Mathematics – Logic
2 + 1 general relativity is a useful toy model to test certain features of the full 3 + 1-dimensional theory. This thesis consists of two major parts—classical and quantum. In the classical part we provide boundary conditions on the variables of the theory with a negative cosmological constant. We consider two boundaries, one at spatial infinity and one internal boundary. The internal boundary conditions are chosen to represent a horizon of a stationary black hole. One of the interesting results in this part are explicit, quasi-local formulae for energy and angular momentum of a black hole as well as for the whole space-time. In the second part, we investigate the canonical quantization of 2 + 1 gravity. We begin with analysis of different classical formulations as points of departure for quantization. Next, we address the issue of constructing the physical Hilbert space of states. For this purpose we construct certain important operators in quantum gravity, including a new regularization of the Hamiltonian constraint. Finally we discuss two strategies for finding its solutions.
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