Classical Aspects on the Road to Quantum Gravity: A Study in the Framework of the New Variables

Doktorsavhandling, 1994

There has been a lot of progress in the quest for the structure of spacetime since a canonical non-perturbative quantization program for general relativity was set up ten years ago. The crucial ideas are two choices of coordinates on phase space. First, Hamiltonian gravity was reformulated in terms of an SL(2,C)-valued connection and a densitized triad, the Ashtekar variables. Then, loops came into the picture. The loop variables, a generalization of Wilson loops, were for the first time used in quantum gravity. This thesis tries to clarify and complement five papers by the author and collaborators concerning three topics in pure classical gravity. The first two, paper I and II, are studies of an earlier found generalization of possible actions for gravity. They prove in some sense that Einstein's cosmological constant is not unique. The moduli space of SL(2,C)-connections up to gauge transformations is in fact non- Hausdorff. The implications for the quantum theory are unknown, classically it implies a non-bijective transformation from Ashtekar to loop variables. Based on an analysis in the vacuum case, paper III and IV investigate the influence of a cosmological constant. The holonomy class is still an observable of the gravitational field but the intersection of the set of the degenerate points with the real slice of the constraint surface changes. The last paper, paper V, illuminates a new classical picture of quantum geometry. A Hamiltonian formulation of a distributional ansatz for the gravitational fields is here found, and the hope, just as for the analogous Regge calculus is to find a good working technique for solving practical problems in classical as well as quantum gravity.