Extended loops: A new arena for nonperturbative quantum gravity

Abstract

We propose a new representation for gauge theories and quantum gravity. Alternatively, it can be viewed as a new framework for doing computations in the loop representation. It is based on the use of a novel mathematical structure that extends the group of loops into a Lie group. This extension allows the use of functional methods to solve the diffeomorphism and Hamiltonian constraint equations. It puts in a precise framework some of the regularization problems of the loop representation. It has practical advantages in the search for quantum states. Making use of it we are able to find a new solution to the Wheeler-DeWitt equation that reinforces the conjecture that the Jones polynomial is a quantum state of nonperturbative quantum gravity.