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.
