Mathematical development and applications of loop quantum gravity .

Abstract

Loop Quantum gravity is a tentative theory to describe the quantum structure of spacetime at the Planck scale, the scale at which both general relativity and quantum theory manifest equally. The theory comes in three versions: The canonical approach, covariant approach and geometric approach. All the approaches use the same Hilbert space, but we do not know whether they actually correspond to the same theory. In this thesis, I will present our main results in the loop quantum gravity program, all of which lie in between the three approaches. We start with describing The canonical and covariant approaches in which the notations and general concepts of the theory are fixed. Then, we discuss our contribution on the length spectrum of space, the length of the tetrahedral edges. After that, we investigate the quantum polyhedra and its relation to loop quantum gravity. More specifically, we discuss the quantum tetrahedron: the 4-node Hilbert space. We finish the chapter by investigating our contribution in the filed the quantum polyhedra: the discreteness of the area of space via Bohr-Sommerfeld quantization. Next, we investigate our deriving to the volume of space spectrum for arbitrary number of faces of the polyhedron. We use the idea of virtual lines together with the fact that the node Hilbert space with valency N can be split into series of connected 4-valent nodes Hilbert spaces. Then, we study the quantum pentahedron in which a nice representation on phase spaces for the pentahedral atoms of space is given. Next, we investigate our works on: (a) Regge and Twisted Geometries in the context of the loop Gravity Hilbert space and (b) Regge and Twisted Geometries in Schwarzschild Spacetime. We discuss the interesting results in which twisted-truncation is included in interpreting the loop gravity graph. Furthermore, the Schwarzschild Spacetime graph is well-studied. Finally, a new quantity called space density is introduced and an interpretation for gravity force is discussed.