Hermitian operators on quantum general relativity loop space

Abstract

The author discusses a more precise mathematical formulation of the Rovelli and Smolin loop space representation of quantum general relativity. Their representation space based on wavefunctions defined on sets of loops can be made more definitive by using either subsets or Cartesian products of the loop space of the 3-manifold Sigma . This enables him to consider the problem of defining an inner product on the wavefunctions. He constructs a 'model' inner product on a subspace of the representation space using a discrete sum instead of an integral over elements of the domain space of the wavefunctions. With respect to this inner product, Hermitian conjugates to the T operators can be constructed. The Hermitian operators he obtains are significant from the point of view of their action on the sets rather than the individual loops.