Loop quantum gravity and its geometrical applications.

Abstract

Loop Quantum Gravity is a non-perturbative, background-independent and quantum field theory of geometry itself. It is based on the quantum implementation of General relativity (GR) by using Dirac quantization program. The kinematical Hilbert space is constructed by cylindrical wave functionals through holonomies defined by the su(2) connection along a system of smooth oriented paths. The gauge invariant Hilbert space is the kernel space of the Gauss constraints; it is constructed by the spin network state which is a collection of oriented curves, a spin number at each curve and an invariant intertwiner at each node. The area and volume operators in LQG has been provided. From their quantum spectrum, the fuzziness and discreteness property of space is predicted; it is shown that spacetime is fundamentally discrete and at the scale of the Planck length. A beautiful interpretation of the space atom in terms of the quantum Euclidean polyhedral is provided. In this thesis, an alternative approach introducing a 3d- Ricci scalar curvature operator given in terms of volume and boundary area as well as new edge length operator is proposed. An example of monochromatic 4-valent node intertwiner state (equilateral tetrahedra) is studied and the scalar curvature measure for a regular tetrahedron shape is constructed. We show that all regular tetrahedron states are in the negative scalar curvature regime and for the semi-classical limit the spectrum is very close to the Euclidean regime.