Quasi-Hopf Symmetry in Loop Quantum Gravity with Cosmological constant and Spinfoams with timelike surfaces

Abstract

In this thesis we study two separate problems concerning improvements to the Loop quantum gravity and spinfoam approach to quantum gravity. In the first part we address the question about the origin of quantum group symmetries in Loop quantum gravity with non-vanishing cosmological constant Λ. Our focus is mainly the 3-dimensional Euclidean case with Λ > 0. We clarify, both at the classical and the quantum level, the quasi-Poisson and quasi-Hopf structures that arise in this case, respectively. This type of symmetry has, until recently, seen not much attention in the Loop quantum gravity literature, despite its importance for the approach. We explain the connection of our work with the Turaev-Viro state sum model, which relies heavily on the notion of twisting. To analyze our q - deformed model, for q being a root of unity, we construct for the first time certain gauge invariant geometric observables for the (restricted) weak quasi-Hopf algebra 𝓤ʳᵉˢq(𝔰𝔩(2,ℂ)) with truncated coproduct, using so-called tensor operators. We show that these tensor operators satisfy the quasi-Hopf version of the Wigner-Eckart theorem and explicitly calculate the action of length- and angle- operators, which confirms the spherical curvature of our quantum geometry.

The second topic investigated in this thesis is the problem of timelike contributions for 4-dimensional Lorentzian spinfoam models, using the twistorial parametrization of Loop quantum gravity. We prove how the cotangent bundle T*SU(1,1) can be embedded into T*SL(2,C) via symplectic reduction by the simplicity constraints for a spacelike normal vector and an area matching constraint. This mathematical result is used to study timelike 2-surfaces in 4D Lorentzian gravity, both at the classical and quantum level. We investigate in particular the spectrum of the area operator for timelike faces and find that it is discrete. Furthermore, building on our results, we propose a new Lorentzian spinfoam model, which allows to include timelike contributions.