| The term 'spin foam models' was invented only a couple years ago by Baez to refer to a new approach to quantization of general relativity that appeared as an offsping of loop quantum gravity. Although this new approach was motivated, both logically and historically, by loop quantum gravity, it became clear by now that the two approaches are rather independent. While loop quantum gravity attempts to give a canonical quantization of general relativity, spin foam model approach is set to make sense of the path integral for gravity. Eventually, the two approaches will probably be shown to be equivalent, but no rigorous result to this effect exists as for now. In this thesis I develop the spin foam quantization of gravity from scratch, referring to results from loop quantum gravity only for comparison. I start from a review of 2 + 1 gravity and discuss different roots to quantize it. While some of them, as, for example, using Chern-Simons theory, only exist in 2 + 1, others can be generalized to higher dimensions. Spin foam models give such a generalization. Developing it, we will encounter, in particular, a deep relation between group representation theory and geometry. I also discuss in some details such related topics as topological field theory, notably BF theory, higher-dimensional Euclidean geometry, and quantum groups. |
