Topological quantum field theory and quantum gravity

TL;DR

This thesis develops triangulation-independent state sum models for fermions and scalars on 1-manifolds, and explores gauge gravity quantization in 2+1 dimensions, with initial steps towards 3+1 dimensions.

Contribution

It introduces simple, triangulation-independent models for matter fields and advances the quantization approach for gauge gravity in lower dimensions.

Findings

State sum models match continuum results using zeta-function regularisation.

Quantization of 2+1D gauge gravity via sum over histories.

Initial Hamiltonian analysis for 3+1D gauge gravity.

Abstract

This thesis is broadly split into two parts. In the first part, simple state sum models for minimally coupled fermion and scalar fields are constructed on a $1$-manifold. The models are independent of the triangulation and give the same result as the continuum partition functions evaluated using zeta-function regularisation. Some implications for more physical models are discussed. In the second part, the gauge gravity action is written using a particularly simple matrix technique. The coupling to scalar, fermion and Yang-Mills fields is reviewed, with some small additions. A sum over histories quantisation of the gauge gravity theory in 2+1 dimensions is then carried out for a particular class of triangulations of the three-sphere. The preliminary stage of the Hamiltonian analysis for the (3+1)-dimensional gauge gravity theory is undertaken.