EXTENDED KNOTS AND THE SPACE OF STATES OF QUANTUM-GRAVITY

In the loop representation the quantum constraints of gravity can be s olved. This fact allowed significant progress in the understanding of the space of states of the theory. The analysis of the constraints ove r loop-dependent wavefunctions has been traditionally based upon geome tric (in contrast to analytic) properties of the loops. The reason for this preferred way is twofold: on the one hand, the inherent difficul ties associated with the analytic loop calculus, and on the other hand , our limited knowledge about the analytic properties of knots invaria nts. Extended loops provide a way to overcome the difficulties at both levels, On the one hand, a systematic method to construct analytic ex pressions of diffeomorphism invariants (the extended knots) in terms o f the Chern-Simons propagators can be developed. Extended knots are si mply related to ordinary knots (at least formally). The analytic expre ssions of knot invariants could be produced then in a generic way. On the other hand, the evaluation of the Hamiltonian over extended loop w avefunctions can be thoroughly accomplished in the extended loop frame work, These two ingredients promote extended loops as a potential reso rt for answering important questions about quantum gravity.