| The program of canonical quantum general relativity in the loop representation has obtained several successes toward the construction of a quantum theory of gravity, but there are still open problems. This thesis focuses on the loop-theoretic frameworks of the program and attempts to gain some insights to solve some of the problems. The problems to which this thesis is related are (I.a) physical interpretations of the knot states, (I.b) how to recover low-energy physics from the loop formalism, (II.a) the inner product of the representation space of loop functionals, and (II.b) relations to other quantum gravity programs.; Correspondingly we describe the following attempts and their results. (I) We develop technologies which allow one to study a "semiclassical" approximation around the "weave," a tangle of loops which approximates a classical metric geometry at large scales. We define "low-frequency constraints" satisfied by low-energy physical states and clarify their physical meaning. We construct low-frequency constraint invariant loop states which have clear physical interpretations in terms of gravitons, low-frequency fluctuations of the gravitational field. Among them are the Poincare invariant vacuum state and the graviton states. The vacuum state could be understood as "quantum flat space-time." We understand that the loop representation has the potential to describe graviton physics, one of low-energy quantum theories. These results suggest that the exact states sensitive to quantum geometry at the Planck scale but representing gravitons at large scales may exist. (II) We reformulate the Ponzano-Regge quantum gravity model, a path-integral model in 3-dimensions (which is known to be equivalent to the Ashtekar formulation in 3-dimensions with SO(3) group), such that its relation to the loop representation is manifest. We discuss how to recover the already known inner product of the (2 + 1) loop representation from this formulation and how to extend this formulation to (3 + 1)-dimensions, suggesting a possibility of constructing an inner product of the (3 + 1) loop representation. In addition, we realize that since this formulation does not include any field variables, it could be seen as a discretization of a genuinely background independent string theory. |
