Topological methods in quantum gravity
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The main technical problem with background independent approaches to quantum gravity is inapplicability of standard quantum field theory methods. New methods are needed which would be adapted to the basic principles of General Relativity. Topological field theory is a model which provides natural tools for background independent quantum gravity. It is exactly soluble and, at the same time, diffeomorphism invariant. Applications of topological field theory to quantum gravity include description of boundary states of quantum General Relativity, formulation of quantum gravity as a constrained topological field theory, and a new perturbation theory which uses topological field theory as a starting point. The later is the central theme of the thesis. Unlike the traditional perturbation theory it does not require splitting metric into a background and fluctuations, it is exactly diffeomorphism invariant order by order, and the coupling constant of this theory is dimensionless. We describe the basic ideas and techniques of this perturbation theory as well as inclusion of matter particles, boundary states, and other necessary tools for studying scattering problem in background independent quantum gravity.