This thesis is a small piece in the giant puzzle of finding a theory of quantum gravity and new insights into the intricacies of quantum gravity are provided. Two new ways of attacking the problem are presented: One approach is from the microscopic side, starting out from the symmetries of the circle, the affine Lie and the Virasoro algebra, a field theory in four dimensions is constructed. The second approach is from the macroscopic side: Investigating the possibility of a phenomenon analogous to the macroscopic phenomena of superfluidity and superconductivity in neutron stars.
In the first approach, one starts with the symmetries of the circle to arrive at the co-adjoint action. By identifying the co-adjoint action in two different ways, one as an infinitesimal transformation law and two as a Gauß law constraint, one is able to construct field theories in any space-time dimension, from algebras describing those symmetries. In particular, the affine Lie algebra lead to a Yang-Mills Lagrangian and the Virasoro algebra leads to a gravitational theory. In addition, a rigid connection between gauge and gravitational sector is observed. A possible link of this gravitational theory to the right-hand side of Einstein's equations is discussed.
The second approach considers macroscopic effects of quantum gravity. Starting in the post-Newtonian approximation to general relativity, first a semi-classical treatment following the one of London is given, then a Ginzburg-Landau treatment of the phenomenon is presented. A gravitational Meissner effect with a London penetration depth of 12km for a neutron star of 1:4 solar masses is predicted, as well as a gravitational Aharanov-Bohm effect, infinite gravitational conductivity, and for the triplet state a gravitational 'ferromagnetic' type phase.
The last part, starts from the gravitational wave expansion to general relativity. The spontaneous symmetry breaking character of the phenomenon becomes apparent. The graviton becomes massive, acquiring three new degrees of freedom in the same way as the photon receives one new degree of freedom in the BCS theory of superconductivity. Again, a Meissner effect is predicted. Furthermore, the consequences for black hole physics are discussed.