Intersecting N loop solutions of the hamiltonian constraint of quantum gravity

Abstract

Following the introduction by Ashtekar of a new set of canonical variables for the gravitational field and a new representation of quantum gravity based on them, Jacobson and Smolin presented a large class of solutions to the quantum hamiltonian constraint of general relativity based on the holonomy of the Ashtekar connection along simple loops and two intersecting loops. For all these solutions the metric is degenerate everywhere, including the point of intersection. This motivated Husain to extend the results to consider the case of three intersecting loops. However, the metric was degenerate as well on the three-loop solutions found since the loops were only allowed to have two independent tangent vectors at the point of intersection. We have developed a computer algebra code capable of generating solutions for an arbitrary number of loops. We explicitly present new four- and five-loop solutions. These are the first solutions known to have three independent tangent vectors at the intersection point. In spite of this they fail, however, to have a nondegenerate metric. We present a general argument that shows that the same situation arises for an arbitrary finite number of loops. Implications of this degeneracy for the interpretation of the connection and loop space representation of quantum gravity are also discussed.