Timelike twisted geometries

Abstract

Within the twistorial parametrization of Loop Quantum Gravity we investigate the consequences of choosing a spacelike normal vector in the linear simplicity constraints. The amplitudes for the $SU(2)$ boundary states of Loop Quantum Gravity, given by most of the current spinfoam models, are constructed in such a way that even in the bulk only spacelike building blocks occur. Using a spacelike normal vector in the linear simplicity constraints allows us to distinguish spacelike from timelike 2-surfaces. We propose in this paper a quantum theory that includes both spatial and temporal building blocks and hence a more complete picture of quantum spacetime. At the classical level we show how we can describe $T^{\ast }SU(1,1)$ as a symplectic quotient of 2-twistor space ${\mathbb{T}}^2$ by area matching and simplicity constraints. This provides us with the underlying classical phase space for $SU(1,1)$ spin networks describing timelike boundaries and their extension into the bulk. Applying a Dirac quantization we show that the reduced Hilbert space is spanned by $SU(1,1)$ spin networks and hence are able to give a quantum description of both spacelike and timelike faces. We discuss in particular the spectrum of the area operator and argue that for spacelike and timelike 2-surfaces it is discrete.

v2: Added some clarifications in Sec. V E. Published version

References in corpus (54)

• 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
• ``Sum over Surfaces'' form of Loop Quantum Gravity
• A New Spin Foam Model for 4d Gravity
• The loop-quantum-gravity vertex-amplitude
• A Lorentzian Signature Model for Quantum General Relativity
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• A "general boundary" formulation for quantum mechanics and quantum gravity
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Spin-Foams for All Loop Quantum Gravity
• From twistors to twisted geometries
• Spin foam model for Lorentzian General Relativity
• General boundary quantum field theory: Foundations and probability interpretation
• The projector on physical states in loop quantum gravity
• Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?
• Spectra of Length and Area in 2+1 Lorentzian Loop Quantum Gravity
• Lorentz covariance of loop quantum gravity
• Spacetime states and covariant quantum theory
• Lorentzian LQG vertex amplitude
• Black Hole Entropy from complex Ashtekar variables
• Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory
• Area spectrum in Lorentz covariant loop gravity
• Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint
• Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory
• A spin foam model for general Lorentzian 4-geometries
• The twistorial structure of loop-gravity transition amplitudes
• 3+1 spinfoam model of quantum gravity with spacelike and timelike components
• A Ponzano-Regge model of Lorentzian 3-Dimensional gravity
• On Spinfoam Models in Large Spin Regime
• Twistor Networks and Covariant Twisted Geometries
• Linking covariant and canonical LQG II: Spin foam projector
• General boundary quantum field theory: Timelike hypersurfaces in Klein-Gordon theory
• Spin foams with timelike surfaces
• The Holst Spin Foam Model via Cubulations
• Twistorial phase space for complex Ashtekar variables
• A note on the Holst action, the time gauge, and the Barbero-Immirzi parameter
• Timelike surfaces in Lorentz covariant loop gravity and spin foam models
• Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity
• Loop Quantum Cosmology with Complex Ashtekar Variables
• Null twisted geometries
• A note on the secondary simplicity constraints in loop quantum gravity
• Twisted geometries, twistors and conformal transformations
• Spectra of geometric operators in three-dimensional LQG: From discrete to continuous
• Unitary irreducible representations of SL(2,C) in discrete and continuous SU(1,1) bases
• The Lorentzian proper vertex amplitude: Asymptotics
• Near-Horizon Radiation and Self-Dual Loop Quantum Gravity
• The Lorentzian proper vertex amplitude: Classical analysis and quantum derivation
• Hamiltonian spinfoam gravity
• 3d Lorentzian loop quantum gravity and the spinor approach
• A one-dimensional action for simplicial gravity in three dimensions
• Wigner-Eckart theorem for the non-compact algebra sl(2,R)
• A note on the spinor construction of Spin Foam amplitudes

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• Analytic Continuation of Spin foam Models
• Gravity as an SU(1,1) gauge theory in four dimensions
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• $SU(1,1)$ Barbero-like variables derived from Holst action
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