In the previous chapter we discussed the basics of the loop representation for quantum gravity. We obtained expressions for the constraints at both a formal and a regularized level and discussed generalities about the physical states of the theory. In this chapter we would like to discuss several developments that are based on the loop representation. We will first discuss the coupling of fields of various kinds: fermions using an open path formalism, Maxwell fields in a unified fashion and antisymmetric fields with the introduction of surfaces. These examples illustrate the various possibilities that matter couplings offer in terms of loops. We then present a discussion of various ideas for extracting approximate physical predictions from the loop representation of quantum gravity. We discuss the semi-classical approximation in terms of weaves and the introduction of a time variable using matter fields and the resulting perturbation theory. We end with a discussion of the loop representation of 2 + 1 gravity as a toy model for several issues in the 3 + 4 –1 theory.
Inclusion of matter: Weyl fermions
As we did for the Yang–Mills case, we now show that the loop representation for quantum gravity naturally accommodates the inclusion of matter. In the Yang–Mills case, in order to accommodate particles with Yang–Mills charge one needed to couple the theory to four-component Dirac spinors. A Dirac spinor is composed of two two-component spinors that transform under inequivalent representations of the group.