Introduction

The problem of background independent Quantum Gravity is the problem of defining a Quantum Field Theory of matter and gravity in the absence of an underlying background geometry (see Chapter 1 by Rovelli). Loop quantum gravity (LQG) is a promising proposal for addressing this difficult task. Its main predictions and underlying mathematical structure are described in Chapter 13 by Thiemann. Despite the steady progress of the field, dynamics remains to a large extend an open issue in LQG. Here we present the main ideas behind a series of proposals for addressing the issue of dynamics. We refer to these constructions as the spin foam representation of LQG. This set of ideas can be viewed as a systematic attempt at the construction of the path integral representation of LQG.

The spin foam representation is mathematically precise in 2 + 1 dimensions, so we will start this chapter by showing how it arises in the canonical quantization of this simple theory (more about 2+1 gravity can be found in Chapter 16 by Freidel). This toy model will be used to precisely describe the true geometric meaning of the histories that are summed over in the path integral of generally covariant theories.

In four dimensions similar structures appear. We call these constructions spin foam models as their definition is incomplete in the sense that at least one of the following issues remains unclear: (1) the connection to a canonical formulation, and (2) regularization independence (renormalizability).