6 - The connection representation

Summary

The quantum theory of the preceding chapter grew out of the ADM formulation of classical (2+1)-dimensional gravity. As we saw in chapter 4, however, the classical theory can be described equally well in terms of geometric structures and the holonomies of flat connections. The two classical descriptions are ultimately equivalent, but they are quite different in spirit: the ADM formalism depicts a spatial geometry evolving in time, while the geometric structure formalism views the entire spacetime as a single ‘timeless’ entity.

The corresponding quantum theories are just as different. In particular, while ADM quantization incorporates a clearly defined time variable, the quantum theory of geometric structures, which we shall develop in this chapter, will be a ‘quantum gravity without time’. Nevertheless, the two quantum theories, like their classical counterparts, are closely related: the quantum theory of geometric structures will turn out to be a sort of ‘Heisenberg picture’ that complements the ‘Schrödinger picture’ of ADM quantization.

The approach of this chapter is commonly called the connection representation, and closely resembles the (3+1)-dimensional connection representation developed by Ashtekar et al. The name comes from the fact that the basic variables – in this case, the geometric structures of chapter 4 – are associated with the spin connection rather than the metric. In particular, the ‘configuration space’ of geometric structures is the space of SO(2,1) holonomies of the spin connection.

Covariant phase space

Our starting point for this chapter is the classical description of (2+1)-dimensional gravity developed in chapter 4.