| Loop representation of Quantum Gravity is studied with particular stress in three areas: The kinematic solutions to the constraints of the theory are analyzed; the continuous limit of the Planck-scale discrete geometry is probed; the problem of time in quantum General Relativity is considered.; First, spaces of intersecting knots, which are solutions to the diffeomorphism constraint of loop gravity, are studied. These turn out to be moduli spaces, thus being continuous (non-discrete) in size. This is attributed to the fact that the information about the embedding of the knot states in the manifold is contained in the parameter space defining the knots. The formula for the upper bound of the dimension of such spaces (as a function of the multiplicity of the intersection) is derived, and it is conjectured to be actually attained.; Second, the weave states are studied. These are eigenstates of the area and volume operators, and they also solve the Gauss and diffeomorphism constraints of loop gravity. They are constructed in order to study the emergence of the continuous characteristics on scales much larger than the Planck scale from the discrete geometry of that scale. Two different weaves are constructed, both approximating a flat metric in the classical limit. One is a collection of pairwise intersecting circles of a size on the order of the Planck scale. The second is arranged in a domain structure, with high symmetry on scales comparable to the Planck length, and generic geometry on large scales due to the random orientation of the domains.; Third, the issue of time in quantum General Relativity is considered from the point of view of Quantum Mechanics. It is demonstrated that the frozen-time formalism can be accommodated in Quantum Mechanics. In particular, an operator corresponding to the measurement of the time of arrival of a particle at a detector gate is defined. The basis of its eigenstates--the time representation--is studied. The general feasibility of approaches to standard Quantum Mechanics which do not depend on a Newtonian notion of time is demonstrated. |
