configuration. It is shown that only at the globally oriented and time-oriented spinfoam critical configuration, the leading order contribution of spinfoam large spin asymptotics gives precisely an exponential of Lorentzian Regge action of General Relativity. At all other (unphysical) critical configurations, spinfoam large spin asymptotics modifies the Regge action at the leading order approximation.

35 pages, no figure

References in corpus (8)

• LQG vertex with finite Immirzi parameter
• Polyhedra in loop quantum gravity
• Area-angle variables for general relativity
• Self-Energy of the Lorentzian EPRL-FK Spin Foam Model of Quantum Gravity
• Zakopane lectures on loop gravity
• Generalized Spinfoams
• Spinfoams in the holomorphic representation
• Semiclassical Analysis of Spinfoam Model with a Small Barbero-Immirzi Parameter

Cited by in corpus (24)

• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Statistics, holography, and black hole entropy in loop quantum gravity
• Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity
• Spinfoam on Lefschetz Thimble: Markov Chain Monte-Carlo Computation of Lorentzian Spinfoam Propagator
• Einstein Equation from Covariant Loop Quantum Gravity in Semiclassical Continuum Limit
• 4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block
• Covariant Loop Quantum Gravity, Low Energy Perturbation Theory, and Einstein Gravity with High Curvature UV Corrections
• Numerical computations of next-to-leading order corrections in spinfoam large-$j$ asymptotics
• Emergent 4-dimensional linearized gravity from spin foam model
• Analytic Continuation of Spin foam Models
• Group Field Theory and Holographic Tensor Networks: Dynamical Corrections to the Ryu-Takayanagi formula
• Complex critical points in Lorentzian spinfoam quantum gravity: 4-simplex amplitude and effective dynamics on double-${\Delta }_3$ complex
• On Spinfoams Near a Classical Curvature Singularity
• Spinfoams and high performance computing
• Area Law from Loop Quantum Gravity
• Spin foam amplitude of the black-to-white hole transition
• Finiteness of spinfoam vertex amplitude with timelike polyhedra, and the full amplitude
• Classical dynamics for Loop Gravity: The 2-vertex model
• Semiclassical Behavior of Spinfoam Amplitude with Small Spins and Entanglement Entropy
• Semi-Classical Holomorphic Transition Amplitudes in Covariant Loop Quantum Gravity
• Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
• A simpler way of imposing simplicity constraints
• A look at area Regge calculus
• A saddle-point finder and its application to the spin foam model