Quantum hyperbolic geometry in loop quantum gravity with cosmological constant

Abstract

Loop Quantum Gravity (LQG) is an attempt to describe the quantum gravity regime. Introducing a non-zero cosmological constant $\Lambda$ in this context has been a withstanding problem. Other approaches, such as Chern-Simons gravity, suggest that quantum groups can be used to introduce $\Lambda$ in the game. Not much is known when defining LQG with a quantum group. Tensor operators can be used to construct observables in any type of discrete quantum gauge theory with a classical/quantum gauge group. We illustrate this by constructing explicitly geometric observables for LQG defined with a quantum group and show for the first time that they encode a quantized hyperbolic geometry. This is a novel argument pointing out the usefulness of quantum groups as encoding a non-zero cosmological constant. We conclude by discussing how tensor operators provide the right formalism to unlock the LQG formulation with a non-zero cosmological constant.

6pages, 1 figure

References in corpus (8)

• From twistors to twisted geometries
• A new Hamiltonian for the Topological BF phase with spinor networks
• Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
• A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
• Spinors and Twistors in Loop Gravity and Spin Foams
• 6j symbols for U_q(sl_2) and non-Euclidean tetrahedra
• On the Asymptotics of Quantum Group Spinfoam Model
• On the q-quantum gravity loop algebra

Cited by in corpus (24)

• Decorated tensor network renormalization for lattice gauge theories and spin foam models
• Towards a phase diagram for spin foams
• Hypercuboidal renormalization in spin foam quantum gravity
• Coarse graining flow of spin foam intertwiners
• Observables in Loop Quantum Gravity with a cosmological constant
• Quantum group spin nets: refinement limit and relation to spin foams
• Renormalization of symmetry restricted spin foam models with curvature in the asymptotic regime
• Compact phase space, cosmological constant, discrete time
• Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
• The Fock Space of Loopy Spin Networks for Quantum Gravity
• Quantum geometry from higher gauge theory
• Topological lattice field theories from intertwiner dynamics
• Symplectic and Semiclassical Aspects of the Schläfli Identity
• 3d Lorentzian loop quantum gravity and the spinor approach
• Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a Cut-off for Quantum Gravity
• Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models
• BTZ Black Hole Entropy and the Turaev-Viro model
• $q$-deformed 3D Loop Gravity on the Torus
• Local Observables in ${\mathrm{SU}}_q(2)$ Lattice Gauge Theory
• A one-dimensional action for simplicial gravity in three dimensions
• On the Hydrogen Atom in the Holographic Universe
• A Curvature Operator for a Regular Tetrahedron Shape in LQG
• Thermofield Double States in Group Field Theory
• Polyhedron phase space using 2-groups: kappa-Poincare as a Poisson 2-group