Quantum hyperbolic geometry in loop quantum gravity with cosmological constant

摘要

Loop Quantum Gravity (LQG) is an attempt to describe the quantum gravityregime. Introducing a non-zero cosmological constant $\Lambda$ in this contexthas been a withstanding problem. Other approaches, such as Chern-Simonsgravity, suggest that quantum groups can be used to introduce $\Lambda$ in thegame. Not much is known when defining LQG with a quantum group. Tensoroperators can be used to construct observables in any type of discrete quantumgauge theory with a classical/quantum gauge group. We illustrate this byconstructing explicitly geometric observables for LQG defined with a quantumgroup and show for the first time that they encode a quantized hyperbolicgeometry. This is a novel argument pointing out the usefulness of quantumgroups as encoding a non-zero cosmological constant. We conclude by discussinghow tensor operators provide the right formalism to unlock the LQG formulationwith a non-zero cosmological constant.