Two-dimensional Hořava-Lifshitz theory of gravity.
In this dissertation, two-dimensional Hařava theory of gravity has been studied on the classical and quantum mechanical levels. The classical solutions of the projectable and nonprojectable Hořava gravity have been found and their spacetime structures are investigated in terms of Penrose diagrams. When quantizing the theory in the canonical approach, the integral Hamiltonian constraint in the projectable case will generate the so-called Wheeler-DeWitt equation which can be exactly solved if the invariant length and its conjugate momentum are used as new variables. On the other hand, for the nonprojectable case, the lapse function is no longer a Lagrangian multiplier but one of the canonical variables. This results in a local and second-class Hamiltonian constraint which can be solved for the lapse function. The quantization of nonprojectable case is carried out by directly dropping the unphysical degrees of freedom, that is, replacing Poisson brackets with Dirac brackets. In the last part of the dissertation, the interactions between two-dimensional Hořava gravity and a non-relativistic scalar field are considered. In the projectable case, the minimal coulping is adopted and canonical quantization is carried out in the same way as we have done for the pure gravity case. In the non-projectable case, we turn to the non-minimal couplings and find both Killing and Universal horizons from classical solutions.